Compute Engine Changelog

Coming Soon

Breaking Changes

• A negative Take/Drop count counts from the end (#414, reported by enumeratio). Take(xs, -n) is the last n elements of xs, and Drop(xs, -n) is xs without its last n elements: Take([1, 2, 3, 4, 5], -2) is [4, 5] (it was []) and Drop([1, 2, 3, 4, 5], -2) is [1, 2, 3] (it was the whole list). A string gives a string: Take("hello", -2) is "lo" and Drop("hello", -2) is "hel". A count past the length is still clamped, now in both directions: over 5 elements, Take(xs, 10) and Take(xs, -10) are all 5 elements, and Drop(xs, 10) and Drop(xs, -10) are []. A count of 0 is unchanged. A negative count needs the length of the collection: when that length is not known or is infinite (Take(Range(1, ∞), -2), a symbol with no value), the expression stays unevaluated. Compiled JavaScript and Python give the same results. In compiled JavaScript, a negative count over an infinite collection is a compile error when the count is a constant, and throws when the count is known only at run time.

• A collection count can be a bigint (#416, reported by enumeratio). The type of expr.count and of the count collection handler is now number | bigint | undefined. A bigint is used only for a finite count that is not a safe integer (larger than Number.MAX_SAFE_INTEGER); a smaller count is always a number, and an infinite count is Infinity. A handler can return a bigint for any count: expr.count converts a safe one to a number. Code that does arithmetic with expr.count must check its type first, because JavaScript throws a TypeError when a bigint and a number are mixed. Count(QuotientRing(Integers, 2^61 - 1)), Length and |\mathbb{Z}_{2^{61}-1}| now give the exact integer 2305843009213693951; they stayed unevaluated. Other counts are exact where they were rounded floats, Infinity or undefined:

  • Range with exact integer bounds: Count(Range(1, 10^20 + 1)) is 100000000000000000001 (it was 1e20).
  • CartesianProduct, Permutations, Combinations and PowerSet: Count(Permutations(Range(1, 200))) is 200! (it was +∞), and PowerSet of a set with 1100 elements has the count 2^1100 and is finite (it was +∞ and infinite). A count that needs more than 10,000 multiplications to compute stays unknown; a finite collection is never reported as infinite.
  • Repeat(x, n) with an n that is not a safe integer.
  • The views of such a collection: Take, Drop, Rest, Most, Zip, Insert, DeleteAt, ReplaceAt and When.

  The count of a CartesianProduct or a PowerSet with an operand of unknown size is undefined; it was NaN. A ring ℤ/nℤ with more than 2^53 elements is not enumerable: Sum, Max and the other operators that walk a collection to its end stay unevaluated. Indexing from the end of a collection whose count is a bigint is not supported yet: Last(Range(1, 10^20)) stays unevaluated (it was 1e20), and Take(Range(1, 10^20), -2) stays unevaluated (it was the wrong [1e20, 1e20]).

Behavior Changes

• simplify() combines same-base powers in a product as it does in a quotient (#415, reported by enumeratio). For a symbol x with no assumptions, x^a / x^b simplified to x^(a−b), but x^a · x^b was left unchanged, because the product rule required a base known to be non-zero. The engine already reads x/x as 1 for a symbol x, so products now follow the same convention: x^a · x^b is x^(a+b), x^a · x^(−a) is 1, x · x^a is x^(a+1) and x^a · x · x^b is x^(a+b+1). evaluate() does not change: it still leaves x^a · x^b as it is. A literal zero base is not affected: 0^1 · 0^(−1) is still Indeterminate.

• The branch of LambertW must be an integer, and it can be named (#418, reported by enumeratio). The branch is the second argument, LambertW(z, k), as in mpmath, SciPy, SymPy and Julia. Mathematica (ProductLog[k, z]; LambertW[k, z] in Wolfram|Alpha), Maple, Sage and MATLAB put the branch first, so input copied from them has the two arguments swapped. The signature is now (z: complex | infinity, branch: integer?) -> number (it was (complex | infinity, number?) -> number):

  • A branch that is not an integer is a type error. LambertW(-1, -0.1) (the Wolfram order) was left unevaluated; it now has an incompatible-type error on -0.1. A swap of two integers cannot be found: LambertW(1, 2) is W₂(1) and LambertW(2, 1) is W₁(2).
  • The branch can be given by name, in any position: lambertW(-0.1, branch: -1) and lambertW(branch: -1, z: -0.1) in Epsil, or ["LambertW", -0.1, ["NamedArgument", "'branch'", -1]] in MathJSON. All give W₋₁(−0.1) ≈ −3.5772.

  The LambertW reference documentation now gives the argument order, the real domain of each real branch, and the named form.

New Features

• The tie rule of Round is an engine setting (#417, requested by enumeratio). ce.roundingTies selects how Round rounds a value exactly halfway between two integers: "away-from-zero" (the default, unchanged: Round(2.5) is 3 and Round(-2.5) is -3), "to-even" (IEEE 754, Python, NumPy and Mathematica: Round(2.5) is 2 and Round(3.5) is 4), "toward-zero", "toward-positive-infinity" (JavaScript Math.round: Round(-2.5) is -2) and "toward-negative-infinity". The rule applies at every precision, to an exact rational (Round(5/2)), to an exact constant that is at a tie under .N(), to the sign of Round, and to the form Round(x, n), which rounds to n decimal places (with "to-even", Round(0.125, 2) is 3/25). An unknown rule is an error. Compiled JavaScript, interval JavaScript, GLSL, WGSL and Python use the rule in effect at compile time; a function compiled before a change keeps its rule, and an automatically compiled Map is compiled again. Remainder does not use the rule: its quotient is still rounded with a tie toward +∞.

Issues Resolved

• Round of a large odd integer in compiled Python and shader code. Python computed sign(x)·floor(|x| + 0.5), and |x| + 0.5 is rounded when |x| ≥ 2⁵²: Round(2⁵² + 1) was 2⁵² + 2. GLSL and WGSL had the same error in f32 from 2²³ (Round(8388609) was 8388610). The code now compares the exact distance to the floor of |x| with 0.5.

• Remainder at a tie in compiled Python, GLSL, WGSL and interval JavaScript. The interpreter rounds the quotient with a tie toward +∞, as JavaScript Math.round does: Remainder(5, 2) is 5 − 2·3 = −1. Python and the shaders rounded the quotient with np.round or round(), which round a tie to even, and gave 1. The interval target rounded a tie away from zero, and gave 1 for Remainder(−5, 2). All of them now give −1.

• Take and Drop with a symbolic count no longer give a wrong answer to Any or All. Any(Take([1, 2, 3], n), x > 0) was False and All(Drop([1, 2, 3], n), x < 0) was True. Both now stay unevaluated.

• Take and Drop with a count past the safe integers or an infinite count clamp to the length. Take([1, 2, 3], 10^20), Take([1, 2, 3], 1e20) and Take([1, 2, 3], ∞) stayed unevaluated, while compiled JavaScript gave [1, 2, 3] and compiled Python raised an OverflowError. All three now give [1, 2, 3].

• Collection views no longer give a wrong answer to Any or All when their walk cannot produce the elements. Reverse, RotateLeft and RotateRight of a collection with more than 2^53 elements, and Insert, DeleteAt, ReplaceAt and Slice without a usable position, reported that they could be walked, so Any gave False and All gave True. These now stay unevaluated. Repeat(x, n) with a negative n past the safe integers is empty in every respect: Element(x, …) was True.

• A Range with exact bounds and step has the exact number of elements. The count of a Range was computed in floats with a small tolerance, so a Range whose exact upper bound is just below a step gave one element too many, above its upper bound: Range(0, 9999999999999/10^12) gave the elements 0 to 10, and Range(1, 10^20, 1/2) counted 200000000000000000000 elements (the count is 199999999999999999999). A Range with a bound past the largest double, such as Range(1, 10^400), was infinite: Length was +∞. With exact bounds and step, the count, the elements, Length, Element, Last and the reducers now all use the exact count. With a float bound or step, the count is computed in floats as before.

• Max and Min of a Range are exact. Min(Range(10^20 + 1, 1, -1)) was 0, Max(Range(1, 10^20 + 1)) was the rounded float 1e20, and Min(Range(1/3, 1, 1/3)) was 0.333…. They are now 1, 100000000000000000001 and 1/3: the first or last element, read exactly.

• Sum, Mean, Median, Variance and PopulationVariance of a Range no longer go through every element. With exact integer or rational bounds and step, they use the closed form of an arithmetic sequence: Sum(Range(1, 10^6)) took 3 s and now takes less than 1 ms, and Sum(Range(1, 10^20)) did not finish and is now the exact 5000000000000000000050000000000000000000. evaluate() gives the exact value, and .N() gives a float for a result that is not an integer.

• Powers of a matrix are no longer combined as if they were numbers (#415). The product of two matrices is the matrix product, but Exp of a matrix and a non-integer power of a matrix are element-wise, so the same-base rules gave wrong answers for matrices: simplify() turned Exp(A)·Exp(B) into Exp(A+B) (also for literal matrices), A^a / A^b into A^(a−b), and √A·√A and A^(1/3)·A^(2/3) into A, and (e^A)^2 was e^(2A). These now stay as they are, except where a matrix power applies: √A·√A is MatrixPower(√A, 2), (e^A)^2 is MatrixPower(e^A, 2), and A·A is MatrixPower(A, 2).

• A definite integral with an irrational bound keeps its exact value. ∫_2^{√5} 4 dx was 0.944…; it is now 4√5 − 8. The same applied to EvaluateAt(f, a, b) whenever f(b) and f(a) are two different exact numbers, as √3 and √2. EvaluateAt(…).N() still gives a float.

• The sum of two lists or matrices keeps exact elements exact. [√2, 2] + [1, 1] was [2.414…, 3]; it is now [1 + √2, 3], and [√2, 2] − [1, 1] is [−1 + √2, 1]. A list plus a number was already exact. .N() still gives floats.

• Determinants of matrices with irrational or symbolic entries.

  • The determinant of [[√2, 1], [1, 1]] was 0.414…; it is now √2 − 1. The same applied to every determinant whose computation subtracts two different exact numbers.
  • From 4×4, a matrix with a symbol or an exact irrational entry has an expanded determinant. The determinant of a 4×4 matrix of 16 symbols was a quotient of 1,900 characters with terms in a⁴; it is now the sum of its 24 terms. This applies up to 6×6 for a matrix with symbols, and up to 10×10 for a matrix of numbers. A singular 4×4 matrix with radical entries has determinant 0, where it was a nonzero float.
  • Determinant(m).N() of a 4×4 or larger matrix chooses the largest pivot of each column. A pivot that is zero in exact arithmetic but about 1e-16 after rounding gave 2048 for a matrix whose determinant is −155.19.
  • CharacteristicPolynomial([[a, b], [c, d]], x) stayed Determinant([[x − a, −b], [−c, x − d]]); it is now x² − (a + d)x + ad − bc.

• PolyLog at a large order past |z| = 1 (#412, reported by enumeratio). For an integer order above 170 and |z| > 1 the value was wrong: PolyLog(250, 4).N() was −0.25 and PolyLog(250, −4).N() was 0.25, where the values are 4 and −4 to double precision. At a larger order the evaluation was slow or did not end: PolyLog(1000, 4) stayed unevaluated after 0.2 s, PolyLog(5000, 4) ran for minutes and PolyLog(2500000, 4) did not return. Now, when the first terms of the power series give the value (the error of the partial sum has a proven bound off the branch cut), they are used: these points answer at once, also for a non-integer order (PolyLog(2500000.5, 4) stayed unevaluated after 2 s). The compiled PolyLog gives the same values: PolyLog(200, −4) was 0.25 and is now −4. A non-integer order between about 70 and 200 past |z| = 1 was also inaccurate: PolyLog(170.5, −4) was off by 0.7%, and PolyLog(200.5, 1e40) was about −1e−40·i. The first is now correct, and the second stays unevaluated, as do the other points where no method gives the value, such as PolyLog(250, 1e100). For an integer order from 8 to 170 past |z| = 1 the series is more accurate than the inversion formula used before: on the measured points where the series answers, the largest relative error went from 5e−14 to 7e−16.

  Inside the unit disk, an integer order of 8 or more now uses the power series. PolyLog(1e9, 0.99) took seconds and now answers at once, and the values are correct to the last digit (PolyLog(2500000, −0.9) was −0.9000000000000012 with precision: 'machine'). Above machine precision, PolyLog(2500000, 0.3) took 1.6 s and PolyLog(1e7, 0.3) 2.3 s; they now answer at once.

• LerchPhi and PolyLog at a very large |z| or a large order. The continuation of the Lerch transcendent past |z| = 1 returned wrong values that passed its own error check: LerchPhi(1e100, 20.25, 1).N() was −0.170, where the value is −1.29e−71, so PolyLog(20.25, 1e100).N() was −1.7e99 instead of −1.29e29. At orders from 80 to 150 and |z| from 1e8 to 1e40, values were off by up to 2.5e−10. The tail integral did not follow the oscillation of the integrand, and several factors underflowed. Now the values are correct within 1e−11, or the expression stays unevaluated: on 12,000 points checked against an independent 25- to 100-digit computation, no value is off by more than 8.8e−13 (before, 1,209 were off by more than 1e−11). For an order with a positive real part, the first terms of the series with a proven bound of the rest now give the value at once where they suffice.

0.148.0 2026-10-05

Behavior Changes

• |S| of a set or a string is its number of elements. |\{1,2\}| was Abs(Set(1, 2)), an incompatible-type error. It is now Count(Set(1, 2)), which is 2, on the LaTeX, MathJSON and ce.function routes: Abs(Set(1, 2)) is Count(Set(1, 2)). |\mathbb{Z}/5\mathbb{Z}| is 5, |\emptyset| is 0, |\mathbb{Z}| is PositiveInfinity, and |\text{abc}| is 3. For a set with no value, |S| is Count(S). A list is unchanged: |[1, -2]| is the element-wise absolute value [1, 2]. A dictionary still gives the type error. Count of a set or a string is written back as |S|. \#S, \operatorname{card}(S) and \operatorname{Card}(S) read as Count(S); \# alone, or followed by an operator (\#+1), is still the symbol hash. When the user defines Abs, |S| keeps that definition and Count is written \mathrm{Count}(S). Compiled Abs of an operand whose type could be a string or a set throws a TypeError for such a value, where it gave NaN.

• A dictionary has a LaTeX form. A dictionary serialized to an empty string, so it disappeared from LaTeX output. It is now written \operatorname{Dictionary}(\operatorname{KeyValuePair}(\text{a}, 1), …), which parses back to the same dictionary.

• Special characters in a LaTeX string are escaped. A string is written \text{…}, with \, {, }, $, %, #, &, _, ~ and ^ escaped, so that it reads back as the same string: \text{$} was read as the unit USD and \text{%} as an empty string. A string whose \text{…} form reads as a unit or a keyword (m, km, and) is written "m". The string "50%" was written as the number 50%. A control character or a private-use character is written as its code point (a tab is \char"0009{}), where it was written as it is: TeX rejects most control characters and reads a tab or a line break as a space (#345). A character value is written with the same escapes as the one-character string: # was written \text{#} and a backslash \text{\backslash }.

• A Solve result list holds all the solutions, on every route. An empty list states that there is no solution. When the solver cannot show that its list is complete, Solve stays unevaluated, expr.solve() returns null, and expr.explain('solve') ends with the step solve.incomplete-… and the unevaluated Solve as its result. Over ℝ, ℂ or with no domain, a periodic trig equation gives its principal roots: a finite list whose translates by the period give every root. When the unknown has no declared type and no domain is given, a polynomial gives all its roots, complex roots included (x^2 + 1 = 0 is [i, −i]), and every other equation gives its real solutions only (e^x = −1 and |x^2 + 1| = 0 are [], |x − 1| = 3 is [−2, 4]). Declare the unknown complex to solve such an equation over ℂ. These results change:

  • expr.solve() and explain('solve') returned [] where Solve stayed unevaluated: e^x = x + 2, x^2 = 2^x, cos x = x, arsinh(x) = 1, (x + e^x)(x − e^{−x}) = 0, x^5 + ax + 1 = 0 now give null (new step solve.incomplete-no-roots). Thus Solve(x = 1 ∨ e^x = x + 2, x) was [1] and now stays unevaluated. An Or whose alternatives all have no root gives [] (e^x = −1 ∨ e^x = −2), where it gave null. An identity (x = x) gives a list with one free parameter, as Solve does. A candidate of a root template that the host added to ce.solveRules that the check rejects no longer gives [], and it does not show that a factor of a product has no root: with the template BesselJ(0, x) → 0, (x − 1)·BesselJ(0, x) = 0 was [1], and now has no answer.
  • A trig function of an argument that is not linear in the unknown gives no answer: sin(e^x) = 0 was [] (ln π is a root), sin(x^2) = 0 was [0], sin(√x) = 0 [0], sin(ln x) = 0 [1], cos(2^x) = 0 one root, sin(1/x) = 0 [+∞, ~∞]. The harmonization rules sin(f) → f, cos(f) → f − π/2 and tan(f) → f, which gave one branch only, are removed. sin(sin x) = 0 (was [0, π]) also has no answer now.
  • An infinite value is never a root: 1/x = 0 was [+∞, ~∞], it is [].
  • A trig function of a linear argument a·x + b gives all its principal roots: sin(2x) = 0 was [0], it is [0, π/2]; cos(2x) = 0 was [π/4], it is [π/4, −π/4]; sin(x + 1) = 0 was [−1], it is [−1, π − 1]; cos(2x + 10) = 0 in degrees was [40], it is [40, −50]. tan(2x) = 1, sin(2x) = 1/2 and cos(3x − 1) = 1/2, which had no answer, now have one. a·sin x + b·cos x = 0 gives its second root: sin x = cos x was [π/4], it is [π/4, 5π/4].
  • An even power under a logarithm or a rational power keeps both signs: ln(x^2) = 2 was [e], it is [−e, e] (also over [−5, 5]); x^(2/3) = 4 was [8], it is [8, −8] (the engine gives the real value (−8)^(2/3) = 4); x^(4/3) = 16 was [8], it is [8, −8]. A logarithm of such a power is solved as it is given, not from its simplified form (2/3)·ln(x): ln(x^(2/3)) = 2 was [e^3], it is [e^3, −e^3] (also over [−30, 30]); ln(x^(4/3)) = 4 is [e^3, −e^3] and ln(x^(−2/3)) = 2 is [e^−3, −e^−3]. ln(x^(3/2)) = 3 is still [e^2].
  • A logarithm or an exponential of a linear argument is solved: ln(x + 1) = 2 is [e^2 − 1], ln(2x + 1) = 1 is [(e − 1)/2], −3ln(x + 1) = 6 is [e^−2 − 1], log_10(x + 1) = 2 is [99], ln(x + 1) + 1 = 0 is [1/e − 1], e^(x + 1) = 2 is [ln(2) − 1] and 10^(x + 1) = 5 is [log_10(5) − 1]; they were []. An exponential with a positive base has no real root when the other side is not positive: e^(x + 1) = −2 is [].
  • A constant times an absolute value is solved: 2|x| = 1 is [1/2, −1/2], 3|x| − 6 = 0 is [2, −2], |x| = e^−1 is [1/e, −1/e] and ln(x^−2) = 2 is [1/e, −1/e]; they had no answer. 2|x| + 1 = 0 is [].
  • A substitution no longer uses a symbol of the equation as its temporary unknown. A symbol with a value was used: with u := 1/2, sin(2x) + u = 0 gave [] from expr.solve(), and it is now [−π/12, 7π/12]; e^(2x) − 3e^x + 4u = 0, x^4 + x^2 − 2u = 0 and x·√(x^2 + 1) = 2u were null or [], and they now have their roots. A type of the same name no longer applies to the temporary unknown: with u: integer, 2√x + 3·⁴√x = 2 gave [], and it is now [1/16].
  • A list that a recursive strategy made from a partial list is not an answer: ln((x − 2)(x + e^x) + e) = 1 was [2]. A polynomial factor with a symbolic coefficient that gives no root is not shown to have none: (e^x − 2)(x^3 + ax + 1) = 0 was [ln 2].
  • Over a bounded domain, a list that is known to be partial is accepted only when the numeric scan resolves each root it finds: sin(x)·(x + e^x − 1 − 10^−8) = 0 over [−1, 1] was [0] (the root near 5·10^−9 was missing) and now stays unevaluated. Each periodic factor of a product gives its family in the domain: (x − 1)·cos(x) = 0 over [0, 5] is [1, π/2, 3π/2], e^x·sin(x) = 0 over [−1, 7] is [0, π, 2π], and x·sin(x) = 0 over [−10, 10] gives the seven multiples of π; they stayed unevaluated.
  • A system of congruences with no solution (x ≡ 1 (mod 4), x ≡ 2 (mod 8)) gives [] from the system solver; before, the [] came from the univariate solver, which now gives null there.
  • Over a bounded domain, the samples of the numeric scan no longer show that a list is complete: they can only find a root that is not in the list. Interval arithmetic must prove that each other part of the domain has no root. sin(x) − 2·e^(−10^12·(sin(x) − 1/2)^2) = 0 over [0.3, 0.8] was []: its two roots near π/6 are in a dip that is narrower than the distance between two samples. It now stays unevaluated. A pole in the domain does not stop the proof: the numerator of the equation written as one fraction (sin(x) − cos(x) for tan(x) − 1) is used near the pole. Only an enclosure of real values is a proof: √(sin x/(sin x − 2)) − 2·e^(−10^12·(sin x + 1/2)^2) = 0 over [3.3, 4] was [], but it has two roots near 7π/6; it now stays unevaluated.
  • In Solve, a parameter (a symbol other than the unknown, with no value) whose type is not declared is real, as in simplify(). A parameter declared complex, or with a type that excludes the reals, can be complex. The unknown keeps its own type or domain. A root whose membership in the type of the unknown is not decided is not removed as a decision: the list has no answer (step solve.incomplete-undecided when a part of the list is kept). These results change:
    • tan(2x) = a and tan(3x) = a were [] (x = 0 is a root for a = 0); they are [arctan(a)/2] and [arctan(a)/3]. cos(2x + 1) = a and sin(2x) + a = 0 were []; they give their two principal roots, with the guard |a| ≤ 1. (x − 1)(tan(3x) − a) = 0 was [1]; it is [1, arctan(a)/3].
    • The guarded roots are the same for a parameter declared real: cos(2x) = a with a: real was [].
    • For an unknown declared real: x = a, 2x = a, (x − 1)(x − a) = 0, sin(x) + a = 0 and 2|x − a| = 2 were []; they are [a], [a/2], [1, a], the two guarded principal roots, and [a + 1, a − 1].
    • With a declared complex, all these equations have no answer.
    • A polynomial that is a product of factors with a coefficient that is not a number is solved factor by factor: (x − 1)(x − a) = 0 is [1, a], not the two roots of the quadratic formula.
    • A decided check still removes a root: for an integer x, 2x = 3 is []; 2x = a has no answer (a/2 is an integer only for an even a).
  • An exponential is positive only for a real exponent. e^(ix + 1) = −e, e^(ix^2) = −1, e^(i(x + 1)) = −1 and e^(ix) = −1 were [] (x = π is a root of the first and the last); they now have no answer. e^(ax) = 2 is still [ln(2)/a] and e^(x + a) = 2 is still [ln(2) − a]; with a declared complex, they have no answer. Roots that are not real are removed: e^(x^2) = 1/2 was [±√(−ln 2)] and 2^(x^2 + 1) = 1 was [i, −i]; both are [].
  • |u| = r is split into u = ±r only when u is real for each real x. 2|x + i| = 2 was [1 − i, −1 − i] and |x + i| = 1 the same (0 is the real root): they now have no answer. |x − a| = 2 is still [a − 2, a + 2], and has no answer for an a declared complex. Roots that are not real are removed: 2|x^2 + 1| = 1 and |x^2 + 1| = 1/2 were four imaginary roots and are []; |x^2 − 1| = 3 was [2, −2, i√2, −i√2] and is [2, −2].
  • An absolute value beside other terms with the unknown is split into its two cases, and each case is solved: |x − 1| = x^5 + x was [] (a root template solved only one case), it is [0.486…]; |x| + x^5 − 3 = 0, which had no answer, is [1.133…]. |x| = x was [0], but each x ≥ 0 is a root: it now has no answer. When a case is not solved, there is no answer: |x − 1| = x + 2e^x was [] (a root near −0.27).
  • For an unknown declared complex, the strategies that suppose a real unknown give no answer: e^x = −1, sin(x) = 2, cosh(x) = 0 were [], and |x| = 1 was [−1, 1].
  • A logarithm with no root template is solved: log(x) = 2 is [100], log(x) = log(7) is [7], log_2(x + 1) = 0 is [0], log(x^2 − 3) = 0 is [2, −2], log_2(x) = 0 is [1]; they had no answer. A root no longer keeps b^(log_b(v)): log_2(x + 1) = log_2(3) was [−1 + 2^(log_2 3)], it is [2], and log_2(x + 1) = log_2(3) + 1 is [5]. A root of a logarithm with a base that is not known to be positive and different from 1 is guarded: log_a(x + 1) = 3 was [a^3 − 1], it is [a^3 − 1] with the guard 0 < a ∧ a ≠ 1; log_a(x) = 3, which had no answer, is [a^3] with the same guard.
  • A root template is built in only when it is one of the template objects of the engine, not when it has the id of one. A wrong copy of solve.linear that keeps its id ({ ...rule, replace: 7 }) no longer makes the rejection of its candidate a decision: with such copies of solve.linear and solve.power, 5x − 10 = 0 was [], it now has no answer.

• #409 The N operator gives an inexact result. A number that N returns is a float, also when its value is an integer. ["N", 2] evaluated to the exact integer 2, thus N(2)/3 was the exact rational 2/3, and ["Mean", ["N", ["List", 1, 2, 3, 4], 30]] was 5/2. Now ["N", 2] evaluates to the float 2, which serializes as {"num": "2.0"} (the same value as ce.box({num: "2.0"})), N(2)/3 is 0.666… and the mean is the float 2.5. The elements of a list or tuple result are floats too: ["N", ["List", 1, 2], 30] is ["List", {"num": "1.0"}, {"num": "2.0"}]. With a precision above machine precision, the float is a big decimal. N of a lazy collection is lazy, and each element is a float when it is read: N(Range(1, 4))/3 was [1/3, 2/3, 1, 4/3], it is now [0.333…, 0.666…, 1.0, 1.333…]. These results do not change: an infinity, and a symbolic result (the 1 in N(x + 1) stays exact). An N in the body of a Map gives inexact elements, also when the Map is compiled: Map(x ↦ N(x^2), [1, 2, 3]) is [1.0, 4.0, 9.0]. The .N() method of an expression is not changed: ce.box(6).N() is still the exact 6, and the elements of the .N() of a lazy Map keep the values of .N(). The serialization of the .N() of a lazy Map changes: the body of its function is in the engine-internal marker NumericApproximation, not in N, for example ["Map", ["Function", ["Block", ["NumericApproximation", ["Sin", "_1"]]], "_1"], ["Range", 1, 200]].

• #391 N(x, p) does not change the precision of the engine. With p above the precision, N(x, p) set ce.precision to p and left it there: after ce.box(["N", "Pi", 30]).evaluate(), ce.precision was 30, and every later evaluation computed at 30 digits. Now N(x, p) computes at p digits plus guard digits (see below) and then restores the precision of the engine and of the big decimals, also when the evaluation throws: ce.precision stays 21. The result is rounded to p digits, and toString(), .json and .latex show its p digits: N(Pi, 30) is 3.14159265358979323846264338328 (before, .json also showed the guard digits, 3.141592653589793238462643383279503). An operation on the result is computed at the precision of the engine, as for any other big decimal, and its result shows that precision: N(Pi, 30) + 1 is 4.14159265358979323846. This applies to each element of a list or tuple result, to each part of a complex result, and to each float of a symbolic result: N(x + π, 50) is x + 3.14159… with 50 digits, and N(x + π, 5) is x + 3.1416 (it was x + 3.14159265358979323846). On an engine at machine precision, the float of a symbolic result is a machine float. A float of the operand does not change: N(x + 0.1, 50) is x + 0.1. A value that does not change with the precision is displayed with the precision of the engine, because its digits past that precision are not correct: with w = (1/7).N()·sin(1).N(), computed at 21 digits, N(w, 40) showed 40 digits, 19 of them wrong, and now shows 21. The same applies to the value of a kernel that computes with machine floats. The elements of a lazy collection result are computed from the operand of N when they are read, each one at p digits by its own N: N(Map(Range(1, 3), x ↦ xπ), 30) gives 30 digits of π, 2π and 3π. The function of a lazy Map is computed by the N of each element: with N(Map(Range(1, 3), x ↦ √(x + 0.5)), 30), the first element was 1.2247448713915890491 (the 21 digits of the engine), and it is now 1.22474487139158904909864203735. This also applies to the elements of another lazy collection, such as Reverse(Map(…)), and to an infinite collection. The function of the Map keeps the variables of its scope: N(Block(c := 2, Map(x ↦ x/c, Range(1, 3)))) gives [0.5, 1, 1.5], also when the caller has a variable c. On an engine at machine precision, N(x, p) with p at or below 15 gives a machine float: N(1/3, 11) gave a big decimal, because it is computed at 16 digits. A function literal remembers its values for each precision, also the values of an exact evaluation (which can be a float): with f(t) := N(π) and g(t) := f(t), N(g(1), 50) after N(g(1)) showed the 21-digit value of f(1). A sequence remembers its numeric terms for each precision, apart from its exact terms: with B_n = π·B_{n−1}/n, N(B_10, 50) after N(B_10) was the 21-digit term 0.025806891390014060012598262 (wrong past the 22nd digit), and it is now 0.025806891390014060012598294252898849657186441048147; an exact evaluation of B_10 after .N() gave the float term. N(x, p) computes with guard digits (5, or p/10 for a large p) and then rounds to p digits. Before, the last digit could be wrong: N(Exp(i), 30) had the real part 0.540302305868139717400936607444, now …607443, and N(Gamma(1/3), 30) ended with …098, now …097. A complex result of N(z, p) with p at or below the precision is rounded as a big decimal: the parts were rounded to machine floats, and N(Exp(i), 20) was (0.5403023058681398 + 0.8414709848078965i). It is now (0.5403023058681397174 + 0.84147098480789650665i). A float above 1.8e308 is rounded as a big decimal too (it was not rounded), and an imaginary part below 5e-324 is kept (it was dropped). An infinite sum with a closed form is computed from the closed form when the precision is above machine precision: N(Sum(1/k^2, k=1..∞), 30) was 1.6449340668482313 (from an extrapolation with machine floats, 14 correct digits), and it is now 1.64493406684822643647241516665. An infinite sum with no closed form is still computed with machine floats, and shows the 17 digits of its value. N(Measurement(v, δ), p) keeps the uncertainty and rounds v: N(Measurement(1.23456, 0.01), 3) was 1.23, it is now Measurement(1.23, 0.01). Thus the numeric value of an integral, N(Integrate(…), p), is a Measurement, not a plain number. The .N() method of a lazy Map whose function reads a variable of another scope now reads that variable: Block(c := 5, Map(x ↦ x/c, Range(1, 300))).N() was [1/c, 2/c, …], it is now [0.2, 0.4, …].

• N(list, p) rounds each element. With a precision p at or below the working precision, N rounded a number result to p significant digits, but not the elements of a list or tuple result: N([1/3, Pi], 4) was [0.333333333333333333333, 3.14159265358979323846]. It is now [0.3333, 3.142], also in nested lists and tuples.

• A set is never paired by position in an element-wise operation. A set has no order, but beside a list it supplied elements by position and the result was cut to the shorter length: Power([1, 2, 3], Set(1, 2)) was [1, 4]. A set at a number parameter now gives the incompatible-type error that Power(Set(1, 2), 2) already gave. At a parameter that is not a number, the set is used whole in each element: String([1, 2], Set(3, 4)) is a list of two strings, and Less([1, 2, 3], Set(1, 2)) is a list of three comparisons that stay unevaluated. Power([1, 2], Interval(0, 1)) gave [1, 1.189…] from the endpoints of the interval, and it is now the same error.

• Numeric functions give a type error for a set operand. Power and Sin gave incompatible-type for a set, but 82 other numeric functions, such as Floor, Mod, Arctan2, Gamma and Zeta, stayed unevaluated, and IsPrime(Set(1, 2)) was False. They all give the error now. A list still broadcasts: Floor([1.5, 2.5]) is [1, 2].

• Solve of a computed collection is a type error. Solve(Range(1, 3), x) and Solve(Linspace(a, 0, 3)) were [], a false statement that there is no solution. A first operand that is a collection but is not a written list, set or tuple of equations (Range, Linspace, Map, a symbol that holds a list) now gives an incompatible-type error. A written list is a system of equations, as before.

• Error functions and trigonometric integrals on the imaginary axis. Erf, Erfc, Erfi, SinIntegral, SinhIntegral, CosIntegral and CoshIntegral of an argument exactly on the imaginary axis (an exact imaginary number such as 2i, or a float complex number with a real part of 0) are computed with the identities erf(iy) = i·erfi(y), erfc(iy) = 1 − i·erfi(y), erfi(iy) = i·erf(y), Si(iy) = i·Shi(y), Shi(iy) = i·Si(y), Ci(iy) = Chi(|y|) ± iπ/2 and Chi(iy) = Ci(|y|) ± iπ/2, from the real kernels:

  • The part that is a constant is exact. Erf(i).N() was 2.22e-16 + 1.6504i, it is now 1.6504i; Erfc(i).N() was 0.9999999999999998 − 1.6504i, it is now 1 − 1.6504i.
  • Above machine precision, the other part of Erf, Erfc and Erfi has the working precision: at 50 digits Erf(i).N() is 1.650425758797542876025337729561362443895679874874i, was a double.
  • A value past the number range stays unevaluated instead of NaN: a complex value that is too large has a direction that no infinity of the engine holds, the rule Gamma already follows. This applies to Erf(27i) and Si(1000i) at machine precision, Ci(1000i), and Erf(10^{10}i) at every precision. A real overflow is still +∞ (Erfi(27)).

• Bessel and Airy functions of a complex argument. BesselJ, BesselY, BesselI, BesselK (integer order) and AiryAi, AiryBi, AiryAiPrime, AiryBiPrime compute a value at a complex argument under .N(), and under evaluate() when the argument is a float. They stayed symbolic: BesselJ(3, 0.7i).N() was BesselJ(3, 0.7i), it is now -0.007367373607628006i. BesselY and BesselK also have a value on the negative real axis, their branch cut: the limit from above, as for Ln(-2) (BesselY(0, -2).N() is 0.5103756726497453 + 0.4477815582824712i). On the imaginary axis J_n(iy) = iⁿ·I_n(y) and I_n(iy) = iⁿ·J_n(y), so one part is exactly 0. Measured against mpmath at more than 2,000 points: a relative error of at most 7·10⁻¹⁴ for J, I and K, 4·10⁻¹⁴ for Y away from its zeros, 5·10⁻¹⁴ for the Airy functions up to |z| = 40 and 3·10⁻¹³ at |z| = 120 (the conditioning of the function there). A non-integer order stays symbolic. The result type of BesselY and BesselK of a real argument that is not positive is now number, not real. Compiled code computes these functions for a real argument only: a complex operand is refused (compiled BesselI(0, z) gave 1 at z = 1 + i).

• LambertW on every branch, and Sinc, FresnelS, FresnelC of a complex argument. .N() (and evaluate() of a float argument) computes W_k(z) for every integer branch k and every complex z, with the branch cuts of mpmath's lambertw (on a cut, the limit from above, as for Ln). A real argument outside the real domain of a branch has its complex value: LambertW(-1).N() is -0.31813 + 1.33724i and LambertW(0.5, -1).N() is -2.25916 - 4.22096i (they stayed symbolic). W_k(0) = -∞ for k ≠ 0. Sinc(1+2i).N() is 1.41700 - 0.87439i. On the imaginary axis Sinc(iy) = sinh(y)/y, FresnelS(iy) = -i·FresnelS(y) and FresnelC(iy) = i·FresnelC(y), with the constant part exact. Against mpmath, the largest relative error is 6.6·10⁻¹⁶ for LambertW (31,000 points, branches −3 to 3) and 4.3·10⁻¹⁶ for Sinc; for the Fresnel integrals it is 4·10⁻¹⁶ for |z| < 1 and 5·10⁻¹⁴ for |z| < 10, and further out it follows the condition number π|z|².

• LambertW(-1/e) and LambertW(-1/e, -1) are exactly -1, under evaluate() as well as .N(), at every precision. evaluate() left them unevaluated. At machine precision .N() gives -1, not the complex value of the double just below -1/e.

• The type of LambertW(x) for a real x is real only when the branch and the range of x prove it (W₀ on [−1/e, ∞], W₋₁ on [−1/e, 0)). Otherwise it is number: LambertW(-1) was typed real.

• solve keeps a root from a LambertW template only when it is real. The templates that solve x·eˣ = b and the related shapes give the real roots: W₀ when its argument is in [−1/e, ∞), and W₋₁ for the second real root when its argument is in [−1/e, 0). Outside these intervals W₀ and W₋₁ now have complex values, which are two of infinitely many complex roots: solve returns the real roots of these equations, as it does for sin(x) = 2. So x·eˣ = 3 still solves to W(3) only, and x·eˣ = -1 has no root.

• A parameter declared as a scalar maps over a tuple argument. Apply((u: real) ↦ 2u, (a, b)) was an incompatible-type error. It is now (2a, 2b), as Sin((a, b)) is (sin(a), sin(b)), and Apply((u: real) ↦ 7, (a, b)) is the tuple (7, 7). A scalar is a number type or boolean, as Not((True, False)) is (False, True): (p: boolean) ↦ ¬p applied to (True, False) is (False, True). A parameter declared string binds a tuple whole. The rule is the same for a function assigned the literal with no declaration, and for a function declared function and then assigned. A signature declared for the function also declares its parameters: k declared (real) -> real and assigned u ↦ 7 gives k((a, b)) = (7, 7). A list of tuples maps over the list, then over each tuple, and a tuple of tuples maps at each level. A component that does not match the declared type is an error in its own cell: (n: integer) ↦ n + 1 applied to (1.5, 2) gives (Error(incompatible-type, …), 3). A tuple with a list component (([1, 2], [3, 4])) is an incompatible-type error, as for Sin. A symbol declared as a tuple with no value holds the call, as a list symbol does, and the call maps when the symbol gets a value. The type of the call is the tuple of the results (tuple<real, real>). A parameter with no declared type (u ↦ 7) still binds the tuple whole. The compiled code computes the same tuple: an array in JavaScript, an array of enclosures on the interval target, and a vecN on GLSL and WGSL (a tuple of booleans is declined there, because a shader vector holds numbers). Before, these targets declined the call, and k((a, b)) for k declared (real) -> real and assigned u ↦ 2u compiled to NaN in JavaScript and to an empty interval on the interval target. A list beside the tuple, at a parameter declared as a collection, is bound whole in each cell: (u: real, w: list<real>) ↦ u + Length(w) applied to (1, 2) and [10, 20] is (3, 4), typed as a tuple, and the JavaScript target compiles it. A failing component carries the same "while applying … element-wise" note as a failing element of a list.

• The norm of a point with list components of different lengths is an error, and the norm of an empty point list is []. With L = [1, 2, 3] and M = [], Norm((L, M)) and Abs((L, M)) are the incompatible-dimensions error that Hypot(L, M) and L + M give. Before, they were [], as if the empty component made the other list not count. The L∞ norm of such a point, which stayed unevaluated for any length mismatch, gives the same error. PointList(L, M) zips to the shortest component and is a list of zero points, so Norm(PointList(L, M)) is now [], one norm per point, as its type list<number> says. Before, it was 0, the norm of an empty vector.

• A function literal applied to a list symbol with no value stays unevaluated. With v declared list<real^2> and no value, Apply(u ↦ 7, v) has the type of a list of two numbers, but it evaluated to 7, and Apply((u: real) ↦ 2u, v) was an incompatible-type error. Both now stay unevaluated, as they do when v is one of several arguments. When v gets a value, the application maps over it: v := [3, 4] gives [7, 7] and [6, 8].

• A declared user function applied to a list with no value stays unevaluated. With f declared (real) -> real and assigned u ↦ 7, and w declared list<real> with no value, f(w) evaluated to 7 while its type was list<real>. A body that reads its parameter (u ↦ u + 1) gave w + 1. The call now stays f(w), as it does for a function assigned with no declaration and for a function literal (Apply(u ↦ 7, w)). After w := [1, 2], it evaluates to [7, 7]. A scalar argument is applied as before.

• A function assigned without a declaration holds a call on an argument that can still become a list. With h := u ↦ 7 and q declared real | list<real> with no value, h(q) evaluated to 7, which is wrong after q := [1, 2] (the call then gives [7, 7]). The call is now held, as f(q) for a function f declared function and Apply(u ↦ 7, q) already were, and it maps when q gets a list value. A q declared real, or with a scalar value, is applied at once, as before.

• A call on a set or a dictionary with no value is no longer held. With s declared set<real> (or dictionary<real>) and no value, Apply(u ↦ 7, s) and f(s) for a function f declared function and assigned u ↦ 7 stayed unevaluated, while h(s) for h := u ↦ 7 gave 7. A call is held only so that it can map over the argument when the argument gets a list value, and a set or a dictionary is never mapped over. The three calls now give 7, and with u ↦ u + 1 or a real parameter they give the same incompatible-type error. An argument that can still be a list (list<real>, real | list<real>, broadcastable<real>, collection<real>) is held on all three routes, as before.

• Apply of a function literal gives the error for an argument that its parameter refuses. Apply((u: real) ↦ 7, "abc") evaluated to the inert Apply((u) => 7, Error(incompatible-type, …)), while h("abc") for h := (u: real) ↦ 7 gave the error itself. The Apply now gives the same Error(incompatible-type, …), for a string, a boolean or a set. A cell of a list mapped by a function declared function and assigned such a literal also gives the error, not the inert Apply.

• A string or a boolean at a declared real parameter makes the call invalid on every route. h("abc") for h := (u: real) ↦ 7 was refused when it was boxed, typed error, while f("abc") for f declared function and Apply((u: real) ↦ 7, "abc") were typed integer until they were evaluated. The three calls are now refused when they are boxed.

• The type of Apply(u ↦ 7, s) for s declared collection<real> or broadcastable<real> is broadcastable<…>. The call is held until s has a value, and it maps over a list value, but it was typed as one value (number). The named calls h(s) and f(s) were already typed broadcastable<…>, and now the three agree.

• Derivative(f) is a function literal also when its closed form holds the derivative of another function. With h(x) := g(x)·x and g not defined, Derivative(h) evaluated to the expression _ · Apply(Derivative(g, 1), _) + g(_) in the hole _, where Derivative(Sin) evaluates to (x) ↦ cos(x). It now evaluates to (x) ↦ x · Apply(Derivative(g, 1), x) + g(x). The derivative of a function literal (Derivative((x) ↦ x·g(x))) gave the body alone, with x free; it now gives the literal too. h'(t), Apply(Derivative(h), t) and their value once g is defined do not change. The derivative of a recursive function (f(x) := x·f(x−1)), whose closed form holds Derivative(f) itself, stays the inert Derivative(f): as a function literal, its body would evaluate Derivative(f) again at each application.

• A short numerator over a long denominator serializes as \frac. \frac{x}{y^2+z^2} serialized as (x)(y^2+z^2)^{-1}, and \frac{1}{(x+1)(x+2)} as ((x+1)(x+2))^{-1}. The default fraction style wrote an inverse power when the numerator had at most two leaves and the denominator more than five. With a numerator other than 1, that output parsed back to Multiply(x, Divide(1, …)), not to Divide(x, …). The default style now writes \frac{x}{y^2+z^2} and \frac{1}{(x+1)(x+2)}. A power -1 of a long base serializes the same way: \frac{1}{x-\frac{\pi}{2}}, not (x-\frac{\pi}{2})^{-1}. The reciprocal fraction style still writes the inverse power when a caller selects it. In that style, the numerator gets parentheses only when it needs them (a sum, a negative value): x\times(y^2+z^2)^{-1}, 2(y^2+z^2)^{-1}, (x+1)(y^2+z^2)^{-1}, (-y)(y^2+z^2)^{-1}, not (x)(y^2+z^2)^{-1} or (2)(y^2+z^2)^{-1}. A symbol or a decimal number before the inverse power gets an explicit \times: x(y^2+z^2)^{-1} parses as a call of x, and 1.5(2)^{-1} as the inverse of the repeating decimal 1.5222…. The output parses back to Multiply(x, Divide(1, y^2+z^2)), which evaluates to the same expression as Divide(x, y^2+z^2).

• A negative number at the start of a difference has no parentheses. -x-1 serialized as (-1)-x, and \frac{-x-1}{y} as \frac{(-1)-x}{y}. They now serialize as -x-1 and \frac{-x-1}{y}: the terms also keep their order, and the prettified MathJSON of -x-1 is ["Subtract", ["Negate", "x"], 1], not ["Subtract", -1, "x"]. A raw ["Subtract", -1, "x"] serializes as -1-x. A negative number after a minus sign keeps its parentheses (x-(-1)). A sum or a difference subtracted inside a sum now has parentheses: Add(y, Negate(Add(a, b))) serialized as y-a+b, which is a different value, and now serializes as y-(a+b). toString() keeps the order too: -x - 1, not -1 - x. Only a positive constant moves in front of a negated term (1 - x).

• The timeLimitMs option of loadIntegrationRules also sets the time limits of the rule driver's sub-searches. The step budget decides when the driver gives up, and the time limits only guard against a hang. But the two sub-searches of the driver (the native rational fallback and the clean-up of an expansion result) had a fixed limit of 5 s. On a slow or loaded machine, that limit could stop a sub-search that the step budget allowed, and a caller could not raise it. Each sub-search now gets one sixth of timeLimitMs: 5 s for the default of 30 s, as before. On a heavily loaded machine, a caller can raise timeLimitMs so that no time limit decides the result. For example, ∫(3+2x)³·csc(1+2x)/(1+sin(1+2x)) dx needs about 64,000 of the 300,000 steps and takes about 4 s on an idle machine; with a load average of 300 on 8 cores, the 30 s limit stopped it and the integral stayed unevaluated.

• The integration rules wait longer before they give up on the clock. The default timeLimitMs of loadIntegrationRules() is now 120 s (it was 30 s), and each of the two bounded sub-searches of the rule driver gets one sixth of it (20 s). The step budget, which does not depend on time, decides when the driver gives up; the clock is only a guard against a hang. With 30 s, a machine 2 to 3 times slower than an idle desktop, or a loaded one, could stop a search that the step budget allows, and the integral stayed unevaluated there while it closed on a faster machine: for example \int(3+2x)^3\csc(1+2x)/(1+\sin(1+2x))\,dx needs about 4 s on an idle machine. A real hang now blocks for up to 2 minutes before the integral is given up; set a lower timeLimitMs to change that.

• In the lenient grammar, a number after √ is the whole radicand. √12 was read as √1·2 and √2.5 as √2·0.5, the TeX reading of a radicand without braces. They are now √(12) and √(2.5), as sqrt12 and sqrt2.5 are. The number ends the radicand: √12x is √12·x. Because the TeX reading differs, a number of more than one token reports ambiguous-radical (√2 does not). The strict grammar keeps the TeX reading: \sqrt12 and √12 are \sqrt{1}2.

• In the lenient grammar, an unbraced subscript of digits is the whole run of digits on every base. π_12 was π_1·2 and (x)_12 was (x)_1·2, while x_12 was x_{12}. They are now π_{12} and (x)_{12}, and 2748_16 is the number 2748 in base 16. On a base that is not a symbol, a run that starts with 0 keeps the one-token reading, because the number value drops the zero: (x)_01 is (x)_0·1, and it reports ambiguous-implicit-subscript. A symbol keeps the text: π_01 is Pi_01, as x_01 is x_01.

• More ambiguous-* parse diagnostics in the lenient grammar. These inputs were read with no diagnostic, although a person can mean a second reading. The reading does not change, and the strict grammar reports none of these codes:

  • ambiguous-letter-decimal: a period and digits after an operand that is not a letter (√b.5, e^f.5, (x+1).5, |x|.5), and after a superscript of digits (x².5, x^{2}.5). The Desmos spellings t^{i}.4 and \left(1-t\right).9 are not reported.
  • ambiguous-implicit-subscript: a Greek base (Δ_a2, as x_a2), and a constant or a group (π_a2, π_1y, (x)_a2). Also an unbraced subscript of two letters that is split because a script or a digit follows it: x_ab^2 is x_a·b^2 and a_kx^k is a_k·x^k, and a person can mean x_{ab}^2, the reading of x^2_ab.
  • ambiguous-radical: a number after √, white space and a number (√1 000 is √1·0).
  • ambiguous-number-notation: a number with a subscript of digits as the argument of a function name with no parentheses (min3_12 is min(BaseForm(3, 12)), sin2_8), as 10111_2 is reported.
  • ambiguous-factorial: a radicand or an exponent with a subscript (√i_1!, as √i!), and a ! after white space after the argument of a function name (αtanπ ! is α·(tan π)!, while tan x! is tan(x!)).
  • ambiguous-radical: a radicand with a subscript then white space (√a_1 b, as √a b), and a run of letters read one letter at a time before √ (xy√i, as y√i).
  • ambiguous-exponent-end: √ after an exponent (e^θ√z, x^√θ√g), and a function name after an exponent and white space when the exponent is not signed (e^x cos(x), e^x sin x). Before, a function name was reported only after a signed exponent (e^-x sin x); the same rule now applies to every operand after the exponent. x^2 sin x and e^{x} cos(x) are not reported.
  • ambiguous-function-subscript: a base of log that is the 0 of a longer run of digits (log_01(x), log012(x)). The base is 0, as in the strict grammar \log_01(x), and a person never means base 0.

  See docs/plans/2026-10-01-lenient-ambiguity-codes.md, section 5.

• In the lenient grammar, a braced -1 on a function name is the inverse function, as the unbraced -1 is and as \sin^{-1} is in LaTeX. sin^{-1}(x) and sin⁻¹(x) (the tokenizer reads ⁻¹ as ^{-1}) were 1/sin(x). They are now arcsin(x), and they report ambiguous-inverse-function, as sin^-1(x) does.

• In the lenient grammar, the inverse of a logarithm is a power, as in the strict grammar. ln^-1(x) was Apply(InverseFunction(Ln), x), which does not simplify, and is now exp(x); log^-1(x) and lg^-1(x) are now 10^x. The code ambiguous-inverse-function is still reported.

• In the lenient grammar, a letter followed by digits that start with 0 keeps the digits. x01 was x_1 and x012 was x_12: the number value of the digits dropped the zero. They are now the symbols x_01 and x_012, as x_01 is. A run with more digits than a JavaScript number holds exactly is kept the same way: x123456789012345678901 is the symbol x_123456789012345678901. These still report ambiguous-implicit-subscript.

• A subscript on cbrt or nPr keeps the meaning of the name, in the lenient grammar. cbrt_2(x) was Apply(Subscript(Root, 2), x), which lost the index 3 of the cube root, and is now Apply(Subscript(Root, 2), x, 3). nPr_2(5, 2) was Permutations_2(5, 2), a subscript on the collection of the arrangements, and is now Subscript(Binomial(5, 2)·2!, 2), a subscript on the count. Both still report ambiguous-function-subscript.

• Argument (Arg) gives the angle in the engine's angular unit, as the inverse trigonometric functions do. With ce.angularUnit = 'deg', Argument(-1) was π (a value in radians) while Argument(i) was 90, and the compiled Argument always returned radians. Now Argument(-1) is 180 in degrees, 200 in grads and 1/2 in turns, for evaluate(), .N() and a float operand (Argument(-5.1) is 180), and AbsArg takes the same angle. The compiled code on the javascript, interval-js, glsl, wgsl and python targets multiplies the radian result by the unit factor, so it agrees with .N(). Radian mode does not change, and neither do Ln, Sqrt and ComplexRoots of a complex value, which compute their angle in radians.

• A Solve over a bounded domain whose list of roots would be too long to give now stays unevaluated, instead of returning a part of the list. When the domain holds more than 1000 periods of a trigonometric equation, the solver does not expand the roots. It returned the principal roots that are in the domain: Solve(sin(x) = 0, x ∈ [0, 10^9]) was [0, π], Solve(cos(3x) = 1/2, x ∈ [0, 10^7]) was [π/9], and Solve(sin(x) = 0, x ∈ [10^8, 10^9]) was [], a false statement that there is no root. These now stay unevaluated. An integer Range that is small enough is enumerated instead: sin(x) = 0 over -10^4..10^4 is still [0]. The same rule applies to an equation that has the unknown also outside a trigonometric function of a linear argument: its list of roots is given only when a numeric scan of the domain finds no other root. Solve(x·sin(x) = 0, x ∈ [-10, 10]) was [0, π] and Solve(sin(x^2) = 0, x ∈ [0, 5]) was [0]; both now stay unevaluated, while Solve(x·sin(x) = 0, x ∈ [0, 3]) is still [0]. Over the real line (x ∈ ℝ) and the complex numbers, Solve still gives the principal roots. Over another domain that is not bounded, the principal roots were given as if they were all the roots: Solve(sin(πx) = 0, x ∈ ℤ) was [0], but each integer is a root. Such a Solve (over ℤ, a half-line such as [0, ∞), or a union) now stays unevaluated, and a finite set is enumerated: Solve(sin(πx) = 0, x ∈ {0, 1/2, 1}) was [0] and is now [0, 1]. An equation with the unknown in a function that the solver cannot invert, and that the numeric scan cannot check, also stays unevaluated over a bounded domain: over [0.5, 20], (x - 1)·Haversine(x) = 0 and (x - 1)·Sinc(x) = 0 were [1], and over [0.5, 10], (x - 1)·BesselJ(0, x) = 0 was [1]. The functions that the solver inverts are the arithmetic operations, powers and radicals, exponentials and logarithms, Abs, the hyperbolic functions and the inverse trigonometric functions. The haversine is now a periodic term, as sin is: (x - 1)·Haversine(x) = 0 is [1] over [0.5, 6], and (x - 1)·Haversine(πx) = 0 over -4..4 was [1] and is now [-4, -2, 0, 1, 2, 4].

• Solve no longer gives a part of the roots as the answer, also without a domain and over ℝ. The principal roots are the answer only for a trigonometric equation, because they represent its periodic families of roots. For a function that the solver cannot invert and that is not periodic, the list of the solver is only a part of the roots: Solve((x - 1)·BesselJ(0, x) = 0, x) was [1], and now stays unevaluated, also with x ∈ ℝ. A product is solved factor by factor, and a factor that gives no root can have roots that the solver does not find: Solve((x - 2)(x + e^x) = 0, x) was [2], but x + e^x = 0 has the root −W(1) ≈ −0.567. Such a Solve now stays unevaluated, without a domain, over ℝ and over [−1, 3]. A factor that is shown to have no real root does not change the answer: a polynomial (x^2 + 1), a power of a constant (e^x), or a function of known sign (e^x + 1). Thus (x - 2)(x + 3), (x - 2)·e^x, (x - 2)(x^2 + 1) and x·ln(x) give [2, −3], [2], [2] and [1] as before. Over a bounded domain, the numeric scan can still show that the list is complete: (x - 1)·Haversine(x) = 0 over [0.5, 6] is [1]. The method expr.solve() follows the same rule and returns null for these equations: .solve("x") of (x - 1)·BesselJ(0, x) returned [1], and of (x - 2)(x + e^x) it returned [2]. For a list of equations in one unknown, Solve keeps the candidates that are roots of every equation; when each equation gives only a part of its roots, it now stays unevaluated: Solve([(x - 2)(x + e^x) = 0, (x - 2)(x + e^x)(x + 5) = 0], x) was [2], but −W(1) is also a common root. expr.solve() of an Or returns null when one of its alternatives has no answer: the union of the other lists is not an answer (x = 3 ∨ (x - 1)·BesselJ(0, x) = 0 was [3, 1]). expr.explain('solve') gives the same result: for these equations its last step is "No complete answer", with the reason (a factor gave no solution, or the unknown is in a function that the solver cannot invert), and its result is the unevaluated Solve. A root template that you add to ce.solveRules is your claim of a solution, and its roots are still returned for a function that the solver cannot invert: with a template BesselJ(1, x) → 0, Solve((x - 1)·BesselJ(1, x) = 0, x) is [1, 0].

• In the lenient grammar, a Greek name written without a backslash takes a subscript into its name, as the backslash spelling does. alpha_{01} was alpha_1 and is now the symbol alpha_01, as \alpha_{01} is. alpha_max was alpha_m·a·x and is now alpha_max. pi_01 was Pi_0·1 and is now Pi_01. The order of the scripts does not change the reading: alpha^2_{01} and alpha_{01}^2 are both alpha_01^2. The rules and the diagnostics are the same as for the backslash spelling: a run of two letters before a script is split (alpha_ab^2 is alpha_a·b^2, and reports ambiguous-implicit-subscript), and white space before the _ stops the join (alpha _{01} is alpha_1). The ASCII names of constants keep the subscript outside the name: oo_{01} is still Subscript(PositiveInfinity, 1). The strict grammar does not change.

• In the lenient grammar, a spelled-out name is a whole unbraced exponent also when a _ or a digit follows it, as the command spelling is. x^alpha_1 and e^alpha_1 were read one letter at a time (x^a·l·p·h·a_1), and are now x^{alpha_1} and e^{alpha_1}, as x^\alpha_1 is. x^alpha2 is x^alpha·2 and x^oo_1 is x_1^∞, as x^\alpha2 and x^\infty_1 are. An operand after the exponent reports ambiguous-exponent-end (x^alpha_1 t), as after x^\alpha_1 t.

• In the lenient grammar, a script, a prime or a digit after a run of letters belongs to the last part of the run when the run holds a spelled-out name, as after a run of single letters (xy_1 is x·y_1, xy2 is x·y_2). xalpha_1 was Subscript(x·alpha, 1) and is now x·alpha_1, as x\alpha_1 is; xalpha^2 was (x·alpha)^2 and is now x·alpha^2; alphax_1 is alpha·x_1. xalpha2 was x·alpha·2 and is now x·alpha_2, as x\alpha2 and alpha2 are, and xalpha01 was x·alpha·1 and is now x·alpha_01. The run still reports ambiguous-letter-run, and a digit subscript reports ambiguous-implicit-subscript.

• In the lenient grammar, a spelled-out Greek name directly after an argument of one letter is read as the name, as after a letter in a run of letters. The argument is an unbraced exponent, radicand or argument of \frac. e^xalpha_1 was e^x·a·l·p·h·a_1 and is now e^x·alpha_1, as e^x\alpha_1 is; \sqrt xalpha_1 is \sqrt{x}·alpha_1; e^xpi was e^x·p·i (with the imaginary unit) and is now e^x·π. The codes do not change: e^xalpha_1 reports ambiguous-exponent-end and ambiguous-letter-run. A bare function name after the argument is read as the function, as after white space: e^xsin(t) was e^x·s·i·n(t) and is now e^x·sin(t), with ambiguous-exponent-end, as e^x sin(t) is. A word that starts at the argument is not split: \sqrt beta stays \sqrt{b}·e·t·a, x^foo stays x^f·o·o and x^acos t stays x^a·c·o·s·t.

• A symbol whose subscript a dictionary entry reads has the same reading in each order of the scripts, in both grammars. \delta^2_{01} was delta_01^2 and \mu^2_0 was mu_0^2; they are now KroneckerDelta(0, 1)^2 and Mu0^2, as \delta_{01}^2 and \mu_0^2 are. \varepsilon^2_0 is VacuumPermittivity^2. The same applies to the set notations with a subscript: \R^2_- was a syntax error and \mathbb{R}^2_{>0} was Subscript(RealNumbers, Error)^2; they are now NegativeNumbers^2 and PositiveNumbers^2, as \R_-^2 and \mathbb{R}_{>0}^2 are. White space before the _ does not change this reading, as LaTeX ignores it: \R^2 _- was a syntax error and \mu^2 _0 was mu_0^2; they are now NegativeNumbers^2 and Mu0^2. (White space before the _ still stops the join of a subscript to a symbol name: x^2 _{01} is x_1^2.) A script or a prime after \delta_{ij} now applies to KroneckerDelta(i, j): \delta_{01}^2 was a syntax error.

• \delta_{11} is KroneckerDelta(1, 1). A subscript of KroneckerDelta that is a run of digits with no separator is one index for each digit, as a run of letters is (\delta_{ij} is KroneckerDelta(i, j)). The run was one number: \delta_{11} was KroneckerDelta(11), which evaluates to 0 (δ₁₁ is 1), \delta_{01} was KroneckerDelta(1) (the 0 was lost) and \delta_{12} was KroneckerDelta(12). They are now KroneckerDelta(1, 1), KroneckerDelta(0, 1) and KroneckerDelta(1, 2). A subscript with a separator keeps its numbers: \delta_{10, 2} is KroneckerDelta(10, 2), and KroneckerDelta(10, 2) serializes to \delta_{10, 2}. One index of two or more digits serializes in a second group, which reads back as one index: KroneckerDelta(11) is \delta_{{11}}. A group of letters and digits is one index for each, in both grammars: \delta_{n0} is KroneckerDelta(n, 0); in the lenient grammar it was KroneckerDelta(n_0), and \delta_{nm} no longer reports ambiguous-letter-run. A command that is a symbol is one index: \delta_{\pi1} is KroneckerDelta(Pi, 1) in both grammars (in the lenient grammar it was KroneckerDelta(Pi_1)). In the lenient grammar, a spelled-out Greek name is one index, as its command is: \delta_{nalpha} was KroneckerDelta(n·alpha) and is now KroneckerDelta(n, alpha), with ambiguous-letter-run; \delta_{pi1} was KroneckerDelta(Pi_1) and is now KroneckerDelta(Pi, 1). The strict grammar reads one index for each letter. In the lenient grammar, an unbraced run of digits is the whole subscript, as x_11 is x_{11}, and is read as the braced run: \delta_11 was KroneckerDelta(1)·1 and is now KroneckerDelta(1, 1); \delta_01 was the symbol delta_01 and is now KroneckerDelta(0, 1). The strict grammar reads one token, as TeX does. \delta_0 was the symbol delta_0 and is now KroneckerDelta(0), as \delta_1 is KroneckerDelta(1).

• evaluate() returns an exact sequence term that is not a number. A term of a sequence declared with ce.declareSequence() or a_n := … was returned only when it was a single number literal; any other term stayed unevaluated. For B_n = π·B_{n−1}/n, B_0 = 1, B_10.evaluate() was B_10 and is now π^10/10! (1/3628800·π^10); for a_n = a_{n−1} + √2, a_0 = 1, a_10.evaluate() was a_10 and is now 1 + 10√2. .N() gives a float, as before. A term with a free symbol is also returned: for c_n = x·c_{n−1} + 1, c_0 = 1, c_3.evaluate() is x(x(x + 1) + 1) + 1. A term with more than 250 nodes stays unevaluated (c_62 is returned, c_63 is not), and so does a term whose type is not a number. ce.getSequenceTerms() returns these terms too: getSequenceTerms('a', 0, 3) was undefined, and it is now [1, 1 + √2, 1 + 2√2, 1 + 3√2]; it is still undefined when a term in the range has no value, or when a term is not a number (Undefined). The terms of a sequence whose name is not one letter (alpha, fib) are read correctly: getSequenceTerms() read alpha_{0} as a product of letters. ce.checkSequenceOEIS() looks up the numeric values of the terms, from the first index of the domain of the sequence. It reports an error when a term has no numeric value (a term with a free symbol), when a term is complex or not an integer (i·Z_{n−1}, a_{n−1} + √2: OEIS holds integer sequences, and the real part was sent), and when a term of a defined sequence has no value (it reported that the sequence was not defined). ce.getSequenceCache() returns only the exact terms, with the index as the key: after F_6.N() it held the terms F_2 … F_6 that .N() computed, and it is now empty. The terms that .N() computes are kept apart, for each precision, and ce.clearSequenceCache() clears them too. Three more changes to the terms of a sequence:

  • A memoized term is computed again when a symbol that the recurrence or a base case reads changes. With c_n = x·c_{n−1} + 1, c_0 = 1 and x := 2, c_3 is 15; after x := 3, c_3 was still 15 (from the memo), and it is now 40. An assumption or a change of the precision also makes the memo outdated; an assignment to a symbol that the recurrence does not read does not. A term with a free symbol, such as x(x(x + 1) + 1) + 1, is now memoized too.
  • A cold read of a large index no longer overflows the call stack: the lower terms are computed on demand, with a stack of the terms whose computation is not complete, so the depth of the calls does not grow with the index. F_500.evaluate() and F_500.N() for the Fibonacci recurrence threw RangeError: Maximum call stack size exceeded on a new engine; F_2000 now gives its 418 digits. This applies to every pure recurrence, also one with two indices (P_{400,200} of the Pascal recurrence) and two sequences that read each other. A recurrence with side effects (an assignment, Print, Random(), a function declared pure: false) is computed recursively, and each term is computed one time, so each side effect occurs one time for each term; a cold read that needs more than 100 nested levels of such a recurrence stays unevaluated (read lower indices first). Only the terms that the recurrence reads are computed: with A_n = If(n = 1000, 7, A_{n−1} + Random()), A_1000 no longer computes A_1 … A_999 and consumes no draw, and with the index If(n < 10, 0, n − 1), A_5 reads A_0 only. A constraint that reads the sequence no longer throws an internal value, and a constraint that reads the term it checks no longer overflows the call stack. A recurrence that reads a symbol whose value has a side effect (a value Assign(c, c + 1) or Random(), evaluated at each read) is treated as a recurrence with side effects. The memoized terms become outdated when a symbol that the constraints read changes, or when a symbol that another sequence the recurrence reads depends on changes: with A_n = B_n and B_n = B_{n−1} + x, after x := 2, A_2 was still 2, it is now 4. A term whose computation changes a symbol that the recurrence reads is computed one time in a read (before, a_10 of such a Fibonacci recurrence took 5157 computations). A read that needs more than 100,000 terms, an index that is not a safe integer (A_{10^20}, which did not stop), and a term that reads itself stay unevaluated. The read stops when the time limit of ce.withTimeLimit() expires.
  • A base value is evaluated when it is read: with b_0 = s, b_n = b_{n−1} + 1 and s := 10, b_3 was s + 3 and is now 13; .N() of p_2 for p_0 = π, p_n = p_{n−1} + 1 stayed unevaluated and is now 5.14159….
  • An exact term that contains a float (h_n = h_{n−1} + 0.1√2) is remembered for one precision: N(g(3), 50) with g(k) = h_k gave the 21 digits of the term, and a later h_3.evaluate() gave the 55-digit term that N computed.
  • A term that stays unevaluated is remembered, so it is computed one time. Before, a recurrence that reads two lower terms took a time exponential in the index when its terms stay unevaluated: G_n = G_{n−1} + y·G_{n−2} took 6.9 s for G_20, and d_n = d_{n−1}(d_{n−1} + 1) with d_0 = w took 3.9 s for d_12. Both now take less than 0.2 s.

New Features

• #391 An accuracy goal and a precision goal for N. N(x, [p, a]) (MathJSON ["N", x, ["List", p, a]]) gives a value of x with p correct significant digits or with an absolute error below 10^-a, whichever goal is met first, as N[x, {p, a}] does in Mathematica. Either goal can be PositiveInfinity: ["N", x, ["List", "PositiveInfinity", 20]] asks for 20 correct digits after the decimal point, however many significant digits that takes. For x = 10^10·(e^100 − e^(999999999999/10^10)), it gives 2.688117141681729591326298974395630530648558808379370953675983977e+43 (64 digits), where N(x) gave 2.6881171417e+43. The engine evaluates x at a precision that it doubles until two successive values agree to within the goal, with two more digits, then rounds the value to min(p, a + ⌊log10 |x|⌋ + 1) significant digits. A value below 10^-a is 0.0: N(Exp(-100), [PositiveInfinity, 20]) is 0.0. A result that is not a number, or a list of numbers, ignores the goal and is the value at the engine precision: N(x + Pi, [PositiveInfinity, 5]) is x + 3.14159265358979323846. When the precision would have to be more than 1000 digits, the call stays unevaluated. A value that does not change with the precision does not meet a goal, because two of its values agree also in their digits that are not correct: N(Zeta(0.5 + 14.134725141734693i), [30, PositiveInfinity]) and N(w, [40, PositiveInfinity]) for a float w with 21 digits stay unevaluated. A value with fewer significant digits than a machine float is exact to its digits and meets the goal: N(0.1, [30, PositiveInfinity]) is 0.1, and N(Cosh(Ln(2)), [30, PositiveInfinity]) is 1.25. An exact value meets it: N(1/2, [30, PositiveInfinity]) is 0.5, and an exact integer keeps all its digits: N(123456789012345678901234567890, [PositiveInfinity, 5]) is 123456789012345678901234567890. A precision goal for a value that is 0, such as Γ(1/3)·Γ(2/3) − 2π/√3, stays unevaluated after three working precisions, because the values are rounding errors (before, the engine computed up to 2000 digits, which took about a minute); with an accuracy goal, the value is 0.0. The precision of the engine does not change. Each number of a list result has its own digits: N([π, 1000π], [PositiveInfinity, 3]) is [3.142, 3141.593]. A goal list that does not have two elements, or that has an element that is not a number, is a type error, and so is a set (N(\pi, \lbrace 30, 5\rbrace)): the elements of a set have no order. A precision goal below 1, or two infinite goals, leave the call unevaluated. A lazy collection of two elements is a valid goal: N(Pi, Range(4, 5)) is 3.142. An exact value that would need more than 1000 digits leaves the call unevaluated. A float 0 from cancelling terms does not meet a precision goal: N(Exp(10^-100) - 1, [30, PositiveInfinity]) is 1e-100. A float of the operand meets a goal that needs no more digits than it has: for a machine float y, N(2y, [PositiveInfinity, 5]) is 6.28319. An impure operand, such as Random(), is evaluated one time. On an engine at machine precision, a value outside the range of a machine float is kept as a big decimal: N(Exp(1000), [5, PositiveInfinity]) is 1.9701e+434, not +∞. A float that is exactly 0 meets a goal only when the exact value is 0, because terms can cancel at the working precision: N(10^100·(e^(10^−100) − 1), [PositiveInfinity, 20]) was 0, it is now 1.0000…. Each float of a symbolic or mixed result is compared at two working precisions and rounded to the goal: for R = e^(π√163) − 262537412640768744, N([R, x], [10, PositiveInfinity]) was [-0.001, x], it is now [-7.499274028e-13, x], and N([π, x], [5, PositiveInfinity]) is [3.1416, x]. An operand that reads an impure stored value, such as a symbol whose value is an unevaluated Random(), is evaluated one time.

• Compiled JavaScript runs without the engine. The new entry point @cortex-js/compute-engine/runtime exports createJavaScriptRuntime(options), the _SYS helper bundle that compile() builds for run(), with no engine behind it: store the code of a JavaScriptTarget result, and run it on a page, in a worker or on a server with runtime.load(result). The runtime type exposes only load, frame, setFrame, iterationLimit, deadline and runtimeVersion; the helpers are internal and may change in any release. The helpers that read engine state take it as options: random is the source of draws outside any WithRandomSeed frame and of the integrals' Monte-Carlo samples (null denies draws, and a draw then throws a CapabilityDeniedError, a different class in each bundle, so test e.name, not instanceof); frame (or runtime.setFrame()) is the frame of an interpreted WithRandomSeed the code is called from ({ seedLo, seedHi, next } or { seed, next }), and runtime.frame.next is the advanced counter after the call; iterationLimit caps the lazy-stream walks (default 1024, and Infinity when set to 0 or less, as ce.iterationLimit) and deadline is an optional time after which the shuffle and choice loops throw. A seeded program gives the same values interpreted, with run() and as stored code: WithRandomSeed(7, RandomShuffle(Range(1, 6))) is the same list in all three. load() evaluates the definitions that read nothing per call (a constant list, a memo) once, as run() does, from the new CompilationResult.preambleOnce, preamblePerCall and callCode, so a At(L, Floor(x)) with a 1,000-element L no longer rebuilds L on every call. load() also applies the input conversions run() does, from the new CompilationResult.entryPlan: a real given to a complex-declared symbol is lifted (z^2 + z at z = 2 is 6 from both, where load() gave a complex NaN), and a Float64Array for a list symbol is copied. The digits a negative base's exponent is read to, (-2)^x at x = 33.3333333333333, are fixed in CompilationResult.reconstructionDigits, since the runtime's own number library cannot tell a machine-precision engine from the default (10822639409.68 from run(), NaN from the runtime before). run() uses the same recorded digits: before, it read the working precision on each call, so code compiled at machine precision gave NaN for this power after ce.precision was changed or after another engine was constructed. CompilationResult.runtimeVersion and runtime.runtimeVersion are the version of the helper set; load() throws when they differ or when the stored code has none. Functions passed in the functions or imports compile options are copied into the code as source (toString()), so a closure loses its enclosing scope and a function given by name must exist where the code runs. The engine's own _SYS (run.SYS) is built from the same factory. (#372, contributed by enumeratio)

• ResidueClass(k, n): an element of ℤ/nℤ (#399, #411, contributed by enumeratio). A residue class is now a value.

  • Canonical form. k is reduced to 0…n−1: ResidueClass(7, 5) is ResidueClass(2, 5) and ResidueClass(-1, 7) is ResidueClass(6, 7). n must be an exact integer ≥ 1. A rational k whose denominator is a unit reads as u·v⁻¹: ResidueClass(1/3, 7) is ResidueClass(5, 7). A float or a symbolic k or n stays unevaluated.
  • Equality. ResidueClass(7, 5) == ResidueClass(2, 5) is True. Classes are not ordered.
  • Arithmetic. Sums, differences, products and integer powers of classes of one modulus are classes: ResidueClass(5, 7) + ResidueClass(4, 7) is ResidueClass(2, 7). An exact integer, or a rational whose denominator is a unit, is read in the ring: ResidueClass(5, 7) + 3 is ResidueClass(1, 7). An inverse, a quotient or a negative power needs gcd(k, n) = 1: 1/ResidueClass(3, 7) is ResidueClass(5, 7), while 1/ResidueClass(2, 4) stays unevaluated. Classes of two different moduli, a float and an infinity are never combined with a class: the expression stays unevaluated.
  • No number rule applies to a class. Canonical forms, evaluate(), .N(), simplify(), Expand, Factor and the .add(), .mul(), .div() and .pow() methods fold an expression that holds a class only with the rules of the ring. So c/c and c − c are never cancelled to the integers 1 and 0 when the class c has no inverse. This also holds for a symbol whose value holds a class. Solve, D and Integrate stay unevaluated for an expression that holds a class.
  • ℤ/nℤ. QuotientRing(Integers, n) lists ResidueClass(0, n) … ResidueClass(n−1, n), without building them all for a large n. Its element type is value. Element(ResidueClass(7, 5), ℤ/5ℤ) is True, a class of another modulus is False, and an integer is not an element: Element(7, ℤ/5ℤ) is False.
  • LaTeX. \overline{k}_{n} is ResidueClass(k, n) when k and n are integer literals, and a class is written back the same way. A bare \overline{7} is still Conjugate(7), and \overline{z}_1 is still the conjugate of z₁. Subscript(Conjugate(3), 5) is now written {\overline{3}}_{5}, so that it reads back as written.
  • Mod is still the remainder: Mod(7, 3) is 1.

• Contour integration by the residue theorem. ce.contourIntegrate(f, z, contour) and the ContourIntegrate operator integrate over a circle, a rectangle or a simple polygon (CircleContour, RectangleContour, PolygonContour), and \oint_{|z-c|=r} now evaluates. The report gives each pole with its location, order, residue and leading Laurent coefficient, the sum of the residues and the value. Supported integrands are rational functions, entire numerators, products of sin/cos of a real affine argument in the denominator, and P(z)·exp(c/(z−a)) with an essential singularity. An input that cannot be certified exactly stays unevaluated, and a pole on the contour gives Indeterminate. A call has a step budget, so the point where it gives up does not depend on the machine. See docs/CONTOUR-INTEGRATION.md. Contributed in PR #410.

• Real integrals by residues. Integrate gives an exact value over (−∞, ∞) for a rational function, or a rational function times cos, sin or exp(i·a·x); over [0, ∞) for such an even integrand; and over [0, 2π] or [0, π] for a rational function of sin x and cos x. For example, ∫ sin(x)/x dx over (−∞, ∞) is π, and ∫₀^{2π} dx/(2 + cos x) is 2π/√3. .N() uses the exact value: before, .N() of the sinc integral was 7.1 ± 7.3. A Cauchy principal value is never assumed; RealLineContour(True) asks for one.

• A pole on the path of a real integral gives +∞, −∞ or no value. ∫ dx/x² over (−∞, ∞) was 0 and is now +∞, and ∫₀^{2π} dx/(1 + cos x) is +∞. ∫ (1/x² − 1/(x−1)²) dx over (−∞, ∞) was 0 and is now Indeterminate: the integrand tends to +∞ at 0 and to −∞ at 1.

Improvements

• More accurate erf, erfc and erfi in doubles. The machine kernels of the error functions, which compiled JavaScript also uses, are now W. J. Cody's rational approximations (erfi through the Dawson integral), evaluated with compensated arithmetic. Measured against mpmath on 7,100 points in [0, 30], the largest error went from 15 to 1 unit in the last place for erf, from 889 to 2 for erfc, and from 502 to 2 for erfi (3 next to its overflow at 26.71). They take about the same time per call as before.
• Erfi of a large argument above machine precision finishes. Erfi(1000).N() at 50 digits did not finish; the big-decimal kernel now uses the asymptotic series once it is accurate to the working precision.
• Complex Erf and SinIntegral near 0. The relative error of Erf(10^{-8}(1 + i)).N() was 10⁻⁸, and of SinIntegral at 10⁻⁶(1 + i) 10⁻¹⁰: the kernels subtracted two values close to each other. Both now use their Maclaurin series for |z| ≤ 1, with a relative error below 2·2⁻⁵².
• SinhIntegral and CoshIntegral near 0. SinhIntegral(10^{-10}).N() was 9.99982e-11, it is now 1e-10. The real kernels of Shi and Chi use their Maclaurin series for |x| ≤ 2.
• More accurate SinIntegral, CosIntegral, SinhIntegral, CoshIntegral and ExpIntegralEi in doubles, mostly as fast or faster. Measured against mpmath on more than 30,000 points from 0.12 to 10³⁰⁰, the largest error is now 0.74 unit in the last place for SinIntegral (21 before), 1.5 for CosIntegral away from its zeros (73 before), and for SinhIntegral, CoshIntegral and ExpIntegralEi 1 for 2 ≤ x < 50 and 2 above (11 before); CoshIntegral below 2 is within 1.6 away from its zero (4 before). CosIntegral gave wrong values past 10¹⁵, and SinhIntegral, CoshIntegral and ExpIntegralEi gave +∞ from 716; they are finite up to their overflow (717.05 and 716.36). SinIntegral and CosIntegral are up to 3 times faster for 2 ≤ x ≤ 16; CoshIntegral below 2 is about 2 times slower. Compiled JavaScript uses the same kernels.
• .N() of a sum or product with a very large exponent gap finishes. (10^{10^9} + 1).N(), (10^{10^9}·i).N() and Erf(10^{10^9}·i).N() did not finish at a precision of 50 digits, or failed with "Maximum BigInt size exceeded": the exact sum aligned the operands to a billion-digit significand. Now they take a few milliseconds. In the arithmetic of approximate values above machine precision, an operand that is more than 10,000 digits (or 4 times the precision) below the other is replaced by a value of the same sign just under that limit, which rounds the same way at every precision up to that limit. Exact arithmetic is not changed.

Issues Resolved

• Limit at infinity of a complex-valued function (#396). The limit at ±∞ chose between +∞ and −∞ from the sign of a term, and a complex value has no sign: \lim_{t\to\infty} e^{it} was +∞, \lim_{t\to\infty} \operatorname{erf}(it) was 1 and \lim_{t\to\infty} e^{it}/t was +∞. A function with a non-real constant is now resolved through its real part and its imaginary part: e^{it}/t is 0, (1+it)/t is i, (2t+i)/(t-3i) is 2, and a limit that does not exist (e^{it}) or is not finite (i·t, Erf(it)) stays unevaluated. Gamma goes to 0 along a vertical line: Limit(t ↦ Gamma(i·t), ∞) is 0, and Abs, Real, Imaginary and Conjugate carry such a limit. Limit(t ↦ Gamma(t), ∞) is +∞; it was unevaluated. .N() and NLimit extrapolate the real part and the imaginary part of a complex-valued function separately; they gave NaN.

• Limit at infinity with a coefficient of unknown sign stays unevaluated. Limit(t ↦ a·t, ∞) and Limit(t ↦ a·eᵗ/t, ∞) were +∞ for a symbol a with no assumption. The answer depends on the sign of a, so the limit is now unevaluated; -2t and π·t are unchanged. A non-integer power of a base that goes to −∞ is also unevaluated: Limit(t ↦ (−t)^{3/2}, ∞) was +∞, and the value is not real. An integer power takes its sign from the parity ((−t)³ is −∞).

• NLimit at −∞ samples negative arguments. NLimit(t ↦ arctan t, −∞) was π/2; it is −π/2.

• No double superscript for a half-integer power deep in an expression (#345). Inside two lists, 2^{(1+x)/2} was written 2^{1/2}^{1+x}, which TeX rejects. It is now 2^{(1+x)/2}. At the top level it is still \sqrt{2}^{1+x}.

• A symbol whose stored value is impure is read one time for each place it appears under .N(). With r assigned an unevaluated random draw, .N() of 2r, r + π, r/3, r − 1 and 2r + 1 evaluated the stored value two times, and the result used the second draw. A stored value that reads an impure symbol (s := r + 1) or calls a function with an impure body is treated the same way. The exact evaluations that .N() does again near a jump (Round, Floor, Equal, a lazy rounding broadcast) and after an overflow (a Sum or Product term of ±∞, a factor of 0) no longer evaluate such a symbol again.

• The memos of functions, lazy collections and stored values follow what a sequence reads. With g := k ↦ B_k and B_n = B_{n−1} + x, g(2) kept its first value after x changed. Now it gives the new value. A term whose read was stopped (more than 100,000 terms, or a recursion that is too deep) is not kept by the memo of a function, of a lazy collection (Map, Tabulate) or of a stored value: a later read gives the term.

• A fused Map no longer disposes the definitions of the scope of its function. The fast route of Map evaluated each element in a frame that was the scope of the function itself, and the end of the frame disposed every definition of that scope: for a function written at the top level, every global definition, after each element. A disposed definition loses its assumptions at a checkpoint restore, and every memo that reads a global symbol was made outdated by each such Map.

• The check that an evaluation has no side effects has one implementation, which follows stored values, the bodies of user functions and the definitions of sequences. It no longer reads every element of a large numeric list made with ce.list().

• Counting set operations.

  • Count(Intersection(Integers, Set(1, 2))) was PositiveInfinity; it is 2, in both operand orders. Intersection(Integers, Set(1, 2)) now evaluates to Set(1, 2).
  • Count(SymmetricDifference({1, 2, 3}, {2, 3, 4})) was 1; it is 2.
  • Count(Union(A, S)), Count(Intersection(A, S)) and Count(SetMinus(A, S)) of sets with no value were 0; they stay unevaluated.
  • Intersection({1, 2, x}, Integers) evaluated to Set(1, 2), dropping x, although x may be an integer. A set operation now stays unevaluated when the membership of an element in an infinite set is undecided, in both operand orders. A set operation with such a membership is not walked by other operators, so it does not give a partial answer.
  • A walk of Intersection(Integers, Set(1, 2)), and a count of a union with an operand such as Intersection(Integers, Interval(0, 3)), never ended. They now end.
  • Count of an intersection of two large ranges no longer copies their elements.

• A dictionary's MathJSON keeps its values. An exact rational value was written as a machine number (1/2 read back as 0.5), and a function value as a bare array (Sqrt(2) read back as a list).

• simplify() of a list quotient or difference keeps the list. [x]/[x] simplified to 1 and [x] − [x] to 0. They now give [1] and [0], as evaluate() does.

• The identities of loadIdentities() that contain an angle apply only in radians. They are written for radians, but they applied in every angular unit. With ce.angularUnit = 'deg' and the identities loaded, Sin(Pi).simplify() was 0, but sin(π°) is 0.0548…, the value of .N(). Argument(i) was π/2, not 90, and Im(ln z) became Argument(z), an angle in degrees. Now an identity with a trigonometric or inverse trigonometric function, Argument or AbsArg in its sides or its conditions applies only when ce.angularUnit is 'rad'. The unit is read each time the identity is tried, so a change of the unit after the load is taken into account. The other identities, which include the identities of the hyperbolic functions, apply in every unit.

• The inverse of a logarithm with a base. \ln_3^{-1}(x) was exp(x): the base 3 was dropped. In the lenient grammar, ln_3^-1(x) was 1/log₃(x), a reciprocal, and log_3^-1(x) was Apply(InverseFunction(Log), x, 3). All three are now 3^x, the inverse of log₃, as \log_3^{-1}(x) already was. The lenient readings report ambiguous-inverse-function.

• The arguments of a logarithm with a base are kept, in both grammars. The arguments after the first were dropped with no error: ln_3(x, y), \ln_3(x, y) and log_3(x, y) were Log(x, 3). In the lenient grammar, log_10(x, y) was Log(x, y), the logarithm in base y. They are now Log(x, 3, y) and Log(x, 10, y), and the canonical form reports y as an unexpected argument.

• A run of digits keeps its digits in the lenient grammar. A long run lost precision: π_123456789012345678901 was Subscript(Pi, 123456789012345680000), (x)_9007199254740993 had the subscript 9007199254740992, and a longer run gave Infinity. The same was true of a base of log (log_12345678901234567890(x)) and of an unbraced exponent (x^123456789012345678901). A run that a JavaScript number cannot hold exactly is now a number string, with all its digits. A leading zero was dropped: π_01 was π_1, [1,2,3]_01 was At([1,2,3], 1), 1_000 was Subscript(1, 0), x^2_01 was x_1^2 and tan_01x was tan_1(x). After _ on a base that is not a symbol, a run that starts with 0 is now the one token 0, as in the strict grammar ([1,2,3]_01 is [1,2,3]_0·1), and the reading reports ambiguous-implicit-subscript (or ambiguous-function-subscript on a function name). A symbol keeps the text: π_01 is Pi_01 and x^2_01 is x_01^2.

• A subscript on a symbol in parentheses is a compound symbol, in both grammars. (x)_0 was Subscript(x, 0) with the type symbol, so a product with it was a Tuple: (x)_0 y was (Subscript(x, 0), y), and (x)_0 \cdot 1 was a type error. The same was true of a base that becomes a symbol: x2_1 + y (Subscript(x_2, 1) + y) was a type error. (x)_0 is now the symbol x_0, as x_0 is, and x2_1 is x_2_1, so (x)_0 y is the product x_0·y. A base that is not a symbol keeps its subscript: (x+1)_0 is Subscript(x + 1, 0).

• Shape and Rank of a lazy collection. Shape(Range(1, 3)) was () and Rank(Range(1, 3)) was 0, the answer for a scalar. A finite indexed collection now gives the answer of the list with the same elements: Shape(Range(1, 3)) is (3) and Rank(Range(1, 3)) is 1. A set, a dictionary, an infinite collection, and a symbol with a collection type and no value stay unevaluated. Compiled Shape and Rank decline to compile for an operand whose type is a set, a dictionary, a tuple or a string, where the compiled answer was different from the evaluated one.

• Solve of a list of equations.

  • A list of several plain expressions is a system of equations, each equal to 0, as the = spelling is: Solve([x + y - 1, x - y], [x, y]) is [(1/2, 1/2)]. It stayed unevaluated.
  • A list of several equations in one unknown gives the roots that all the equations share: Solve([x = 1, x^2 = 1], x) is [1] and Solve([\sin(2x) = 0, \cos(x) = -1], x) is [π, -π]. Both were []. For periodic equations, when no principal root is common, Solve stays unevaluated instead of saying that there is no solution.
  • A list of inequalities with no equation stays unevaluated. Its solution set is a region: Solve([x < 1, y > 0], [x, y]) was [(1, 0)], a point that does not satisfy x < 1.
  • A domain given as an argument applies to a list of equations: Solve([x + y = 3, x - y = 1], x ∈ 0..5, y ∈ 0..5) is [(2, 1)]. It was [].
  • Solve([x = 1, 2x = 2], x) threw a TypeError (roots is not iterable). It is now [1].
  • expr.solve() of a system of linear inequalities gives null for an unbounded region, as documented. It gave a corner point of the region.

• \bar over a number. \bar{7} was an unexpected-command error. It now parses to Conjugate(7), as \overline{7} does, and 0.\bar{3} is the repeating decimal 1/3, as 0.\overline{3} is. \bar{x} is still Mean(x).

• #405 The error of a numeric integral covers its true error.

  • A semi-infinite oscillatory integral took as its error the difference of the last two accelerated values of its lobe sums. When the lobes have a part of one sign (the lobes of an integrand with two frequencies whose ratio is rational repeat a pattern), the accelerated values move slowly toward the integral and two of them agree long before they reach it: ∫₀^∞ sin(t/2)·cos 2t/t dt = 0 was -0.00000000681 ± 0.00000000089, and ∫₀^∞ sin t·cos 3t/t dt = 0 was 0.00001183017 ± 0.00000000075. The error now also covers the values from the last four lobe sums and from the first half of the lobe sums, and the Levin u-transform of the sums of one period of lobes gives a second value. The two integrals are now 0.0000000000000 ± 0.0000000000059 and 0.0000000033 ± 0.0000000071, and ∫₀^∞ sin t·(2 + cos t)/t dt = 5π/4, which was 4.037 ± 0.065, is 3.926990818 ± 0.000000010.
  • When the lobes shrink but no method finds the integral, the result is NaN, not a Monte-Carlo value: ∫₀^∞ sin t·cos(√2·t)/t dt = 0 was 0.072 ± 0.018.
  • The lobes are integrated by Gauss–Kronrod instead of adaptive Simpson, and the values are more accurate with fewer evaluations: ∫₀^∞ sin t/t dt was 4.2e-11 from π/2 with an error of 2.5e-10, after 9634 evaluations. It is now 1.570796326795 ± 0.000000000012, 7e-16 from π/2, after 2387 evaluations.
  • The error of a lobe whose panels are not resolved (a singularity inside the lobe) is added to the error of the result. ∫₀^∞ sin t/(t − 5)² dt, which diverges, was -217023352.094 ± 0.0013. It is now NaN. A divergent integral whose lobes have a part of one sign that decreases too slowly to have a sum (∫₁^∞ sin t·(1 + sin t)/√t dt, which was -0.14277 ± 0.000069) is also NaN.
  • A lobe with an integrable singularity is integrated again in two parts, with the singular point at an end of each part, and a lobe ends at a pole of the integrand, not 10⁻¹⁴ before it. ∫₀^∞ sin t/√|t − 5| dt was -1.96685088 ± 0.00000000066, 2.7e-4 from the integral (-1.9671167915329387). It is now -1.967116791534 ± 0.000000000046. ∫₀^∞ sin t/|t − π|^1.5 dt was 1.25699002 ± 0.0000000008, and is now 0.144840978411 ± 0.000000000065 (the integral is 0.14484097841008…). ∫₀^∞ sin t·ln|t − 5|/t dt is now 2.929478525094 ± 0.000000000014 (the integral is 2.92947852509409…).
  • ∫₀^∞ sin t·cos 6t/√t dt and other integrals whose lobes repeat a pattern with several lobes of the same width were NaN. They now have a value: -0.04339550 ± 0.00000011 (the integral is -0.04339545…).
  • The error of the adaptive Gauss–Kronrod quadrature is at least four units in the last place of the sum of the magnitudes of its panels: the integrand is evaluated in doubles. ∫₁² ln³t/(t − 1) dt was 0.1425141979357109345283 ± 0.0000000000000000000031, with a true error of 1.9e-17, and is now 0.14251419793571093 ± 0.00000000000000013.

• #409 Exact values of Arcosh, Arcoth, EllipticK and EllipticE at 0. Under evaluate(), Arcosh(0) and Arcoth(0) are now iπ/2, and EllipticK(0) and EllipticE(0) are now π/2. They stayed unevaluated, and only .N() gave the value. More generally, Arcosh(x) for an exact x in [−1, 1] is i·Arccos(x) when Arccos(x) is a special angle: Arcosh(1/2) is iπ/3, Arcosh(−1) is iπ, Arcosh(√3/2) is iπ/6 (they stayed unevaluated). simplify() gives the same values for Arcosh and Arcoth. A float argument (Arcosh(0.0), EllipticK(0.0)) still gives a float.

• #409 Inverse trigonometric functions of other spellings of the special values. The table of special angles holds one spelling of each value, for example \frac{\sqrt{10-2\sqrt5}}{4} for sin(π/5). A value written another way stayed unevaluated. Now Arcsin, Arccos, Arctan, Arccot, Arcsec and Arccsc also find the special angle when the square of the argument is exactly the square of a value of the table: Arcsin(√2/4·√(5 − √5)) was unevaluated, it is now π/5; Arcsin(√(10 + 2√5)/4) is 2π/5; Arctan(√(1 − 2√5/5)) is π/10; Arcsin(√(2 − √3)/2) is π/12; Arccos(−√2/4·√(5 − √5)) is 7π/10. The test is exact: a value that is only near a special value (Arccos((1 + √5)/4 + 10⁻³⁰)) stays unevaluated.

• #409 A term whose radical coefficients cancel is dropped from a sum. (√2/4·x).sub(√2/4·x) was the product 0·x, not 0. This also made the difference of two equal radical expressions such as √2/4·√(5 − √5) read as not zero.

• QuotientRing(Integers, n) is a finite collection of n residue classes, not a set of integers (#399, contributed by enumeratio). \mathbb{Z}/5\mathbb{Z} and \mathbb{Z}_5 had no count (count was undefined, and Count(…) stayed unevaluated), and their type was set<integer>, although 7 and 2 are the same element of ℤ/5ℤ. For a positive integer literal n, the collection now has the count n, is finite and is not empty, and Count evaluates: Count(\mathbb{Z}/5\mathbb{Z}) is 5. A symbolic, zero or negative modulus, or a base other than Integers, stays inert. The type is set<value>: its elements are the classes ResidueClass(k, n) (see New Features).

• A finite collection whose elements cannot be computed no longer reads as empty. Linspace(a, 1, 3) with a symbolic a has the count 3 but no elements that can be computed, and so does ℤ/5ℤ. Many operators checked only that a collection was finite and then walked its elements, so they read it as empty and gave a wrong answer with no diagnostic: Union(Linspace(a, 1, 3), {1}) was Set(1), Unique(…) was [], Tally(…) was ([], []), Intersection(…, {1}) was EmptySet, SetMinus(Set(1), …) was Set(1), First(…) was NaN, Flatten([[1], Linspace(a, 1, 3)]) was [1], Equal(Linspace(a, 1, 3), [1, 2, 3]) was True, and Sort(…) gave an internal-error. These operators now stay unevaluated, and Equal, IdenticallyEqual and .isEqual() are undecided. An empty collection still gives a definite answer: Union(Linspace(a, 1, 0), Set(1)) is Set(1). An element-wise function of such a collection gives the lazy form that it already gives for Range(1, n): Sin(Linspace(a, 1, 3)) is Map((_) => sin(_), Linspace(a, 1, 3)).

• EvaluateAt with one operand no longer throws. EvaluateAt(x^2) threw a TypeError. It now stays unevaluated.

• BesselY and BesselJ of a real argument are accurate. BesselY computed Y₀ and Y₁ from a series that loses about x/2.3 digits, then recurred: BesselY(12, 40).N() was -1.83 instead of -0.0236, and BesselY(30, 200).N() was -5.5·10⁷⁶ instead of -0.0224. BesselJ started its backward recurrence too low for a large argument: BesselJ(12, 95) had a relative error of 10⁻⁸, and BesselJ(80, 80) was 1.98 instead of 0.104. Y₀ and Y₁ now come from Steed's continued fractions for 2 < x ≤ 30, and the recurrence of J starts high enough. Against mpmath at orders 0 to 80 and x up to 240, both are within 1·10⁻¹⁴ of the size of the function. The asymptotic expansions also reduce the phase x − (2n+1)π/4 exactly (BesselY(0, 1000) had an error of 3·10⁻¹⁴). Compiled JavaScript uses the same kernels.

• FresnelS and FresnelC of a real argument between 1.6 and 4.5 are accurate to 7·10⁻¹⁶. One coefficient of the auxiliary function was wrong: the relative error was up to 4.5·10⁻¹² (FresnelS(2.5) was 0.61918175581901, it is 0.6191817558195929). The GLSL and WGSL lowerings had the same coefficient (with no visible effect in 32-bit floats); it is corrected there too.

• LambertW next to -1/e is accurate. The real kernels returned exactly -1 for an argument within 10⁻¹⁵ of -1/e, also above machine precision: at 50 digits, LambertW(-1/e + 10^{-20}) had an error of 2·10⁻¹⁰. They now use the series about the branch point, with 1/e held to more than double precision, and above machine precision Halley's iteration with extra digits.

• In Epsil, f(x) := body defines f as f(x) = body does (#400, contributed by enumeratio). As a statement, f(x) := x^2 + a parsed to Assign(f(x), x^2 + a), which evaluates to itself with no diagnostic and leaves f undefined, so f(3) was f(3). It now parses to DefineFunction(f, Function(x^2 + a, x)), and f(3) is a + 9. The two spellings are synonyms for every function head: typed, rest and wildcard parameters, a return type, a definition inside a block, and literal-pattern clauses. So f(0) := 1 followed by f(n) := n*f(n-1) defines the factorial, and f(5) is 120. Assign itself is unchanged.

• The GLSL and WGSL targets compile the prime of a function. \sin'(t) is Apply(Derivative(Sin, 1), t), and the shader targets declined it ("Could not compile Apply"), also inside a function body (h(x) := x^3 + \sin x, then f := t \mapsto h'(t) and f(x)). A shader has no function values, so the argument is now substituted into the closed form of the derivative: \sin'(t) compiles to cos(t), and f to float _fn_f(float t) { return 3.0 * (t * t) + cos(t); }. The interval target uses the same closed form. A derivative with no closed form, or with a closed form too large to write out, still declines, with a message that says so.

• The GLSL and WGSL targets compile a Map whose source has a length known at compile time. Min(Map(k \mapsto k^2, x + [0, 4, 2])) declined on the shader targets ("Could not compile Map"). The Map is now written out element by element: min(min((x * x), _gpu_pow2(x + 4.0)), _gpu_pow2(x + 2.0)). This applies under Min, Max, Sum and Product, and a Map alone is a vector value (vec3(…) in GLSL, vec3f(…) in WGSL). The source can be a literal list, a literal range, or a list built from them by element-wise arithmetic, with up to 64 elements. A Map over a list of unknown length still declines.

• The norm of a point whose component is a list compiles on the interval-js, glsl and wgsl targets. \left|(x+[\frac12,1],y)\right| is one point per element of the list, so its norm is one number per element. The javascript target compiled it to the list of norms; the other targets declined it ("the interval target requires a fixed-arity point operand", "component 1 is collection-valued … scalar components only"). They now compile it as √(c₁² + … + cₙ²), element by element, with the same code as the norm written by hand: on glsl, sqrt(_gpu_pow2_v2(x + vec2(0.5, 1.0)) + (y * y)), and on interval-js an array of one enclosure per element. This applies when the list components have one length that is known at compile time. When the lists have different lengths, or a length known only at run time, interval-js zips the points to the length of the shortest list, as javascript does (|(x+[1,2],y+[3,4,5])| is two norms), and glsl and wgsl decline with a message that gives the lengths. A point of scalars compiles as before (length(vec2(x, y)), _IA.hypot(_.x, _.y)). On interval-js, Abs of a PointList of scalars also compiles now; it declined before.

• The compiled norm of a point with a restricted list component is one number per element. A restricted list L\{c\} has the type list<…> | missing, and its missing part hid the list from the tests that decide whether a coordinate is a list. On the javascript target, |(L\{0<t\}, y)| with L = [1, 2, 3] and y = 3 was √23, one norm of the vector (1, 2, 3, 3). It is now [√10, √13, √18], as in the interpreter, and NaN when the condition fails. Hypot of such a point is corrected in the same way. The Abs of a restricted list of points, |([1,2]\{0<t\}, y)|, was [[1, 3], [2, 3]] (the absolute value of each coordinate). It is now [√10, √13], and undefined (the interpreter's Missing) when the condition fails; interval-js now compiles it too. The type of such a norm is list<number> | number, not number.

• Arctan2 has a symbolic derivative. D(Arctan2(y, x), x) is -y / (x^2 + y^2) and D(Arctan2(y, x), y) is x / (x^2 + y^2), with the chain rule for composite arguments (D(Arctan2(t^2, cos t), t) has a closed form). The multi-index form also has a closed form: Apply(Derivative(Arctan2, 1, 0), y, x) is x / (x^2 + y^2). In a non-radian angular unit the partial derivatives are divided by the unit factor, as for Arctan: in degree mode, D(Arctan2(y, x), x) is -180y / (π(x^2 + y^2)). Before, the derivative stayed Apply(Derivative("Arctan2", 0, 1), y, x): the JavaScript target compiled it to a numeric derivative, and the interval-js, GLSL and WGSL targets did not compile it. The closed forms now compile on all targets.

• Derivative(F, k₁, …, kₙ) of a library operator has a closed form. The multi-index form applies the same rules as D: Apply(Derivative(Log, 0, 1), x, b) is -ln(x) / (b * ln(b)^2), Apply(Derivative(Power, 1, 0), x, n) is n * x^(n - 1) and Apply(Derivative(Mod, 0, 1), x, m) is -floor(x / m). Before, these stayed unevaluated although D of the same expression had a closed form. A partial derivative with no closed form (BesselJ in its order) stays unevaluated. Evaluating Derivative(F, k₁, …, kₙ) no longer declares the symbols _1, _2, … in the caller's scope.

• D of Mod in the modulus, of Round with a digit count and of the upper incomplete gamma function. D(Mod(x, m), m) is -floor(x / m), and when both operands depend on the variable, D(Mod(u, c), v) is u' - floor(u / c) * c'. D(Round(x, 2), x) is 0, as for Round(x). D(Gamma(s, z), z) is -(e^(-z) * z^(s - 1)); the partial derivative in s has no closed form and stays symbolic. Before, these derivatives stayed unevaluated.

• A univariate Derivative of a two-argument operator with a derivative key stays symbolic. Derivative(F) for a user operator F(x, y) declared with a derivative key threw "Not canonical". It now stays unevaluated, as Derivative(Arctan2) does.

• The L∞ norm of a point with list components is one maximum per point. With L = [1, -5, 3] and K = [-4, 2, 1], Norm((L, K), "Infinity") is [4, 5, 3], as the compiled JavaScript code already answered and as the other orders give one norm per point. Before, the interpreter left it unevaluated.

• The string order "Infinity" of Norm compiles to JavaScript. Norm(v, "Infinity") compiles as Norm(v, +∞), for a vector, a matrix, a point, a point with list components and a list of points. Before, it did not compile ("the "Infinity" norm has no compiled form"), while the interpreter answered it. The GLSL, WGSL and interval targets compile only the default order, and refuse both spellings with the same message.

• The GLSL and WGSL targets check the arguments of an applied function literal against the declared parameter types. An argument whose type does not match the declared type of its parameter is an incompatible-type error under evaluate(): Apply((u: real, w: list<real>) ↦ u + Length(w), V, V) for V: list<real^2>. The shader targets now decline such an application, and the message names the parameter and the two types. (A mismatch that the types already prove, as for V here, now makes the application invalid when it is boxed, so it is declined as an invalid expression.) A parameter with no declared type is not checked. A tuple at a parameter declared as a scalar is not a mismatch: the application maps over its components (see "A parameter declared as a scalar maps over a tuple argument" under Behavior Changes).

• The GLSL and WGSL targets compile the square of a vector. A structural Square([1, x]) declined because the scalar helper _gpu_pow2 was chosen for a vec2. It now compiles to _gpu_pow2_v2(vec2(1.0, x)), as [1, x]^2 does.

• An applied derivative of a function of several arguments compiles on all targets. Apply(Derivative(F, k₁, …, kₙ), a₁, …, aₙ) compiles from the closed form of the derivative on the interval-js, GLSL and WGSL targets: Apply(Derivative(Power, 0, 1), x, n) is log(x) * pow(x, n) in GLSL, and the partial derivatives of Arctan2 and of a user function h(u, w) compile in the same way, also in degree mode. Before, these targets declined the application ("only a function-literal callee compiles on the interval target"); the JavaScript target already compiled the closed form. A derivative with no closed form (BesselJ in its order) is still declined. A derivative of a user function of several arguments whose body is large is declined before it is differentiated, with the size limit that a function of one argument has (the size of the body and the total order k₁ + … + kₙ): the mixed derivative (2, 1) of a four-deep nested radical in two variables takes about 24 seconds to compute and has a closed form of more than 20,000 nodes. The decline messages now give the true reason on all four targets: the derivative has no closed form, its closed form is too large to write out, or the number of arguments is not the number of parameters of the closed form. Before, the GLSL and WGSL targets said "no closed form small enough to write out" in all three cases, and the JavaScript target said that the Derivative compile handler had no lowering.

• The JavaScript target declines an Apply with the wrong number of arguments. The interpreter curries ((a, b) ↦ a + b)(x) into a function of one argument and rejects (a ↦ a + 1)(x, y). The compiled code answered NaN for the first and x + 1 for the second. The application is now declined, as on the other targets, and the compilation falls back to the interpreter.

• The JavaScript target declines an Apply whose argument does not match a declared parameter type. In a strict engine, an argument whose type does not match the declared type of its parameter (a string, a boolean or a set at u: real, or an element of such a type in a mapped list) is an incompatible-type error under evaluate(). The compiled code did not make this check. The application is now declined with the reason, as on the GLSL and WGSL targets, which use the same check. A tuple at a parameter declared as a scalar is not a mismatch: the application maps over its components (see Behavior Changes).

• The text form of a power keeps the parentheses of its base. ce.box(['Power', ['Power', 't', 2], ['Rational', -3, 10]]).toString() was t^2^(-3/10), which reads as t^(2^(-3/10)). It is now (t^2)^(-3/10). The same fix wraps other bases that are not one unit: a rational or radical number ((3/4)^x, not 3/4^x), an imaginary number or a number in exponent notation ((2i)^x, (1e+30)^x), a quantity ((5 m)^2, not 5 m^2), a measurement, a sum, an integral, a limit and a restriction ((x {0 < x})^2). A power with the exponent 1 that is not in canonical form keeps the parentheses of its base in a product: (a + b) * c, not a + b * c. The LaTeX form was already correct.

• Evaluating a Derivative declares nothing in the caller's scope. Derivative(Sin) evaluates to (x) => cos(x). To make it, the handler declared the symbols x and _ in the caller's scope, with the type number. A later use of x was then checked against that type: x && True gave an incompatible-type error. The multi-index form Derivative(Arctan2, 1, 0) declared its parameters x_1, x_2 in the same way. The symbols are now declared in a scope that is discarded, and a caller binding of the same names (such as x_1 declared as a string) does not change the result.

• A derivative with respect to a function stays symbolic. D(f(x), f) was 0, and Derivative(Apply, 1, 0) evaluated to (x_1, x_2) => 0: the rules took the application f(x) as a constant, because the variable was its function and not one of its arguments. There is no closed form, so they now stay unevaluated.

• Derivative(Exp) has no ln(e) factor. It evaluated to (x) => ln(e) * e^x. It is now (x) => e^x.

• A compiled call of a named function checks the declared parameter type. With k assigned (u: real) ↦ 2u, a call whose argument does not match real (a string, a boolean, a set) is an incompatible-type error under evaluate(). Such a call is now declined with the reason on the JavaScript, GLSL, WGSL and interval targets, with the check that the compiled Apply of a function literal uses. A tuple argument maps over its components instead (see Behavior Changes).

• The cells of Apply over a list are typed with the declared parameter type. Apply((u: integer) ↦ u + 1, [1.5, 2]) was typed vector<real^2>, from the type of the elements, while the named calls were typed vector<integer^2>. The three agree now. Each failing cell of such a map carries the "while applying 'Apply' element-wise" note, as on the named routes.

• The JavaScript target binds a tuple whole at a parameter with no declared type. The compiled code mapped a function over any array at run time, and a point is an array, so Apply(u ↦ 7, (a, b)) gave [7, 7] and g(P), Apply(u ↦ 7, P) and Map(g, P) for a list of points P gave [[7, 7], [7, 7]], where evaluate() gives 7 and [7, 7]. They now give the same values as evaluate().

• The JavaScript target compiles h(P) for a list of points P. With h := (u: real) ↦ 2u, the call was declined as "a broadcastable head over a possibly list-valued operand", while f(P) for f declared function and Apply of the literal compiled. It now compiles to the value of evaluate(), [[2, 4], [6, 8]] for P = [(1, 2), (3, 4)].

• A block that declares a function, assigns it, and calls it now compiles. ["Block", ["Declare", "k", "'function'"], ["Assign", "k", ["Function", ["Multiply", 2, "u"], "u"]], ["k", "x"]] — also the parse of k(u) \coloneq 2u; k(x) — evaluated to 6 for x = 3, but the JavaScript and interval targets declined it with Unknown operator `k` . The call now compiles the same way as for const k = (u) => 2u: with the run-time broadcast over a list argument, the arity check, recursion, and the fold of a call with constant arguments. An Assign with no Declare compiles too. A call of a block-local function that is assigned again now uses the function the interpreter uses at that point: let h = (k) => k + 1; h := (k) => k + 2; h(3) compiled to 4 (the old function), where the interpreter gives 5, and a reassignment inside a loop body gave 4 for 13. On the JavaScript target such a call (after a second assignment, an assignment inside a branch or a loop body, or from a function body) reads the variable at run time and is not folded. It declines with the reason when the assigned functions do not have the same signature, or when the name is also assigned a value that is not a function. The interval target declines a call of a function that is assigned more than once, with the reason. The arity decline now states the interpreter's behaviour correctly: a call with fewer arguments than parameters is a partial application (a function value), and a call with more is an error.

• The index form of a partial derivative costs no more than nested D. Apply(Derivative(Arctan2, 3, 3), y, x) took about 2.2 times as long as the same derivative written as six nested D. The index form now evaluates that nested D over fresh symbols and gives the same expression. The time went to deriving the types of the intermediate expressions, not to the differentiation rules. The same evaluation runs for .N() of an applied partial derivative.

• A partial derivative is computed once across evaluation points. Apply(Derivative(f, k₁, …, kₙ), x, y).N() sampled at many points, as a plot does, computed the partial derivative again at each point (about 0.5 s at each point for Derivative(Arctan2, 3, 3)). The result is now cached per function and order vector, as Derivative(f, n) already was. A new definition of the function or of a function that it calls, an assignment to a free symbol of its body, and a change of the assumptions or of the configuration make the next request compute it again.

• A product whose text ends with a symbol keeps a \times before a parenthesized factor. Multiply(InvisibleOperator(2, v), Add(x, 1)) serialized as 2v(x+1), and a new engine read it as 2 times a call of v (and declared v a function). The same happened with a nested Multiply(0.03, v), with -2v, with uv before a tuple, and with the Desmos shape V_{einColor}\cdot0.03v_{einO}\cdot\sum…, in the raw and structural forms. A bare symbol already got the sign. Now any factor whose text ends with a symbol name gets it: 2v\times(x+1). A decimal number before a group that starts with a digit gets it too: Multiply(1.5, Delimiter(2)) serialized as 1.5(2), which reads as the repeating decimal 1.5222…, and now serializes as 1.5\times(2).

• A lazy collection that is the value of a block or of a function call reads the locals of that block or call. A Map or a Filter computes its elements when they are read, and it found its function and its sources by name at that time. After the block or the call had ended, its locals did not resolve: let r = do { let h = (k) => k + 1; map(h, [1, 2]) } stored [h(1), h(2)], and the same with function f() { … }; f(), with a local source list, or with an element of an infinite source read after the block. The compiled code gave [2, 3]. When the collection leaves the block or the call, each local that it reads is now replaced with its value, and the collection stays lazy: the value is Map(k ↦ k + 1, [1, 2]), which gives [2, 3]. A function or a source of an enclosing scope is not replaced, and the collection continues to read it by name.

• An assignment stores a lazy collection inside a list literal as the list of its elements. With let r = [map(h, [1, 2])] followed by h = (k) => k * 10, r[1] gave [10, 20], while the compiled code and the list written with calls ([[h(1), h(2)]]) gave [2, 3]. An assignment already stored a finite lazy collection that reads a variable as the list of its elements; this now also applies to such a collection in a list or tuple literal, at every depth.

• A constant factor in the numerator of a quotient no longer stops an integral. ∫ a·sin(t)/t dt stayed unevaluated, while ∫ a·(sin(t)/t) dt gave a·Si(t). The integrator now moves the factors of the numerator that do not depend on the variable out of the integral, as it already did for a product and for the denominator. ∫ a·sin(t)/t dt is now a·Si(t), ∫₋₁¹ a·sin(t)/t dt is a·Si(1) − a·Si(−1), and ∫ a·eᵗ/t dt, ∫ a·cos(t)/t dt, ∫ a/ln(t) dt and ∫ −a·sin(t)/t dt also close. This applies when the integration rules package is not loaded; with it, these integrals already closed.

• An integral with an xˣ factor no longer runs without end. ∫ eˣ·xˣ dx overflowed the stack, ∫ cos(x)·xˣ dx did not finish in minutes, and DSolve of y'' + y = xˣ did not finish in 300 s. Integration by parts chose u = xˣ, whose derivative xˣ + ln(x)·xˣ contains xˣ again, and recursed with no progress. It also accepted a result that was a sum with an unevaluated integral in it, and the evaluation of that integral started the same integration by parts again. Integration by parts now declines both. These integrals stay unevaluated in less than a second, and y' + y = xˣ gives the quadrature y = (c₁ + ∫ eˣ·xˣ dx)/eˣ.

• The derivative of PolyLog(s, z) in z has a closed form. D(PolyLog(s, z), z) stayed Apply(Derivative("PolyLog", 0, 1), s, z), which N() cannot evaluate. It is now PolyLog(s − 1, z)/z, which reduces where the engine has a closed form: for s = 1 it is 1/(1 − z), and for s = 2 it is −ln(1 − z)/z. The chain rule applies to the second argument: D(PolyLog(2, x²), x) is −2 ln(1 − x²)/x. The derivative in s has no closed form and stays Apply(Derivative("PolyLog", 1, 0), s, z).

• simplify() cancels the terms of a sum that differ only by the sign of the argument of an odd or even function. simplify() moved the sign out of sin(-x) and cos(-x), but not out of a negative number such as sin(-1), and not for the other odd functions. Also, in a sum it did not simplify a hyperbolic or special function term. So sin(1) - sin(-1), Si(1) - Si(-1), sinh(x) + sinh(-x) and Erf(-x) + Erf(x) stayed as they were. Now simplify() uses f(-x) = -f(x) for the odd functions Sin, Tan, Cot, Csc, Sinh, Tanh, Coth, Csch, Arcsin, Arctan, Arccsc, Arsinh, Artanh, Arcsch, SinIntegral, SinhIntegral, Erf, Erfi, FresnelS and FresnelC, and f(-x) = f(x) for the even functions Cos, Sec, Cosh and Sech, when the argument is -x or a negative real number. sin(1) - sin(-1) simplifies to 2sin(1), Si(1) - Si(-1) to 2Si(1), cos(-2) - cos(2) and Erf(-x) + Erf(x) to 0, and the value of ∫₋₁¹ a·sin(t)/t dt simplifies to 2a·Si(1). evaluate() does not change: it keeps sin(-1) and Si(-1) as they are.

• The lenient grammar no longer drops a subscript on a function name. Only log used its subscript (its base, log_2(8)). On another function name the subscript was dropped with no diagnostic: tan_1x was tan(x), ln_3(x) was ln(x) and sqrt _110theta was sqrt(theta). The subscript is now kept as the strict grammar keeps it: ln_3(x) is Log(x, 3), as \ln_3(x) is, and tan_1x is Apply(Subscript(Tan, 1), x), as \tan_1 x is. A person can mean a name such as tan_1, so the reading reports the new code ambiguous-function-subscript.

• A ^ after a postfix operand is no longer dropped, in both grammars. x^g.5^ was read as (x^g)^{0.5} and x^g.5^2 as (x^g)^{0.5}·2: the parser read the two factors of the product x^g·0.5 as a base and an exponent, and dropped the ^. The same fault dropped an operator that had no right operand: g=^Δ was g^{(missing)}·Δ, with no =. A script after a postfix operator now applies to the whole result, as it does when the operand has no script: x^g.5^2 is (x^g·0.5)², as x.5^2 is (x·0.5)², and x^2!^3 is (x^2!)^3 (it was (x^2)^{(missing)}·3). A ^ with no exponent holds an Error (x^g.5^, g=^Δ). The scripts after an unknown LaTeX command apply to its error, and the parse continues after them: \foo^2 + y is Power(Error, 2) + y, and \intop_0^1\sin x dx keeps the integrand.

• More Unicode superscript characters are read as a superscript, in both grammars: ⁺, ⁽, ⁾ and the modifier letters ˣ and ʸ. e⁻ˣ was e^{-} (the postfix Superminus) times the string ˣ; it is now e^{-x}, as e⁻ⁿ is e^{-n}. x⁻⁽¹⁾ and x⁽ⁿ⁺¹⁾ were errors.

• simplify() no longer applies a library identity to a user function that has a library name. With Sin and Cos defined by the user as t ↦ t + 1, sin(0) simplified to 0, sin(π/6) to 1/2 and sin(x)² + cos(x)² to 1, while evaluate() gave 1, π/6 + 1 and 2(x + 1)². The same fault changed the value of a user Ln, Log, Exp, Abs, Sqrt, Root, Tan, Csc, Cot, Sinh, Tanh, Arsinh, Factorial or Binomial: ln(1) → 0, √4 → 2, |−2| → 2, (n + 1)!/n! → n + 1. simplify() now replaces each call of a shadowed library name by an opaque symbol before the rules apply, and puts the calls back after. So no library identity applies to such a call, also when the call is in a sum or a product, and the fu strategy does the same. A rewrite that uses the calls only as values still applies (Sin(x)·Sin(x) is Sin(x)²), and √x·(√x + 1) keeps its value with a user Sqrt. The trig strategy does not write cos θ + i·sin θ where a user definition, a parameter or a block local shadows Cos or Sin. The same is true for a function parameter or a block local with a library name: ((Sin) ↦ Sin(0) + Sin(x))(t ↦ t + 1) simplified its body to Sin(x). When no library name is shadowed, simplify() does not change.

• A user value of Pi is no longer read as π by evaluate(). With ce.declare('Pi', { value: 3 }), sin(π) evaluated to 0, cos(π) to -1, sin(π/6) to 1/2 and e^{iπ} to -1, while .N() gave sin(3), cos(3) and sin(1/2) (and e^{iπ}.N() was also -1). The code that finds a multiple of π in an angle now recognizes only the library constant, so these are sin(3), cos(3), sin(1/2) and cos(3) + i·sin(3).

  An exact result that holds the library π is also no longer returned when a user binding gives Pi another value. Its .N() was correct, but its MathJSON names Pi, and boxed again it had the user value: with Pi equal to 3, Argument(-1) evaluated to π and boxed again to 3, and Degrees(30) gave π/6, which boxed again to 0.5. These calls now stay symbolic under evaluate(), and .N() gives the same value as before: Argument of a negative number or of i, Arctan2 when its exact value holds π, Degrees and DMS of an exact angle in radians, the values at an infinite argument of Arctan, Arctan2, Arccot, Arcsec, Artanh, Arsech, SinIntegral and CosIntegral, ComplexRoots when a root is not a closed form (ComplexRoots(1, 4) is still [1, i, −1, −i]), and e^{iθ} in degrees, grads or turns. The degree conversion does not use the user value: 30° is the angle π/6 whatever the name Pi holds. Also, ce.Pi.mul(0) is 0 again: with a user value of Pi it was held as the product 0·π, as for a variable with a value. With Pi equal to 3, LogGamma(1/2) evaluated to ln(3)/2 and now stays symbolic, as do EllipticK(0) and EllipticE(0) (π/2, which boxed again to 1.5), ClausenCl(2, Pi) was 0 (the value at π) and is now ClausenCl(2, 3), and a periodic equation solved over a bounded interval (Solve(sin(x) = 0, x ∈ [0, 10])) gave only some of its roots ([0]) and now stays unevaluated. A Solve over a domain also stays unevaluated when a root in the domain holds π: Solve(x·sin(x) = 0, x ∈ [0, 4]) was [0, π] and Solve(sin(x) = 0, x ∈ ℝ) was [0, π], which boxed again to [0, 3]. A root list with no π is still given: Solve((x - 1)·sin(x) = 0, x ∈ [0.5, 2]) is [1], and in degrees Solve(sin(x) = 0, x ∈ [0, 360]) is [0, 180, 360] (it stayed unevaluated).

• The order of the scripts no longer changes the reading of a subscript. A subscript before the superscript joins the symbol name and keeps its text: x_{01}^2 is x_01^2. A subscript after the superscript was read as a number, so x^2_{01} was x_1^2, in both grammars. In the lenient grammar, \alpha^2_01 and θ^2_01 were alpha_0^2·1 and theta_0^2·1, while \alpha_01^2 is alpha_01^2, and x^2_max was x_m^2·a·x. A subscript after a superscript now joins the symbol name as it does before it, on a letter, a Greek letter, a command (\operatorname{speed}), a constant and a primed symbol (x'^2_{01} is (x_01')^2), with each spelling of the prime marks: x^{\prime}^2_{01} (the spelling of the serializer) was Subscript(x', 1)^2 and is now (x_01')^2, as are x^\prime^2_{01} and x^{\doubleprime}^2_{01} (with x_01''). It reports the diagnostics of the other order. The same rules apply: a base that evaluates its subscript (\gamma) and an indexed collection keep the subscript as before. The constant π now has the reading of \pi: π_{01} was Pi_1 and is Pi_01, as \pi_{01} is. In the lenient grammar, π_01 was π_0·1 with ambiguous-implicit-subscript, while \pi_01 was Pi_01; both are now Pi_01, as x_01 is x_01. The strict grammar reads one token after an unbraced _ in each order, as TeX does: \pi_01 is π_0·1. White space directly before the _ stops the join in each order: x _{01}^2 and x^2 _{01} are both x_1^2, as x _{01} is x_1. White space before the superscript does not: x ^2_{01} is x_01^2. A spacing command before a script is white space, in both grammars: x\,_{01} was Nothing_1·x (the space was the base of the subscript) and is now x_1, as x _{01} is; the same applies to \;, \quad, \hspace{…} and to x\,^2, which is now x^2. A subscript after prime marks reports the joined name: x'_{01} reported undeclared-symbol for x, and now reports x_01, as x_{01}' does.

• A trigonometric equation solved over a bounded interval gets all its roots. The solver added multiples of the period of the equation to its principal roots, but the roots can be closer together than that period. Solve(sin(x) + cos(x) = 0, x ∈ [0, 7]) was [7π/4], without 3π/4: the period is 2π, but the roots are π apart. The same error gave [0, π, 2π] for sin(2x) = 0 (without π/2 and 3π/2), [0, 2π] for sin(x)cos(x) = 0, and [2π - 1] for sin(x + 1) = 0. The roots are now expanded with their own spacing (π for sin(x) + cos(x), because f(x + π) = -f(x)), and a numeric scan of the interval checks the list. When the scan finds a root that is not in the list, Solve stays unevaluated: sin(13x)cos(x) = 0 over [0.05, 2] was [π/2], one of its nine roots. The scan also finds a root where the function touches zero next to an end of the interval, and a root at the edge of a region where the function is not real: Solve(x·√(cos(x) + 99/100) = 0, x ∈ [3.1, 7]) was [], but 2π − arccos(−0.99) ≈ 3.2831 is a root. Near a large x, two values are the same root only when they are much closer than the samples of the scan: Solve(sin(13x)cos(x) = 0, x ∈ [10^8, 10^8 + 7]) was a list of 3 of its roots. These now stay unevaluated. When the scan finds no root, the answer is the empty list: Solve(sin(x) = 0, x ∈ [0.1, 0.2]) and Solve(sin(x) = 2, x ∈ [0, 7]) stayed unevaluated, and are now []. Solve(sin(x) = 0, x ∈ [10^17, 10^17 + 64]) did not stop, and now stays unevaluated. Two principal roots that are very near each other are no longer taken as one family of roots: over [0, 7], (sin(x) - 1/2)(sin(x) - 1/2 - 10^-10) = 0 was a list of 3 of its 6 roots, because π/6 and arcsin(1/2 + 10^-10) are only 1.15·10^-10 apart. Two roots are in the same family only when their difference is an exact multiple of the spacing; when the numeric values cannot show it, Solve stays unevaluated. A root of high multiplicity no longer makes the scan reject a complete list: Solve(sin(x)^10 = 0, x ∈ [0, 2π]) stayed unevaluated and is now [0, π, 2π], and Solve((x - 1)^9·sin(x) = 0, x ∈ [0.5, 4]) is now [1, π].

• Solve gives the roots of a trigonometric equation in the angular unit of the engine. In degree mode, Solve(sin(x) = 1/2, x) was [30]: the second root was π − arcsin(1/2), which is not a root when the angle is in degrees. It is now [30, 150] (in grads [100/3, 500/3], in turns [1/12, 5/12]). Solve(cos(x^2) = 0, x) was [] in degree mode, and is now [3√10, −3√10]. Over a bounded interval, the period of the equation is now in the angular unit: Solve(sin(x) = 1/2, x ∈ [0, 720]) was [30] in degree mode, and is now [30, 150, 390, 510].

• Integrate no longer integrates a user function as the library function of the same name. With Sin := t ↦ t + 1, ∫₀¹ Sin(x) dx evaluated to 1 − cos(1) (.N() gave 1.5) and ∫ x·Sin(x) dx to −x cos x + sin x. The body of the user function now replaces the call, as D does: these are 3/2 and x³/3 + x²/2. When the body cannot replace the call (an evaluate handler, a block local or a parameter with a library name, a body that reads a symbol with the name of the integration variable), the integral stays unevaluated.

• Sinc gives the same value in every angular unit. Sinc(x) is sin(x)/x with x read as a number, not as an angle, but its exact branch computed Sin of the operand, which reads the operand in the current angular unit. In degree mode, Sinc(10⌊π⌋) evaluated to 1/60; the value is sin(30)/30 ≈ −0.0329. The exact branch now converts the operand from radians to the current unit before it computes Sin, so the exact values Sinc(π) = 0 and Sinc(π/2) = 2/π are the same in every unit.

0.147.0 2026-10-02

Behavior Changes

• Totient and NPartition evaluate for zero and negative integers. Totient(0) is 0 and Totient(-n) is Totient(n) (Totient(-12) is 4); NPartition(n) is 0 for a negative integer n. These are the values of Mathematica's EulerPhi and PartitionsP, and of the Fungrim identities the engine already loads (Totient(-n) = Totient(n), NPartition(-n) = 0). Before, these inputs stayed unevaluated (Totient(0), NPartition(-3)).

• arctan and arccot are odd on their branch cuts. The cut of arctan is the imaginary axis outside [−i, i], the cut of arccot the imaginary axis inside it. Both halves of each cut took the side right of the axis, so arctan(−z) was not −arctan(z) there. Each half now takes the side it is continuous with, as in mpmath and Mathematica, and as arsinh already did on the same axis. The values that change, in the interpreter (N(), and evaluate() of a float argument), compiled JavaScript and Python, and for arctan only in GLSL and WGSL (the shader targets compile arccot of a real argument only):

  • arctan(iy) for y < −1: the real part changes from π/2 to −π/2 (arctan(−2i) was π/2 − 0.549i, it is now −π/2 − 0.549i; arctan(−1.5i) was π/2 − 0.805i, it is now −π/2 − 0.805i).
  • arccot(iy) for 0 < y < 1: the real part changes from π/2 to −π/2 (arccot(0.5i) was π/2 − 0.549i, it is now −π/2 − 0.549i).
  • arctan(iy) for y > 1, arccot(iy) for −1 < y < 0, and artanh and arcoth on their cuts on the real axis do not change.

• The lenient grammar reads ** as ^. ** was an infix operator that bound looser than a prefix minus and than a ^ in its exponent. It now has the same reading as ^ (ce.parse(text, { strict: false })):

  • -e**θ is -(e^θ), was (-e)^θ; -x**2 is -(x²), was (-x)².
  • 1**x^M, a**b^c and x^2**3 are an unexpected-superscript error, as 1^x^M is. Before, they were 1^(x^M), a^(b^c) and (x²)³.
  • x**y! is (x^y)!, as x^y! is, was x^(y!).
  • x**2y is x²·y and reports ambiguous-exponent-end, as x^2y does.
  • A chain of ** alone is read from the right, as in programming languages, and does not change: 2**3**2 is 2^(3^2).

• White space after ^ does not change the reading of the lenient grammar. x ^ pi is x^π, was x^p·i; x^ -y is x^(-y), was an error. White space after the - of an exponent on a letter is also skipped: e^- x is e^(-x), was Superminus(e)·x. A^+ x (the pseudo-inverse of A times x) and \R^- x do not change, and the strict grammar does not change.

• x^^2 and x^² are a double-superscript error, in both grammars. The tokenizer dropped the second ^ of a ^^ that was not followed by two hexadecimal digits, so these inputs were read as x². A TeX ^^ character code such as ^^41 is still read.

• The factorial of a real non-integer follows ce.precision. x! for a real non-integer x is Γ(x + 1). It was computed in doubles at every precision, so Factorial(2.5).N() was 3.3233509704478448, a double, Factorial(200.5).N() overflowed to +oo and Factorial(-200.5).N() underflowed to 0. It is now computed as Gamma computes it, to ce.precision digits: Factorial(2.5).N() is 3.32335097044784255118, Factorial(200.5).N() is 1.1174203734326765163e+376 and Factorial(-200.5).N() is 5.63699519017856675668e-374 at the default precision of 21 digits. At precision: "machine" the result is the double, as before. A compiled (1/2 - 1)!, whose constant is folded from .N(), is now 1.772453850905516, the double nearest √π, not 1.7724538509055159.

• The factorial of an exact non-integer stays symbolic under evaluate(). Factorial(5/2).evaluate(), Factorial(-1/2).evaluate() and Factorial(1 + i).evaluate() gave a float, against the rule that evaluate() keeps an exact argument exact. They now stay (5/2)!, (-1/2)! and (1 + i)!, as Gamma(7/2), Gamma(1/2) and Gamma(2 + i) do; .N() and a float operand (2.5!) give the float. Integer factorials are unchanged.

• A factorial too large for the exact product is a big decimal under .N(). Past the exact digit cap (10⁶ digits), Factorial(n).N() and Factorial2(n).N() overflowed to +oo. Above machine precision they are now computed from Γ to ce.precision digits: Factorial(10^6).N() is 8.26393168833124006238e+5565708, Factorial(10^7).N() is 1.20242340051590345614e+65657059 and Factorial2(10^15).N() is 6.83517080822812242208e+7282852759048381. Under evaluate() they stay symbolic, as before. At precision: "machine" they are +oo, since a double cannot hold them. A value whose decimal exponent is past ±2⁵³ ≈ ±9·10¹⁵, the largest exponent a big decimal holds exactly, has no big-decimal value either (the big-decimal exp saturates to infinity or 0 there). A real value that overflows past that range is +oo, as before and as a double overflow is (Factorial(10^15).N(), Gamma(10^15 + 0.5).N()). A real value that underflows past it now stays unevaluated, since a 0 reads as an exact zero and Γ has no zeros: Gamma(-10^15 - 0.5).N() and Factorial(-10^15 - 0.5).N() (decimal exponent about −1.5·10¹⁶) were 0. A complex Gamma, Factorial or BarnesG value past that range, in either direction, stays unevaluated: a complex value that is too large has a direction that no infinity of the engine holds. The big-decimal Γ at a high precision now stops at the time limit (ce.withTimeLimit): the Bernoulli table it builds (about 30 s at ce.precision = 3000) checks the deadline at every step instead of every 256 steps, and the Stirling series checks it at every term.

• A small part of a complex result of a float input is kept. The machine kernels of the interpreter (.N() at machine precision; and at every precision the kernels of the special functions and a complex power, root or exponential with a machine operand) set to 0 a part below 10⁻¹⁴ times the modulus of the result, and the compiled complex helpers set to 0 a part below 10⁻¹⁴ and below 10⁻¹⁴ times the modulus. That part is the value at the float input, and is now kept: at machine precision e^{3.141592653589793i}.N() is −1 + 1.22·10⁻¹⁶i (was −1) and 2^{10^{-100}i}.N() is 1 + 6.93·10⁻¹⁰¹i (was 1); compiled with folding off, e^{\ln(−2)} is −2 + 2.45·10⁻¹⁶i (was −2). The Chop operator removes such a part. An exact input has no such part (next entry): e^{iπ} and e^{\ln(−2)} evaluate to −1 and −2.

• An exact multiple of π is reduced exactly under .N() and in compiled code. The multiple of π is reduced in half-turns with bigints, so no rounding of π reaches a part whose value is 0 or ±1/2. At machine precision, sin(π).N() is 0 (was 1.22·10⁻¹⁶), cos(π/2).N() is 0 (was 6.1·10⁻¹⁷), cos(2π/3).N() is −0.5 (was −0.4999999999999998) and sin(π/6).N() is 0.5 (was 0.49999999999999994). At every precision and in every angular unit, e^{2iπ/3}.N() is −0.5 + 0.866i with a real part of exactly −0.5 (was −0.4999999999999998 at machine precision and −0.500000000000000000001 at 21 digits), e^{i·10^{20}π}.N() is 1 (was 0.919 − 0.394i at machine precision), sinh(iπ).N() and cosh(iπ/2).N() are 0 (were 1.22·10⁻¹⁶i and 6.1·10⁻¹⁷), and tanh(iπ/2).N(), coth(iπ).N(), csch(iπ).N() and sech(iπ/2).N() are ~oo (were NaN, ±8.2·10¹⁵i and 1.6·10¹⁶). Compiled JavaScript, Python, GLSL and WGSL lower sin(πu), cos(πu), tan(πu) and e^{a + iπu} with the angle u in half-turns (JavaScript also over a list, sin(πL)), so the value at an integer or a half-integer u has an exact zero part, +0: compiled sin(πx) at x = 2 is 0 (was −2.45·10⁻¹⁶), e^{iπx} at x = 1 is the plain real −1, and e^{x + iπ} is −e^x. A compiled sin(πx) costs about 25% more per call in JavaScript than Math.sin(Math.PI * x).

• A negative real base with a float exponent has the interpreter's real root in compiled code. The interpreter reads the rational p/q that the double came from, and a q that is odd gives the real root: (−8)^{0.4} is 2.297 and (−8)^{0.3333333333333333} is −2 at machine precision (−1.99999999999999986137 at 21 digits). The compiled complex power gave the principal value (0.710 + 2.185i and 1 + 1.732i), the compiled JavaScript real power x^y of two real variables gave NaN, and Python gave the principal value or nan. Now JavaScript and Python give the real root (−2 for the example) for a constant or a variable exponent (Python compileFunction through a helper _ce_pow; a bare lambda of compileLambda has no place for it and keeps x ** y, the principal value, for a variable exponent); GLSL and WGSL give it for a constant exponent, and keep pow for a variable one, which a shader holds as an f32 and cannot read to 17 digits. An even q, or no rational ((−3)^{√2}), keeps the principal value (in Python with an exact angle: (−1)^{0.5} is i, was 6.1·10⁻¹⁷ + i), and NaN on the JavaScript and shader real lanes.

• An exact complex number divided by an exact number is an exact number literal, and a float complex literal stays inexact. The canonical form and its MathJSON and LaTeX change:

  • \frac{i}{3} was ["Divide", ["Complex", 0, 1], 3]; it is now the literal (1/3)i, ["Complex", 0, ["Rational", 1, 3]], as \frac{1}{3}i already was. Its LaTeX is now \frac{1}{3}\imaginaryI.
  • \frac{3+i}{2} is 3/2 + (1/2)i, \frac{i}{\sqrt2} is (√2/2)i, and \frac{i}{3}\cdot 3 is i.
  • A complex divisor folds too: \frac{2}{i} is -2i and \frac{3+i}{1-i} is 1+2i.
  • A float complex literal is no longer read as an exact value when its parts are integers. \frac{1.0i}{3} and \frac{3}{1.0i} keep the Divide, 1.0i\cdot 3 and (2.0+1.0i)\cdot 3 keep the Multiply, and 1.0i+3 is the inexact 3.0+1.0i. Before, the products and the sum folded to the exact 3i, 6+3i and 3+i. This is the rule for a real float, where \frac{1.0}{3} keeps the Divide.
  • The same rule applies to arithmetic on number values: a sum, product or quotient of an exact value and a float complex value is a float, also when the parts of the float are integers. \frac{1}{3}+1.0i, \frac{1}{3}\cdot(2.0+1.0i), \frac{3}{1.0i} and \sum_{k=1}^{3}\frac{k}{2}\cdot 1.0i evaluate to floats (they were exact), as do ce.number([1,3]).add(ce.parse('1.0i')) and the product of (1+i).N() with the exact 1-i.

  An imaginary factor inside a Divide was not seen by the exact reduction of an imaginary multiple of π, so \tanh(\frac{i}{3}\cdot\frac{3}{2}\pi) and \coth(\frac{i}{3}\cdot\frac{3}{2}\pi) gave NaN under .N(). They now give ~oo and 0, as \tanh(\frac{i\pi}{2}) and \coth(\frac{i\pi}{2}) do. A symbolic divisor (\frac{i}{x}) keeps the division.

• A complex number from the host with integer parts is exact; a complex float is never read as exact. ce.number(new Complex(2, 3)), ce.number({ re: 2, im: 3 }) and ce.box(new Complex(2, 3)) are now the exact 2+3i (isExact is true, the MathJSON is ["Complex", 2, 3]; it was ["Complex", {num: "2.0"}, {num: "3.0"}]), as ce.number(2) is the exact 2. Integer-valued BigDecimal parts are exact too. A part with a fraction (new Complex(2.5, 3)) gives a float, as before. In return, a float whose parts are integers is no longer taken for an exact value in a computation, so a float operand makes the result a float:

  • \frac{1}{3}\cdot 1.0i and \sqrt2\cdot 1.0i evaluate to floats (they were the exact (1/3)i and √2·i).
  • (1.0+1.0i)^2 is the float 2.0i (it was the exact 2i); (1+i)^2 stays the exact 2i.
  • \Gamma(1.0i) and \cos(1.0i\pi) evaluate to floats under evaluate() (they stayed symbolic, as \Gamma(i) does).
  • A float multiple of π that is a special angle stays exact: e^{1.0i\pi} is -1, as before.

  The complex results of the numeric routines stay floats: the conjugate of a float, an even root of a negative number under .N(), the complex kernels (\Gamma(1.0+1.0i)), and the complex entries of a matrix that holds a complex float (Trace, Transpose, MatrixMultiply, Determinant of [[1.0+1.0i, 0], [0, 1.0+1.0i]]).

  The results of the numeric routines are now floats also when they are real integers (they were exact): NIntegrate(1, 0, 2) is 2.0 (it was 2), as are the values of NLimit, ND, NDSolve (the inner grid points too; an end point that is an exact limit stays exact), the fitted parameters of FindFit and FindRoot, and the roots that ComplexRoots computes for a float (ComplexRoots(1.0, 4) starts with 1.0). For a matrix with a float entry, the results of Eigenvalues, Eigenvectors, SingularValues, SVD, QRDecomposition and the spectral norm are floats (SingularValues([[3.0, 0], [0, 4.0]]) is [4.0, 3.0], it was [4, 3]); a matrix of exact entries keeps the results it had.

  The eigenvalues of an exact matrix larger than 3×3 (computed by the QR algorithm) are checked with exact arithmetic: an eigenvalue close to a rational number that makes A − λI singular is returned exact, and Eigenvectors then gives exact vectors. A double eigenvalue of a defective matrix, which the QR algorithm gives with an error of about 10^{-8}, is now exact: the eigenvalues of [[5,4,2,1],[0,1,-1,-1],[-1,-1,3,0],[1,1,-1,2]] are [4, 4, 2, 1] (they were [4.0000000258, 3.9999999742, 2, 1]), with exact eigenvectors.

New Features

• More reading choices of the lenient grammar report an ambiguous-* code (the reading does not change):

  • ambiguous-denominator for ÷: 13÷2x is 13/(2x), as 13/2x is.
  • ambiguous-digit-groups for white space before a decimal separator: 3 .5 is read as 3.5.
  • ambiguous-range for ... directly followed by a digit after a decimal number: .5...5 is the range from 0.5 to 5, and can be .5.. and .5. 1...5 is not reported.
  • ambiguous-function-argument for an argument with no parentheses that starts with +: ln+1 is ln(1). sin -x is not reported.

• Wolfram Language aliases EulerPhi, PartitionsP and Det. They are aliases for Totient, NPartition and Determinant: the canonical form uses the CE name (["EulerPhi", 12] becomes ["Totient", 12], which evaluates to 4), and an invalid operand gives the same error as the CE operator. Before, EulerPhi(12) stayed unevaluated with no error. Det has only the one-operand form: Det[m, Modulus -> n] has no CE equivalent and its second operand is an unexpected-argument error. Tr gets no alias, because Mathematica's Tr of a vector or of a rank-3 tensor has a different meaning than Trace. See docs/MATHEMATICA-NAMES.md.

• More ambiguous-* parse diagnostics in the lenient grammar. These inputs were read with no diagnostic, although a person can mean a second reading. The reading does not change, and the strict grammar reports none of these codes. One code is new and eight codes report more inputs:

  • ambiguous-missing-base (new): a _ with no base is read as the symbol _ (-_1).
  • ambiguous-name-then-number: a run of letters, white space, then a number (xy 0.5, was reported only as a letter run), and ∞ directly followed by a digit (∞2).
  • ambiguous-implicit-subscript: an unbraced subscript of letters directly followed by a digit (M_max0.5, x_a2), or that ends with a function name (A_maxsin t).
  • ambiguous-delta: Δ after a sign (-Δα, -Δx).
  • ambiguous-exponent-end: a one-letter exponent directly followed by a letter (x^xy, e^xy; these also report ambiguous-letter-run, as before), and a function name after a signed exponent and white space (e^-x sin x).
  • ambiguous-factorial: a ! directly after an exponent or a radicand (M³!, 2^-k!, √i!), and a superscript directly after a ! (n!²).
  • ambiguous-radical: a Latin letter directly before √ (t√y), and a one-letter radicand followed by white space and an operand (√a b).
  • ambiguous-function-argument: a power of e after the first factor of an argument without parentheses (sin2e^a), and a function name with no argument (y = min, sin*x).
  • ambiguous-constant-name: ii, read as the imaginary unit.

  See docs/plans/2026-10-01-lenient-ambiguity-codes.md, section 5.

Issues Resolved

• sin\prime(x) in the lenient grammar is the derivative of sin, as sin'(x) and \sin\prime(x) are. The \prime was read as the start of the argument of sin, and gave an unexpected-command error.

• toString() wraps the operand of a factorial when needed. The factorial of 5/2 printed as 5/2!, which reads as 5/(2!), and the factorial of x/2 as 1/2 * x!. An operand that is not a name, an unsigned decimal number or one function call is now in parentheses: (5/2)!, (1/2 * x)!, (n + 1)!, (-3)!; 5!, n!, 2.5! and sin(x)! are unchanged. A factorial used as the base of a power prints as (n!)^2: n!^2 did not parse back, since !^ is read as one operator.

• A complex BarnesG value outside the double range is no longer 0 or left unevaluated. BarnesG(0.5 + 30i).N() was 0, which reads as an exact zero of G (G has none off the non-positive integers), and BarnesG(30 + 0.5i).N() stayed unevaluated: a complex argument is computed in doubles, and exp(ln G) underflowed or overflowed. When it leaves the range of a double, G is now formed from ln G as exp(Re ln G)·(cos Im ln G + i sin Im ln G) with a big-decimal modulus: BarnesG(0.5 + 30i).N() is 1.48501924951e-362 + 2.52832839715e-362i (mpmath: 1.48501924950746e-362 + 2.52832839715190e-362i) and BarnesG(60 + 10i).N() is 1.1631308292e+1883 - 2.10110905e+1881i. The value keeps only the digits that ln G in doubles gives: its error is about 2⁻⁵² times |ln G| relative to the modulus, so 12 digits at 0.5 + 30i and 10 at 0.5 + 300i, and a part smaller than the modulus keeps fewer (each part is rounded at the last correct digit of the modulus), at any ce.precision, as for the other complex results of the special functions. At precision: "machine" the engine has no number that holds such a value, and the result is unchanged (0 on underflow, unevaluated on overflow), as for e^{-1000+i}, which is 0 there.

• A complex Gamma or Factorial value outside the double range is no longer 0 or ~oo. Gamma(0.5 + 1000i).N() and Gamma(-200.5 + 0.5i).N() were 0, and Gamma(200.5 + 0.5i).N() was ~oo, which reads as a pole, while Γ has no zeros and its only poles are the non-positive integers. As for BarnesG above, the value is now formed from ln Γ with a big-decimal modulus, keeping the digits that ln Γ in doubles gives: Gamma(200.5 + 0.5i).N() is -4.90792586263e+373 + 2.63318721817e+373i (mpmath: -4.907925862634e+373 + 2.633187218170e+373i) and Gamma(0.5 + 1000i).N() is 1.57066061e-684 + 1.6251473018e-682i. A complex Factorial takes the same route: Factorial(300 + 2i).N() is 1.22674475486e+614 - 2.78179033693e+614i. At precision: "machine" the result is unchanged (0 on underflow, ~oo on overflow), as for e^{-1000+i}.

• The compiled StieltjesGamma is accurate to a double up to order 30. The JavaScript kernel summed the Euler–Maclaurin form in doubles, where the partial sum and the subtracted power of the logarithm are up to 10⁹ times larger than the result and cancel: the compiled StieltjesGamma(30) was 0.003557728327971422 (7 correct digits), and StieltjesGamma(30, 0.5) was -0.0035240594494571897 (6 correct digits). For a real a between 10⁻¹⁵⁰ and 10¹⁵⁰ the sum is now taken in double-double arithmetic (about 32 digits): 0.003557728855573161 and -0.0035240626522715247, as mpmath gives. Measured against mpmath for every order up to 30 at a = 1, 0.5, 2.5, 10 (and up to 20 at a = 0.001), the largest relative error is 2.2e-16. 1000 values of StieltjesGamma(30, a) take about 35 ms.

• The inverse trigonometric and inverse hyperbolic functions of an argument beyond the double range have a value. \arcsin(10^{400}) was NaN at every precision, and is now π/2 − 921.72718437817821891…i. \operatorname{arcsec}(10^{-320}) was computed from the subnormal double nearest 10⁻³²⁰, which keeps 5 digits (737.5203880715337i), and is now 737.52037693865456…i. At machine precision, \operatorname{arsinh}(10^{400}) was +∞ and \operatorname{arcsch}(10^{-400}) was ~oo; both are 921.7271843781782. A real argument, exact or a big decimal, outside the range of a normal double now gives a value at the working precision for the twelve heads (Arcsin, Arccos, Arctan, Arccot, Arcsec, Arccsc, Arsinh, Arcosh, Artanh, Arcoth, Arsech, Arcsch), with the side of each branch cut that the engine takes at ±10³⁰⁰ and ±10⁻³⁰⁰. A complex argument with a part beyond the double range gives a value with the digits of a double, as the complex kernels do at every precision, and keeps a part of the value too small for a double (above machine precision, \arctan(10^{400}(1+i)) is π/2 + 5·10⁻⁴⁰¹i). A complex argument with a part too small for a double next to the other (\arcsin(2+10^{-400}i)) takes the side of the branch cut that the sign of that part selects: it was π/2 − 1.317i, below the cut, and is now π/2 + 1.317i. \arcsin(10^{400}+i) was NaN and is π/2 + 921.727i. At a logarithmic branch point the part that grows as ln(1/δ) keeps its value: \operatorname{artanh}(1+10^{-400}i) is 460.8635921890891 + 0.785i. An exact real argument next to ±1 is no longer read as ±1: \arcsin(1+10^{-400}) was the real π/2, and is π/2 − 1.414·10⁻²⁰⁰i; at machine precision \operatorname{artanh}(1-10^{-400}) was +∞ and is 460.86359218908911. At machine precision, Arctan of a complex argument whose double is ~oo (\arctan(10^{400}(1+i))) was an incompatible-type error and is π/2; an argument that is ~oo stays an error. A float argument whose double is subnormal is unchanged at machine precision: the kernels read that double.

• A trigonometric function of an operand with a random draw draws once under evaluate(). \arcsin(1+\operatorname{Random}()) evaluated the operand a second time, to look for a special angle, and used up two draws of a WithRandomSeed frame. An operand that is not pure is no longer read again.

• Above machine precision, the inverse trigonometric and inverse hyperbolic functions of a real argument with a complex value have the working precision. \operatorname{arcosh}(1/2), \arcsin(2), \operatorname{artanh}(2), \operatorname{arsech}(-1/2), \operatorname{arcoth}(1/2) and the others on a branch cut of the real axis had a complex value computed in doubles (16 digits) at every precision. At 50 digits, \operatorname{arcosh}(1/2) was 1.0471975511965979i and is now 1.047197551196597746154214461093167628065723133125i. A float argument is read as its shortest decimal, as for a real value: \operatorname{arcoth}(0.999999) at 21 digits is 7.25432861926204720674 − 1.57079632679489661923i (from the double nearest 0.999999 it was 7.2543286192476694). The sides of the cuts are those of the double kernels.

• A complex value with a NaN part from a kernel is NaN. At machine precision \sin(10^{400}+i) reached the double kernel with an infinite real part, and the NaN imaginary part of its value failed an internal assertion. The value is now NaN.

• Compiled JavaScript keeps a small part of a complex result, as the interpreter does. The compiled complex functions set to 0 a part below 10⁻¹⁴ and below 10⁻¹⁴ times the modulus of the result, so arcoth(10⁻¹⁰⁰) was −(π/2)i compiled and 10⁻¹⁰⁰ − (π/2)i in the interpreter, 2^{10⁻¹⁰⁰i} was 1 compiled and 1 + 6.93·10⁻¹⁰¹i in the interpreter, and sin(π + i) at the double π lost its real part 1.9·10⁻¹⁶. Every compiled complex function now keeps every part: the kernels give an exact 0 for a part whose value is 0, so a real argument in the real domain still gives a plain real number. The complex power is computed by repeated squaring for a small integer exponent, and with the angle cos(πt) + i·sin(πt) exact at the multiples of π/2 for a base on an axis or a diagonal, so it has an exact 0 part without a removal ((−2.0)^3 is −8, was −7.999999999999998). e^z compiles to the exponential, as the interpreter computes it: Math.E^z lost digits (e^{40} had a relative error of 2·10⁻¹⁵). The compiled complex square root no longer loses a small argument: √(10⁻³⁰⁰·i) was 0 and is 7.07·10⁻¹⁵¹·(1 + i).

• In degrees, an inverse trigonometric value keeps a small imaginary part. The conversion of the angle to the angular unit set to 0 an imaginary part below 10⁻¹⁴: arcsin(10^{-200}i).N() in degrees was 0, and is 5.73·10⁻¹⁹⁹i; arctan(10^{-50}i) was 0, and is 5.73·10⁻⁴⁹i.

• The compiled ~oo is ∞ + ∞i on the complex lane, at run time and in a folded constant. arctan(±i) and arccot(±i), which are ~oo in the interpreter, gave 0 ± ∞i at run time (JavaScript, Python and GLSL/WGSL) and the real Infinity when the argument was a constant. A constant that evaluates to ~oo on a node whose value is complex folded to Infinity too. Both now give ∞ + ∞i, the value cot 0 and csc 0 already gave on the complex lane. The outputs that change: arctan(±i), arccot(±i), arctan(i)^2 and (1 + i) + ~oo compiled with a constant argument (were Infinity). A ~oo on a node whose value is real still compiles to Infinity (1/0, Γ(−2)).

0.146.0 2026-10-02

Behavior Changes

• A definite integral with only one bound stays unevaluated. A limit that has an upper bound and no lower bound, or a lower bound and no upper bound (\int^2 y^2\,dy, \int_0 y^2\,dy, Limits(y, Nothing, 2), the flat ["Integrate", f, "x", 10]), is neither an indefinite nor a definite integral, and no default is chosen for the missing bound. \int^2 y^2\,dy evaluated to 7/3, as if the lower bound were 1, and \int_0 y^2\,dy to 1/3; with a second limit the result could contain 1 - Nothing^2. They now stay unevaluated, under evaluate() and N() alike, and compile() falls back to the interpreter. The indefinite integral (no bounds) and the definite integral (two bounds) are unchanged.

• A sum or product with only one bound stays unevaluated, as an integral with only one bound does. No default is chosen for the missing bound, under evaluate() and N() alike, and compile() falls back to the interpreter.

  • \sum_{k=1} k evaluated to 1: the subscript k=1 was read as the UPPER bound, Limits(k, Nothing, 1). The subscript is now the lower bound, Limits(k, 1, Nothing), and the sum stays unevaluated. The same applies to each index of \sum_{k=1, j=2}, and to \prod.
  • \sum_{k}^{10} k evaluated to 55, as if the lower bound were 1. It now parses to Limits(k, Nothing, 10) and stays unevaluated. So Sum(k, Limits(k, Nothing, 10)), which evaluated to 55 and serialized as \sum_{k}^{10}k, now stays unevaluated and round-trips through LaTeX unchanged. The flat ["Sum", "k", "k", 10] is the same sum.
  • \sum^{10} k and \prod^{5} k parsed to ["Sum", "k"] and ["Product", "k"]: the bound was lost. The bound is now kept with no index, as for \sum_1^9 k: Sum(k, Limits(Nothing, Nothing, 10)), which stays unevaluated and serializes back to \sum^{10}k.
  • A lower bound with no upper bound was read as a sum to infinity under N(): Sum(1/k^2, Limits(k, 1, Nothing)).N() gave 1.6449…. It now stays unevaluated: write the upper bound +∞ (\sum_{k=1}^{\infty}) for an infinite sum.

  A sum or product with two bounds, over an indexing set (k \in S), or with no bounds at all (\sum_k f) is unchanged, and so is \sum_{i \le 10} i, which keeps an implied lower bound of 1.

• A definite integral over a pole whose position is a free symbol stays unevaluated. Integrate((y − a)⁻², Limits(y, 0, 4)).evaluate() gave -1/a - 1/(4 - a), which is wrong for 0 ≤ a ≤ 4 (the integral is +∞ there); it now stays unevaluated. When the declared type of the symbol or an assumption proves the pole outside the bounds, the integral is evaluated as before: with ce.assume(a > 4), or a declared real<..<0>, the result is -1/a - 1/(4 - a). The poles examined are the roots of a denominator of degree 1 or 2 (factor by factor for a product), the zero of a logarithm of a linear argument in a denominator, and the poles of tan, cot, sec, csc, coth and csch and the zeros of a circular or hyperbolic denominator, of a linear argument. A quadratic denominator whose discriminant is not proven negative may have a real root, so ∫₋₁¹ dt/(t² + m) now stays unevaluated (its roots ±√(−m) are inside for −1 ≤ m ≤ 0), and so does ∫₀¹ dt/(t² + m²) (a pole at 0 for m = 0) unless the sign of m is known (with ce.assume(m > 0) it is arctan(1/m)/m). In an iterated integral, the integration variable of an enclosing integral is examined over its range: Integrate((y − x)⁻², Limits(y, 0, 10), Limits(x, 3, 4)).evaluate() gave the wrong partial result int_(0)^(10)(-1/(y - 3) + 1/(y - 4) dy) and now stays unevaluated (the pole x = y is in [3, 4] for 3 ≤ y ≤ 4), while Integrate((y − x)⁻², Limits(y, 5, 10), Limits(x, 3, 4)) gives ln(2) + ln(6) − ln(7) (ln(12/7)). The same is true when the iterated integral is written as nested integrals: \int_5^{10}\int_3^4 (y-x)^{-2}\,dx\,dy gives ln(2) + ln(6) − ln(7). .N() is unchanged.

  A symbol with no declared type is taken to be real when the position of a pole is proven, as the antiderivatives already assume (∫ dt/(t − a) is ln|t − a|). So ∫₀¹ dt/(t + a² + 1) is ln|a² + 2| − ln|a² + 1|, ∫₁² dt/(t(t² + a² + 1)) and ∫₀¹ dt/(t² + a² + 1) have their closed forms, and ∫₀¹ eᵗ/(eᵗ + a² + 1) dt is ln|a² + 1 + e| − ln|a² + 2|. A symbol whose type was only inferred from its uses is taken to be real too, so the result does not depend on what the engine computed before. A symbol declared by the user (complex, number) is not taken to be real. A free symbol beside a pole at a number keeps the closed form when the pole is removable for every value of the symbol: ∫₀² a(t² − 1)/(t − 1) dt is 4a, ∫₋₁¹ a·sin(t)/t dt is a·Si(1) − a·Si(−1), and ∫₋₁¹ (sin(t)/t + a) dt is 2a + 2Si(1).

• A definite integral with no value is Indeterminate. When the integrand changes sign across a pole inside the bounds (∫₋₁¹ dt/t, ∫₀² tan t dt, ∫ from ½ to 2 of dt/ln t), or has poles at both bounds that diverge with different signs (∫₀¹ (1/t − 1/(1 − t)) dt), the integral has no value, not even an infinite one. .evaluate() kept these integrals unevaluated; it now gives Indeterminate, or NaN when a bound or a number in the integrand is a float (∫₋₁¹ 1.5/t dt). .N() and NIntegrate give NaN, as before. A pole with one sign is still +∞ or −∞, and a divergence whose sign is not established still stays unevaluated.

• A constant NIntegrate inside a compiled expression is folded to its value. compile(y + NIntegrate(x ↦ x, 0, 1)) gave success: false, since NIntegrate has no compiled form and the compiler declined to evaluate it ahead of time; it now compiles to _.y + 0.5. The compiler evaluates a constant NIntegrate like a one-dimensional Integrate, within the same cost limit, so a nested integral (a user function whose body is another NIntegrate) still declines.

• NIntegrate uses quadrature first. NIntegrate(f, a, b) now uses the same methods, in the same order, as Integrate(f, x, a, b).N(): on a semi-infinite interval the oscillatory quadrature, then adaptive Gauss–Kronrod quadrature (for a compiled integrand and, with a smaller panel budget, for an interpreted one). It used to go straight to Monte Carlo sampling. NIntegrate(x ↦ x², 0, 1) was 0.333422 and is now 0.3333333333333333. A divergent integral is now +∞, −∞ or NaN (NIntegrate(x ↦ 1/x, 0, 1) is +∞, and a proven pole strictly inside the bounds gives +∞, −∞ or NaN), as for Integrate(…).N(); before, both gave a sampled finite number. Monte Carlo is still the fallback when the quadrature does not converge, so NIntegrate still reads the WithRandomSeed frame. A seeded NIntegrate gives different digits than before when the quadrature succeeds, because no sample is drawn. The result is still a plain number (real or complex), not a Measurement.

• PrimitiveRoot ignores the sign of n. The unit group mod −n is the unit group mod n, so PrimitiveRoot(-7) is now 3; it stayed unevaluated before. This agrees with the new PrimitiveRootList, which also gives [0] for n = 1, as PrimitiveRoot(1) is 0. PrimitiveRoot(0) stays unevaluated and PrimitiveRootList(0) is [].

• MultiplicativeOrder ignores the sign of n, as PrimitiveRoot does, because the unit group mod −n is the unit group mod n: MultiplicativeOrder(3, -7) is 6; it stayed unevaluated before. This applies to the form with a list of residues too. n = 0 still stays unevaluated.

New Features

• StirlingS2(n, k) is an alias for Stirling(n, k), the Stirling number of the second kind, under its Mathematica name. Its canonical form is Stirling, the preferred name: StirlingS2(6, 3) evaluates to 90, and StirlingS2(n, k) is Stirling(n, k). Stirling is unchanged. (#395, contributed by enumeratio)

• Modular roots, primitive roots and discrete logarithms. PowerModList(a, s/r, m) lists every x in [0, m) with x^r ≡ a^s (mod m) in order: PowerModList(3, 1/2, 11) is [5, 6], PowerModList(1, 1/3, 7) is [1, 2, 4], PowerModList(-1, 1/2, 625) is [182, 443], and an a^s that is not an r-th power gives []. PowerMod accepts the same rational exponent and returns the least root, as Mathematica does: PowerMod(4, 1/2, 7) is 2, where it was an incompatible-type error, and it stays unevaluated when there is none (PowerMod(3, 1/2, 7)). PrimitiveRootList(n) lists every primitive root: PrimitiveRootList(25) is [2, 3, 8, 12, 13, 17, 22, 23], and [] when n has none. MultiplicativeOrder(k, n, [r1, r2, …]) is the least m ≥ 1 with k^m ≡ r_i (mod n) for some i: MultiplicativeOrder(5, 7, [3, 11]) is 2; it stays unevaluated when no r_i is a power of k. RationalReconstruction(a, m) recovers the fraction p/q with p ≡ a·q (mod m) and |p|, q ≤ ⌊√((m − 1)/2)⌋ (Wang's algorithm): RationalReconstruction(6, 11) is 1/2. All five work over bigints (PowerModList(2, 1/3, 2^89 − 1) lists its three roots) and stay unevaluated rather than answer wrongly when the modulus cannot be factored, a discrete logarithm is out of reach, or a list would pass 100000 entries. The integer exponent forms of PowerMod and the two-argument MultiplicativeOrder are unchanged. When there are too many roots to list, PowerMod still finds the least root if it is below 100000: PowerMod(0, 1/2, 2^200) is 0 (#395, contributed by enumeratio).

• DirichletEta and DirichletBeta, the two alternating cousins of ζ, both entire (#395, contributed by enumeratio). DirichletEta(s) is η(s) = Σ (−1)ⁿ⁻¹/nˢ = (1 − 2¹⁻ˢ) ζ(s) and DirichletBeta(s) is β(s) = Σ (−1)ⁿ/(2n+1)ˢ = 4⁻ˢ (ζ(s, ¼) − ζ(s, ¾)). Exact values at the integers: DirichletEta(1) is ln 2, DirichletEta(2) is π²/12, DirichletEta(0) is 1/2, DirichletEta(-3) is -1/8; DirichletBeta(1) is π/4, DirichletBeta(2) is CatalanConstant, DirichletBeta(5) is 5π⁵/1536, DirichletBeta(-4) is 5/2 (Euler numbers). A real s is answered at ce.precision digits, also next to the pole of ζ: N(DirichletEta(1/2)) is 0.604898643421630370247, and η(1 + 10⁻³⁰) keeps every digit instead of cancelling the pole. A complex s is answered in doubles: DirichletBeta(0.5 + 14i) is 1.5371154384 + 1.3434514269i. Both are 1 at +∞ and Indeterminate at -∞. A real s too close to 1 for the digits to be reached stays unevaluated. Both compile to JavaScript.

• StieltjesGamma(n, a) is the generalized Stieltjes constant γₙ(a), the Laurent coefficient of the Hurwitz zeta function at its pole, ζ(s, a) = 1/(s − 1) + Σₙ (−1)ⁿ γₙ(a)(s − 1)ⁿ/n!, as in Mathematica and mpmath; StieltjesGamma(n) is γₙ = γₙ(1). StieltjesGamma(0) is EulerGamma, StieltjesGamma(0, a) is −PolyGamma(0, a), StieltjesGamma(2, 1) is StieltjesGamma(2), and a nonpositive integer a is a pole (StieltjesGamma(2, -1) is ComplexInfinity). StieltjesGamma(1).N() is -0.0728158454836767248606, StieltjesGamma(2, 1/2).N() is 0.968864475220290711422, and N(StieltjesGamma(2, 3/4), 40) is 0.1193766260185842196972365071220126165487: a real a > 0 follows ce.precision by Euler–Maclaurin on lnⁿ(x)/x with exact integer derivatives, the remainder bounded in ball arithmetic so that the digits returned are certified. A complex a or a negative non-integer a is computed in doubles (StieltjesGamma(2, 1 + i).N() is 0.1703042014685874 + 0.3731771957574952i). Orders past 30 stay unevaluated. The head threads over lists and compiles to JavaScript for a real value (#395, contributed by enumeratio)

• LogGamma, BarnesG and LogBarnesG are new, and PolyGamma(-1, z) is LogGamma(z). LogGamma(z) is the analytic continuation of ln Γ with its branch cut on (−∞, 0], as in Mathematica and mpmath's loggamma. It is not GammaLn, which is the principal logarithm of Γ(z) and jumps by 2πi across the zeros of Im Γ: GammaLn(-2.5 + 1.5i) is -3.7175 - 1.4299i, LogGamma(-2.5 + 1.5i) is -3.7175 - 7.7131i, and LogGamma(-1.5) is 0.86005 - 6.28319i. GammaLn is unchanged. LogGamma(5) is ln(24) exactly, LogGamma(1/2) is ln(π)/2, and the poles at the non-positive integers are +∞; a real argument follows ce.precision, a complex one is a double. BarnesG(z) and LogBarnesG(z) are the Barnes G-function, with G(z+1) = Γ(z)·G(z), and its logarithm continued with LogGamma as in Mathematica. BarnesG(5) is 12, BarnesG(10) is 5056584744960000 (the superfactorial, an exact integer), BarnesG(0) is 0 and LogBarnesG(0) is −∞. A real z is computed to ce.precision from the asymptotic series of ln G at an argument of about twice the digits asked for (the difference of two of its values, so the constant ζ′(−1) is not needed), walked to z by G(u+1) = Γ(u)·G(u), with every product rounded to the working precision: N(BarnesG(1/2)) is 0.603244281209446206191…, and 1000 digits take under a second. Up to 1000 steps from 1 (N(LogBarnesG(120.5)) is 23552.5384297242683359); farther, a real z stays unevaluated at ce.precision above 15 digits. A complex z, and a real z more than 60 from 1 at machine precision, use an asymptotic series in doubles (about 3e-12 relative for G, measured against mpmath). LogBarnesG(-2.5) is -2.5747 + 18.8496i. PolyGamma(-1, z) follows Mathematica's convention, PolyGamma(-1, -5/2 + 3i/2) is LogGamma there; the other negative orders stay unevaluated (#395, contributed by enumeratio).

• ClausenCl(n, θ) is the Clausen function Clₙ(θ). For an integer order n ≥ 1 and real θ it is Im Liₙ(e^{iθ}) = Σ sin(kθ)/kⁿ when n is even and Re Liₙ(e^{iθ}) = Σ cos(kθ)/kⁿ when n is odd (mpmath's clsin and clcos; Mathematica writes them as Im/Re of PolyLog). N(ClausenCl(2, 1)) is 1.0139591323607684, N(ClausenCl(3, 2.5)) is -0.7606561109685137 and N(ClausenCl(2, 3.14159)) is 1.8393282835451e-6 (the expansion of Liₙ at the unit circle, DLMF 25.12.12, with θ reduced mod 2π and moved off π by the duplication formula so the even orders keep their relative accuracy there). A real θ is computed to ce.precision digits for orders up to 40 and |θ| up to 10¹², the ζ(n − k) being bignum zeta values and exact Bernoulli rationals; outside that, or where the value cancels against its own terms, the head stays unevaluated. The exact points are closed: ClausenCl(2, π/2) is Catalan's constant and ClausenCl(2m, π/2) is DirichletBeta(2m), ClausenCl(3, 0) is ζ(3), ClausenCl(2, 0) and ClausenCl(2, π) are 0, ClausenCl(1, 0) is +∞. A non-integer or non-positive order and a symbolic θ stay unevaluated. The compiled JavaScript lane agrees with .N() (#395, contributed by enumeratio).

• DirichletCharacter(k, j, n) and DirichletL(k, j, s) give the Dirichlet characters modulo k and their L-functions, in Wolfram's indexing (j = 1 the principal character, j up to φ(k)). DirichletCharacter(5, 2, 2) is i, DirichletCharacter(7, 3, 3) is e^(2πi/3) and a character is 0 where gcd(n, k) > 1. DirichletL sums k^(−s) Σ χ(r) ζ(s, r/k) through HurwitzZeta, so a real value follows ce.precision (DirichletL(3, 2, 1.5) is 0.703968244868733261668 at 21 digits, DirichletL(5, 3, 1.01) keeps every digit next to the pole) and a complex value is at double precision; the odd character mod 4 is DirichletBeta (DirichletL(4, 2, 1) is π/4); the principal character is ζ(s) Π (1 − p^(−s)) over the primes dividing k, so DirichletL(5, 1, 1) is ~oo; at a nonpositive integer the value is exact from the Bernoulli polynomials (DirichletL(5, 2, 0) is 3/5 + i/5, DirichletL(8, 2, -3) is 11); within 1/4 of s = 1 a non-principal character with a complex value is summed from its Laurent series in the Stieltjes constants at double precision, since the Hurwitz terms there have poles that cancel (DirichletL(5, 2, 1.01)); a real character at a real s takes guard digits for the cancellation instead (and, closer to 1 than they reach, its Laurent series in certified Stieltjes constants). L(1, χ) of a real character is exact, from the class number formula: DirichletL(3, 2, 1) is sqrt(3)/9 * pi (π/(3√3)), the quadratic character mod 5 gives 2/5sqrt(5) * ln("GoldenRatio") (2·ln φ/√5), DirichletL(8, 2, 1) is sqrt(2)/2 * ln(1 + sqrt(2)); it stays symbolic for a modulus above 1000. A character with complex values at a real s is answered in doubles, its terms computed for a double at any ce.precision. A modulus above 1000 stays symbolic for DirichletL (it sums k Hurwitz values). Both heads are listable. (#395, contributed by enumeratio)

Issues Resolved

• The GLSL and WGSL inverse trigonometric functions of a complex argument are accurate. The shader helpers of Arcsin, Arccos, Arctan, Arsinh, Arcosh and Artanh (also used by Arccsc, Arcsec, Arsech and Arcoth) used the logarithm formulas in 32-bit floats, and gave \arcsin(1000) a real part of 1.660 instead of π/2, \arcsin(10^4) a real part of π, \operatorname{arsinh}(-10^4) the value -∞, \operatorname{arcosh}(-1.5) a negative real part, and \operatorname{artanh}(10^{20}) a NaN real part. On the real axis outside the domain, \arcsin, \arccos and \operatorname{artanh} took the other side of the branch cut (\arcsin(1.5) was π/2 + 0.962i, the interpreter gives π/2 - 0.962i). The helpers now use the same formulas as the interpreter, written for 32-bit floats, and the side of each cut is the interpreter's. With correctly rounded built-in functions, each part has a relative error below 3·10^{-7} (a few units in the last place) for a modulus from 10^{-40} to 10^{30}. A GPU can be less accurate: GLSL and WGSL allow log an absolute error of 2^{-21} on [0.5, 2] and 3 units in the last place elsewhere, and atan an error of 4096 units in the last place. The helpers do not call log on an argument near 1 (ln(1 + x) for a small x is a series). Arccsc, Arcsec, Arsech and Arcoth have their own helpers, which do not compute w = 1/z and then the function of w: near z = ±1 the rounding of 1/z was amplified (\operatorname{arcoth}(-1 + 10^{-4}i) had a relative error of 6·10^{-5}), 1/z overflowed for a modulus above 10^{19}, and a subnormal z gave NaN (\operatorname{arsech}(10^{-40}) is 92.80). \operatorname{arsech}(0) is +∞, as in the interpreter; it was NaN.

• Compiled Python takes the interpreter's side of each branch cut. For a complex argument, Arcsin, Arccos, Arctan, Arsinh, Arcosh, Artanh and their reciprocals Arccsc, Arcsec, Arccot, Arcsch, Arsech and Arcoth compiled to cmath or NumPy routines, which pick the side of a cut from the sign of a zero part: \arcsin(2) of a complex variable gave π/2 + 1.317i, the interpreter gives π/2 - 1.317i. The compiled code now gives a zero part the sign that selects the interpreter's side, on every cut of the twelve functions. Also fixed: Arccot of a complex argument with a negative real part was off by π (it compiled to π/2 - \arctan(z), not \arctan(1/z)); Artanh of a complex variable with a real value outside [-1, 1] and Arsinh on its cut gave nan; \arctan(\pm i) and \operatorname{artanh}(\pm 1) raised a ValueError (they are now infinite, as in compiled JavaScript); the reciprocal functions raised a ZeroDivisionError at 0; and a list of complex values takes the same code on each element. A complex-typed argument now always gives a Python complex for these twelve functions, also when the value is real: Artanh, Arsinh and Arcosh returned a float before (Arcsin, Arccos and Arctan already returned a complex).

• A definite integral over a zero of eᵗ − a, sin t − a or (t − a)^(−n) stays unevaluated. ∫₀¹ eᵗ/(eᵗ − a) dt gave ln|e − a| − ln|1 − a|, wrong for 1 ≤ a ≤ e, where the integrand has a pole. ∫₀¹ cos t/(sin t − a) dt gave ln|sin 1 − a| − ln|−a|, wrong for 0 ≤ a ≤ sin 1, ∫₀¹ cos t/(sin t − a)² dt gave a finite value where the integral is +∞, and ∫₀¹ (t − a)^(−n) dt gave a closed form for every n. These integrals now stay unevaluated, unless the declared type of a or an assumption puts the zero outside the interval: with ce.assume(a > 3), ∫₀¹ eᵗ/(eᵗ − a) dt is ln(a − e) − ln(a − 1). The zero of a divisor g(t) − c, where g is monotone on the interval, is located by comparing c with g at the bounds. Any other divisor with a free symbol whose zeros cannot be located keeps the integral unevaluated.

• A definite integral with a free symbol in a bound and a pole at a number stays unevaluated. ∫ₐ¹ dt/t gave −ln|a|, ∫₋₁ᵇ dt/t gave ln|b| and ∫₀ᵇ dt/(t + 1) gave ln|b + 1|: each is wrong when the pole is between the bounds (a < 0, b > 0, b < −1). ∫₀ᵃ dt/t gave +∞, wrong for a < 0. They now stay unevaluated unless an assumption puts the pole outside the interval: with ce.assume(a > 0), ∫ₐ¹ dt/t is −ln(a). A removable or integrable singularity is not a pole: ∫₀ˣ sin(t)/t dt is still Si(x) and ∫₀ᵇ dt/√t is still 2√b. A pole of tan, cot, sec or csc (or a zero of a sin or cos divisor) with a free symbol in a bound keeps the integral unevaluated: ∫₀ˣ tan t dt was ln|sec x|, wrong for x ≥ π/2.

• ∫ ln(ax + b) dx is correct. ∫ ln(x + 1) dx gave (x + 1)·ln(x) − x + 1, and ∫₀¹ ln(x + 1) dx gave +∞. They are now (x + 1)·ln(x + 1) − x and 2ln(2) − 1. ∫ ln(2x + 3) dx is (2x + 3)·ln(2x + 3)/2 − x.

• .N() of an integral over a pole agrees with evaluate() when the integrand has a constant such as π. .N() of \int_{-1}^1 \frac{\pi}{t}\,dt gave a random Monte Carlo estimate (0.0 ± 3.4); it now gives NaN (evaluate() gives Indeterminate). .N() of \int_{-1}^1 \frac{\pi}{t^2}\,dt gave 2.1e154 ± 8.8e143; it now gives +∞, as evaluate() does.

• ∫₋₁¹ dt/(eᵃ·t) is Indeterminate, since eᵃ is never zero and the integrand changes sign across the pole at 0. It was 0.

• The flat spelling of several integration variables or indexes reads a repeated name as a bound. ["Integrate", ["Multiply", "x", "y"], "x", 0, "y", "y", 0, 1] read as Limits(x, Nothing, 0), Limits(y, Nothing, Nothing), Limits(y, 0, 1), with y an index twice. A name that would be an index twice is now the upper bound of the index before it: Limits(x, 0, y), Limits(y, 0, 1). The same applies to Sum and Product: ["Sum", "k", "k", 1, "n", "n", 1, 10], which had three missing error operands, is Sum(k, Limits(k, 1, n), Limits(n, 1, 10)).

• GammaLn at machine precision is accurate next to 0, 1 and 2. GammaLn(1e-10) had 8 correct digits (23.02585084714237, against 23.025850929882735), and GammaLn(1 + 10⁻¹⁰) had about 10: the double kernel formed z − 1 + i, which loses the low digits of a small z, and its absolute error of about 10⁻¹⁶ is a large relative error next to the zeros of ln Γ at 1 and 2. For 0 < z ≤ 7/2 the kernel now uses the Taylor series of ln Γ(1 + e) about 1 with ζ(k) − 1 as its coefficients, and is within a few units in the last place of mpmath there. The new LogGamma uses the same kernel for a positive real argument.

• Inverse trigonometric functions are accurate at large and small arguments. \arcsin(-1000000) gave -1.5707963267948966 + 14.50865012405984i: the imaginary part was correct to six digits only (the correct value is 14.508657738523969). \operatorname{arccosh}(-1000000) had the same error. \arcsin(10^{300}) gave ~oo instead of 1.5707963267948966 - 691.46867507877365i, and \arcsin(10^{-300}i) gave 0 instead of 10^{-300}i. The complex values of Arcsin, Arccos, Arctan, Arsinh, Arcosh and Artanh, and of Arccsc, Arcsec and Arccot, now come from formulas that do not cancel and do not overflow, in the interpreter and in compiled JavaScript (\operatorname{arcsec}(10^{-320}) was NaN and is 737.52i, and an argument near the largest double, such as 1.7·10^{308}(1 + i), no longer gives an infinite part). Each part of the result has a relative error below 5·10^{-16} over the whole finite plane. The side of each branch cut is unchanged. At moderate arguments only the last digit of some values changes, and a residual real part such as the -5.55e-17 of \operatorname{arsinh}(0.5i) is gone.

• Arcsch, Arsech and Arcoth are accurate at large and small arguments. For a real argument, \operatorname{arcoth}(10^{100}) gave 0 instead of 10^{-100}, \operatorname{arcsch}(-10^{-100}) gave -\infty instead of -230.95165647996451, \operatorname{arcsch}(-10^{6}) had 15 correct digits of 21, and \operatorname{arsech}(-10^{-8}) gave \infty + \frac{\pi}{2}i instead of 19.113827924512311 + \pi i. For a complex argument, the values lost digits near ±1 and ±i and for a large or small modulus (\operatorname{arsech}(3 - 10^{-10}i) had a relative error of 10^{-6} in its real part), and \operatorname{arcsch}(10^{-300}i) was NaN. The three functions now use the same formulas as Arsinh, Arcosh and Artanh, written with (1 ± z)/z in place of 1/z ± 1, in the interpreter, with big decimals at a precision above machine precision, and in compiled JavaScript. The side of each branch cut is unchanged.

• Compiled JavaScript keeps a small complex result. The complex helpers of compiled code set to 0 any part smaller than 10^{-14} in magnitude, so \operatorname{arcoth}(x) compiled to 0 at x = 10^{20} instead of 10^{-20}. A part is now removed only when it is also not larger than 10^{-14} times the modulus of the result. The constant folding of these helpers uses the same rule and the same kernels as the run-time helpers.

• More inverse trigonometric functions compile for a complex argument. Arsinh of a complex argument compiled to Math.asinh, which gives NaN (\operatorname{arsinh}(2i)). It now compiles to a complex helper, as Arcosh and Artanh do. The compiled Arccsc, Arcsec and Arccot of a complex argument now use the same kernels as the interpreter: \operatorname{arcsec}(-10^{-6}) had an imaginary part correct to 5 digits and \operatorname{arccsc}(10^{-300}) was NaN.

• A restriction by an infinite list of conditions no longer runs out of memory. Reading the elements of When([1, 2, 3], Range(1, ∞)) read the whole condition list on the first element, and the process ran out of memory. The conditions are now read only up to the element requested: the result is the three restricted elements, and an element of When(Range(1, ∞), Range(1, ∞)) can be read by its index.

• FromDigits and ChineseRemainder no longer round a non-integer element. FromDigits([1.5, 2]) gave 22 and ChineseRemainder([5/2, 3], [3, 5]) gave 3; both now stay unevaluated. A float with an integer value (2.0) is still accepted.

• An infinite list no longer exhausts memory in ChineseRemainder, FromDigits and FromContinuedFraction. ChineseRemainder(Range(1, ∞), [3, 5]) read every element of the range and crashed the process. These heads now stay unevaluated when a list operand is not known to be finite or has more than 1,000,000 elements.

• An iterated integral with a slowly integrable singularity has the right value and error. Integrate(y^(−0.999), Limits(y, 0, 1), Limits(x, 0, 2)).N() gave 0 ± 1.3e+289; it is now 2000.000000002 ± 0.000000024 (exact: 2000). With y^(−1/2) it gave 0 ± 9.4e+136; it is now 4.000000000000 ± 0.000000000015 (exact: 4). The error of the inner levels was averaged over the nodes of the outer level and multiplied by its range, but the nodes are packed next to the singularity, where the inner values and their errors are largest. When the inner values keep one sign, the error added is now their relative error times the magnitude of the result.

• ∫₋₁¹ dt/(a·t) with a free a is Indeterminate. It was 0. The pole at t = 0 could not be confirmed by sampling while a is free. The integrand is now split into a constant factor and a part with only the integration variable ((1/a)·(1/t)), and the pole of that part is confirmed. For every a ≠ 0 the integrand changes sign across the pole, and for a = 0 it is undefined everywhere, so the integral has no value. ∫₋₁¹ dt/(a·t²) was −2/a; it now stays unevaluated, and is +∞ when a > 0 is assumed and −∞ when a < 0. ∫₋₁¹ a/t dt was 0; it now stays unevaluated (it is 0 for a = 0). ∫₁² dt/(a·t) is still ln(2)/a. An integrand with a pole at a number and a free symbol that is not a constant factor (1/(t·(t² + a² + 1)) over [−1, 1]) also stays unevaluated.

• .N() of an iterated integral with a moving pole and an infinite bound is +∞. Integrate((y − x)⁻², Limits(y, 0, +∞), Limits(x, 3, 4)).N() gave 5140000000000000 ± 440000000000000; it is now +∞: for every x in (3, 4) the inner integral has a pole at y = x. The scan for such poles now places its points on an infinite range with the transform of the quadrature, and cuts an infinite range to a finite one for the pole check.

• Stirling, StirlingS1 and Eulerian outside their triangle are 0. Stirling(3, 5), StirlingS1(3, 5) and Eulerian(3, 5) stayed unevaluated (and their compiled form gave NaN); they are now 0, as Binomial(3, 5) already was. Eulerian(0, 0) is now 1. A negative or symbolic operand still leaves the expression unevaluated.

• Log and Ln check their operand count in strict mode. Log and Ln take one or two operands (the second is the base; Ln(3, 4) is Log(3, 4)). With ce.strict = true, Log(8, 2, 3) was valid and evaluated to 3, and Ln(3, 4, 5) became Log(3, 4, 5). Now the extra operand is an error, as for Sqrt(4, 5): Log(8, 2, 3) is Log(8, 2, Error("unexpected-argument", "3")), and Ln(3, 4, 5) is Ln(3, 4, Error("unexpected-argument", "5")). This applies to ce.box(), ce.function() and ce.parse() (\log(8, 2, 3)). In non-strict mode the extra operand is kept, as before.

• A redeclared If, Sum, Block, … compiles as the user definition. With ce.declare('If', { ...ce.lookupDefinition('If').operator, evaluate: () => ce.number(100) }), If(x > 0, sin(x), cos(x)) gives 100 in the interpreter, but compile() gave success: true and the code of the library If (0.644… at x = 0.7). The compilation now fails closed (success: false, the result falls back to the interpreter, which gives 100), on every target. The check for a user definition that shadows a library operator ran only for the heads that a target lowers through its function or operator table. It now also runs for the heads that the compiler lowers by name (If, Which, When, Match, Loop, Sum, Product, Block, Function, …), for a node that a loop body or a block lowers as a statement, and for the bodies of the user functions that the expression calls. A user definition of Which with its own compile handler uses that handler. An unchanged copy of the library definition still compiles as the library operator.

• A chain of a copy under another name keeps its name. With a copy MyNotEqual of NotEqual (const { name, ...def } = ce.lookupDefinition('NotEqual').operator; ce.declare('MyNotEqual', def)), MyNotEqual(x, y, z) boxed to x != y && y != z, with library NotEqual nodes, so an evaluate handler of the copy did not run. It now boxes to MyNotEqual(x, y) && MyNotEqual(y, z): each node that the library handler builds with the library name gets the name of the copy. An operand that is a library NotEqual keeps its name, also when the handler returns it (MyAnd(And(A, B)), with a copy MyAnd of And, gives And(A, B)). When the handler flattens a library operand (And(And(A, B), C) is And(A, B, C)), the call is kept unflattened under the name of the copy: Nand2(And(A, B), C) stays Nand2(And(A, B), C), so the library And is not evaluated by the handlers of Nand2.

• A missing operand gives an Error("missing") operand. ce.box(['Annotated', 'x']) logged error canonicalizing `Annotated`: Cannot read properties of undefined and gave Annotated(x); it now gives Annotated(x, Error("missing")). ce.box(['Subscript', 'x']) gave Subscript(x, [undefined]) or threw Cannot read properties of undefined, and now gives Subscript(x, Error("missing")). The same fix applies to Complex() (threw Expected one or two arguments; an extra operand is now an unexpected-argument error operand), Apply(), CanonicalForm(), Factorial(), Lb(), Lg(), Log2(), Log10(), Lucas() and PrimeNumber(), which threw or built a node with an undefined operand. An extra operand of Factorial, Lb, Lg, Log2, Log10, Lucas and PrimeNumber was dropped, also in strict mode: Lb(8, 3) gave Log(8, 2), which is 3. In strict mode, it is now an unexpected-argument error operand (Log(8, 2, Error("unexpected-argument", "3"))); otherwise it is kept, as for Sin(1, 2), after the base of Lb, Lg, Log2 and Log10 (Lg(8, 3) gives Log(8, 10, 3)). CanonicalForm(x, y), with a name that is not a canonical form, logged Invalid canonical form and now gives CanonicalForm(x, Error("unexpected-argument", "y")).

• A pole whose position comes from an assigned symbol is found. With q := 1, NIntegrate(y ↦ (y − q)⁻², 0, 2) gave a sampled finite number that changed at each call (13931349, then 36076471), and Integrate((y − q)⁻², y, 0, 2).N() gave 1010000000 ± 910000000. Both now give +∞, as they do when the integrand is written with 1. Integrate((y − q)⁻², y, 0, 2).evaluate() gave -2 (the antiderivative differenced across the pole) and now gives +∞. The pole check now substitutes the values of the assigned symbols of the integrand, which the quadrature and the antiderivative already read; a symbol with no value is not substituted, so its integral is unchanged.

• A pole at a constant such as π is found. ∫₀⁴ (y − π)⁻² dy is +∞. Integrate((y − π)⁻², Limits(y, 0, 4)).evaluate() gave -1/π - 1/(4 - π) (the antiderivative differenced across the pole: a negative value for a positive integrand), .N() gave a sampled value that changed at each call (7000000 ± 3500000), and NIntegrate(y ↦ (y − π)⁻², 0, 4) gave 13898520. All three now give +∞, and so do the multi-limit form and (y − e)⁻². A sign change across such a pole (1/(y − π) on [0, 4]) gives no value, as for a pole at a number. The pole check now reads a coefficient with a known real value (π, e, a constant declared with a value) as that number. A pole outside the bounds is unchanged: ∫₀³ (y − π)⁻² dy is 1/(π − 3) − 1/π.

• A pole at a bound of an integral gives +∞ or −∞. ∫₀¹ t⁻² dt is +∞, but .evaluate() gave ~∞ (ComplexInfinity, the antiderivative −1/t evaluated at 0) and .N() and NIntegrate(t ↦ t⁻², 0, 1) gave NaN (the quadrature found a divergence, with no sign). All now give +∞. The same applies to ∫₀^π (y − π)⁻² dy (~∞ → +∞), ∫₀¹ −t⁻² dt (~∞ → −∞), ∫₋₁⁰ t⁻³ dt (~∞ → −∞) and the multi-limit form. ∫₀¹ dt/t gave +∞ under .evaluate() and NaN under .N(); .N() now gives +∞ too. The integral is the one-sided limit from inside the interval, so the sign is the sign of the integrand next to the bound, inside the interval (negated for reversed bounds): ∫₋₁⁰ dt/t is −∞. Poles at both bounds with different signs (∫₀¹ (1/t − 1/(1 − t)) dt) give no value: unevaluated under .evaluate(), NaN under .N(). An integrable singularity at a bound keeps its value: ∫₀¹ t^(−1/2) dt is 2, ∫₀¹ t^(−0.95) dt is 20, ∫₋₁¹ dt/√(1 − t²) is π. The sign is given only when samples of the integrand very close to the bound confirm a pole of order 1 or more; otherwise the result stays as before (unevaluated under .evaluate(), NaN under .N() and NIntegrate).

• The quadrature no longer takes a logarithmic factor for a divergence. ∫₀¹ t^(−0.95)·ln t dt is −400 and ∫₀¹ t^(−0.9)·ln²t dt is 2000, but .N() and NIntegrate gave NaN: the adaptive Gauss–Kronrod quadrature took the slow growth of the logarithm for a divergence at 0. They now give −399.999999912 ± 0.000000039 and 1999.99999980 ± 0.00000020. The divergence test now also requires the growth of the integrand toward the bound to stop slowing down. A pole is still found: ∫₀¹ dt/t, ∫₀¹ t⁻² dt and ∫₀^{π/2} tan t dt are +∞, ∫₋₁⁰ t⁻³ dt is −∞.

• A logarithm in a denominator is a pole site. ∫₀¹ dt/(t·ln²t) gave ~∞ (ComplexInfinity) under .evaluate() and NaN under .N(); the integrand is positive and behaves as 1/(t − 1)² at t = 1, so all routes now give +∞ (∫₀^½ dt/(t·ln²t) keeps its value 1/ln 2). The pole check now reads the zero of a logarithm of a linear argument in a denominator, and the zeros of each factor of a denominator that is a product. So ∫ from ½ to 2 of dt/ln t, whose integrand changes sign at the pole t = 1, now stays unevaluated (it was LogIntegral(2) − LogIntegral(1/2), about 1.4238, the principal value) and is NaN under .N(). A ~∞ difference of an antiderivative for a real integrand over a real interval now leaves the integral unevaluated instead.

• An iterated integral with a pole that moves with another variable. Integrate((y − x)⁻², Limits(y, 0, 10), Limits(x, 3, 4)).N() gave a large finite number such as 2710000000000000 ± 170000000000000; for each y in (3, 4) the inner integral has a pole at x = y, and the value is +∞. It is now +∞ (−∞ for reversed bounds of y, and NaN for (y − x)⁻³, which changes sign across the pole). The check runs before the quadrature, so the long quadrature run of such an integral is also gone.

• A nested integral with a divergent inner integral. \int_0^2\int_0^2 (y-1)^{-2}\,dy\,dx stayed unevaluated under .evaluate() and gave NaN under .N(), while the multi-limit form gave +∞. Both routes now give +∞ (−∞ for reversed outer bounds). The integral of an infinite constant over an interval of nonzero length is now that infinity, negated for reversed bounds: Integrate(+∞, Limits(x, 0, 2)) stayed unevaluated under .evaluate() and gave NaN under .N(), and now gives +∞. An interval of unknown length ([0, a]) keeps the integral unevaluated.

• The multi-limit form with reversed outer bounds and a divergent inner dimension. Integrate((y − 1)⁻², Limits(y, 0, 2), Limits(x, 2, 0)).N() gave +∞, while .evaluate() gave −∞. .N() now gives −∞: each dimension with reversed bounds negates the integral.

• NIntegrate of a function given by its name. NIntegrate(Sin, 0, 2) was NaN, because each sample read the function itself instead of its value. It is now 1.4161468365471424 (1 − cos 2). The interior-pole check also applies to such a function: NIntegrate(Tan, 0, 3) and NIntegrate(Sec, 0, 2) are NaN (the integrand changes sign at π/2), and with g := x ↦ 1/x², NIntegrate(g, -1, 2) is +∞.

• A copy of a library operator definition is differentiated and compiled as the library operator. After ce.declare('Sqrt', { ...ce.lookupDefinition('Sqrt').operator }), the overloading idiom of the guide, D(sqrt(sin(x)), x) gave cos(x) * Apply(Derivative(sqrt, 1), sin(x)) and now gives cos(x) / (2sqrt(sin(x))); compile() of \sqrt{\sin x} failed (success: false, with a fall back to the interpreter) and now succeeds, with the same code as without the copy. The same applies to every library operator (Sin, Add, Power, Abs, Floor, …), and to a copy used as a compiled callback (Map(Sqrt, xs)). The rule: a definition of a library name is the library operator when its evaluate, canonical, compile, derivative and evaluateAsync handlers are the same function objects as those of the library definition of that name (or are absent in both), and its lazy and broadcastable flags have the same values, so a copy that changes only description counts too. A copy that changes one of these, such as a copy of Sin with broadcastable: false, changes how the operator is evaluated or compiled, and is not the library operator. A copy that replaces one of the handlers, a user function (function Sin(x) { … }, Sin(x) := …), a parameter named like a library operator, and a copy declared under another name are not the library operator, as before. A copy made from the definitions of another engine bundle has other function objects, so it is a user definition.

• A copy of a library operator definition under another name keeps its name. After const { name, ...sin } = ce.lookupDefinition('Sin').operator; ce.declare('MySin', sin), ce.box(['MySin', 'x']) gave sin(x) and now gives MySin(x); MySin(x).evaluate() gave sin(x) and now gives MySin(x). The canonical and evaluate handlers of most library operators (the trigonometric and hyperbolic functions, Integrate, Matrix, the distributions: about 180 of the 241 library operators that have a canonical handler) build their result with the name of their own operator, so the copy turned into the library operator, and a copy with its own evaluate handler (ce.declare('MySin', { ...sin, evaluate })) never ran it. Now, when such a handler returns an expression with the name of the library operator that holds it, the result gets the name of the copy. A rewrite to another operator (Rational to Divide) and a value (MySin(π) is 0) are kept. As a result, D differentiates MySqrt(sin(x)) as a user function, cos(x) * Apply(Derivative("MySqrt", 1), sin(x)); before, it saw the Sqrt that the handler returned and used the rule of Sqrt.

• A compiled callback that names a host-declared operator falls back to the interpreter. With ce.declare('MySqrt', { signature: '(real) -> real', evaluate: ([x]) => ce.number(100) }) and ce.declare('rs', 'list<real>'), compile(ce.box(['Map', 'MySqrt', 'rs'])) gave success: true, and run({ rs: [1, 4] }) threw TypeError: _f is not a function. It now gives success: false, and run() gives [100, 100], the value of the interpreter. The same applies to a definition of a library name with its own evaluate (ce.declare('Sin', { ...sinDef, evaluate })), and to the other callback positions (Filter, Reduce, …). An operator whose evaluate is a function literal, and a library operator, still compile. An operator that the compiled code can apply, because its definition has a compile handler or the functions option of compile() maps it, compiles as (p) ↦ MySqrt(p), as Map(Function(MySqrt(t), t), rs) does: with functions: { MyOp: (x) => 10 * x }, Map(MyOp, rs) compiles and run({ rs: [2, 3] }) gives [20, 30].

• Derivative() with no operand logged "error canonicalizing Derivative" and stayed Derivative(); it is now Derivative(Error("missing")), the standard form of a missing required operand.

• The flat MathJSON form of a definite integral reads its bounds for any bound expression. ["Integrate", ["Power", "y", 2], "y", 0, "Pi"] read Pi as a second integration variable and evaluated to -1/3 * pi; it is now Integrate(y², Limits(y, 0, Pi)), which evaluates to 1/3 * pi^3. The Epsil call form ∫(y^2, y, 0, π) gives this MathJSON. After the variable, an operand that is not a variable name (a number, a constant such as Pi, or an expression) starts the bounds, and the next operand is the upper bound, whatever expression it is. A variable name after the variable is still the next integration variable: ["Integrate", f, "x", "y", "z"] is the triple indefinite integral, as before.

• NIntegrate declares the bounds it reads. The signature of NIntegrate was (function, limits:(tuple|symbol)?) -> number, but only the form NIntegrate(f, lower, upper) was evaluated: NIntegrate(x ↦ x², (0, 2)) stayed unevaluated. The signature is now (function, lower:number, upper:number) -> number, and a Tuple of bounds or a missing bound is an error operand (incompatible-type, missing). For a multiple integral, use Integrate(f, Limits(x, …), Limits(y, …)).N().

• A numeric integral with a slowly integrable singularity at a bound is correct, with an error that covers the true error. A large part of such an integral is closer to the bound than a floating-point number can be (∫₀^δ x^(−0.999) dx is about 500 for δ = 1e-300), so the quadrature gave a wrong value with a tight error. The value of that part is now found by extrapolation of the integrals over the shells that the quadrature cuts next to the bound (Wynn's ε-algorithm and the Levin u-transform). The extrapolation is used only when the shells decrease as the terms of a convergent series do, two extrapolations agree, and the error is at most 1e-3 of the value. A converged result next to a singular bound is checked against the tanh-sinh (double-exponential) rule, and is kept when tanh-sinh does not converge (a smooth integrand with a narrow peak at a bound, such as ∫₀¹ dx/(10⁻¹² + x²) = 1570795.3267948966, keeps its value). The compiled Integrate and the levels of a multiple integral use the same correction. Integrate(…).N() and NIntegrate (compiled integrand / integrand that does not compile):

  • ∫₀¹ x^(−0.999) dx = 1000: 508.24896298688 ± 0.00000000053 / 18.9 ± 4.4 → 1000.0000000014 ± 0.0000000061 / 1000.0000000009 ± 0.0000000045.
  • ∫₀¹ x^(−0.99) dx = 100: 99.922310890258 ± 0.000000000098 / 89.72 ± 0.86 → 100.00000000000 ± 0.00000000011 / 100.000000000005 ± 0.000000000093.
  • ∫₀^½ dx/(x·ln²x) = 1/ln 2 = 1.4426950408889634: 1.44131049670385 ± 0.00000000000014 / 1.43831 ± 0.00016 → 1.44269504088899 ± 0.00000000000021 (both).
  • ∫₀¹ (1 − t)^(−0.95)·ln(1 − t) dt = −400: NaN (both) → -399.9999998 ± 0.0000021.
  • ∫₀¹ (1 − t)^(−0.95)·ln(10⁻⁶·(1 − t)) dt = −676.3102111592855: -380 ± 110 (1.3 s) / -236 ± 84 → -676.3102109 ± 0.0000043 (2 ms).
  • ∫₀¹ t^(−0.995)·ln t dt = −40000: the integrand that does not compile gave -41.1 ± 6.9 and now gives -39999.99995 ± 0.00025. The compiled one still gives NaN.
  • ∫₀¹ dx/√(1 − x²) = π/2: 1.5707963111 ± 0.0000000073 → 1.57079632679493 ± 0.00000000000047.
  • ∫₁^∞ x^(−1.01) dx = 100: the integrand that does not compile gave 2.71 ± 0.44 and now gives 100.000000003 ± 0.000000036, as the compiled one does.

  A quadrature result that did not converge and is kept instead of the Monte-Carlo estimate now reports an error of at least the Monte-Carlo standard error, |estimate|/√samples: ∫₀¹ sin(1/x) dx gives 0.50407 ± 0.00016 (was ± 0.000051); the same floor applies to a level of a multiple integral. Smooth integrands converge before any of this runs, and take the same time as before.

  Next to a singular bound that the extrapolation could not resolve, where the integrand keeps one sign, the quadrature result is kept with its widened error, and is never replaced by a Monte-Carlo estimate, which under-weights the neighborhood of the singularity: ∫₀^0.01 dx/(x·(−ln x)·ln²(−ln x)) = 1/ln(ln 100) = 0.6548… gave 0.3220 ± 0.0063 and now gives 0.503 ± 0.018. When the shells decrease as a power of their index (1/(x·(−ln x)^q)), only the Levin u-transform is used: ∫₀^½ dx/(x·(−ln x)³) = 1/(2·ln²2) = 1.0406844905… gave 1.040684324 ± 0.000000012 and now gives 1.040684485 ± 0.000000050. A part of the interval where the integrand has no value makes the error of the quadrature infinite, so the estimate of the other parts is not kept as a value. A compiled integral that runs inside another quadrature has no Monte-Carlo fallback (it would draw 1e7 samples at each node of the enclosing quadrature): .N() of \int_0^{10}\int_3^4 (y-x)^{-2}\,dx\,dy, whose inner integral has no value for y in [3, 4], ran for more than 200 s and now gives NaN in about 2 s.

  An integral whose integrand oscillates without a limit next to a bound has no value, and .N(), NIntegrate and the compiled Integrate now give NaN for it: ∫₀¹ sin(ln x)/x dx gave -0.11112857219 ± 0.0000000014 with a compiled integrand, and a Monte-Carlo value otherwise. The same applies to ∫₁^∞ sin(ln x)/x dx, ∫₀¹ sin(ln(1 − x))/(1 − x) dx and sin(ln x)/x^p on [0, 1] for p ≥ 1.

• ∫₀^∞ sin x/x dx is π/2 to 1e-11. The oscillatory quadrature skipped [0, 1e-8], where sin x/x is 0/0, and gave 1.57079631675 ± 0.00000000025, wrong by 1e-8; it now integrates the first lobe from 0 with the adaptive quadrature, which resolves a singularity at 0, and gives 1.57079632675 ± 0.00000000025. ∫₀^∞ sin x/x^1.5 dx = √(2π) = 2.5066282746310 was 2.50658840820 ± 0.00000000013 (true error 4.0e-5) and is now 2.50662827461 ± 0.00000000014.

• A compiled semi-infinite oscillatory integral uses the oscillatory quadrature. The compiled Integrate (_SYS.integrate) used only the adaptive quadrature, which does not converge on such an integral: with a parameter p, ∫₀^∞ sin x/x^p dx gave NaN for p = ½ and 2.5237 for p = 1.5. It now gives 1.2533141372613 (√(π/2)) and 2.5066282746098 (√(2π)), as .N() does.

• A semi-infinite oscillatory integral with reversed bounds has the right sign. ∫_∞^0 sin x/x dx gave +π/2 under .N() and NIntegrate (the infinite lower bound was read as −∞), and NaN when compiled. All three now give −π/2.

• The numeric integral of a small integrand has a relative error. The adaptive quadrature stopped when its error was below an absolute tolerance of 1e-12, so an integral of a small integrand was "converged" with any relative error: ∫₀¹ 10⁻³⁰⁰·x^(−0.999) dx = 10⁻²⁹⁷ gave 9.76e-300 ± 8.1e-300. The absolute tolerance is now at most 1e-10 times the sum of the magnitudes of the panel values, and NIntegrate gives 9.9999999997e-298.

• A pole at a bound of large magnitude is found. Next to a bound such as 10⁶, the quadrature reaches the spacing of the doubles after about 29 bisections, too few for its divergence test: ∫ dx/(x − 10⁶) on [10⁶, 10⁶ + 1] gave 22.7 ± 0.58. It now gives NaN, and a convergent integral there (∫ (x − 10⁶)^(−½) dx = 2) keeps its value.

• The flat form of Sum and Product is read like the flat form of Integrate. ["Sum", "k", "k", 1, 10] gave Error("missing"), as did the Epsil call ∑(k, k, 1, 10), which produces it. It is now Sum(k, Limits(k, 1, 10)), which is 55; ["Product", "k", "k", 1, 5] is 120, and ["Sum", "k", "k", 1, "n"] is Sum(k, Limits(k, 1, n)), with n free (its simplification is (n² + n)/2). An index followed by one bound (["Sum", "k", "k", 10]) has that bound as its upper bound, and several indexes can follow each other (["Sum", f, "i", 1, 3, "j", 1, 2]). The Limits, Element and Tuple forms are unchanged.

0.145.0 2026-10-01

Behavior Changes

• A vars-mapped symbol with a declared type compiles as if it had no value. The compiled code reads the input, but the compiler used to analyze the expression with the symbol's stored value. With a declared real and a := 4, compile(\sqrt{a}, { vars: { a: '_.a' } }) gave the real-only Math.sqrt(_.a), which returns NaN for a negative input. It now gives _SYS.csqrt(({ re: _.a, im: 0 })), the code for an a with no value. This applies on every target (javascript, python, glsl, wgsl, interval-js); on a direct custom target (compile(expr, { target })) the value is no longer folded into the code. The value is hidden for the duration of the compilation and restored afterwards, also when the compilation throws. A value put in force by an assumption is hidden the same way: with assume(a = 4), \lfloor a\rfloor + x compiled to _.x + _.a (the assumption typed a as an integer) and now compiles to _.x + Math.floor(_.a). A symbol whose type was inferred from its value (ce.assign('u', 4) with no declaration) keeps its value during the compilation, but it is no longer folded on a direct custom target, where \sqrt{u} with vars: { u: '_.u' } compiled to 2.

• An integral over a vars-mapped input keeps its closed form. \int_0^x a t\,dt with a mapped by vars: { a: '_.a' } compiled to the quadrature _SYS.integrate((t) => (_.a * t), 0, _.x); it now compiles to the closed form (0.5 * _.a * (_.x * _.x)), as it does for an unmapped a with no value. The input stays live: the closed form reads _.a at run time. The closed form is still skipped when the integral reaches a mapped symbol that has a visible value (one whose type was inferred from its value), when the mapping is not a plain read (an identifier such as u_a or a member read such as _.a), or when it reaches a function overridden with the functions option. A host that used a vars mapping to force quadrature must now map the symbol to source that is not a plain read, for example { a: '(_.a)' }.

• The shader and Python targets check a vars mapping to a bare identifier for reserved words. On GLSL, { in: 'in' } compiled x + \sin(\mathrm{in}) to x + sin(in), which no driver accepts, while the free symbol in declined. The mapping now declines with the same error, and so does a mapping to a WGSL reserved word (loop) on WGSL. A mapping to another name ({ in: 'u_in' }) or to source that is not a bare identifier (u.in) is unchanged.

• The Python target declines a free symbol named after a Python keyword. x + \sin(\lambda) compiled to x + np.sin(lambda), which is a Python syntax error; it now declines with "lambda" is a reserved word in python. The same applies to the other keywords (in, if, class, yield, …), to a vars mapping to one, and to a keyword used as a Sum/Product index, a comprehension variable or a function-literal parameter. Map the symbol to another name with vars ({ lambda: 'lam' }) to compile it.

• GLSL: struct, class, lowp, mediump and highp are reserved words. A free symbol with one of these names compiled to invalid GLSL; it now declines like in or sample.

• Four parse diagnostic codes of 0.144.0 are renamed into the ambiguous-* group, so that a host refuses every reading with a second common reading on one prefix: letter-run-split is now ambiguous-letter-run, implicit-product-in-denominator is now ambiguous-denominator, spaced-digit-groups is now ambiguous-digit-groups, and letter-before-decimal is now ambiguous-letter-decimal. A host that matched one of the first names must use the new one.

• A name in parentheses is a factor, never the head of a function application, in the strict and the lenient grammar. (k)(x - 1) is now k·(x - 1) (["Multiply", "k", ["Add", "x", -1]]), was the application ["k", ["Add", "x", -1]]. (a)(b) is now a·b, was ["a", "b"]. y = (m)(x) + b is now y = m·x + b. (k)(x, y) is now k·(x, y), was ["k", "x", "y"]. (g)(3)^2 with g undeclared is now 9g, was g(3)^2. A name that the library or the host declared a function keeps the application reading ((\sin)(x) is sin(x), and (f)(3)^2 is f(3)^2 when f is declared a function); a function type that the engine only inferred from an earlier f(x) does not count. k(x - 1), with no parentheses around k, keeps the rules it had.

• In the lenient grammar, +- is ± and -+ is ∓ when there is no white space between the two signs. x = 2 +- 0.1 is now ["Equal", "x", ["Measurement", 2, 0.1]] (as 2 ± 0.1 and 2 \pm 0.1), was 2 + (-0.1). a+-b is now Measurement(a, b), was a + (-b). y = +-sqrt(x) is now Measurement(0, √x), was -√x. a -+ b is now ["MinusPlus", "a", "b"] (as a \mp b), was a - b. The strict grammar keeps + then -.

• A prefix ∓ has a reading. \mp 1 and ∓1 are MinusPlus(0, 1), as a prefix \pm 1 is Measurement(0, 1). Strict x = \mp 1 was an unexpected-command error. In the lenient grammar a prefix -+ is read the same way: -+x is MinusPlus(0, x), was Negate(x).

• The lenient grammar reads five more function names before a parenthesis: mod (Mod), pow (Power), trunc (Truncate), Re (Real) and Im (Imaginary). These inputs change:

  • mod(x, 2) was an application of an unknown function mod, and mod(7, 3) evaluated to itself. It is now Mod(x, 2), and mod(7, 3) evaluates to 1.
  • pow(x, 2) was an application of an unknown function pow; it is now x^2.
  • trunc(x) was an application of an unknown function trunc; it is now Truncate(x), and trunc(2.5) evaluates to 2.
  • Re(z) and Im(z) already had the canonical form Real(z) and Imaginary(z); only the raw form (form: 'raw') changes, from ["InvisibleOperator", "Re", ["Delimiter", "z"]] to ["Real", "z"] (and the same for Im).

  Without a parenthesis these words keep their reading as letters (7 mod 3 is a product). The strict grammar is not changed.

• A call through a field of a constant that names an operator is the direct call, also without named arguments. With bob declared with isConstant: true and the value {S -> bob_S}, bob.S(3) (Apply(Field(bob, "S"), 3)) was kept as Apply(Field(bob, "S"), 3); it is now bob_S(3), and prints and serializes as bob_S(3) (in Epsil too). The operator's argument checks now apply when the call is built, not when it is evaluated, as for bob_S(3) written directly. A call through a field that holds a function literal is unchanged without named arguments; with named arguments its callee is the literal.

• A named call through a variable whose declared type gives no parameter names is an error. The names of a call such as alias(3, factor: 5), where alias is a variable that holds a function, are now matched only against the DECLARED type of alias, never against the function it holds now. When the declared type gives no parameter names (the bare function type, or a signature with no names such as (number, number) -> number), the call is the error argument-names-unavailable, with a message that suggests positional arguments or a declared signature with parameter names. Example: with bob_S declared (x: number, factor: number) -> number and let alias: function = bob_S, alias(3, factor: 5) was 15 and is now the error; the call was stored as alias(3, 5), so after alias = other, with other declared (factor, x) and computing factor - x, a stored call gave -2 while the names ask for 2. With alias: (number, number) -> number, the error code was argument-name-unknown and is now argument-names-unavailable. A variable declared with a signature that names its parameters, a variable with no declaration (whose inferred type is used), and a function defined with f(x) := … or function f(x) {…} are unchanged. A program that used names through a variable declared function must call it positionally or declare the variable with a signature that names the parameters.

• LaTeX serialization uses standard spellings for LCM, Log2, Log10, the number sets and Degrees (#345, contributed by enumeratio). LCM(a, b) was \lcm(a, b) and is now \operatorname{lcm}(a, b); Log2(x) and Log10(x) were \mathrm{Log2}(x) and \mathrm{Log10}(x) and are now \log_{2}(x) and \log_{10}(x); Degrees(30) was 30\degree and is now 30^{\circ} (a compound operand is parenthesized, (x+1)^{\circ}); Integers, RationalNumbers, RealNumbers, ComplexNumbers, NonNegativeIntegers and the sign-restricted sets were \Z, \Q, \R, \C, \N, \R_{>0}, … and are now \mathbb{Z}, \mathbb{Q}, …, \mathbb{R}_{>0}. \lcm, \degree and the short set commands (which only MathLive defines) are still read, and each new spelling parses back to the same expression. A call with the wrong number of arguments is written as a function call, \mathrm{Mod}(a, n, 1) and \mathrm{Interval}(\bigl\lbrack0, 1\bigr\rbrack), where Mod(a, n, 1) was "" and Interval(List(0, 1)) ended in a stray comma. The imaginaryUnit serialization option, which only the parser honoured, now also sets how ImaginaryUnit and complex numbers are written, and the new exponentialE option does the same for ExponentialE and Exp; both default to \imaginaryI and \exponentialE.

New Features

• The lenient grammar reports each reading choice that has a second common reading. The list of reading choices, their codes, and the choices that have one common reading is docs/plans/2026-10-01-lenient-ambiguity-codes.md; a reading choice with a second common reading and no code is a defect. The codes added in this release.

• The lenient grammar reports more reading choices that have a second common reading. With ce.parse(text, { strict: false, diagnostics: true }), each of these choices now reports a parse diagnostic. The reading does not change, and the strict grammar reports none of them:

  • ambiguous-exponent-end: where an unbraced exponent ends. e^2pi is e^2·π and x^2y is x^2·y (an operand directly after the exponent); e^i pi, e^-x y and e^2 pi i (an operand after white space when the exponent is a name or is signed, or the base is e); e^x/2, e^-x/2, x^pi/2 and x^1/2 (a / after an exponent that is a name, is signed or is 1). x^2 y and x^3/2 are not reported. detail: { exponent }.
  • ambiguous-implicit-subscript: x2, θ2, α1 are read as x_2, θ_2, α_1, and x_1y is read as x_1·y. atan2, log2 and log10 are not reported.
  • ambiguous-name-digits: letters and digits that are not a library function, before a parenthesis: atan3(y) is read as arctan(3·y).
  • ambiguous-function-argument: a function name without parentheses and an argument of more than one factor: sin x y is sin(x·y), also sqrt 2 x, ln 2 x, exp 2 x, abs 2 x; and log 2 x, read as log_2(x). sin 2x is not reported.
  • ambiguous-function-without-parentheses: a symbol declared as a function, followed by an operand: f x and 2 f x are products.
  • ambiguous-name-then-number: a name, white space, then a number: x 2 and θ 2 are products.
  • ambiguous-delta: Δ or Delta followed by a letter: Δx, Delta x and Q = m c ΔT read Δ as a factor.
  • ambiguous-constant-name: a library constant alone on the left of =: e = 1.6e-19, i = V/R, pi = 3.14, inf = 2.
  • ambiguous-lookalike-letter: a Greek letter that looks like a Latin letter (the capitals Α Β Ε Ζ Η Ι Κ Μ Ν Ο Ρ Τ Υ Χ, and ο).
  • ambiguous-unknown-character: a character that is not math, read as a string: y = ж.
  • ambiguous-radical: the extent of √ without braces or parentheses: √2π, √2x, √xy (√2·π), √x² and √x²+y² ((√x)²), and 3√8 (3·√8).
  • ambiguous-absolute-value: bars that pair two ways: |x|y|z| is read as |x|·y·|z|.

• More reading choices of the lenient grammar report an ambiguous-* parse diagnostic. With ce.parse(text, { strict: false, diagnostics: true }), these inputs keep their reading and now also report a code, so that a host can refuse a line that a person could have meant another way:

  Code	Example	Reading	Second common reading
  ambiguous-equation-number	y = x^2 (2), x = 4 (m)	a product	an equation label or a unit
  ambiguous-group-product	(x)(1,2)	a scalar times a point	a call
  ambiguous-factorial	5!=120	5 ≠ 120	5! = 120
  ambiguous-arrow	x <- 2	x < -2	an assignment arrow
  ambiguous-equal-chain	x = y = 0	nested equations	an assignment to several names
  ambiguous-element	y = x in [0,1]	(y = x) ∈ [0,1]	y = x for x ∈ [0,1]
  ambiguous-range	1..10..2	first, second, last	first, last, step
  ambiguous-percent	y = 50%	y = 50 (% starts a comment)	a percentage
  ambiguous-comma	1,5	a sequence	the decimal number 1.5
  ambiguous-list-label	1. y = x, (1), (iv), a) y = x	math	the label of a list item
  ambiguous-number-notation	1_000, 0x10	a subscript and a product	a number
  ambiguous-date	2026-10-15, 555-1234, 7-11	a difference or a quotient	a date, a phone number or a range

  Ordinary math is not reported: 3/4, 24/7, 2-1, x = 2, 1..10, [0,1], f(x, y), (1, 2), 0 < x < 1, x != y, 5! = 120, x in [0,1]. The strict grammar reports none of these codes.

• Three more ambiguous-* parse diagnostics in the lenient grammar, and a wider ambiguous-constant-name. With ce.parse(text, { strict: false, diagnostics: true }), these readings now report a diagnostic. The reading does not change, and the strict grammar reports none of them.

  • ambiguous-log-base: log with two arguments in parentheses. log(x, 2) is Log(x, 2), the base second as in Python and in spreadsheets, and other tools put the base first. Also the name lg (lg(x) is base 10 by ISO 80000-2, and base 2 in computer science). detail: { name }.
  • ambiguous-engine-operator: a one-letter library operator written as a plain letter before a parenthesis, read as a call of the operator: N(x) (numeric evaluation), D(x) (derivative). A person writing plain text usually means a function of their own. \operatorname{N}(x) is not reported. detail: { name }.
  • ambiguous-interval: after in, \in, ∈ or \notin, a bracket pair followed by an operator, which is then read as a list or a tuple, not as an interval: M in [0,1]^2 is Element(M, Power(List(0, 1), 2)). Also M in (0,1)^2, M in [0,1] + 1 and a range, M in [0..1]^2. M in [0,1] and x in [0,1) are read as intervals and are not reported.
  • ambiguous-constant-name also reports a library constant followed by a parenthesized group on the left of =: pi(x) = x is read as π·x = x, and a person can mean the definition of a function pi. f(pi) = 3 is not reported.

• Two more ambiguous-* codes. ambiguous-inverse-function: in plain text, sin^-1(x) is read as the inverse function, and a person also means 1/sin(x) (\sin^{-1}(x) is not reported). ambiguous-range also reports a range with one .. next to an operation: 1..5/2 is the range from 1 to 5/2, and a person can mean (1..5)/2; -1..5 is not reported.

• The onAmbiguity parse option. ce.parse(text, { strict: false, onAmbiguity: 'error' }) puts an Error node in place of the smallest expression that holds the source span of each parse diagnostic whose code starts with ambiguous-. The error code is the diagnostic code, and the error holds the source text of the span: ce.parse('y = 2 3', { strict: false, onAmbiguity: 'error' }) is ["Equal", "y", ["Error", "'ambiguous-digit-groups'", ["LatexString", "'2 3'"]]]. The option works without diagnostics: true, and for every ambiguous-* code. When the parser cannot find an expression with that span, the Error node replaces the whole result. The default, 'report', keeps the reading and reports the diagnostic when diagnostics: true. In strict mode the option has no effect.

• The ambiguous-sign parse diagnostic (lenient grammar only). It reports a prefix ± (also spelled \pm, \plusmn or +-) with no left operand: x = ±1 is read as Measurement(0, 1), and a person often means the two values 1 and -1. It also reports two signs in a row: --x, x - -y, a<--b, a + -b, -+x, and a -+ b (read as MinusPlus(a, b)). detail: { signs }. a +- b with no white space is read as Measurement(a, b) and is not reported. The reading does not change.

• #393 LaTeX notation on a running engine. LatexSyntax.addEntries(entries) adds LaTeX dictionary entries after construction; the next parse or serialization uses them. With IntegerModRing declared, ce.box(['IntegerModRing', 5]).latex was \mathrm{IntegerModRing}(5); after ce.latexSyntax.addEntries([entry]) it is \mathbb{Z}/5\mathbb{Z}, and parsing \mathbb{Z}/5\mathbb{Z} gives ["IntegerModRing", 5]. The default dictionary and the array given as the dictionary option are not modified. An engine created without the latexSyntax option has its own LatexSyntax, so the entries apply to that engine only; an instance given to several engines is shared, and the entries apply to all of them. LaTeX notation stays separate from library definitions: a LibraryDefinition still has no LaTeX entries.

• #393 Load a library on a running engine. ce.loadLibrary(library) takes the same LibraryDefinition as the libraries constructor option and declares its definitions in the global scope, as ce.declare() does. An expression boxed before the call uses the new definitions: ce.box(['Sq', 3]).evaluate() gave Sq(3), and after loading a library that defines Sq as x ↦ x² it gives 9. Each library in requires must already be loaded. The call throws, and declares nothing, for a library name that is already loaded, a missing required library, a standard library (select those with the libraries option), or a definition name that is invalid, repeated, defined by another library, or already declared in the global scope. A definition that ce.declare() rejects (for example for an unknown key) also makes the call throw, and the definitions declared before it are removed: for any failure, the call declares nothing. (The exception is a call made while an evaluation or a declaration batch is in progress, where the definitions before the rejected one stay declared.) When a library in requires is a standard library, the error says to select it with the libraries option. A checkpoint taken before the call (ce.checkpoint()) undoes it.

• #393 ce.libraryOf(name) returns the name of the library that defines name: 'trigonometry' for Sin, the name of a caller library given to the constructor or to ce.loadLibrary(), and undefined for a name declared with ce.declare(), a name nothing defines, or a library name that a declaration in the current scope shadows.

• #394 ce.declare(name, patch, { extend: true }) extends the operator definition that is visible for name instead of replacing it. Before, a redeclaration replaced the whole definition: ce.declare("SetMinus", { signature: "(value, value*) -> set" }) made SetMinus(5, 2) valid, but removed the evaluate and collection handlers and the description, so SetMinus(Set(1, 2, 3), Set(2)) no longer evaluated. In extend mode:

  • The engine builds a new definition from the visible one and the fields of the patch. A field the patch does not name keeps its value. A field the patch names replaces the old value: when two extensions give the same handler, the later one wins. With the patch above, SetMinus(5, 2) is valid and SetMinus(Set(1, 2, 3), Set(2)) still evaluates to Set(1, 3).
  • signature replaces the signature. The new field addSignature adds an overload: the signature becomes old & new. The new signature must be a subtype of the old one, so that every call that was valid stays valid: (any*) -> any cannot replace (value+) -> value, and the call throws. This does not apply when the old signature was only inferred (an operator declared without a signature, or a function whose signature is inferred from its body).
  • An extension of a standard-library operator that keeps its evaluate, canonical, compile and derivative handlers is still the library operator: after ce.declare("Sin", { description: "…" }, { extend: true }), D(Sin(x), x) is cos(x) and Sin(x) compiles. An extension that replaces one of these handlers is a user definition with the library name, as with a plain ce.declare().
  • A name that is already declared in the same scope can be extended: two libraries that extend Basis both succeed, and the second one sees the fields of the first. A plain ce.declare of that name still throws.
  • It is an error to extend a name with no operator definition, a name declared as a value, or a function defined by clauses.
  • The old definition is not changed: an expression boxed before the extension keeps it. A checkpoint restore undoes an extension like any declaration.
  • The third argument of ce.declare() can now be an options object { scope?, extend? }. A scope as the third argument works as before.

• #393 An operator definition can give its own derivative with the new derivative key. Before, the key was refused (unexpected key "derivative"), and the derivative of an operator whose evaluate handler answers only for numbers stayed symbolic: D(Sq(x), x) gave Apply(Derivative(Sq, 1), x). The key gives the partial derivative with respect to each argument, and D applies the chain rule:

  ce.declare('Sq', {
    signature: '(number) -> number',
    evaluate: ([x]) => (isNumber(x) ? x.mul(x) : undefined),
    derivative: [['Function', ['Multiply', 2, 'x'], 'x']],
  });
  ce.box(['D', ['Sq', 'x'], 'x']).evaluate(); // 2x
  ce.box(['D', ['Sq', ['Power', 'x', 2]], 'x']).evaluate(); // 4x^3
  ce.box(['Sq', 'y']).evaluate(); // Sq(y), not unfolded

  The value is either an array with one function literal for each argument (each literal takes all the arguments as parameters), or a handler (ops, { engine, argument }) => Expression | undefined that returns the partial derivative with respect to argument number argument at ops. A partial that the key does not give stays symbolic. Derivative(Sq), Sq'(3), Apply(Derivative(F, 1, 0), 2, 3) and the compiled form of D(Sq(x^2), x) use the key too. The key has precedence over the body of a function-literal evaluate handler; an operator of the standard library keeps its own derivative rule.

• #393 The options passed to an evaluate handler have a new precision field: the number of significant digits requested. Inside N(x, p) it is p, for the evaluation of x and of everything evaluated inside it, also when p is lower than ce.precision. Under any other numeric approximation (x.N(), N(x)) it is ce.precision. In an exact evaluation it is undefined. A handler can then compute to the requested digits without reading ce.precision. N(x, p) with p above ce.precision still raises ce.precision and leaves it raised, as before.

Issues Resolved

• A pure imaginary factor in a product was serialized in parentheses that are not necessary: i·x was (\imaginaryI)x and e^{iπ} was \exp((\imaginaryI)\pi). They are now \imaginaryI x and \exp(\imaginaryI\pi), and 2i·x is 2\imaginaryI x. A factor with a leading sign ((-2\imaginaryI)x) or with a real part ((1+2\imaginaryI)x) keeps its parentheses. The old and the new output read back as the same expression. Two related power-base spellings were wrong or misleading: Power(Complex(0, 1.0), x) was 1.0\imaginaryI^{x}, which reads back as 1.0·i^x, and is now (1.0\imaginaryI)^{x}; and Power(Complex(1/2, 0), x) was \frac{1}{2}^{x} and is now (\frac{1}{2})^{x}, as for a plain 1/2.

• #390 A call through a field of a record or a dictionary ignored the declaration of the function in the field. With bob_S declared (x: number, factor: number?) -> number, the call bob.S(3, factor: 5) (MemberCall(bob, "S", …), or Apply(Field(bob, "S"), …)) gave the error argument-names-unavailable, while bob_S(3, factor: 5) gave 15. Two kinds of receiver now take named arguments:

  • A constant receiver: bob declared with isConstant: true and the value ["Dictionary", ["Tuple", "'S'", "bob_S"]], or const bob = {S -> bob_S} in Epsil. Its field cannot change, so the call is made the direct call: bob.S(3, factor: 5) was argument-names-unavailable, and is now bob_S(3, 5), which evaluates to 15. The operator's lazy flag applies too: with y := 10 and a lazy bob_H that returns Hold of its argument, bob.H(y * z) gave Hold(10z) and now gives Hold(y * z), as bob_H(y * z) does. A field that holds a function literal is applied with the names matched against the literal's parameters. The canonical form of such a call is the direct call, so it prints as bob_S(3, 5).
  • A receiver whose type is a record that gives the field a signature with named parameters (record{S: (x: number, factor: number?) -> number}). The names are matched against that signature: bob.S(factor: 5, x: 3) was argument-names-unavailable, and is now Apply(Field(bob, "S"), 3, 5), which evaluates to 15. The receiver can be a variable, because the match reads its type, not its value. lazy is not honored on this route, since a signature type has no lazy flag: bob.H(x: y * z) gives Hold(10z).

  A variable typed dictionary<function> gives no parameter names, so a named call through it is still argument-names-unavailable, and its value is not read: lazy is honored only through a constant receiver.

• Inside the call of a user function, a free symbol of the caller with the name of a parameter is an unknown. With f = (x, y) ↦ x = y, f(y, x) answered False instead of staying y = x, x ≠ y answered True, and If(x = y, 1, 0) answered 0. The cause was .unknowns: it looked up each symbol by name, and during the call the name x also denotes the parameter, which has a value. .unknowns now resolves a symbol the way evaluation reads its value, so the x of the argument is an unknown and the parameter x used in the body is not. The same defect caused a performance regression since 0.142.0: the evaluate() of a call whose arguments contain a Floor (also Ceil, Round, Truncate) was 300 to 6,000 times slower than in 0.141.0. With f = (x, y) ↦ \{\sqrt{x^2+y^2} \le 1: 1, 0\}, the call f(⌊31x⌋/31, ⌊31y⌋/31) took about 60 s, and now takes about 10 ms: each sign of ⌊31x⌋ tried to order the free 31x and 1 as two constants, with a higher precision and a symbolic proof.

• A library list with core but without control-structures can build function literals. With new ComputeEngine({ libraries: ['core', 'arithmetic'] }), every function literal, and every caller library with an evaluate formula, printed "Cannot read properties of undefined (reading 'bindings')": a function literal is made canonical with a Block body, and Block was in the control-structures library. Block is now in the core library, and the library reference lists it under core.

• ComputeEngine.getStandardLibrary() with a list of categories threw for every category except core: getStandardLibrary('physics') gave "Library "physics" requires "arithmetic", which is not available", because the list held only the requested libraries. Each requested library now comes with the libraries it requires, directly or indirectly: getStandardLibrary('physics') gives core, arithmetic, units and physics, in load order.

• A ce.declare() of an operator that threw (for example a canonical handler together with the commutative flag) left the name bound to a placeholder value of type function. The name is now restored to the binding it had before the call, or is not declared at all.

• The numeric value of a definite integral whose integrand does not compile now uses adaptive quadrature. Before, such an integrand (for example an operator declared with only a JavaScript evaluate handler) went directly to a Monte-Carlo estimate with 1e4 samples. With Sq declared so that Sq(x) is x² for a number x, Integrate(Sq(x), x, 0, 1).N() was 0.3356 ± 0.0030; it is now 0.3333333333333333 ± 6e-21, with 240 evaluations of Sq instead of 10 000. The integrand is now integrated with the adaptive Gauss–Kronrod quadrature that a compiled integrand uses, with a smaller panel budget (at most 9 660 evaluations), and Monte Carlo is used only when the quadrature does not converge and its error bound is not better than the estimate of Monte Carlo. A divergent integral is now NaN, as for a compiled integrand: with Inv(x) = 1/x, Integrate(Inv(x), x, 0, 1).N() was 10.9 ± 1.4 and is now NaN.

• #394 Redeclaring an arithmetic operator with a copy of its definition changed its canonical forms and types, even when no field of the copy changed. This is the idiom the guide documents to extend a standard operator: ce.declare('Sqrt', { ...ce.lookupDefinition('Sqrt').operator, evaluate }). It affected Add, Multiply, Negate, Square, Sqrt, Exp, Ln, Log, Power, Root and Divide. After ce.declare('Add', { ...oldAdd }): Add(2, x, 5) gave 2 + x + 5 and now gives x + 7; Add(1, "s") was valid and is now invalid; Add(1, ImaginaryUnit) was typed integer and is now typed complex; in 1 + w, w stayed unknown and is now inferred number. After the same copy of Sqrt: Sqrt(8) stayed sqrt(8) and now gives 2√2, and √2·√2 now gives 2. The definitions of these operators now have a canonical handler, and a copy takes it along. The standard canonical form applies only to a copy declared under the SAME name whose signature, lazy flag and associative/commutative/idempotent/ involution flags are unchanged; a copy that changes only evaluate or metadata (description, wikidata, examples) is such a copy. A copy with another signature or another flag, or a copy declared under another name, does not use the standard canonical form: the copy keeps its own head and its own evaluate. With MyPower declared as a copy of Power with evaluate: () => 5, MyPower(x, 3) gave x^3 (the head was replaced by Power, which had a canonical handler before this change) and now stays MyPower(x, 3) and evaluates to 5. A definition that is not a copy (function Add(a, b) { … }, or a declaration with its own canonical handler) still replaces the standard operator, as before.

0.144.0 2026-10-01

Behavior Changes

• The attributes entry of About is a list, and it says when an operator is lazy. It was one string with the flags separated by spaces: About(Add) gave "commutative associative idempotent". It is now a list of strings, the algebraic flags and then lazy when the arguments of the operator are passed to it unevaluated: About(Add) gives ["commutative", "associative", "idempotent", "lazy"], and About(Hold), which had no attributes entry, gives ["lazy"]. A program that read the entry as a string must read it as a list.

• #397 Solve answers an identity with a free parameter, and stays unevaluated when it finds no candidate root. Solve(x = x, x), Solve(0 = 0, x) and Solve(2(x + 1) = 2x + 2, x) were [] ("no solution"); they are now [t], one solution for every value of the fresh parameter t, the same form as the parametric answers of the integer and congruence solvers. An equation for which no strategy of the solver gives a candidate root (a·x⁵ + x + 1 = 0, sin(x) = x³ + eˣ, which has a real root) and an equation whose answer depends on another unknown (Solve(a = 0, x)) were also []; they now stay unevaluated. A contradiction (x + 1 = x + 2) and an equation whose candidate roots are all rejected (√x = -1, sin x = 2, eˣ = 0) are still []. The .solve() method is unchanged. A program that read [] as "no solution" for these equations gets the unevaluated Solve or [t] instead.

• #397 Range with an exact rational bound or step enumerates exact values. Range(0, 1, 1/3) was [0, 0.333…, 0.666…, 1]; it is now [0, 1/3, 2/3, 1], as in Mathematica, and the two-sample form [1 + 4/d, 1 + 8/d...5] with d = 500 starts at 126/125, not 1.008. A float bound or step, and a step that is a constant expression (π/4), still give floats, and .N() of the range gives floats. Sum(Range(0, 1, 1/3)) is now 2, not 1.9999999999999999.

• #397 simplify() combines or splits logarithms only when their arguments are provably non-negative. ln(x) + ln(y) was ln(xy) and ln(x/y) was ln(x) − ln(y) for any unconstrained x and y, which is wrong for negative values: at x = y = -1, ln(x) + ln(y) is 2πi but ln(xy) is 0. They now stay as they are, as does ln(1/x), which was −ln(x). An argument is used when it is non-negative by its value (ln(2) + ln(3) is still ln(6)), by its type, by an assumption (with assume(x > 0) and assume(y > 0), ln(x) + ln(y) is ln(xy)), or because it is an absolute value. The same applies to log_c. Solve still combines the logarithms of an equation, and checks each root against the original equation: Solve(ln(x + 1) + ln(x − 1) = 0, x) is [√2]. The policy is in docs/SIMPLIFY.md.

These changes apply to non-strict parsing only (ce.parse(latex, { strict: false })). Strict parsing is unchanged.

• An unbraced run of letters after _ is the whole subscript. x_max was x_m·a·x and is now the symbol x_max, the same as x_{max}; T_max is T_max, and y = x_1 + x_max is y = x_1 + x_max. The run ends at the first token that is not a letter. A run of two letters followed by _, ^ or a digit keeps the previous reading, so that adjacent indexed symbols stay apart: a_nb_n is still a_n·b_n and a_kx^k is still a_k·x^k. Input that changes: x_ij was x_i·j and is now the symbol x_ij, and a_nx was a_n·x and is now the symbol a_nx (write a_n x for the product).
• An unbraced exponent is one whole operand. As a run of digits already was (x^12), the exponent is now a number with its decimal part (x^2.5 is x^{2.5}, was x^2·0.5), a word read as one symbol (x^pi is x^π, was x^p·i; x^theta is x^θ), or a bare function call (e^sin(x) is e^{sin(x)}, was e^s·i·n·(x)). The result is the same as the braced spelling. White space ends the exponent (x^2 y is still x^2·y), and where the extent is not clear the reading is unchanged: x^2y is x^2·y, e^2pi is e^2·π, x^ab is x^a·b.
• A sign directly after ^ starts the exponent. The exponent is the sign and one operand: e^-x is e^{-x} (was Superminus(e)·x), 2^-x is 2^{-x}, e^-sin(x) is e^{-sin(x)}, e^-(x) is e^{-(x)}, and e^+x is e^{+x} (was PseudoInverse(e)·x). A second superscript is the same double-superscript error as in e^{-x}^2. When no operand follows the sign directly, the reading is unchanged: \Z^+ is still the positive integers, A^+ is still the pseudo-inverse, and x \to 0^+ is still a one-sided limit point. x^-2 is still x^{-2}.
• The words in and infinity are read as math. M in [0,1] is M \in [0,1] (Element(M, Interval(0, 1)), was M·i·n[0,1]), and infinity is PositiveInfinity like inf (was the product of its letters). in is read this way only as a separate word between two operands: index, ink, int and xin keep their previous reading.

New Features

• A parse diagnostic for a letter run read as a product. In non-strict mode, with diagnostics: true, a run of two or more letters that is not a known word and is read as a product of its parts now gives one letter-run-split diagnostic: eps (e·p·s), sinx, a word of prose, or xpi (x·π). Its span is the run, and detail is { run, parts }, for example { run: "eps", parts: ["e", "p", "s"] }. An explicit product (a*b*c) and a run read as one name (sin(x), alpha, foo(x)) give no such diagnostic, and neither do the differentials of dy/dx. An unbraced exponent or subscript that takes only the first letter of a run is reported too: e^xy is e^x·y and (x)_ab is (x)_a·b. The diagnostic does not change the parse result.

• Three new opt-in parse diagnostics for non-strict input. With ce.parse(latex, { strict: false, diagnostics: true }), the result's parseDiagnostics now also reports:

  • implicit-product-in-denominator: the denominator of a / is an implicit product, which binds tighter than /. 1/2x, 1/2 x, pi/2x and x/2 y are read as 1/(2x), π/(2x) and x/(2y), which is not what every reader intends. 1/2 * x, 1/(2x), dy/dx and a call such as 1/f(x) are not reported.
  • spaced-digit-groups: white space between digits was read as one number (2 3 is 23, 1 000 is 1000). The digit group separators \, and {,} are not reported.
  • letter-before-decimal: a symbol directly followed by .digits (x.5) is read as the product x \cdot 0.5.

  The readings do not change, and strict mode reports none of these.

Issues Resolved

• Solve found no root of a linear equation whose coefficient of the unknown is a sum: x − ax + a = 0, πx + x = 1 and x = e(x − 1) gave []. Only the shape ax + b was recognized. They are now solved from their coefficients (1/(π + 1), e/(e − 1)).

• Solve found no root of an equation with one logarithm of a non-linear argument, or with several logarithms of one base: ln(x² + 2x) = 3, log_2(x) + log_2(x + 2) = 3 and ln(x) − ln(x − 1) = 1 gave []. They now give −1 ± √(1 + e³), [2] and [e/(e − 1)]. The roots are checked against the original equation, so a root where a logarithm is not defined is rejected (log_2(x) + log_2(x + 2) = 3 has no root −4).

• #397 Thirty-seven Wikidata ids, in the library and in OPERATORS.json, named an unrelated item: PlanckConstant was Q524 (Mount Vesuvius), Nor was Q189561 (narcolepsy). They now name the concept of the head, and a test checks that OPERATORS.json agrees with the library.

• #397 LerchPhi at a negative integer order and exact operands stayed unevaluated where its value is 0: LerchPhi(-1, -1, 1/2) is now 0, also under .N(), as in Wolfram. For s = -n (n ≤ 12) and exact z ≠ 1 and a, LerchPhi(z, s, a) is now the exact rational value of its closed form aⁿ/(1 − z) + Σⱼ C(n, j)·aⁿ⁻ʲ·Li₋ⱼ(z): LerchPhi(3, -4, -5/2) is -5725/32.

• Solve(eˣ = 3, x) gave the float 1.0986… instead of ln(3), and Solve(e^(2x) = 5, x) gave []. The Solve operator replaced the constant ExponentialE by its float value before solving. A constant that keeps its value until .N() (ExponentialE, Pi) now stays symbolic, and the answers are [ln(3)] and [ln(5)/2].

• #398 About now reports the examples and keywords of a definition, each as a list of strings: About(Sin) includes "examples" -> ["Sin(Pi / 6)", "Sin(1)", "N(Sin(1))"] and "keywords" -> ["sine"]. The boxed operator and value definitions did not keep the examples of the definition they were made from. They now keep them as a list of strings; a definition that gives one string is stored as a list of one string.

• #388 A function literal in compiled JavaScript was wrapped in a broadcast dispatch even where no argument could be a list. Two cases now compile to the bare arrow function. A callback fed the elements of a Range whose start and step are literal numbers (Map(q ↦ …, Range(1, Length(s)))) receives a finite number at every call, so the wrapper and the NaN and absence tests on its parameter (q === q, typeof q === 'number') are gone. An Apply of a literal to arguments that are scalars by construction (a number, a declared scalar input, a loop index) no longer builds the wrapper and its closure at each evaluation. A list argument still broadcasts, as in the interpreter.

• #387 On the JavaScript target, a compiled Range is built in a loop, and a Map, a Fold/Reduce or a Sum(Map(…)) over a finite Range builds no range at all. A Range compiled to Array.from({length: n}, (_e, i) => a + i * s), which is about 18 times slower on V8 than a preallocated array filled by a counted loop. The compiled code now calls a run-time helper, _SYS.range(a, b, s), that fills the array in a loop. A Map over a finite range, a Reduce over one (Fold is its canonical form) and the Sum or Product of a Map over one now walk the range with the counted loop that predicates over a range already use (issue #373). Fold((acc, k) ↦ acc + k, 0, 1..n) at n = 1000 went from about 38 µs to under 1 µs. The values are unchanged: the element count is the interpreter's own, element i is a + i × s, the elements are folded in range order, a seedless fold over an empty range is NaN, and an infinite bound at run time still throws a RangeError. A range that is shared by common-subexpression elimination, or has a non-finite bound, keeps the array lowering.

• #393 A library given in the libraries constructor option with no requires list loaded before the standard libraries listed before it. The libraries were sorted so that every library with no dependencies came first, whatever its place in the list. A caller library whose definitions use a function literal (evaluate: ["Function", …]) then made Block a plain symbol before control-structures could define it: the engine printed "Duplicate operator definition: Block" and stayed broken (x := 2; x + 1 gave {2; 3}). The libraries now load in the order of the list, and each library loads after the libraries in its requires list. The order of the standard libraries does not change.

• #393 D did not differentiate an operator defined by a library given in the libraries constructor option: with Sq defined as x ↦ x², D(Sq(x), x) gave Apply(Derivative("Sq", 1), x). Such a library is installed in the same scope as the standard library, so D took its operators for built-in ones. D(Sq(x), x) now gives 2x, as it does when Sq is declared with ce.declare().

• D used the library rule for a user function with the name of a library function: after \operatorname{Sinh}(x) := 3x, Sinh(2) is 6 but D(Sinh(t), t) gave cosh(t). It now gives 3.

• #394 SetMinus, Length, Count, IsEmpty, Contains, Append and Slice validated their operands against a copy of their signature text, not against the signature of their definition. A host that redeclared one of them with a wider signature and kept its handlers (ce.declare('SetMinus', { ...ce.lookupDefinition('SetMinus').operator, signature: '(value, value*) -> set' })) still had the old validation: SetMinus(5, 2) stayed an incompatible-type error. These operators now read the signature of the definition in effect. A stock engine gives the same results as before.

• The resolveApplication hook sees every name before a parenthesis in non-strict mode. A multi-letter name (foo(x)) or a spelled-out Greek letter (gamma(x), alpha(x+1), theta(x)) was read as a name before the hook could see it, so a host could not choose the reading. The hook is now called for these names with the same precedence rules as for f(x): explicit declarations, function parameters and resolveSymbol facts take precedence, and a name with a library definition (Gamma(x), pi(x), sin(x)) is not submitted. When the hook returns undefined, the reading is unchanged. Reported by a host.

• Parsing a deeply nested exponent in non-strict mode is fast. Each e^{…} was read twice, so the time doubled with each level of nesting in e^{-(e^{-(…)})}. Each exponent is now read once.

• The bare names sign and sgn are the sign function. In non-strict mode, sign(x) and sgn(x) were read as an undefined function Sgn. They are now Sign: sign(-2) evaluates to -1.

• A chain of <=, >= or != parses as a chain. 0.1 <= M <= 5 gave LessEqual(0.1, Equal(Less(M, Error(missing)), 5)): the right operand of the first <= read the < of the second <= as Less, because < binds tighter than <=. The parser now considers only the longest operator that matches the input, and stops the operand when that operator binds too loosely. 0.1 <= M <= 5 is LessEqual(0.1, M, 5), the same as 0.1 ≤ M ≤ 5 and 0.1 \le M \le 5, and 5 >= M >= 0.1 is GreaterEqual(5, M, 0.1), the same as 5 ≥ M ≥ 0.1. This applies in strict and non-strict mode. A chain of \ge, \geq, \geqslant, > or \gt is also one flat expression in the raw form now (5\ge M\ge 0.1 is GreaterEqual(5, M, 0.1), it was GreaterEqual(5, GreaterEqual(M, 0.1))), as a chain of \le or < already was. The canonical form is unchanged.

• More plain-text operator spellings in non-strict mode. x ÷ 2 is Divide(x, 2), as x / 2 and x \div 2 are. a =< b is LessEqual(a, b) and a <> b is NotEqual(a, b). In strict mode these spellings are not operators, as before.

• √(x+1) is Sqrt(x+1) in non-strict mode, as sqrt(x+1) is. It was the juxtaposition of the Sqrt function and the parenthesized group. Strict mode is unchanged.

• Parentheses in a fraction or an exponent are not written twice. A structural or raw expression that kept the parentheses of its source, such as y = 1/(1+x^2), was written y=((1+x^2))^{-1}. It is now written y=\frac{1}{1+x^2}: the \frac arguments and the superscript braces already group their content, so x^(1/2) is written x^{\frac{1}{2}}, not x^{(\frac{1}{2})}. Parentheses elsewhere are kept.

• \operatorname{arccot}(x) is the inverse cotangent. It was a call of an undefined function arccot; only the spelling arcctg was read as Arccot. Both spellings are now Arccot, also as \mathrm{arccot}.

• Scientific notation with the base written {10}. 2\times{10}^{-1} is the number 0.2, the same as 2\times10^{-1}, and 4.35\times{10}^2, 2\cdot{10}^{3} and a/2\times{10}^3 read the same as the spellings without braces. {10}^{3} with no mantissa, a symbolic exponent (2\times{10}^{n}) and another braced base (2\times{11}^{3}) are unchanged.

0.143.0 2026-10-01

Behavior Changes

• A symbol declared with a nested list type is the matrix it describes. list<vector<integer^3>^2> (two rows of three integers) was not a subtype of matrix<integer^(2x3)>, so a function declared (x: matrix<T^(MxN)>) -> N where T, M, N admitted it without reading its lengths (the call was typed integer<1..> instead of 3), and a square constraint matrix<T^(NxN)> admitted a 2×3 list. The nested spelling is now read as its flat shape at a target with two or more dimensions. A value changes with it: with x declared list<list<integer^2>^2>, x^2 is now the matrix power ([[7,10],[15,22]] for [[1,2],[3,4]]), as it is for a matrix literal, where it was the square of each element.

• Sum of a list of points or rows has the type of a point or a row. Sum(pts) with pts declared list<tuple<real, real>> was typed number, but its value is a point: Sum([(1, 2), (3, 4)]) is (4, 6). It is now typed integer | tuple<real, real> (a list with no length may be empty, and the sum of an empty list is 0). A list of rows sums to a row and a matrix to the row of its column sums: Sum([[1, 2], [3, 4]]) is typed vector<integer^2>. A consumer that used the number type to read the compiled value as a scalar was wrong for such a sum.

• Max and Min of a dictionary are an incompatible-type error. They walked the dictionary as a collection and compared its keys with its values: Max({"a" -> 3, "b" -> 5}) was max(5, "a", "b"). An entry is not a number, so Max, Min, Supremum and Infimum of a dictionary that has an entry now give the error that Sum and Mean give, which names the first entry: Error(incompatible-type, number, tuple<string, integer>). A program that read the unevaluated max(…) result gets the error instead. An empty dictionary contributes no value, as an empty list does. A dictionary that is an ELEMENT of a list gives the error that names the whole dictionary, as Sum([1, d]) does. The error is the answer whatever the order of the operands: Max(NaN, d) was NaN while Max(d, NaN) was the error.

• Sum of one element that is not a number is an error. A sum of one element answered the element itself: Sum([{"a" -> 3}]) was {"a" -> 3}, Sum([True]) was True, and Sum({"a" -> 3}) (a dictionary of one entry) was the entry ("a", 3), while two such elements gave an incompatible-type error. One element now gives the same error as two. A point or a row is still summed: Sum([(1, 2)]) is (1, 2).

Issues Resolved

• Registering a chain of functions that each call the next one twice took a time that doubled with each level: the effects inference walked the body of a called function once per call, so the last function of a chain of 12 was walked 2¹² times. A function already walked for the same definition is now skipped, and the work grows polynomially with the depth: 5 666 reads of the declared signatures at depth 12 (from 89 794) and 20 962 at depth 20.

• The compiled Sum and Product of a list of points, rows or matrices with complex entries gave wrong values behind success: true: the sum of the points [(1+i, 2), (3+i, 4)] was ["0[object Object][object Object]", 6] and a product of complex rows or matrices had NaN entries. They now match evaluate(). The compiler chooses a real-only, a complex, or (for a shape known only at run time) a dispatching form of the element-wise helper from the type of the operand, as for the other linear-algebra helpers.

• A compiled expression over the Sum of a list of REAL points read each coordinate as a complex number: Sum(P) + (1, 1) gave NaN coordinates where evaluate() gives (5, 7). It is now correct.

• Product of a list of rows or square matrices was typed number; it is now typed as a row or a matrix (Product([[1, 2], [3, 4]]) is [3, 8]), and a compiled Product(Q) + 1 no longer adds 1 to the matrix as if it were one number. Sum and Product of an abstract collection or a set of rows or matrices, and of a matrix whose size is not known, are typed by shape too, and so is the sum or product of elements typed broadcastable<T>.

• Shapes the JavaScript target cannot compute now refuse to compile instead of giving NaN or a string: a scalar times the Sum of points (2·Sum(P)), a complex parent over a fold of complex rows or matrices, Multiply of two complex rows or matrices read with At, Product of matrices known not to be square, and Sum over elements that are abstract collections. With the default fallback, evaluate() answers them.

• Sum() and Product() with no operand logged an internal exception during canonicalization. The operand is now reported missing, as for Mean().

• #386 ReplaceAt, DeleteAt and Insert did not compile to JavaScript: compile(ReplaceAt(s, 2, 9)) failed with "target 'javascript' has no lowering for it", so a Fold whose step replaced one element of a list did not compile. Each now compiles to a copy of the array with the interpreter's index rules (1-based, a negative index counts from the end, Insert takes positions 1 to n + 1). An index for which the interpreter leaves the expression unevaluated (zero or out of range) throws a RangeError at run time. A string operand is walked as its characters, as in the interpreter: DeleteAt answers a string, ReplaceAt and Insert a list of characters.

• #386 ReplaceAt, DeleteAt and Insert also compile to Python, with the same index rules. An index for which the interpreter leaves the expression unevaluated raises an IndexError. A string operand does not compile to Python, as for the other list operators.

• Compiled statistics of a list of lists gave wrong values. On the JavaScript target Max, Min, Mean, Median, Variance, Mode, Quartiles and the other statistics of [[1, 2], [3, 5]] answered null or a wrong list, and Mean([2, 3], [5, 7]) answered NaN; on the Python target Mean and Median of a list of lists answered a number. The interpreter answers 5 for Max, 17/4 for the two-list Mean, and an incompatible-type error for Mean([[1, 2], [3, 5]]). These no longer compile, so the interpreter answers. A JavaScript Reduce or Scan with Add or Multiply over a list of lists now adds or multiplies the rows element by element, as Sum does (Reduce(s, Add) joined two rows as the string "1,23,5"); a built-in fold over points, strings, sets or dictionaries, and every built-in fold over rows on the Python target (where + joins two lists: Scan(s, Add) gave [[1, 2], [1, 2, 3, 5]]), no longer compiles. A fold whose step was typed for a list of numbers by its uses (Sum(acc)) but whose seed is a list of lists no longer compiles, because the step's arithmetic would join rows as strings. A row type given through a type name (list<Row>) is recognized by all these checks. A fold step whose element parameter is named i was compiled as if i were the imaginary unit: Reduce(L, (acc, i) => acc + i, [0, 0]) over rows answered [{re: null}, {re: null}]; it now answers [4, 7].

• .N() of a Reduce with no initial value over a list of lists was NaN: Reduce([[1, 2], [3, 4]], Add).N() is now [4, 6], as evaluate() gives. The compiled code of the same expression, which used that value, was the constant NaN.

• #386 A compiled Fold whose step replaced elements of a list accumulator copied the whole list at each step, so a fold that visits each element once took time proportional to the square of the length (730 ms for 40,000 elements). When the step can only return the accumulator or a chain of ReplaceAt on it, and reads it nowhere else except through an element read or an aggregate (At, Length, Sum, …), the compiled fold now copies the seed once and updates that copy in place (1.7 ms for 40,000 elements). A step that swaps two slots, ReplaceAt(ReplaceAt(acc, i, acc[j]), j, acc[i]), reads both old values before it writes. Any other step keeps the copying form. The result is the same, and the caller's list is not changed. This applies to the JavaScript and Python targets.

• The compiled call of a function with an annotated parameter did not check an absent argument. function f(p: tuple<number, number>) { p[1] + 1 } called with first(filter([(1, 2)], c => c[1] > 9)) (no element passes, so the argument is absent) is an incompatible-type error in the interpreter, but the compiled JavaScript answered NaN; a list<number> parameter threw a TypeError from inside the body. The compiled call now stops the run with a TypeError that names the parameter and its type, for a parameter annotated with a type that is not numeric and has no missing member. A numeric parameter still reads an absent value as NaN, as the interpreter does.

• HurwitzZeta(s, a).N() of an integer order at an a more than about 10⁶ left of the imaginary axis stayed unevaluated: HurwitzZeta(2, −10¹² + i) is now −0.0739998067554724…, computed with the polygamma reflection ζ(s, a) = (−1)^s·ψ⁽ˢ⁻¹⁾(a)/(s − 1)! for an order from 2 to about 10⁴, and with the Bernoulli polynomial ζ(−n, a) = −Bₙ₊₁(a)/(n + 1) for an order −n ≤ 0. The cost of both does not depend on a. Zeta(s, a) of an even order follows. An order that is not an integer still stays unevaluated there. PolyGamma(1, −5 + 10²⁰i).N() ran for minutes, and now answers at once.

• #385 The JavaScript compilation of Max and Min over an operand typed as an abstract collection (collection, collection<integer>, a set, or collection<any> | number) gave Math.max(w), which is NaN for a list, with success: true. evaluate() gives the maximum. Max, Min, Length, Count, and the collection form of Sum and Product need neither the positions nor the order of the elements, so such an operand now compiles. At run time it accepts a list, a JavaScript Set or a numeric typed array, and a value that is none of these stops the run with a RangeError. Max and Min also stop with a TypeError on an element that is not a real number (a complex value has no order). A single number is read as a collection of one element where the interpreter accepts one: at a parameter typed collection<any> | number, or whose type collection was inferred from its uses (k(L) := Sum(L), k(4) is 4). Length and Count of such an operand were refused before; they now compile. At, Reverse and the other operators that read positions or order still refuse an abstract collection, and all of them refuse a dictionary.

• The compiled Sum and Product of a list of complex values whose type is not known at compile time (the result of a function typed unknown) combined the elements with +, and the sum of [1+2i, 3+4i] was the string "0[object Object][object Object]". They now add and multiply complex values.

• w.mul(ce.Zero) and ce.Zero.mul(w) folded to 0 for a variable w with an assigned value, while w.mul(0) and ce.box(['Multiply', 0, 'w']) keep the product 0w. With w := NaN, the folded 0 hid the NaN. All the spellings now keep 0w, which evaluates with the value w holds then: NaN for w := NaN, 0 after w := 4. A free symbol still folds: 0x is 0.

• Join (or Append) of a dictionary and a value that is not a key-value entry reported an error that named the internal symbol ContinuationPlaceholder: Join(Dictionary(x: 1), [2, 3]) gave Error(incompatible-type, tuple<string, unknown>, "symbol ContinuationPlaceholder"). The error now names the element: Error(incompatible-type, tuple<string, any>, 2).

• simplify() left Max(x, NaN) and Min(x, NaN) unchanged, while evaluate() gives NaN. simplify() now gives NaN too.

• A function declared with a parameter that admits an absent value could not be assigned a function literal with a bare parameter: ce.declare('f', { signature: '(string | missing) -> unknown' }) followed by ce.assign('f', ce.box(['Function', ['IsMissing', 's'], 's'])) threw "not compatible". A bare parameter accepts an absent value, so the assignment is now accepted, and f(Missing) is True.

0.142.0 2026-09-30

Behavior Changes

• A rounding function, Fract, Mod, Sign, Heaviside or a comparison of an exact constant expression is decided by raising the precision. Floor(π·10³⁰) stayed unevaluated and its .N() was the 21-digit float 3.14159265358979323846e+30; it is now the exact integer 3141592653589793238462643383279, as in Mathematica, and Fract(π) is π − 3. A comparison of exact constants was decided on 21-digit values, which cancellation can make wrong: √(10⁶⁰ + 10⁴⁰) − 10³⁰ > 5·10⁹ was True and is now False (the value is 4.99999…·10⁹). The engine computes the operand with an error bound at the working precision plus 10 digits, then with more digits, up to 100 more, until the result is certain. When it is still not certain, the value is very probably at the jump: evaluate() leaves a rounding function, Sign or Heaviside unevaluated, a comparison is equal, and .N() uses the value at the jump (Floor((√2 + √3)² − 2√6).N() was 4 and is now 5).

• .N() of a rounding function, Fract, Mod, Sign, Heaviside or a comparison uses the exact value of an exact operand. .N() approximated the operand first, and near a point where the result jumps, a tiny error changed the result by a finite amount: Floor((25! − 1)/24!).N() was 25 (the value is just below 25, and its 21-digit approximation is 25), and (1/2 − 10⁻³⁰ < 1/2).N() was False. They are now 24 and True, as in Mathematica, where N[Floor[(25!-1)/24!]] is 24.. .N() promises a result that is correct to the working precision, not that every intermediate step is a float. The operand is still approximated first; its exact value is used when it is an exact number (an integer, a rational, or a rational multiple of a square root), or when its approximation is within about 10⁻¹⁰ (relative) of a point where the result jumps. The result of a rounding function under .N() is a float, as before: Floor(25! − 1).N() is 25! − 1 approximated to the working precision. Fract((25! − 1)/3).N() was 0 and is now 0.666666666666666666667. A comparison with a float operand is still decided at the precision of the float: (1/2 − 10⁻³⁰ < 0.5).N() is False.

• Two exact numbers are compared exactly, with no tolerance. The comparison operators and the methods .isEqual(), .isLess(), .isLessEqual(), .isGreater() and .isGreaterEqual() compared two numbers whose difference was less than ce.tolerance (10⁻¹⁰) as equal, also when both were exact, but Less and Greater sometimes compared exactly: Equal(1/2 − 10⁻³⁰, 1/2) and Less(1/2 − 10⁻³⁰, 1/2) were both True. Two exact numbers (integers, rationals, rational multiples of a square root) are now equal only if they have the same value: Equal(1/3, 3333333333333/10^13) was True and is now False, and exactly one of <, = and > holds for two exact numbers, as in Mathematica. The tolerance still applies when a float is involved: Equal(1/10, 0.1) is True. .is() keeps the tolerance for exact numbers too.

• The interval target encloses Binomial and Factorial as functions of real operands. It used to round each operand to the nearest integer, so its enclosure did not contain the value the interpreter and the JS target give at a non-integer point: Binomial(5.5, 2) gave [15, 15] instead of 12.375, Factorial(2.5) gave [6, 6] instead of Γ(3.5) ≈ 3.323, and Factorial(-0.5) was empty instead of √π. A plot of Binomial(x, 2) or x! showed steps. Now Factorial(x) is enclosed as Γ(x + 1), with a pole at each negative integer. Binomial(n, k) is enclosed through the falling factorial n(n−1)⋯(n−k+1)/k! for an integer k ≥ 0. Where n + 1, k + 1 and n − k + 1 are positive over the box, C(n, k) is monotone in n and log-concave in k, so the enclosure is the true range (Binomial(n, k) over n ∈ [4, 6], k ∈ [1, 3] is [4, 20]). Elsewhere it is the product of enclosures of Γ(n+1), 1/Γ(k+1) and 1/Γ(n−k+1). For a negative integer k the result is a jump: C(n, k) is 0 except at the negative integers n ≥ k. Integer points stay exact. What is lost: an interval of n that contains a root of the polynomial gets a wider enclosure (Binomial(n, 2) over [0, 3] is [−1.5, 3]; the true range over the reals is [−0.125, 3]), the Γ product is wider than the true range (Binomial(5, k) over k ∈ [2, 7] is [−67.8, 67.8], true range [−0.022, 10.87]). An n interval that contains a negative integer, together with a k interval or a non-integer k, is reported as a pole: C is unbounded there. The interval enclosure of the falling factorial (k up to 4096) no longer excludes the value when a partial product overflows: Binomial(n, 1000) over n ∈ [1100, 1100.5] gave [1.8e308, ∞] for a value of 1.4e144.

• evaluate({ materialization: true }) gives every element of a finite lazy collection (#380, reported by enumeratio). The option was documented as "if true, and the collection is finite, it is fully materialized", but true gave the display preview: the first five and the last five elements with a ContinuationPlaceholder between them. Range(1, 20) evaluated to [1, 2, 3, 4, 5, …, 16, 17, 18, 19, 20] and now evaluates to the 20 elements. A finite collection with more elements than ce.maxCollectionSize stays in its lazy form, and an infinite collection, or one whose finiteness is not known, still gives the preview. toString(), .latex and toLatex() without a materialization option still print the preview. To get the preview from evaluate(), pass materialization: [5, 5]. toLatex({ materialization: true }) now prints every element, as its documentation says.

• A function that annotates some of its parameters enforces only the annotated ones at a call. With function k(p, n: number) { p[1] + n }, the type of p in the reported signature, indexed_collection<number>, is inferred from the use p[1]. One annotation made the whole derived signature a contract, so that inferred type was enforced at every call, while the same function with no annotation is not validated at all. A call such as fill(a, b, c, 2, xs) with c typed missing | tuple<number, number, number> (a first(filter(…)) result) was refused at an inferred slot although the interpreter runs it. A bare slot now admits any argument, as it does when nothing is annotated; an annotated slot is enforced as before, and the reported signature is unchanged. What is lost: an argument that cannot fit a bare slot of a partly annotated function (k(5, 1)) is reported when the call runs, not when it is boxed. Annotate the parameter to keep the static check. The same rule holds for a function declared with placeholder slots and then assigned its body, which is how a host declares a function before its body is known: with f declared (unknown) -> unknown and assigned p ↦ p[1] + 1, the reported signature is (indexed_collection<number>) -> number, and f(First(xs)), whose argument is typed missing | tuple<…>, was refused at boxing. It is now admitted, answers the value when it is present and NaN when it is absent, as the same body does with no declaration. A slot declared with a type ((unknown, number) -> unknown) is enforced as before.

• An argument whose components are typed unknown is admitted at a declared parameter. A whole argument typed unknown was always admitted, since it claims nothing a declaration can refute; a composite with unknown components was refused. [(A[1], A[2], A[3], 1)], typed list<tuple<unknown, unknown, unknown, integer>> while the type of A is not known, is now accepted at a parameter declared list<tuple<number, number, number, number>>, and checked when the call runs. A component typed any is still refused (it admits the absence markers), and so is a known component that does not fit. A generic signature ((tuple<T>) -> T where T: number) admits the same arguments as the ground one; signature subtyping is unchanged.

• An absent argument at an annotated parameter of a function is an error. With function f(p: tuple<number, number>) { p[1] + 1 }, the call f(first(filter(xs, c => c[1] > 9))) has the argument Missing when the filter finds nothing. It answered NaN (Missing when the body returns a point): a function whose parameters are all numbers or collections answered the absence marker without running its body. It now answers Error(ErrorCode("incompatible-type", "tuple<number, number>", "missing"), "Missing"), as the same function declared with ce.declare("f", { signature: … }) already did. The rule applies to a function literal with at least one annotated parameter (an Epsil function, a lambda (p: T) ↦ …). An absent value at a parameter annotated with a type that has no missing member is an error. A parameter annotated with a numeric type (number, integer, real) reads the absent value as NaN, the absence marker of a numeric domain, so f(x: real) = x + 1 still answers NaN for f(Missing). At a parameter with no annotation the body runs with the absent value, as before. The type of such a call no longer has a missing member (list<…>, not list<…> | missing), so Count and Length over its result compile. What is lost: a call that relied on the quiet NaN or Missing for an absent point, list or string argument. To accept absence, annotate the parameter T | missing and test isMissing(p), or remove the absent case at the call (first(…) ?? fallback). At boxing, an argument typed missing | T is still admitted at a parameter annotated T; the literal Missing at a parameter that is not numeric is an incompatible-type error at boxing. Library operators are not affected (Sin(Missing) is NaN).

• A user function accepts NaN at a parameter typed integer or real. function f(n: integer) { n + 1 } called with a NaN value (a restricted number whose condition fails, first(filter(xs, p)) over numbers when the filter finds nothing) answered Error(ErrorCode("incompatible-type", "integer", "NaN"), NaN), while f(n: number) answered NaN, and so did the same body declared with ce.declare("f", { signature: "(integer) -> unknown" }). The literal f(NaN) was refused at boxing on both routes. A user function now accepts NaN (and Indeterminate) at every parameter typed with a numeric type, and its body computes with it: all of these answer NaN. A number that is not NaN is checked as before (f(1.5) is an error for n: integer). Library operators keep their own NaN policy. What is lost: a call that relied on the error to detect a NaN argument at an integer or real parameter; test isNaN(n) in the body.

• The Epsil static check reports an argument that may be absent at an annotated parameter. With f(p: tuple<number, number>), the call f(first(filter(xs, p))) is reported before the program runs: "expected tuple<number, number>, got missing | tuple<…>". The program still runs. Nothing is reported at a parameter with no annotation, at a parameter annotated T | missing or with a numeric type, or when the absent case is removed with ??. The engine (ce.box, ce.parse, compilation) admits the argument as before. A function literal bound with let or const is checked as a function definition is, and the inferred type of its bare parameters is no longer enforced by the static check: let k = (p, n: number) => p[1] + n followed by k(5, 1) is reported when the call runs.

• An assignment statement stores the elements of a lazy collection that reads a variable. Filter, Map, Scan, TakeWhile and a comprehension are lazy, and their value kept by name each variable it reads. So after let ys = filter(xs, c => c > 1), a later xs = [7, 8] changed ys; xs = filter(xs, c => c > 1) made xs read itself and every later read (xs, length(xs), first(xs)) stayed unevaluated; xs = [c * 2 for c in xs] overflowed the stack; and in for k in 1..3 { xs = filter(xs, c => c > k) } the predicate kept k by name after the loop. An assignment statement (Assign, Declare with a value: the Epsil = and let, the LaTeX :=) now stores the list of the elements (a set for a collection over a set) when the collection is finite and reads a variable, computed with the values the variables have at that statement. This is what compiled code already did. Two cases stay lazy: a collection with no last element (filter(1..oo, p), still a live view of the variables it reads), and one that reads no variable (map(f, 1..10^6)). The host function ce.assign() is not changed: it stores the value it is given, so a host that defines one name from another keeps the live view. A lazy collection over a dictionary also stays lazy (it is a dictionary, not a list). What is lost: a program that relied on ys following xs after let ys = filter(xs, p); a large finite lazy collection that reads a variable is computed at the assignment, not at its first read; and a callback with an effect, in such a collection, runs for every element at the assignment and not at each later read. A lazy collection over a literal source (let s = map(x => f(x), [1, 2, 3])) reads no variable and keeps the documented laziness: the callback runs only for the elements that are read.

• A declared scalar parameter is checked against the evaluated argument. For a function declared with ce.declare(name, { signature }) and then assigned its body, an argument was checked at boxing, against its static type. When that type is not known before evaluation, a scalar parameter accepted any value: f declared (integer) -> unknown with body x ↦ x + 1 answered 2.5 for the value 1.5, a declared (real) accepted 1 + 2i, a declared (string) received Missing and a declared (boolean) received 5. The evaluated arguments are now checked against the scalar parameter types of the declaration, and a value that does not fit answers Error(ErrorCode("incompatible-type", "integer", "1.5"), 1.5), as an Epsil function with the same annotation does. In a broadcast over a list the error is in the cell that does not fit. Not changed: a list, a range or a point at a scalar parameter is still broadcast (f([1, 2, 3]), f((1, 2))); a symbolic argument is left alone; a slot declared unknown or any checks nothing; NaN and an absent value at a numeric parameter answer NaN. What is lost: a call that relied on a declared scalar type being advisory.

• First, Second, Third, Last and At with a literal index are typed without an absent member when the element exists. First([(1, 2), (3, 4)]) was typed missing | tuple<integer, integer> and First([7, 8, 9]) was integer | nan. They are tuple<integer, integer> and integer: the absent member is stated only when the access can find nothing. The element is proved to exist from a list type with a length (list<T^3>, which is the type of a list literal), from a string literal, or from a Range with literal bounds. First(Filter(xs, p)), First(xs) for xs declared list<T>, xs[i] with a variable index and an index past the end (At([7, 8, 9], 99)) keep the member. What to check: a pinned printed type of such an access. type.matches("number") answers as before.

• A parenthesized derivative operand is delimited by its parentheses (#336). \frac{d}{dx}(x)+1 parsed as D(x + 1, x): the operand of a Leibniz fraction took the whole sum that followed it, so the +1 moved inside the derivative, and a product or quotient written after the parentheses (\frac{d}{dx}(x)\cdot 2) was absorbed the same way. An operand that starts with a parenthesis, plain or sized ((…), \left(…\right), \bigl(…\bigr)), now ends at that parenthesis: \frac{d}{dx}(x)+1 is D(x, x) + 1 and evaluates to 2, \frac{d}{dx}(x)\cdot 2 is 2·D(x, x). A superscript, prime or factorial attached to the parenthesis stays inside the operand (\frac{d}{dx}(x+1)^2 is still D((x+1)^2, x)). The rule covers \frac{\partial}{\partial x}(…) and the higher-order \frac{d^n}{dx^n}(…) forms. What changes meaning: a factor juxtaposed after the parenthesis is now outside the derivative, \frac{d}{dx}(x+1)(x-1) is D(x+1, x)·(x-1) where it was D((x+1)(x-1), x); write \frac{d}{dx}((x+1)(x-1)) for the derivative of the product. An operand without a parenthesis keeps its term extent: \frac{d}{dx}x^2+1 is still D(x^2 + 1, x), the \int … dx integrand convention, and so does an operand in another enclosure (\frac{d}{dx}|x|+1 is D(|x| + 1, x)).

• A Range, a Linspace, a comprehension and a Tabulate are typed list<T>; they were indexed_collection<T>. A symbol declared list or list<number> refused every one of them: ce.declare("L", "list") followed by ce.assign("L", ce.box(["Range", 0, 5])) threw a type error, and the only declaration that accepted a document's ranges was indexed_collection, which also admits a tuple and a string. Range(0, 5) and Range(0, n) with n an integer are now list<integer>, Range(0, 1, 0.1) is list<real>, Linspace(0, 1, 5) is list<real> (it was a bare indexed_collection with no element type), a comprehension is a list of its body type and a tabulation is a list of its generator's result. An index span (Range(1, 5), the range type) is now a subtype of list<integer>, so it is accepted by the same declarations; it was a sibling of list. The list type makes no claim about the length: Range(1, +oo) is a lazy list<integer>, exactly as the lazy Map over it, Repeat(x), Cycle(xs) and Iterate(f, x) already were; whether a collection is finite is a property of the value (isFiniteCollection), never of the type. What changes for a host: a check such as type.matches("indexed_collection") still holds for every one of these values (list <: indexed_collection), while a check that a value is NOT a list, or a pinned printed type such as indexed_collection<integer>, reads differently. (Tycho row 337.)

New Features

• A type variable in a collection's length slot (#364, proposed by enumeratio). A where variable may now name a length: (a: vector<real^N>, b: vector<real^N>) -> real where N rejects a call with mismatched lengths at the call site (dot([1,2,3], [4,5]) reports expected vector<real^3>, got vector<integer^2> on the second operand), and (matrix<T^(MxN)>, matrix<T^(NxP)>) -> matrix<T^(MxP)> where T, M, N, P states the shape of its result. The same variable at a parameter or result position is the integer with that value, bound from a literal argument: (n: N, x: vector<T^N>) -> T where T, N accepts foo(3, [1,2,3]) and rejects foo(4, [1,2,3]), and (list<T^N>) -> N types len([1,2,3]) as 3. A length variable is solved by equality across positions, may carry an integer bound (where N: integer<2..>), and an operand whose type states no length is admitted as it is at a literal length. A nominal type may take a length parameter: type permutation<N> = list<integer^N> makes permutation([2,1,3]) a permutation<3>, unrelated to permutation<4>; an unsolved length prints as the family permutation<integer<1..>>. In the Epsil language the same spellings work on a function head, with a trailing where clause or the <N> binder. Not supported yet: arithmetic between lengths (^(M+N)). The parameter kind is a general value kind, so boolean and string parameters can follow once those literals carry singleton types. One reading changed with it: a dimensioned pattern now binds its element variable after peeling as many axes as it states, so (x: matrix<T>) -> T where T on a 2×2 matrix binds T to the scalar element (integer) rather than to a row; no built-in signature was affected.

• More spellings of the arc-minute and arc-second markers in DMS angles (#338, contributed by yelliver). After a degree marker (°, \degree, ^{\circ}, ^\circ), the minutes now accept ', \prime, ^{\prime} and ^\prime, and the seconds accept ", \prime\prime, \doubleprime, ^{\doubleprime}, ^{\prime\prime} and ^\doubleprime (and, added alongside, the ASCII ''). The superscript spellings were parsed as products of primes (9^{\circ}30^{\prime} was Degrees(9) · Prime(30)); every spelling of 9°30'15" now parses to the exact 2281/240 degrees. The siunitx angle units \arcminute and \arcsecond are accepted as markers too (9^\circ 30\arcminute 15\arcsecond); they gave an unexpected-command error. The siunitx commands \minute and \second are deliberately not accepted: in that package they are the time units, not the angle units.

• ce.conformsTo(type, protocol): a public conformance query (#362, contributed by enumeratio). Answers whether type conforms to protocol, inheritance and conditional conformance included, without calling one of the protocol's members and reading protocol-implementation-missing as "no". type may be a plain type string, parsed the way ce.type() parses one; an unknown protocol name answers false rather than throwing.

Issues Resolved

• .N() of an exact constant raised to a large integer power is correct to the working precision. The error of the approximated base was multiplied by the exponent: (π^1000000).N() was 7.45923232449144786258e+497149, wrong from the 18th digit; it is now 7.45923232449144786349e+497149. The base is computed with about log₁₀|k| more digits for an exponent k.

• HurwitzZeta and the two-operand Zeta are accurate at a complex order with a large imaginary part, stay unevaluated where the double-precision result would be wrong, and no longer hang on an extreme argument. HurwitzZeta(-0.93+28.85i, 0.1334).N() was 1.5e8 + 5.4e9i and is now -10.482 - 5.546i; HurwitzZeta(-0.22+12.28i, -6.93-4.78i).N() was off by 1e−9 and now stays HurwitzZeta(…); HurwitzZeta(0.7+1e7i, 0.3).N() took two seconds and HurwitzZeta(1e9, 0.5).N() more than a minute, and both now return at once.

• LerchPhi past the unit circle evaluates in many cases that stayed symbolic: a real a with Re(s) ≥ 1/2 at a large |z| (LerchPhi(-187652.08, 2.5913, 1).N() is 0.00097720268138322), a very negative order with a negative a, and a complex a (on a random sweep with Re(s) from −30 to 1 and |z| up to 1e7, 980 of 1197 points now evaluate, up from 594). Every value is within 1e−11 of the reference, and the slow cases are up to ten times faster.

• The GLSL and WGSL HurwitzZeta, LerchPhi and PolyLog are more accurate. HurwitzZeta(s, a) with s < 1/2 used a sum whose terms cancelled: HurwitzZeta(1/2, 0.3) was 2.3e−5 off and HurwitzZeta(-23, 1/4) returned 0.016 for −0.000215. Every value is now within 1e−5, or NaN next to a zero of the function where single precision cannot resolve it (some values that were returned before, inaccurately, are NaN there now). LerchPhi(z, s, a) and PolyLog(s, z) with z < 0 and s ≤ −3 return a value in most cases that were NaN, for example LerchPhi(-1, -6.5, 1.3) is 0.10992.

• A Pipe stage that uses no topic slot and whose value is not a function is an error. The piped value would be ignored. [1, 2] |> 10 + 3 was already an incompatible-type error, but [1, 2] |> 10 + [3, 4][1] gave 13 and [1, 2] |> 10 + ([3, 4] |> _ * 2) gave [16, 18]: the stage was evaluated, was not a function, and was applied as a constant. Both are now the same incompatible-type error, on the MathJSON, LaTeX and Epsil routes. So is 5 |> Max(3), a complete call whose value 3 is not a function (it gave 3). A stage that is a function is unchanged: a function symbol (xs |> Sum), a call missing its first argument (xs |> Take(1)), a lambda, a stage with a topic slot (xs |> _ * 2), and a body whose unknowns become its parameters (5 |> y + 1 is 6). A _ inside a nested Pipe stage belongs to that stage, not to the enclosing one. Apply keeps treating a value that is not a function as a constant (Apply(3, 5) is 3).

• Binomial(n, k).N() for a huge n and a non-integer k. From about n = 10²⁰, Γ(n + 1) overflows, and the quotient Γ(n+1)/(Γ(k+1)·Γ(n−k+1)) gave NaN: Binomial(1e300, 0.5).N(), Binomial(1e20, 0.5).N(), Binomial(1e15, 0.25).N(). The value is now computed in log form with the Stirling series of ln Γ(a) − ln Γ(a − d): Binomial(1e300, 0.5).N() is 1.1283791670955125739e+150. With ce.precision = 'machine', Binomial uses the same kernel as the compiled JavaScript target.

• The interval enclosures of Gamma and GammaLn contain the true value. They widened the scalar kernel's value by one ulp, but the kernel is off by up to about 300 ulps (Γ(148.6)), and more close to a pole on the negative axis, so an enclosure could miss the value: Gamma(3.5) gave [3.3233509704478443, 3.323350970447845], above the true 3.32335097044784255…. Each value is now widened by a measured bound on the kernel's error. Two other GammaLn defects are fixed: an interval that contains the minimum of Γ near 1.4616 was reversed ([0.5, 2] gave lo = 0.572, hi = 0, while the true minimum is −0.1215), and every interval on the negative axis gave NaN. A Gamma interval that contains several poles now reports the first one.

• Compiled Binomial and Choose give the interpreter's values outside 0 ≤ k ≤ n (#384, reported by enumeratio). The JavaScript target's runtime was a Pascal-triangle table lookup: Binomial(3, -1) and Binomial(2, 5) gave undefined instead of 0, and a negative or non-integer n (Binomial(-3, 2), Binomial(5.5, 2)) threw. The runtime now follows evaluate(): 0 for k < 0 or k > n ≥ 0, the extension to a negative n (Binomial(-3, 2) = 6, Binomial(-3, -5) = 6), the Γ ratio for a non-integer operand, 0 or Infinity at the poles of the Γ factors, the limits at an infinite operand, and NaN for a NaN operand. Integer results below 2^53 are exact. The interval target used the same table and threw for an interval that contains a negative n; it now uses the same kernel. For a very wide n interval, its fallback bound built a table with n²/2 entries; it now computes the one bound it needs, and gives the whole real line when the interval contains a negative n. An enclosure that contains a value above the largest double now extends to Infinity; before, the overflowed grid values were skipped and the upper bound was too low.

• HurwitzZeta(-n, a) at a non-positive integer order is the Bernoulli polynomial −Bₙ₊₁(a)/(n+1) for a symbolic or a complex a, not only for a rational a. HurwitzZeta(-1, a) stayed unevaluated and now evaluates to -1/2 * a^2 + 1/2 * a - 1/12, for orders up to 12. The two-operand Zeta(-1, a) gets the same polynomial only when a is known to be positive: for a negative a its terms with k + a < 0 are |k + a|^n, so Zeta(-1, -5/2) is 109/24 where the polynomial gives -107/24. At a complex a, HurwitzZeta(-60, 2+3i).N() had a real part of -9.1894…e+29, wrong from the third digit, and now gives -9.19246731053263160872e+29. A float a with many digits is also fast: HurwitzZeta(-100, 1e-300).N() took about a minute and now takes a few milliseconds, and HurwitzZeta(-50, 1e-200000).N() did not finish and now takes about 10 ms. A tiny a is no longer taken for a pole: HurwitzZeta(2.5, 1e-2000).N() was ~oo and is now 1e+5000. (#374, contributed by enumeratio)

• Floor, Ceil, Round, Truncate and Fract of an exact number are exact (#382, reported by enumeratio). They rounded a numeric approximation of the operand (21 digits by default, a double with precision: 'machine'), so a large exact operand lost digits: Floor((25! − 1)/24!) was 25 and is now 24, Floor(25! − 1) was 25!, Floor(√2·10^40) was correct only to 21 digits, and Fract((25! − 1)/3) was −1/3 and is now 2/3. A rational or a rational multiple of a square root is now rounded with integer arithmetic; Round still rounds a tie away from zero.

• A Function literal with no parameter list no longer takes the _1, _2 … slots of a nested Function literal as its own parameters (#381, reported by enumeratio). ["Function", ["Add", "_1", [["Function", ["Multiply", "_1", "_2"]], 2, 3]]] had the parameters _1 and _2, so applying it to 5 gave a function and not 11. The search for slots now stops at a nested Function with no parameter list, whose slots belong to it. A nested literal with a parameter list owns only the names it declares, and its other slots still refer to the enclosing literal. The same correction applies to the shorthand callbacks of Map, Filter and Apply, to a Pipe stage (Pipe([1, 2], Map(Function(_ + 1))) gave a function and is now [2, 3]), to the JSON serializer and to the _ slot of the Epsil |> operator.

• serializeEpsil takes linear time in the nesting depth (#379, reported by enumeratio). The formatter computed the cost and the layout of each nested block again every time its parent asked, so the time grew by a constant factor at each nesting level: a 10-deep Add chain took 42 s. Each block now keeps these values per starting column. A 200-deep chain serializes in less than a millisecond, and the output is unchanged.

• PolyLog, the hypergeometric functions, AppellF1, JacobiTheta, DedekindEta and EisensteinE thread over a list operand, as Zeta, LerchPhi and the elliptic integrals already did. PolyLog(2, [0.1, 0.2]) was an incompatible-type error and is now [0.1026177910993911, 0.2110037754397048]; a scalar call is unchanged (#374, contributed by enumeratio).

• LerchPhi and PolyLog answer to the engine precision for real arguments with |z| < 1. They answered in machine precision whatever ce.precision was, where HurwitzZeta gave the requested digits. With ce.precision = 50, PolyLog(2, 1/3).N() was 0.3662132299770634 and is now 0.36621322997706348761674629766426276380206341558968. A value outside the machine range now also gets a numeric value: LerchPhi(0.5, -159.5, 1).N() stayed unevaluated and is now 2.62784929555984310206e+309. Outside the disk |z| < 1, or for a complex argument, the answer keeps machine precision (#374, contributed by enumeratio).

• A collection operand that can be absent as a whole compiles on the JavaScript target (issue #383). With x: list<list<integer>>, the row read At(x, i) is typed list<integer> | missing (the index can be past the end), and Length(At(x, i)), Count(At(x, i)) and every other collection operator over it refused to compile ("operand is not an indexed collection"), where the same read of a tuple<list<integer>, list<integer>> compiled. The same held for a restricted list xs{c} and for a symbol declared list<integer> | missing. The compiled code now binds the operand once and tests it for the absent value: Length(At(x, i)) runs to the row's length, and to NaN when the row is absent, as the interpreter answers NaN for Length(Missing); Reverse, Take, Sort, Join, Map, Filter, Reduce, Any, Dot, Cross, ListFrom, Last and the other collection operators follow the same rule, with the absent object (undefined) as the marker of a non-numeric result. Sum and Product of such an operand answered undefined and now answer NaN. The rule applies on the JavaScript target only; the Python and shader targets keep their decline.

• Sum of a list of points, or of a list of lists, compiled to a complex NaN. Sum(pts) with pts: list<tuple<real, real>> folded each point as a number and answered { re: NaN, im: NaN }, where the interpreter answers the sum point (4, 6). The compiled sum and product now combine the elements element-wise, as the interpreter does; Product of a list of points declines, as the interpreter reports an error (there is no product of two points).

• Lazy collection operators over an operand that can be absent as a whole stayed unevaluated in the interpreter. Filter(At(x, 7), p), Reduce(At(x, 7), Add, 0), Any(At(x, 7), p) and Scan(At(x, 7), f), with x a list of lists that has no seventh row, stayed unevaluated, where Length(At(x, 7)) is NaN; Map(f, At(x, 1)), Sum(Map(f, At(x, 1))) and Product(xs{c}) stayed unevaluated for a present row as well. They now answer the marker of their codomain (Missing, NaN) for an absent operand and compute over the present collection. Reduce(At(t, 1), f, 0) and Length(Filter(At(t, 1), p)) with t a tuple of lists also evaluate now. Under evaluate({ materialization: true }), a view over such an operand was materialized as a Set built from the walk of the unevaluated view (Append(At(x, 1), 2) gave Set(3, 1, 2), the duplicate lost; Reverse of an absent restricted list gave Set(NaN)): the view is now evaluated first, so the result is the list, or Missing.

• LerchPhi with a complex a outside the unit circle returned a wrong value in part of the a plane. The continuation past the circle took the principal branch of an incomplete gamma function, which is the wrong sheet when the argument crosses its cut; for a real z > 1 every a with a positive imaginary part was affected. LerchPhi(3, 2.5, 1.38+0.86i).N() was −0.23906 − 0.28351i and is now −0.71263 − 0.72673i. On 7000 random points, 575 values were wrong; none is now. (mpmath's lerchphi has the same defect, so do not check these values against it.)

• LerchPhi and PolyLog no longer return wrong digits inside the unit disk, next to the circle, next to z = 1, or for a very negative order past the circle. Inside the disk the direct series lost up to 10 digits for a large negative order (LerchPhi(0.9693+0.1294i, -5.84-1.357i, 6.235+0.793i), 2.5e−10 relative); next to the circle PolyLog(-0.9, -0.9999).N() was −0.27647181622385925 and is now −0.27647181621858134; within 1e−2 of z = 1 the logarithm of z had only 8 correct digits (1178 of 3000 values off by up to 1.4e−7); and past the circle a very negative order returned wrong values where the previous version stayed unevaluated (PolyLog(-19.5168, -941.72), wrong in the fifth digit). Every value that LerchPhi returns is now within 1e−11 of the reference on about 20 000 points, and every PolyLog value within 1e−12 on 21 000 points.

• LerchPhi, PolyLog and the incomplete Gamma(s, x) now return a value next to a pole of the gamma function, and LerchPhi with a negative order returns a value for most arguments outside the unit circle. LerchPhi(2, 3.000001, 3.5).N(), PolyLog(3.000001, 2).N() and Gamma(-0.999999, -3).N() stayed unevaluated and now evaluate (0.0091 − 0.0667i, 2.7621 − 0.7547i, 3.2386 + 3.1416i); LerchPhi(-900, -2.5, 0.5).N() is −4.6136259e−5 (unevaluated before). For |z| from 200 to 10⁴ with a real a, 3 of 400 random points answered before and all do now. PolyLog inside the unit disk answers every point the kernel computes accurately (it declined about half of them), and past the disk PolyLog(2.5, -1000000).N() is −220.36048147359768 (unevaluated before).

• Numeric Zeta, HurwitzZeta, PolyLog and LerchPhi are accurate next to a negative order close to 0 or to a negative even integer, left of the critical strip, and for a complex a with a negative real order. The compiled Zeta(s) at s = -1e-9 was −0.4999999577 and is now −0.4999999990810612; at s = -1e-17 it was −Infinity. HurwitzZeta(-11.801, 0.4265-1.271i).N() was 2.2188 + 50.186i and is now 5.6234 + 46.077i; HurwitzZeta(-12, 1.5).N() was −0.00024414062499909618 and is now exactly −2^-12; HurwitzZeta(-200.5, 0.3).N() was off by 1176 units in the last place and is now within 5. On 10 000 random points with a complex a and a negative real order, about a third were off by more than 1e−12 (some by 20 orders of magnitude); none is now, and a call costs the same or less.

• WGSL shaders that use NaN or infinity now compile in browsers. Every WGSL shader that contained a NaN or infinity constant, including those for Gamma, Zeta, HurwitzZeta, LerchPhi and PolyLog, was rejected by the browser's shader compiler (value nan cannot be represented as 'f32'). Both shader targets now spell them as _gpu_nan() and _gpu_inf(). GLSL shaders that use LerchPhi or PolyLog did not compile either (a GLSL keyword was used as a variable name); they do now.

• The GLSL and WGSL targets compute LerchPhi and PolyLog past the unit circle, and more accurately for a negative order. For real operands (z below −1, z = −1 with s < 0, z close to 1) they return the real value where they returned NaN: LerchPhi(-3, 1.5, 0.7) NaN → 1.10658, PolyLog(2, -1000000) NaN → −97.0791, PolyLog(2, -1e30) NaN → −2387.50. Every value returned is within 1.5e−5 of the reference, measured on the GPU; the previous helpers returned values off by up to 5.6e−4 for a negative order (LerchPhi(-0.645, -10, 8.99): 1155547136 → 1154901726).

• ListFrom, SetFrom and TupleFrom of a very large collection were an internal-error. ListFrom(Range(1, 300000)) answered Error(ErrorCode("internal-error", "Maximum call stack size exceeded", …)): the elements were passed to one function call as separate arguments, and a few hundred thousand of them exceed the argument limit. The elements are now added one at a time.

• A list variable reassigned in a loop could be typed integer | list<…>. In for gap in [(A, C), (A, C)] { circles = g(gap[1], 2, circles) }, a first reading of the call's type, taken before the types of the loop were settled, was never. never matches every type, so the type recorded for circles gained an integer member that no assignment gives it, and the JavaScript target declined Length(circles) and Count(circles, …) ("operand is not an indexed collection") while the interpreter computed the value. A value typed never now records nothing.

• The JavaScript target answered a wrong value for a function declared in the compiled program and called with a list of points. With function f(p: tuple<number, number>) { p[1] + 1 } inside the program and xs a list of two points, the interpreter applies f to each point and answers [2, 4]. The compiled call passed the list whole and answered the string "1,21". The compiled call now maps over the points, as it already did for a function defined on the engine (ce.assign). When the parameter has no annotation and its type is only inferred as a point, the call fails to compile with a message that names the argument, and the interpreter answers.

• A recursive Epsil program that builds a list through a local copied from an untyped parameter compiles. function fill(p, q, r, depth, acc) { let out = acc; for c in … { out = [...out, c]; out = fill(…, out) } … } declined on the JavaScript target with "Could not compile ListJoin: operand 1 is not an indexed collection", and the smaller function f(acc) { let out = acc; out = [...out, 1]; out } compiled but broadcast f([0]) over the list. Two causes: the local copied from the parameter kept the parameter untyped, so a use of the local as a collection taught the parameter nothing; and a self-call inside a function body was typed as a broadcast (broadcastable<unknown>) from the bare function type the recursion knot declares, which widened the local past the list its spread gives. Now the first use that narrows such a local narrows the parameter it was copied from, and a self-call types unknown while the clause canonicalizes and installs. The natural recursive spelling of the Apollonian gasket compiles and computes its 224 circles in a few milliseconds; the interpreter takes about 16 seconds. A self-recursive function whose base clause reads its parameter whole now derives that clause's result (-> number) instead of -> broadcastable<unknown>. (Tycho row 339.)

• A self-recursive function that builds a list is typed list<T>. With F declared (unknown, unknown) -> unknown and assigned (n, K) ↦ { n = K - 1: [n], otherwise: join([n], F(n + 1, K)) }, the function derived -> collection<number>, since the recursive call read the declared result unknown while the body was typed, and an unknown operand of Join may hold a set. The derivation now re-types the body under the hypothesis that the result is a list and keeps it only when that pass reproduces it, so F derives -> list<number> and F(0, 5) compiles; a base clause that builds a set still derives -> set. (Tycho row 338.)

• A predicate over a finite Range compiles to a counting loop on the JavaScript target; the range is no longer built. Count(Filter(1..n, k ↦ k mod 3 = 0)) compiled to an Array.from of the whole range, a .filter into a second array, and a .length: about 27 ms at n = 10⁶, where a loop that counts runs in about 2 ms. The compiled code now walks the range with a counted for loop and calls the predicate on each element, allocating nothing. The same walk serves Length(Filter(range, p)), Count(range, p), CountIf(range, p), Any(range, p) and All(range, p), which stop at the first decisive element, the collection form of Sum/Product over a filtered range, and a bare Filter(range, p), which pushes the selected elements into one list. The values are unchanged: the element count is _SYS.rangeCount, the interpreter's own, and element i is start + i × step, so an empty, reversed, zero-step or float-step range answers what the array lowering answered. The loop is not used when the range or the Filter is re-mapped by the caller, shared by common-subexpression elimination, or has a non-finite or non-number bound; those keep the array lowering. A Range consumed any other way (Map, Sum(range), indexing) still materializes. (Issue #373.)

• A list of numbers plus a point is typed error. With L := [0, 1, 2, 3], L + (1, 1) was typed tuple<integer, integer> | vector<integer^4> (earlier indexed_collection<integer>) while it evaluates to a list of four incompatible-type errors, as 2 + (1, 1) is one such error. The Add type handler sent the pair to its tuple branch and widened the two operand types together. A point, or a list of points, added to or subtracted from a collection of numbers at any depth is now typed error on the expression and the descriptor routes, as [1, 2] + "a" already was; a matrix literal plus a point and a list of points plus a list of numbers are covered by the same check. A genuine point broadcast ([(1, 2), (3, 4)] + (1, 1), [0, 1, 2, 3] · (1, 1)) is unchanged, and a symbol later assigned a list of points types as one again. Values are unchanged.

• .N() of a convergent series whose tail is not an integer power of 1/N evaluates. \sum_{n=1}^\infty 1/n^{1.5} (= ζ(1.5) = 2.6123753487), \sum 1/n^{2.5} and \sum \ln(n)/n^2 stayed unevaluated under .N(): the Richardson tableau that accelerates the partial sums extrapolated in integer powers of 1/N only, so a tail in N^{1-p} for a non-integer p, or with a logarithmic factor, never certified. When that tableau does not certify, the tail exponent is now fitted from the partial sums (a repeated exponent absorbs a ln N factor), and the values are accurate to about 1e-13. Acceptance stays gated: the fitted route needs an error estimate at or under max(1e-12, 1e-11·|v|), agreement between two independent sample sequences and a positive fitted exponent, so \sum 1/n, \sum 1/\sqrt{n}, \sum (-1)^n and the like still stay unevaluated. The infinite product takes the same route (\prod (1 + k^{-1.5}) evaluates). Series that already evaluated give the same values as before.

• A destructuring let inside a loop body compiles. let n = 0; while n < 4 { let (a, x) = (2, 3); n = n + a }; n declined on the JavaScript target with "Could not compile a destructuring declaration in value position" wherever the let was in the body, although the interpreter answered 4: the loop-body lowering desugared a destructuring assignment but not a destructuring declaration. A loop body's statements now go through the same desugaring as a block's, in statement position.

• Complex keeps the number contract of its components on the literal route. Complex("str", 2) boxed to the float NaN, where Add("str", 2) is an incompatible-type error: a component that is not a number literal is now built as the sum re + im·i through the Add operator, which checks the contract. A symbol component (Complex(x, 2) is x + 2i) and two number literals (one complex literal) are unchanged.

• A set literal that spreads an absent operand is absent. {...Missing, 0} was Set(Missing, 0) where [...Missing, 0] is Missing (an operator over an absent collection is absent, decision of 2026-09-26): SetFrom, ListFrom and TupleFrom now propagate an absent collection (SetFrom(Missing) was an incompatible-type error), and the set literal reads an absent spread operand as a segment, as the list literal does.

• Length(Range(0, n)) with n typed integer is typed integer. It was integer | signed_infinity: the Length type handler admitted the infinite length for any bound that is not a number literal. A bound or step typed finite (integer, real) cannot make the range unbounded. Static type only; no value changes.

• A connective over an operand that failed is that error. Or(Sin(P) < [1, 2], Sin(P) > 0) with P = [5, 6, 7] was a list of three incompatible-dimensions errors, the error of the first operand copied into every cell of the second. And, Or, Nand, Nor and Implies now answer the error itself, unless another operand decides the result by itself (And(False, e) is still False). Xor and Not already did.

• PolyGamma of a very high order stays symbolic instead of running for minutes, and the big-decimal kernel is correct at high orders. PolyGamma(100000, 2.5).N() did not finish at the default precision (the big-decimal kernel had no order limit and built the exact factorial), and answered NaN at machine precision (the double factorial overflows above 170). The real kernels now share the order limit of the complex kernel (10 000) and stay symbolic above it; the machine kernel reaches the complex kernel, which carries the factorial in scaled form, when its own intermediates overflow (PolyGamma(200, 300.5) at machine precision is now a value, not NaN). Two wrong values found on the way are fixed: the big-decimal kernel shifted the argument only to the working precision, so its asymptotic series diverged when the order was larger (PolyGamma(1000, 150) was wrong by a factor 10⁶), and its tail stopped on an absolute tolerance, so a small result kept 4 digits (PolyGamma(200, 300) is −2.03430·10⁻¹²³, was −2.03471·10⁻¹²³). Order 10 000 at x = 2.5 now takes 0.2 s (was 0.7 s to 2.5 s).

• A NaN literal pattern matches a NaN subject. match NaN { NaN => 1, _ => 2 } answered 2: a number pattern compared with isEqual, which follows IEEE. A pattern or subject that is NaN or Indeterminate now compares structurally (NaN matches NaN, Indeterminate matches Indeterminate, neither matches the other). The MathJSON spelling of the pattern (MatchCase(NaN, …), also PositiveInfinity, NegativeInfinity and ComplexInfinity) never matched either, because the raw pattern keeps the name as a symbol while the subject is the number; those five names are now compared as their number.

• ce.number([n, 0]) is ~oo for n ≠ 0, as ce.box(["Rational", n, 0]) is; it was NaN. ce.number([0, 0]) is Indeterminate.

• simplify() propagates a NaN or Indeterminate operand. ce.box(["Sin", "NaN"]).simplify() stayed sin(NaN) where evaluate() answers NaN. The simplifier now applies the operand's propagate NaN policy as evaluation does, with the same exclusions (lazy operators, user functions, a broadcast over a collection, a reject position), reading number literals only. Binomial(NaN, 0).simplify() is NaN.

• An exact 0 times a float is the exact 0 in both operand orders of .mul(). ce.number(2.5).mul(ce.Zero) was the float 0.0 while ce.Zero.mul(ce.number(2.5)) was the exact 0, as the Multiply operator answers in both orders.

• Integrate of a NaN or Indeterminate integrand has no value on both routes. Integrate(0/0, x) was NaN under evaluate() but stayed unevaluated under .N(), and Integrate(NaN, x, 0, 1) stayed unevaluated on both. Integrate is lazy, so the NaN-policy step of evaluation does not run for it; its handler now answers Indeterminate for an Indeterminate integrand with exact bounds, and NaN otherwise and under .N(), for indefinite, definite and multiple integrals.

• A list bound whose length does not match is the same error under .N() as under evaluate(). Two list bounds of different lengths, or a list integrand and a list bound of different lengths, gave the incompatible-dimensions error under evaluate() but left the integral unevaluated under .N(); both routes now give the error, naming the lengths in the same order.

• At with an infinite index is NaN. At([1, 2], +oo) and At([1, 2], ~oo) stayed unevaluated where At([1, 2], 1.5) and At([1, 2], NaN) answer NaN (a read with no position). An infinite index names no position either.

• BoxedSymbol.mul(0) no longer reads a variable's assigned value at canonicalization. With w := NaN, ce.symbol("w").mul(0) was the NaN literal, frozen across a later w := 4. For a variable that holds a value the product now stays Multiply(0, w) and evaluate() reads the value the variable holds at that time (Indeterminate while w is NaN, 0 after w := 4). A constant with an infinite value still gives NaN; a symbol with no value still folds to 0.

• Remainder is typed with nan when its divisor may be zero, and the statistics that subtract the mean answer the indeterminate form for an infinite datum. Remainder(5, 0) evaluates to Indeterminate but was typed integer; it now has a type handler modelled on Mod's: a divisor that is provably zero gives nan, one that may be zero gives the operands' kind joined with nan (Remainder(n, r) with r: real is nan | real), and a nonzero literal divisor leaves the polytype result. Variance([+oo, 1]), StandardDeviation, PopulationVariance, PopulationStandardDeviation, Kurtosis and Skewness answered NaN from the machine kernel where the deviations ∞ − ∞ are the indeterminate form: they now answer Indeterminate for an infinite datum with no float datum, NaN with a float datum or under .N(), as Mean([+oo, -oo]) does.

• src/math-json/OPERATORS.json and CATEGORIES.json are regenerated from the current library (the tracked files were behind by the operators added since their last generation and by the examples fields).

• The materialization option no longer truncates the operands of an eager operator. Length(Range(1, 5000)).evaluate({ materialization: true }) was 11, and IndexOf(Range(1, 5000), 4000) and Contains(Range(1, 5000), 4000) under the same option were 0 and False: the option, which describes the result ("if the result is a lazy collection, materialize it"), was forwarded to the evaluation of each operand, and a lazy operand with no evaluate handler was materialized to the display preview (the first five and the last five elements with a placeholder between them) before the operator's handler read it. Operands now reach the handler in their lazy form on both the synchronous and the asynchronous route, and a lazy view a handler answers (Insert(Range(1, 200), 2, 99), Partition(Range(1, 300), 3)) is materialized after the handler instead, and the option still descends into a container literal ((Take(xs, 3), 1) is a pair holding the materialized list). A held conditional over a list ([5, 10] {0 < t}) is left as the held form.

• A Fold that builds a list compiles, and so do Length and At of its result (#369, reported by enumeratio). Fold((acc, i) => Join(acc, [2 p[i]]), [], 1..Length(p)) evaluates to [6, 2, 4] for p = [3, 1, 2] but declined on the JavaScript target with "Could not compile Join: operand 1 is not an indexed collection" (fixed in 0.141.0 by the run-time array check of an inferred collection parameter), and Length or At of the fold, or of a block local holding it, still declined: the engine types the fold collection<any>, from the bare accumulator's inferred type. A seeded fold whose seed is an array and whose combiner returns one is now read as an array, and At of a block local typed as an inferred collection reads it through the same run-time check Length uses. An ANNOTATED accumulator ((acc: list<integer | nan>, i) => …) declined as "the type of the value it receives is not provable" on both the JavaScript and the Python targets: the accumulator receives the seed, then the combiner's own result, so it is now judged against the join of the two types, and admitted when both provably satisfy the annotation. An annotation the body does not provably return is still declined, and the diagnostic now names the argument type (list<integer | nan>: p[i] may be out of range). A fold whose combiner is a function defined in the same program (function add(a, x) { a + x } then Fold(add, 0, xs)) declined as "the combiner has no compiled function form": the combiner is now resolved through the compiled block's own bindings, as a call of that function already was.

• A fold's accumulator is typed from its seed and its combiner's result (user decision 2026-09-29, from the investigation of #369). A bare accumulator was typed from its uses only, so Fold((acc, i) => Join(acc, [2 p[i]]), [], 1..n) was typed collection<any> (a Join also accepts a set), and a view over it followed the kind of a value the fold node did not hold: Map(x => x + 1, fold) evaluated to Set(7, 3, 5) and Filter(fold, x => x > 3) to Set(6, 4) where the fold's value is the list [6, 2, 4] and the compiled result was the array [7, 3, 5]. The accumulator is now typed to the fixpoint of the seed's type and the combiner's result type (list<integer | nan> here), as an INFERRED type: the printed literal is unchanged, nothing is enforced at apply time (a fold whose accumulator changes type mid-fold, 1 → 1/2 → 1/6, still evaluates), and the same views now evaluate to the lists [7, 3, 5] and [6, 4]. Scan and the seedless folds are typed the same way. A fold whose accumulator already infers a precise type (a numeric fold) is not re-canonicalized, so its cost is unchanged; a list-building fold costs about one extra canonicalization of its combiner. What changes for a caller: the type of such a fold, and a Map/Filter/Take over it answers a list instead of a set.

• IndexOf and IndexWhere no longer answer 0 for a collection they cannot search (#368, reported by enumeratio). IndexOf(xs, 0) for a symbol xs with no value, for an unknown function f(1, 0) or for Range(1, n) answered 0, a claim that the element is absent, while Contains(xs, 0) and IndexWhere stayed unevaluated: the handler read the "cannot search" answer of the element scan as "not found". Both operators now stay unevaluated unless a finite source was walked to its end. A set is refused as incompatible-type, as First and At refuse it, instead of answering 0 for every value. An unbounded source still answers a match (IndexOf(Repeat(5), 5) is 1) and a refutation (IndexOf(Repeat(6), 5) is 0); a search that finds nothing within the iteration limit stays unevaluated instead of walking forever.

• A DMS angle may omit the minutes. 9°30" (9 degrees 30 seconds) was an expected-closing-delimiter error, 9°30'' and 9°30\prime\prime parsed as the derivative of 9°30', and 9°30\doubleprime as Degrees(9) · Prime(30, 2): the parser tested the minute marker first, and ' and \prime are the first half of '' and \prime\prime. The second marker is now tested first, and every spelling parses to the exact 1081/120 degrees; x°30" is x deg + 30 arcsec.

• A protocol member's Self binds to the conformance target dispatch selects, not the first argument's static type (#362, contributed by enumeratio). With Compare: "(Self, Self) -> number" implemented on real, Compare(5, 1/2) and Compare(3, 2.5) were rejected as incompatible-type while Compare(1/2, 5) and Compare(2.5, 3) passed — acceptance depended on which argument happened to already have the exact declared type. Self now binds to the most specific conformance edge the receiver's type admits (real here) at every call, statically and at runtime, so both argument orders and any mix of integer, rational and real are accepted alike; a -> Self result type binds the same way. A nominal type that conforms to nothing is still refused in either position. When one applies, the binding reads only the edges dispatch can select — those carrying an implementation of the member — so a block-less type integer is PartialOrder that inherits real's implementation does not narrow Self to integer for the static check while the run time serves the call through real; when none is selectable yet, it reads every declared conformance, pending ones included.

• JavaScript compilation: a Range with a computed start or step no longer captures a user variable named i or _e (#367, reported by enumeratio). The runtime-length range spliced its compiled start and step into an Array.from callback whose index parameter was i and whose element parameter was _e, so a start that read a user variable of either name read the callback's parameter instead: a Map over i whose body counted the j > i with p_i > p_j over Range(i + 1, Length(p)) compiled to [0, 0] for p = [2, 1] where evaluate() gives [1, 0], while the same program over k compiled correctly. Every operand is now passed as an argument of a small function, outside the callback, the shape the impure-operand branch and the Python target already used.

• A protocol member call no longer re-parses the member's signature. A signature that names Self bypasses the type parser's shared cache, so every dispatched call parsed it twice, and the first attempt threw and built an error on the way. The parsed signature is now kept per protocol, member and receiver type until the declarations change. A member call on a scalar receiver went from about 12× the cost of an equivalent plain function to about 1.3× (#363, contributed by enumeratio).

0.141.0 2026-09-29

Behavior Changes

• A restriction with a list of conditions pairs an ordered collection of infinite or unknown length over the conditions' length (user decision 2026-09-29). When(Range(1, ∞), [True, False]) was [Range(1, ∞), Missing], the whole collection in each cell, and is now [1, NaN]; only the elements needed are read, with one walk. A collection that ends before the conditions do is truncated, as a finite one was. When the elements cannot be read, the restriction stays unevaluated where it repeated the whole collection: a range with a free bound, and a symbol declared list<number> with no value ([P, Missing, P] is now the held restriction). The static type of a restriction over a carrier declared indexed_collection<T> is list<missing | T> (list<number> for a numeric T), where it was list<indexed_collection<T> | missing | T>. A tuple held by a symbol declared as an ordered collection of numbers is paired element by element (P = (1, 2) gives [1, NaN]); a value held under a type that admits a point or a string (tuple<number, number> | list<…>) is still one value. A collection<T> or set<T> carrier keeps the wider type, because it can hold an infinite set.

• A list literal with a spread is always a list (user decision 2026-09-29). Its canonical form is the new operator ListJoin: [...a, 0] is ListJoin(a, [0]), where it was Join(a, [0]). With g(a) = [...a, 0], g({9, 8}) is [9, 8, 0], where it was Set(9, 8, 0). ListJoin is lazy, keeps the elements of each operand in their order, and keeps duplicates that come from different operands. .json, toString() and LaTeX show the new head; Epsil writes it back as [...a, 0]. A stored Join(a, [0]) keeps its meaning: the result takes the kind of its operands. [...Missing, 0] is Missing, where it was [Missing, 0].

• Join and Append over an operand that may hold a set have a type that admits it, and their value no longer depends on the materialization option (user decision 2026-09-29). With P declared collection<number>, Join(P, [5]) is typed collection<number>, where it was list<number>; with P holding Set(3, 1), evaluate({materialization: true}) gives Set(3, 1, 5), as evaluate() does, where it gave [3, 1, 5]. The kind of the result follows the values the SOURCES hold: every operand of Join, the first operand of Append (Append(acc, s) with s holding a set stays a list when acc holds a list). Compiled JavaScript that reads such a join (Length(Join(a, [0])) in a function body) still compiles.

• A user function that passes its parameter, alone, to an operator that reads a collection whole receives a list argument whole (user decision 2026-09-29). It applies to Mean, Median, Variance, StandardDeviation, Mode, Quartiles, Flatten, SetFrom and TupleFrom, when the bare parameter is the only operand. Before, the function was applied to each element of the list. function h(xs) { mean(xs) } applied to [1, 2, 3] gives 2, where it gave [1, 2, 3]; flatten over [[1], [2, 3]] gives [1, 2, 3]; mean(xs) + xs gives [3, 4, 5], where it gave [2, 4, 6]. A function that only calls such a function is bound whole too, and once one parameter is bound whole a list given to another parameter is bound whole as well (h(xs, k) = mean(xs)·k applied to [1, 2, 3], [1, 10] gives [2, 20]). A scalar argument gives what it gave (h(5) is 5), and the function accepts every argument it accepted. Unchanged, still applied element by element: a use with several operands (mean(x, 1) over [1, 2, 3] is [1, 3/2, 2]), a use through another operator (mean(xs^2)), norm(p) (one norm per point over a list of points), String(p), max(xs), and every parameter with a declared type. The compiled JavaScript agrees with the interpreter on each of these.

• An element-wise operator over an operand declared with an abstract collection type is typed broadcastable<R> (user decision 2026-09-29). With P declared collection<number> and no value, Sin(P), P + 1 and P > 1 were typed number, collection<number> | integer and boolean; they are broadcastable<number>, broadcastable<number> and broadcastable<boolean>, the scalar or an indexed collection of it, because P can hold a list, which the operator maps over, or a set, which it does not. A P that holds a value is typed from that value. What was accepted stays accepted: a symbol declared number can still be assigned Sin(P), and a condition such as Sin(P) > x is still read as a condition by Element, Solve and the rules. The full suite showed one pinned type and no value change. One evaluation does change: Sin(P) = [1, 2] with P valueless stays unevaluated, where it expanded to [sin(P) == 1, sin(P) == 2] on the assumption that Sin(P) is a number.

• A user function that passes its parameter to a collection operator with a callback types that parameter as a collection. function f(xs) { filter(xs, x => x > 1) } applied to [1, 2, 3] returned [Filter(1, …), Filter(2, …), Filter(3, …)] and returns [2, 3]; the same body with reduce returned a list of Reduce(…) and returns 6. These operators hold their operands, and a held operand was never checked against the operator's signature, so the parameter stayed unknown and a function with an unknown parameter is applied to each element of a list argument. An untyped symbol at the collection operand of Map, Filter, Reduce, Scan, Fold, Any, All, TakeWhile, DropWhile, FlatMap, MaxBy, MinBy, ArgMax, ArgMin, Dedup or Differences is now typed from that operand's declared type, as it already was for Find, Count or Sort. The function's type changes from (unknown) -> … to (collection<unknown>) -> …. A symbol that has a declared type or a value is not changed. The same holds for a pipe stage: [1, 2, 3] |> (_1 ↦ Map(k ↦ k², _1)) applied the stage to each element and ran the inner Map over an integer (typed list<collection<number>>); the stage now receives the whole list and the result is the Map of squares over it (typed collection<number>). The inference is limited to those sixteen operators: the base of a compound subscript a_{n+1} stays untyped.

• A collection operator over a symbol declared with an abstract collection type follows the value the symbol holds. With P declared collection<number> (also collection<any> or bare collection) and holding a list or a range, Map, Filter and Scan over P were sets: Map(x ↦ 2x, P) for P = [3, 1, 1] was Set(6, 2), so order and duplicates were lost. The result kind was read from the static type of the lazy node, and an abstract collection<T> is not indexed. It is now read from the value each collection operand holds, so the result is [6, 2, 2]. Such a result also takes part in broadcasts (Map(x ↦ 2x, P) + 1 is [7, 3, 3], it stayed symbolic) and is accepted where an indexed collection is required (Last(Map(x ↦ 2x, P)) is 2, it was a type error). A symbol that holds a set still gives a set, and the static type is still collection<T>.

• A derivative with several variables keeps every variable in LaTeX. D(s^2t^2, s, t) serialized as \frac{\mathrm{d}(s^2t^2)}{\mathrm{d}s}, dropping t, and D(x^3, x, x) serialized as a first derivative, so a round trip through LaTeX changed the value (Tycho item 333). A D with two or more variable operands now uses the partial-derivative spelling: \frac{\partial^{2}(s^2t^2)}{\partial s\,\partial t} and \frac{\partial^{3}f}{\partial x^{2}\,\partial y}. Variables keep their order, and only consecutive repeats are grouped (\partial x\,\partial y\,\partial x). A D with one variable keeps the \mathrm{d} spelling, and a nested D is folded into its order only when it has the same single variable (D(D(f, x, y), x) lost y). A variable operand that is not a symbol falls back to \operatorname{D}(…). The parser now accepts \, between \partial terms, and a differentiand after \partial^{n} in the numerator that is not a plain symbol (\partial^{2}x^3, \partial^{2}(xy), \partial^{2}f(x,y)); those inputs gave a missing error or a Divide.

• An element read by a literal index from a collection whose length is not known stays unevaluated instead of giving NaN. With n declared integer and no value, [0...n][1], At(Range(1, n), -1), First(Range(0, n)), Last(Range(0, n)), a gather such as At(Range(0, n), [1, 2]), and the same reads through Take, Drop, Reverse, Map, a broadcast such as (0...n)^2, or a PointList of such ranges gave NaN (At(Linspace(0, 1, n), 1) gave Missing), although the element exists for most values of n. They now stay symbolic under evaluate() and .N(), as the read by a symbolic index already did, and give the element once n has a value. Reads that are provably out of range are unchanged: index 0, an index past a known length ([0...5][7] is NaN), the end of an infinite collection, and a FINITE walked collection that runs out (Tycho item 334). One more read changes with the same rule: a read that gives up at the iteration limit over a source not known to be finite stays unevaluated, where it answered the absence marker. First(Filter(Range(1, ∞), x ↦ x < 0)) was NaN and Second(Dedup(Cycle([1, 1]))) was Missing; a walk that stops at the limit does not prove the element absent (with the predicate x > 10^6 the first element exists beyond the limit, and was answered NaN too). Both calls still return after one bounded walk.

• A closure created in a nested block keeps that block's locals, and a closure created in a loop captures the loop variable of its own iteration. function mk(n) { if n > 0 { let k = n * 10; () => k } else { 0 } } gave closures that answered the symbol k; a closure created in a for loop saw only the loop's last value (inside a function, the escaped i became the imaginary unit); a counter declared in a nested block did not count. A closure now captures the nested block's binding when it is created (the binding, not a copy of the value: a later write in the same run of the block is visible, and two closures made in one run share the local), and a loop or Sum index is captured per iteration, as a comprehension index already was: for i in [1, 2] { fs = [...fs, () => i] } gives closures that answer 1 and 2, where they answered 2 and 2. Code that relied on a loop closure reading the loop's final value now reads the value of its own iteration (found with Tycho item 330).

• The name Indeterminate is reserved. It is now the library constant for the Indeterminate value (see New Features): ce.box("Indeterminate"), \operatorname{Indeterminate} and the Epsil word Indeterminate are that value, where they were a free symbol typed unknown (and in LaTeX a product of letters). A host symbol of that name changes meaning, an Indeterminate := 2 assignment is refused as for any constant, and in Epsil the word is a literal like NaN (let Indeterminate = 2 is a reserved-word error; the verbatim form `Indeterminate` still names a binding). The type grammar reads Indeterminate as the value type of that literal, and NumberFrom("Indeterminate") reads it back.

• An exact indeterminate form evaluates to Indeterminate, not NaN. An exact question whose answer has no value now answers the exact value Indeterminate (see New Features); NaN stays the answer of a floating-point computation that failed. The printed answer changes from NaN to Indeterminate for:

  • the arithmetic forms, at canonicalization and at evaluation: 0/0, ∞/∞, 0·∞, 0·~oo, ∞ − ∞, 0^0, ∞^0, 1^±∞, (−1)^±∞, i^∞ (an exact base on the unit circle other than 1, to ±∞), ∞^(imaginary), 0^(imaginary), Root(0, ∞), Root(∞, ∞), Log(1, 1) and Log(∞, ∞). The ones that fold at canonicalization (["Divide", 0, 0], 0^0, ∞/∞, ∞^0, 1^∞) now have the canonical form "Indeterminate";
  • Mod(x, 0) and Remainder(x, 0) (Mod(5, 0), and Mod(x, 0) for a symbol x) and Fract(±∞);
  • the special functions at an infinite point where the limit does not exist: Gamma, GammaLn, Factorial, Factorial2, Digamma, Trigamma, PolyGamma and Zeta at −∞ and ~oo; the Bessel functions, AiryAi, AiryBi and their derivatives at ~oo, and AiryAiPrime, AiryBiPrime at −∞; Beta at −∞ or ~oo; the upper incomplete Gamma(s, z) at z = ~oo, s = ~oo or two infinite operands; GammaRegularized at a = −∞, at ~oo or at two infinite operands; the points of Binomial and Pochhammer with no limit; ErfInv at every infinity; SinIntegral, CosIntegral, SinhIntegral, CoshIntegral, ExpIntegralEi and LogIntegral at ~oo; Real, Imaginary and Argument of ~oo (so AbsArg(~oo) is (+∞, Indeterminate)); Beta(+∞, −∞), Beta(+∞, ~oo) and Beta(−∞, +∞) (two infinite operands with no limit), while Beta(+∞, +∞) is now 0 (it was NaN; B(a, b) ≤ 1/a for b ≥ 1);
  • three library answers that reach one of these forms: the sample Variance([5]) and StandardDeviation([5]) of one datum (0/0), the cells of MatrixPower that compute 0·∞ (MatrixPower([[∞, 0], [0, 1]], 2) is [[+∞, Indeterminate], [Indeterminate, 1]]), the cells of a broadcast ([0, 1]/0 is [Indeterminate, ~oo], [0, 1]·∞ is [Indeterminate, +∞], [∞, 1] + [−∞, 2] is [Indeterminate, 3]), and Mean([+∞, −∞]). The .add() method follows the Add operator: ce.PositiveInfinity.add( ce.NegativeInfinity) is Indeterminate.

  What does not change: a float operand anywhere in the form gives NaN (0.0/0.0, 0/0.0, 0.0·∞, 0·2.5·∞, Mod(5.0, 0), 1.0^∞); an absent operand gives NaN (Missing + 2, Missing·0, Length(Missing)); a NaN operand gives NaN; .N() and evaluate({numericApproximation: true}) of every form above give NaN, including the cells of a list or tuple result; the compiled routes answer the target's IEEE NaN; a pole is unchanged (5/0 is ~oo); a point where the value has a direction that depends on the other operand keeps NaN (Gamma(2, −∞), Beta(∞, −1/2)); an anonymous infinity such as ∞ + i gives NaN; ce.number([5, 0]) and ce.number([0, 0]) are still NaN. Code that tested the answer with x.isSame(ce.NaN) no longer matches an exact indeterminate form: use x.isNaN, which is true for both values, or x.isIndeterminate. IsMissing(0/0) is now False and Coalesce(0/0, 5) is Indeterminate: an indeterminate form is a value, not an absent entry. The operators whose handler runs without the evaluation gate (the lazy operators other than Add and Multiply, and the operators with an inert or handle policy other than the reducers) answer NaN for an Indeterminate operand. The expectations changed in the test suite: 67 tests in 13 files, 19 inline snapshots.

• 0/∞ through .div() is 0. ce.Zero.div(ce.PositiveInfinity), .div(ce.ComplexInfinity) and .div(Infinity) answered NaN, where the Divide operator answers 0; they now answer 0. A float numerator over an infinity is the float 0: Divide(2.5, ∞) and Divide(0.0, ∞) were the exact 0 and are 0.0. A float base 1.0 to ±∞ answered 0 and is now NaN (the float twin of the form 1^∞).

• An integer-valued float is written with a fraction part, so it reads back as a float. A float whose value is an integer serializes as {num: "2.0"} in MathJSON and 2.0 in LaTeX (a large one as 1.0e+800 and 1.0\cdot10^{800}); it was 2, which reads back as the exact 2, so Sin(2.0) numericized while its round trip Sin(2) stayed symbolic (user decision 2026-09-28). An exact integer is still written 2. A float in LaTeX with a negative exponent now keeps its fraction part too (123.0\cdot10^{-3}), which read back as an exact rational before. With fractionalDigits: 0 (LaTeX) or digits: {fractional: 0} (MathJSON) the integer spelling is kept, as requested.

• An Epsil decimal literal keeps its value, and a literal with a fraction part is a float. The Epsil parser summed the fraction digits one float at a time, so 0.75 read as 0.7500000000000001, 0.3 as 0.30000000000000004 and 0.000001 as 0.0000010000000000000002. It now reads the decimal exactly. It also dropped the decimal point while normalizing, so 1.0, 2.0 and 1.5e3 were the exact integers 1, 2 and 1500; they are now floats, as on the LaTeX route (the 0.139.0 rule that a literal with a fraction part is a float). 1e3, 3 and 2. (a point with no digit after it) stay exact. A literal a double cannot hold keeps its digits: 1e-400 was the exact 0, and 5e-324 lost digits.

• The negation of a product folds its sign into the numeric factor. -(2x) (MathJSON ["Negate", ["Multiply", 2, "x"]]) canonicalizes to Multiply(-2, x); it was Negate(Multiply(2, x)). -\frac{x}{2} is now Multiply(-1/2, x), the same expression as -\frac{1}{2}x, so the sign of a fraction written in front of it reads back unchanged (#345).

• A float Measurement error evaluates. Measurement(5, 0.2) + 3 was 8 ± sqrt(0.2^2) under evaluate() and is now 8.00 ± 0.20. An exact error stays exact (Measurement(5, 1) * Measurement(2, 1) is 10 ± √29).

• Rounding a float gives an exact number. Round, Floor, Ceil and Truncate of a float now give an exact integer under evaluate(): Round(2.5) was the float 3.0 and is the integer 3; Floor(2.7) is 2, Ceil(2.2) is 3, Truncate(-2.7) is -2. Round(x, n) of a float gives an exact rational: Round(3.14159, 2) was 3.14 and is 157/50, and Round(1234.5, -2) is the integer 1200. A list of floats rounds to a list of exact integers too. This follows Mathematica (Round[3.14159, 10^-2] is 157/50) and reverses, for the rounding family only, the 0.140.0 rule that a float operand makes a numeric result a float. Under .N() the result is still a float, and ±∞ and NaN are unchanged. (#351)

Issues Resolved

• The MCP compile tool resolves the library spellings of an Epsil program. sqrt(2) (or pi, sin(x)) declined with "Unknown operator sqrt": the tool boxed the parsed program without turning the Epsil spellings into the library names, as evaluate and check do. It now compiles sqrt(x) + pi to Math.sqrt(x) + Math.PI.

• A function a program defines is not one of its free symbols. For a Block holding a DefineFunction statement (the Epsil f(x) = x + 1), unknowns and a compilation's freeSymbols listed f beside the genuine inputs, and defines did not list it; the generated code declares f as a local. Block(DefineFunction(f, …), f(z)) now reports z as its only unknown and free symbol, and f as defined.

• A cancellation's cause is no longer rendered as the site of the error. A program stopped by its deadline was reported as "Runtime error: Timeout exceeded at timeout": the second operand of the error value, ["Error", "Timeout exceeded", "timeout"], is the machine-readable cause, which the description read as a site. It reads "Runtime error: Timeout exceeded" now, for every cancellation cause.

• A broadcast call of a user function reports its element count without evaluating. With S := x ↦ (x, x²) and L a counted list, S(L) and d·S(L) typed list<tuple<number, number>> yet answered count: undefined, while d·L answered the length; a caller deciding whether to evaluate had to walk the call to learn its size (Tycho item 326). The count is now the mapped argument's, read through the same participant rule the type derivation uses, for a function literal with scalar parameters called with a collection. A numeric-tuple argument, participants of different lengths, a lone scalar-or-collection union, a declared broadcastable<T> or point-typed slot, a generic signature, and an argument count the signature does not admit still answer undefined.

• Definite integrals of |u| and sgn(u) with u linear (#352). The antiderivative of such an integrand can have a sgn(u) term, which jumps where u changes sign, and the value of the integral included that jump. \int_{-\pi}^{\pi} |x|\cos x\,dx evaluated to -2; it is now -4. \int_{-1}^{1} |x|\cos x\,dx had the value 2(\cos 1 + \sin 1); it now has the value 2(\cos 1 + \sin 1) - 2 (printed as -2 - \sin(-1) + \sin(1) + \cos(-1) + \cos(1)). \int_0^{\pi} |x|\cos x\,dx was -1; it is now -2. The integral is now split at the points where u changes sign. This also gives a value to integrals that stayed unevaluated before: \int_0^2 |x-1|\,x\,dx = 1, \int_{-1}^{1} \operatorname{sgn}(x)\,x\,dx = 1, and the nested \int_{-1}^{1}\int_{-1}^{1} |x-y|\,dx\,dy = 8/3. When a bound is symbolic and the sign change could be between the bounds, the integral now stays unevaluated instead of returning a wrong value: \int_{-a}^{a} |x|\cos x\,dx was \sin(a)|a| + \cos(a)\operatorname{sgn}(a) - \ldots.

• |u| with a non-real coefficient. The antiderivative u|u|/2 of |u| is valid only for a real u. \int_{-1}^{1} |x+i|\,dx gave \sqrt2; the value is \sqrt2 + \operatorname{arsinh}(1), and the integral now stays unevaluated (.N() gives 2.29559). \int |x+i|\,dx also stays unevaluated.

• Antiderivative of a function of an argument that does not depend on the variable. \int x\sin(0x+1)\,dx gave \tilde\infty\cos(1), \int x\cos(2x-2x)\,dx gave \operatorname{NaN} and \int\cos(2x-2x)\,dx gave 0: the coefficient of the variable was 0 (or only the last linear term was read) and was used as a divisor. They now have the values \sin(1)\,x^2/2, x^2/2 and x.

• .N() of a definite integral when the integration variable has a value. With x := 5, \int_{-\pi}^{\pi} |x|\cos x\,dx gave about 0 under .N(), because the integrand was compiled with x read as 5, which removed the |x|. It is now -4, as evaluate() already gave.

• Integration by parts with a sine or cosine of a linear argument. \int x\sin(-x)\,dx and \int x\sin(2x)\,dx stayed unevaluated; they are now x\cos x - \sin x and -\frac12 x\cos(2x) + \frac14\sin(2x).

• #354: Hypergeometric2F1 with an integer b − a or c − a − b is accurate to about 12 digits. When b − a is an integer, the formulas that continue ₂F₁ outside the unit disk through 1/z and 1/(1−z) have a removable singularity; when c − a − b is an integer, so do the ones through 1 − z and 1 − 1/z. When no other formula applied (typically with both differences integral), the value was the average of two evaluations at nearby parameters, which kept as few as 4 correct digits: Hypergeometric2F1(-0.5, 0.5, 1, 1.17 + 0.45i) was 0.67816275 − 0.23183000i (relative error 5e-6) and is now 0.67816104159303 − 0.23183311013602i. These cases now use the logarithmic limit of those formulas (DLMF 15.8.8–15.8.11). Over a grid of such parameters and arguments outside the unit disk, the largest relative error against mpmath went from 3e-4 to 1.2e-14. A difference up to 400 is evaluated this way (relative error below 5e-13 at 400); a difference that rounding moved off the integer (2.3 − 0.3 is 1.9999999999999998) counts as that integer.

• Hypergeometric2F1 returns NaN (stays symbolic) instead of wrong digits. Every formula it uses is a sum whose terms can cancel, and none was checked: a result is now rejected when its largest term (for a formula with two parts, the larger part) exceeds it by more than a factor of 1e3 (1e6 for the machine series with a real z from −1 to 1/2), and another formula is tried; when no formula meets that bound, the best one is used if its factor is at most 1e4 (about 11 correct digits), and the result is NaN otherwise. Over 1,000 random real arguments (a, b from −25 to 25, c from −25 to 45, z from 0.5 to 1 or below −1), 936 have a value, 5 of them with a relative error above 1e-12 (the largest 2e-11). Hypergeometric2F1(14.31, 4, 42.19, 0.752) had 7 correct digits at machine precision. A series whose third parameter is negative was also stopped at a small term just before n = −c, after which its terms grow again: Hypergeometric2F1(6.066, -9.19, 38.584, 0.5193) had 6 correct digits. Hypergeometric2F1(0.5, 60.5, 1.5, 2.5 + i) was 1.3e-18 + 1.3e-18i and is now 0.0132129072203 + 0.0686091095322i. A series that shrank and then grew again (a negative third parameter, or large parameters with z near 1) was stopped at its first small term; with real arguments Hypergeometric2F1(102, 1, 2, 0.9) was 1.15e87 at machine precision and 2.05e98 at the default precision, instead of 1.10e99. The big-decimal series adds working digits when its terms cancel, and so does the big-decimal connection formula when its two parts cancel (Hypergeometric2F1(26.91, 7.09, 38.36, 0.689) at precision 30 had 23 correct digits). The average of two evaluations at nearby parameters, used when no formula applies, kept as few as 4 correct digits when a parameter difference is a nonzero integer: it is now used only when that difference is 0, and the result is NaN otherwise. A polynomial of degree above one million (such as Hypergeometric2F1(-1e20, 0.5, 1.5, 2)) is NaN instead of a computation that does not end.

• A vars-mapped input reached through an assigned value is no longer baked into compiled code. The compile-time folder stopped only at a subtree that mentioned a mapped name itself. With a := sin(y_0) and y_0 mapped in vars while also holding a value, cos(a) folded to a number, the body of u := L ↦ 2L − a under the call u(1) folded, and so did a definite integral or a limit over such a value: the input was in the argument bag but changing it changed nothing (a slider-dependent document definition drew outdated geometry; Tycho item 328). The guard now follows assigned values and user-function bodies at any depth, on every target, so the value is emitted as code reading the input (const _val_a = Math.sin(_.y_0); inline sin(u_y0) on GLSL/WGSL). The same guard makes a function the caller overrides through the functions option run the caller's code when it is reached through a symbol's value (q := g(3) with functions: { g } ran the engine's g). A multi-clause function (function w(x) {…} twice in Epsil) is read clause by clause for both purposes, and freeSymbols now lists a symbol that only a clause body reads; it was missing.

• Reading the type of a product or a sum of calls to declared user functions no longer takes exponential time when a function body reads an undeclared name. With h and r declared (real) -> unknown and assigned bodies that read a free T, the type of h(1) r(1) cost 5 173 signature derivations and a 400-term sum of such calls did not finish in 60 s (Tycho item 329). Each derivation boxed the body again in a fresh scope, declared a new T there, and narrowed it by its use; that narrowing advanced the engine-wide definition version, which is part of every derived-signature memo key, so each derivation threw away the memo of every other declared function. A narrowing by a use no longer advances that version (it can only leave a memoized result wider, never wrong; the rollback of a narrowing still advances it). The sum now types in about 30 ms with 251 derivations, the same count as with T declared, and the reported types are unchanged.

• Assigning a function that nests calls to functions declared with an unknown result no longer takes exponential time. With W_1, W_2 declared (unknown, …) -> unknown, their bodies applying undeclared names (b(x, y)), and E assigned a body that nests them (W_1(W_2(…))), the one ce.assign took 2 ms at depth 1, 3.5 s at depth 2 and more than 60 s at depth 3, with 706 258 signature derivations (Tycho item 336, a regression since 0.137.0; three consumer documents did not open). Boxing a function literal that applies an undeclared name declares that name in the literal's own scope; the engine then re-derived every stored definition waiting on that name, although none of them can see that scope, and each rebuild invalidated the memoized signature of every other function. A definition is now re-derived only when the new binding is on its scope chain, and the declarations a signature derivation makes in its own temporary scope no longer invalidate other memos. Depth 6 takes about 5 ms and the derivation count grows linearly; the reported types are unchanged, and a real declaration or assignment still updates them.

• A recursive call no longer overwrites the caller's loop variables, big-operator indices, or inner-block locals. A for loop variable, a Sum/Product index, or a let inside a nested block of a function body lives in a scope created once when the body is canonicalized, so every application of the function shared it. After a recursive call returned, the caller read the callee's last value: in for c in [n*10, n*10+1] { …f(n+1)…; out = [...out, c] } the caller's c read 21, not 10, and a recursive tree walk visited only the first branch below every node (Tycho item 330). A re-entrant application now saves those scopes on entry and restores them on exit; a call that is not re-entrant is unchanged. The same holds for a recursive call made while a lazy comprehension built by the function is read.

• A function that spreads a parameter into a list or set literal splices the argument's elements. g(a) = [...a, 0] gave [[9, 0]] for g([9]) and h(a, x) = [...a, x] under reduce([1, 2, 3], h, []) gave [] (Tycho item 331). The body canonicalized to Join(a, [0]) without running Join's own canonical handler, so the parameter was never typed from its use and stayed unknown, and a user function with an unknown parameter is applied to each element of a list argument. The parameter is now typed as a collection, as it is for a body written Join(a, [0]), so g([9]) is [9, 0] and the reducer returns [1, 2, 3]. The set literal {...a, 0} had the same defect.

• A block-local let with a literal value is typed from that value on the routes that never run the Declare, and a nested block reads the hoisted binding. let queue = [[…], …] followed by while i <= Length(queue) { … queue = [...queue, g] … } declined to compile to JavaScript with "operand is not an indexed collection": the hoisted binding stayed unknown until the Declare ran, so the first use (Length, or the Join behind a spread) inferred the callee's loose parameter type (collection, collection<any>) onto it, and the later assignment could only widen that. Two fixes (Tycho item 332, programs b and c). The Block canonical hoist now records the type of a closed literal initial value (a number, a string, a List/Tuple of such), widened through the assignment table as Assign does, so let xs = [] hoists list<never> and let i = 1 hoists integer, not the singleton 1. And a reference to a hoisted local from a NESTED block (an if branch, a loop body) now finds the hoisted binding one or more scopes up instead of declaring a second unknown copy in the nested scope and caching that copy as the name's binding for the rest of the block, which left the hoisted binding, and any declared type on it, unused. With the locals typed this precisely, one convention had to be narrowed: a list literal reads an unknown bare symbol as a number (the generic-symbol fold, [x, y] is a vector<2>), which typed [c] as vector<1> for a block local c still waiting for its let to run, so out = join(out, [c]) made out a list<number> and a later stringJoin(listFrom(out)) was refused although out only ever held characters (the JSON parser example of the Epsil documentation). A hoisted block local is no longer read as a generic number by that fold: [c] is a list until c is typed. A destructuring let (x, y) = p from a block-local tuple literal now compiles too (the arity is statically known); it declined before.

• A compiled for loop whose list holds complex values no longer reads the loop variable as a real number. The JavaScript target shaped the loop variable from its declared type (number, read as real) while the for (const k of …) loop handed it the {re, im} objects the list produced, so every comparison on k was false and every division gave NaN: an Apollonian-gasket program returned 4 circles instead of 224 (Tycho ask 332). When every element of the list is complex the variable is now shaped complex; when only some are, or their lanes cannot be told apart, the compilation is a lane mismatch, which the default auto mode answers by recompiling in complex mode and strict mode refuses. The check that a local first assigned a real value is never later assigned a complex one now also enters the braced bodies of if/for/while (r = r + 1/k inside a loop put an object into r, and the next pass turned it into a string). Max/Min of real operands now keep the operands' bounds (Max(0, x) is real<0..>, it was real), so √(Max(0, x)) compiles to a real square root instead of the complex route.

• A let with an initial value is typed from that value, on every route, and the recorded type follows later widening. Only Assign recorded the static type of its value on the local at canonicalization; a let did not, so let cands = fourth(a, b) or let gap = queue[i] stayed unknown on the compile route, and filter(cands, …) or gap[1] failed the compiler's shape gates although the callee's declared result and the list's element type were known. Declare now records the initializer's type as assignment evidence, as Assign does; a declared type still wins. Reading the type once, at the statement, was too narrow when a later statement of the same loop widened the source (let e = circles[j] before circles = [...circles, (k, x, y, depth)] with k: number): in the complex lane e[1] - k then read a {re, im} object as a number, the de-duplication never fired, and the compiled gasket program never ended. The block now re-reads each let/Assign value type and each for index type after its statements are canonicalized, until they stop changing; a type that keeps growing is capped, and a local that never settles falls back to unknown. An element read of a union of tuple types now gives the types at that index only. With type annotations on its helpers, Tycho's gasket program a compiles and answers 224 in about 18 ms; the interpreter takes 21 s (Tycho item 332).

• An Epsil program whose helper functions have no type annotations compiles to JavaScript. Tycho's gasket program a, as written, compiles and answers (224, [4, 6, 18, 54, 86, 28, 8, 8, 4, 4, 4, 0]) in about 4 ms; the interpreter takes 21 s (Tycho item 332, user decision 2026-09-29). Four changes. A parameter that the body only counts or measures (function seen(acc, c) { count(acc, e => …) > 0 }) is typed collection by inference, which also admits a set, and Count, Length and the other list operators refused it. Such a parameter now compiles with a check at run time that the value is an array; a set or a dictionary that arrives there stops the run with a RangeError. A DECLARED collection type still declines, so that the interpreter evaluates it. Max(0, x) counts as non-negative in the compiler when x is read as real, also when x may be NaN, so √(Max(0, x)) keeps the real square root. A sum and a square root of operands that may be NaN keep their range (p² + q² for p, q: nan | real is nan | real<0..>, and its square root nan | real, where they were nan | real and complex | nan), so the inlined programs b and c now compile in the real lane and run in about 5 ms. The types recorded for a local from its assignments are joined by structure: two lists join their element types and two tuples of one length join slot by slot, where the union of the two list types reduced to a bare list and every local read from it became unknown.

• Count(xs, p) with a predicate compiles to JavaScript. It lowers as CountIf does; only the value form Count(xs, v) still declines, since it needs the interpreter's structural element equality.

• A restriction with a list-valued condition has the right static type over every ordered carrier. P\{P.x > 1\} with P declared indexed_collection<tuple<number, number>>, collection<…> or range was typed list<indexed_collection<tuple<number, number>> | missing>: the whole collection type stood where the element type belongs, because the type handler recognized only a list carrier while the evaluation zips any finite collection. The cells are now typed from the element type (list<missing | tuple<number, number>>) when the carrier is finite by type (list, a vector or matrix, range, a dictionary or record). A carrier whose finiteness the type does not decide (indexed_collection<T>, collection<T>, which admit Range(1, ∞)) types its cells as the element type joined with the carrier type (list<indexed_collection<T> | missing | T>), because the evaluation puts the WHOLE value in the cell when it is not finite. A condition that is a set or a collection<boolean> is a mask too, as it is at evaluation. For the same reason PointX/PointY/PointZ over a declared collection or set of points is typed as a list of coordinates, not as a point, and Which/If with a list condition over a range type their cells integer, not integer | range. The evaluated values were right and are unchanged (Tycho item 335).

• Narrowing a union type meets it arm by arm. narrow('missing | vector<real^4>', 'indexed_collection | dictionary') was never, because the union was tested only as a whole; it is vector<real^4>. A local typed from an element read, which admits absence, was narrowed to never by its first indexed use and the expression built on it became a type error.

• Beta, Zeta and Lb write conventional LaTeX when applied, and the sign of a numeric fraction moves in front of it. Beta(2, 3) wrote \Beta(2, 3) (capital beta is roman, not a separate glyph — MathLive renders it as an error) and Zeta(3) wrote \Zeta(3); both came from the fallback that spells an unrecognized function head as its symbol's notation, which for these two names is the Greek-letter entry. They now write \mathrm{B}(2, 3) and \zeta(3); the old spellings still parse, and a bare \mathrm{B} is still the upright letter B. Lb(x) wrote \lb(x), not a standard LaTeX command, and now writes \log_2(x) (which already parsed to Lb). A negative Rational, or a fraction with a number denominator, wrote its sign inside the numerator or the denominator — Rational(-1, 2) as \frac{-1}{2}, Divide(x, -4) as \frac{x}{-4} — or, for Negate of a fraction, with a redundant parenthesis (Negate(Rational(3, 4)) as -(\frac{3}{4})); all three now write the sign in front: -\frac{1}{2}, -\frac{x}{4}, -\frac{3}{4}. A fraction with a symbolic denominator keeps the sign in the numerator (\frac{-1}{x}), since -\frac{1}{x} reads back as a different expression (#345, contributed by enumeratio).

• A negative big number keeps its digits when its sign moves. The LaTeX serializer removed the sign of a negative literal in a sum through a JavaScript double, so Add(x, -9007199254740993) wrote x-9\,007\,199\,254\,740\,992. The sign is now removed from the digit string.

• \operatorname{rank}(A) parses to MatrixRank, and MatrixRank(A) writes \operatorname{rank}(A). It parsed to the free symbol rank applied to A, the same gap \operatorname{lcm} (→ LCM) already covered for a different head (#345, contributed by enumeratio).

• Zeta(s, a) and HurwitzZeta(s, a) honor ce.precision for real s and a (part of #340, contributed by enumeratio). Both were machine precision only at every engine precision; HurwitzZeta(3, 1/2).N() at ce.precision = 50 now returns 50 correct digits (8.4143983221171599977981671305801499353549040463835) instead of a double's ~16. The digits are significant digits at every magnitude: HurwitzZeta(200, 10) (about 1e-200) and HurwitzZeta(-400.5, 0.3) (about 7.75e549, past the double range) are correct to the last digit. A value the kernel cannot reach within its limits (s below about −1279) falls back to the double kernel, and stays symbolic where the double overflows. A complex operand still evaluates at machine precision — the complex special-function kernels do, at every engine precision.

• a * 2n parses in Epsil. The right operand of an explicit * or / refused an invisible multiplication, so a * 2n parsed as a * 2 with an unexpected-symbol diagnostic for n — and the serializer writes that form. a * 2n is now a·(2n) and a / 2n is a/(2n).

• Exact 2×2 eigenvalues. Eigenvalues([[1, 2], [3, 4]]) was [5.372…, -0.372…]; it is now [(5 + √33)/2, (5 − √33)/2].

• AdjugateMatrix and PseudoInverse evaluate. Both stayed unevaluated for every matrix. AdjugateMatrix is the transposed cofactor matrix, for any square matrix. PseudoInverse is computed for a full-rank matrix: the inverse of an invertible square matrix, (A*A)⁻¹A* with full column rank, A*(AA*)⁻¹ with full row rank; a rank-deficient matrix stays unevaluated.

• Trigonometry. InverseFunction(Csc) returned an invalid-symbol error, and InverseFunction(Cot), (Coth) stayed unevaluated; every circular and hyperbolic function now maps to its inverse. Arccot has exact special values (Arccot(1) is π/4, Arccot(-1) is 3π/4). Sinc(Pi) is exactly 0 (it was 1.2e-25 under N) and Sinc(Pi/2) is 2/π. TrigExpand(Sin(x + Pi/2)) is cos(x) (it left cos(π/2) and sin(π/2) unreduced), and TrigExpand(Tan(x + Pi/2)) is -cos(x)/sin(x).

• Arithmetic. Log(1/8, 2) and Lb(1/8) are -3 (they stayed symbolic, while Log(8, 2) was 3). ComplexRoots(1, 4) is [1, i, -1, -i], exactly and under N (it was [1, 6.1e-17 + i, …]), and the roots of an exact real are exact (ComplexRoots(8, 3) is [2, -1 + √3 i, -1 - √3 i]). Supremum and Infimum of an open interval are its endpoints (they stayed symbolic). Interpret(1 + 2 + … + n) works from Epsil, whose left-nested sum the recognizer did not match. PreIncrement(5) is 6 and PreDecrement(5) is 4 (they had no evaluate handler). Sum(2^(-k), (k, 0, oo)) is 2, as Sum((1/2)^k, …) was: the geometric-series rule did not read a negated index.

• Exact 3×3 eigenvalues. A matrix of exact rationals whose characteristic polynomial has a rational root has exact eigenvalues: Eigenvalues([[2, 0, 0], [0, 3, 4], [0, 4, 9]]) is [11, 2, 1] (it was [11, 1.000000000000003, 2.0000000000000018]), and Eigenvalues([[2, 1, 0], [1, 2, 1], [0, 1, 2]]) is [2 + √2, 2, 2 − √2]. The exact eigenvalues are ordered by decreasing real part. N(Eigenvalues(…)) gives their values.

• N(Arcosh(x)) for a real x in [−1, 1] is purely imaginary (N(Arcosh(1/2)) was 5.6e-17 + 1.047…i).

• The upper incomplete gamma Gamma(s, x) for Re(x) < 0 (#353). On the negative real axis .N() dropped a term once |x| grew: Gamma(-1, -20) was 1357393.643181685, it is now 1357392.893567075 + 3.141592653589793i; Gamma(1/2, -20) was -111433110.2i, it is now 1.7724538509055159 - 111433109.93704489i. Off the axis the error grew with |x| wherever Re(x) < 0 (Gamma(-0.7, -12+5i) had 9 correct digits). The complex kernel now picks a method without cancellation for each region; measured against mpmath, the relative error is below 1e-13 for |x| up to 150 in every direction. The same kernel gives E₁, Ei, Si, Ci, Shi, Chi, erf and erfi for complex arguments, and they gain the same accuracy (Si(4+6.9i) had 5 correct digits). For a negative integer s and a complex x in the right half-plane the old kernel was also inaccurate: Gamma(-9, 13+7.5i) was 3.5e-18 - 1.1e-17i, it is now 2.449e-18 + 7.84e-20i. The real kernel had the same defect for a real x > 0: Gamma(-15, 60) was 2.389e-55, it is now 2.457e-55, and Gamma(-9, 30) had 8 correct digits, now 15. The complex Gamma(s) is now accurate near its poles (Gamma(-1.999999 + 10^{-9}i) had 10 correct digits, now 15). Where the kernel cannot certify about 12 digits, .N() leaves Gamma(s, x) unevaluated: some s close to 0, -1, -2, … with x near the negative real axis (how close depends on x), and some points with |Im s| > 10 (on 2500 random points with |Re s|, |Im s| ≤ 30 and 0.01 ≤ |x| ≤ 600 it declines at 87 and answers the others with an error below 1.7e-13). A value or a factor outside the range of doubles no longer spoils the result: Gamma(-5, -712).N() was NaN, it is now 1.2778259067196765e+292; Gamma(-170, -100).N() was 7.4e-308, it is now 3.925357618957481e-299; a value above the range of doubles (Gamma(-5, -2000)) stays unevaluated, where it was ~oo (the pole of the one-argument Gamma at -5 was applied to the two-argument form). The complex Gamma(z) had the same kind of defect: Gamma(-2+300i).N() was NaN, it is now 3.448337914830325e-211 - 8.293880091137963e-212i, and Gamma(150+i).N() was ~oo, it is now 1.1034056813344657e+260 - 3.632309144571929e+260i.

New Features

• Indeterminate: the exact answer to an indeterminate form. A second number with no value beside NaN, ce.Indeterminate (MathJSON "Indeterminate", LaTeX \operatorname{Indeterminate}, Epsil Indeterminate), is the answer an EXACT computation gives to a form such as 0/0, where NaN stays the result of a floating-point computation that failed and the absence marker. Its double value is NaN, so isNaN is true and its type widens to nan (its own value type prints Indeterminate); the new isIndeterminate property is true for it only. It is not the same value as NaN (isSame and .is() are false between them, and each has its own hash); Equal is False for it as for NaN. An operation forwards it when no operand is inexact: Sin(Indeterminate), Indeterminate + 1, 2·Indeterminate, Max(1, Indeterminate) and Mean([1, Indeterminate]) are Indeterminate, while a float or NaN operand gives NaN (Indeterminate + 1.5, Max(Indeterminate, NaN), Max(1.5, Indeterminate)), and so does an operator that does not forward it (Hypot, GCD). .N() of any result is NaN, and compiled code spells it as the target's NaN. It is a value, not an absent entry: IsMissing(Indeterminate) is False and Coalesce(Indeterminate, 5) is Indeterminate. No computed answer changes yet: 0/0, 0^0 and the other exact indeterminate forms still evaluate to NaN; a later release switches them to Indeterminate (docs/plans/2026-09-28-indeterminate-value.md, GitHub issue #355).

• epsil --compile: run an Epsil program as compiled JavaScript. The CLI compiles the program (or, with --from latex, the LaTeX expression) to JavaScript and runs the generated code instead of interpreting it, and writes the value as it writes an interpreted result, under every output option. Compiled arithmetic is machine arithmetic, so 2 + 1 is the float 3.0 and sqrt(2) is 1.4142135623730951; a symbol with no value is a runtime error, since compiled code has no symbolic values; and a construct the JavaScript target declines is a runtime error naming it, never an interpreter fallback. See "Running a Compiled Program" in the CLI guide.

• ce.withStepBudget({ steps, label }, fn): a deterministic hang guard. Runs fn with at most steps steps of engine work, where a step is one of the engine's cooperative cancellation checks: the count of a computation on one engine state is the same on every machine, unlike a wall-clock limit, so a host can bound a decision without letting the clock decide it (Tycho item 326). The step is an opaque unit — deterministic, not a measure of cost, and not comparable across engine versions — so a budget is tuned empirically, with a withTimeLimit span outside it as the guard of last resort. A spent budget throws a CancellationError with the new cause: 'step-budget', the span's label as its attribution and the active labels in spans; a time expiry keeps cause: 'timeout'. CancellationCause gains 'step-budget', and isTimeoutCancellation answers true for both causes. An Epsil program stopped by a spent budget answers ["Error", "Step budget exhausted", "step-budget"], as a timeout answers ["Error", …, "timeout"].

• PolyGamma(m, z), Digamma(z) and Trigamma(z) evaluate at a complex z (#340, contributed by enumeratio). PolyGamma(1, 1+2i).N() is 0.1249311621409446 − 0.4778255501472298i: ψ⁽ᵐ⁾(z) = (−1)^(m+1) m! ζ(m+1, z) for m ≥ 1 (DLMF 5.15.2), and an asymptotic series for the digamma. Left of the imaginary axis the reflection formula is used, so the cost does not depend on Re(z) (PolyGamma(8, -10^12+i) answers at once). The order is limited to 10 000. A value outside the range of a double stays unevaluated, not ~oo or 0. The compiled JavaScript PolyGamma is now marked real-only, as Zeta is: a complex argument there ran the real kernel.

• LerchPhi is new (#340, contributed by enumeratio). LerchPhi(z, s, a) is the Lerch transcendent Φ(z,s,a) = Σ zᵏ(k+a)^(−s), generalizing HurwitzZeta (LerchPhi(1, s, a) reduces to it exactly). LerchPhi(0.5, 2, 1).N() evaluates by direct summation, LerchPhi(3, 2, 1).N() continues past the unit disk through the incomplete gamma function, and LerchPhi(-1, 1, 1) is ln 2 on the disk's rim, where direct summation alone would need an impractical number of terms. A base point a at a non-positive integer is the pole ComplexInfinity when Re(s) > 0 and z is provably nonzero, as for HurwitzZeta. Values are computed at machine precision only, even when precision is higher, and are within about 1e−11 relative of the true value (measured against mpmath). Where the kernel cannot vouch for that accuracy, LerchPhi stays symbolic under N() instead of returning a wrong value: past the unit disk, for some s within about 10⁻² of a positive integer when z is near the real axis (the incomplete gamma function it needs is inaccurate there), when the terms of a sum cancel too far (for example LerchPhi(-0.999, -6, 1) or LerchPhi(-2, -3.5, 1.5)), and when the base point is further left than a = −10⁶. A float operand gives a float result (LerchPhi(1.0, 2, 1) is 1.6449…, not π²/6). LerchPhi compiles to JavaScript, GLSL and WGSL for real operands. The JavaScript lane runs the interpreter's kernel and is NaN where the value is complex (real z > 1, or a < 0 with a non-integer s) or the kernel declines. The GPU lane is NaN past the unit disk (except for s = 0, −1, −2, where Φ is a rational function of z), and inside it wherever its f32 sums cannot converge within their term budget (|z| closer to 1 than about 0.995) or have cancelled too far.

• PolyLog now evaluates at a non-integer or complex order (#340, contributed by enumeratio). PolyLog(s, z) previously answered only an integer order s ≥ 2; PolyLog(2.5, 0.5).N() is now 0.5549972787175124 and PolyLog(1.5+0.5i, 0.5).N() is 0.6126403889001154 - 0.05103210425890372i, both by Liₛ(z) = z·Φ(z,s,1) through the LerchPhi kernel at base point a = 1. Past |z| = 1 that kernel's continuation answers, and where it declines Jonquière's inversion formula takes over: PolyLog(1.5, -3).N() is -1.6790897305048254 (a real order gives a real value everywhere on the real axis below -1). PolyLog(s, 1) and PolyLog(s, -1) now reduce exactly to Zeta(s) and the Dirichlet eta identity (2^(1-s) - 1)·Zeta(s) for every order, not only an integer one (the kernel answers instead for an order within 1e-6 of 1 at z = -1, where the identity cancels). A negative integer order from -2 to -12 at a number z uses its rational closed form, so PolyLog(-2, 1/2) is the exact 6 and PolyLog(-2, -2) the exact 2/27. Every existing integer-order and elementary-form result is unchanged. The widened kernel declines (stays symbolic) rather than answer a value it cannot certify to 1e-12: within 1e-3 of the z = 1 branch point, where Re(s) is too negative for the series inside the disk, and past the disk where both the continuation and the inversion formula decline (some orders within about 1e-2 of a positive integer, on or past the unit circle, for example PolyLog(3.000001, 2)). numerics/polylog.ts records the measured boundaries. PolyLog compiles to JavaScript, GLSL and WGSL for real operands; it was not compilable at all before. The JavaScript lane answers every order the way the interpreter does. The GPU lane answers the orders 1, 0, -1 and -2 to -12 in closed form and other orders only for |z| < 1 and at z = 1.

• PolyLog of a real order and a real z is typed real only when z is known to be at most 1 (or the order is an integer ≤ 0). The value is complex on the cut z > 1 (PolyLog(1.5, 2) is 1.549 - 2.951i), and the type used to say real. A float order now makes a closed-form result a float: PolyLog(0.0, 1/2) is the float 1.

• Examples for the arithmetic, trigonometry and linear-algebra libraries, with a hand-written introduction for each reference page: 97, 84 and 40 examples, each executed when the pages are generated.

0.140.0 2026-09-27

Behavior Changes

• The spelling rule holds at machine precision, and a float operand makes a numeric result a float. On a precision: 'machine' engine, ce.parse('2.0') was exact, so \sin(2.0) stayed symbolic there while it evaluated at the default precision; a machine number is now always inexact, and exact integers are the exact class, as ce.number(2) already was. 2.0^2, \ln(1.0), 2.0!, \sinh(0.0), 4^{0.5} and similar results of a float operand are floats at every precision (user decision 2026-09-27). \max(2.0, 3) still returns its operand 3 as given, and \operatorname{sign}(2.0) is still the integer 1, as in Mathematica. A consequence at machine precision: (1.5+1.5)·(1/7) is the float 0.42857142857142855, not the exact 3/7. Also: x^{0.5} stays Power(x, 0.5) (only an exact 1/2 becomes \sqrt{x}), as x^{1.0} already stayed; a float 0 is no identity in a sum (0.0 - 2 is the float -2); \gcd(4.0, 6) is the float 2; the statistics of float data are floats (Mean([1.0, 3.0]) is 2.); a list of more than 100 numbers computed from floats keeps its integer-valued elements as floats. isMachineNumeric is now false for a number or list that holds an integer-valued float such as 2.0, because re-boxing it with ce.list(...) would make it exact.

• Integrate of a tuple integrates each coordinate. \int_0^1 (x, 2x)\,dx stayed unevaluated; it is (1/2, 1), as \sum_{n=1}^3 (n, 2n) is (6, 12) (user decision 2026-09-27). The indefinite integral gives (x^2/2, x^2), and a list bound gives one tuple per element of the bound (\int_0^{[1,2,3]} (x, 2x)\,dx is [(1/2, 1), (2, 4), (9/2, 9)]).

• Implies, Nand, Nor, Xor and Equivalent follow the three-valued rules of And and Or for an absent operand (user decision 2026-09-27). An operand that decides the result wins (Implies(Missing, True) is True; it was Missing); beside an unknown symbol the expression stays symbolic (Nand(A, Missing) was Missing), as And(A, Missing) does; the result is Missing only when the other operands are decided and do not decide it (Nand(True, Missing)). The same rule decides an implication beside an unknown: Implies(A, True) and Implies(False, A) are True (they stayed unevaluated).

• Compiled JavaScript spells an absent point, tuple or list undefined in arithmetic. 2·P and P + Q with an absent point P gave NaN and [NaN, NaN]; they give undefined, as a read of the absent point already did and as the interpreter fallback does (user decision 2026-09-27). A coordinate read (PointX(P)) stays NaN.

• A user binding shadows a capitalized library name. let Pi = 3; Pi was Pi (π) and is now 3; so for ExponentialE, Missing, Undefined, True, False, All and None. function Square(x) { x + 100 }; Square(3) was 9 and is now 103; so for a user Sqrt, Negate, Exp, Ln, Log, Power, Root, Divide, Add and Multiply, on the Epsil and the box routes. These names are interned constants, or heads that canonicalization folds by name, and both paths returned the library definition without looking the name up. Shadowing Add (or Multiply, Divide, Power, Negate) also changes the operator that builds it (+, *, /, ^, unary -) within the binding's scope, as shadowing Subtract already did. Compiled code does not use a shadowed library operator: compile() of such a call fails closed and falls back to the interpreter (it emitted the library operator before, y * y for a user Square, and so for a user Abs or Sin).

• A binding of Nothing, Missing or Undefined is an error. These absence markers are recognized by their name (an operand named Missing is read as absent whatever it is bound to), so they cannot be rebound. let Missing = 3 and the other binding forms (an assignment, a function, a parameter, a loop variable, a match pattern) report absence-marker-binding, and the declaration evaluates to that error. Before, let Nothing = 3 was silently ignored, and let Undefined = 3; Undefined + 0 was NaN.

• Length of an infinite collection is +oo. Length(Integers), Length(Repeat(5)), Length(Cycle([1, 2])) and Length(Interval(0, 1)) stayed unevaluated; they are now +oo, the answer Count gives. Only an unbounded Range answered +oo before. A collection whose size is not known (a Filter over an infinite source) still stays unevaluated. The reported type follows: Length of a tuple, a string, a literal list or set, a dimensioned list or a Range with finite literal bounds is integer, and Length of any other operand, including a symbol typed list<…>, is integer | infinity, because a list value can be an infinite lazy list. A function declared to return integer whose body is Length(xs) keeps its declared type.

• StringFrom of a string, a character or a symbol returns its text. StringFrom("hi") returned "\"hi\"" and StringFrom(True) returned "\"True\"": the printed form of those values carries quotes, and StringFrom used the printed form. It now returns "hi" and "True", as its description says, and a symbol such as integer converts to its bare name. Other values keep their printed form (StringFrom(x + 1) is "x + 1", StringFrom(Pi) is "pi").

• Timing declares an unnamed tuple. Its result type was tuple<time: number, result: value>, but the value carries no element names, so Timing(expr).result was an incompatible-type error. The declared type is now tuple<number, value>; read the parts as Timing(expr)[1] and Timing(expr)[2].

• The imaginary part of a numeric result has the working precision. .N() of √2 + √2i printed 1.4142135623730950488 + 1.4142135623730951i (21 digits on the real part, 16 on the imaginary part); both parts now have 21 digits, or the engine's precision. Ln, Exp, Power, Root and Sqrt of a complex argument compute both parts at the working precision above machine precision (Ln(1.1+1.1i).N(), e^{1+2i} at 50 digits). The other complex transcendental functions (Sin, Gamma, Zeta, …) keep machine precision. The rounding noise a complex kernel removes is now relative to the working precision (10^{2−precision} times the modulus, it was 10^{-14}), and a component that is small but real is kept: e^{i·10^{-800}} is 1 + 10^{-800}i.

• The Epsil serializer writes the lowercase spelling of a library name. serializeEpsil(["Sin", "x"]) is sin(x), "Pi" is pi, and ["Map", "Sin", "xs"] is map(sin, xs); the epsil format command, the --epsil output mode of the CLI and the MCP server, and the snippet a diagnostic quotes follow. Before, the serializer wrote the MathJSON names (Sin(x)). A library name keeps its MathJSON spelling where the lowercase one would read back as something else: when the expression binds or mentions the spelling (let sin = 3, a parameter named pi), and when the new isBound option reports it bound outside the expression (the CLI passes the session's bindings). The new option libraryNames: 'mathjson' restores the previous output. A round trip through parseEpsil alone now yields the raw lowercase head (["sin", "x"]); resolveLibraryNames, which executeEpsil and the CLI already run, reads it back to Sin.

• Epsil library reference pages. src/epsil/docs/reference/<category>.md, one page per library category (19 pages), lists every definition under its Epsil spelling with its MathJSON name, its signature, its full description and its executed examples; the Standard Library page links each category to its reference page. npm run doc generates them (scripts/build-library-reference.ts); a hand-written introduction in reference/<category>.intro.md is spliced in when present.

• A repeated constant in a call resolves. The Epsil resolution pass read a symbol operand that also occurs in another operand of the same call as the call's variable, for every operator: [pi, pi], max(pi, 2 * pi) and (pi, pi) left pi unresolved, and so did series(x + pi, x, pi, 3). The rule now reads only the operand positions that hold a variable: the positions whose parameter is typed symbol in the signature (D, series, factor, …), and a listed position for the operators that type their variable loosely (limit, solve, jacobianMatrix, characteristicPolynomial, findRoot, findFit, and the trailing variable of linearRegression and polynomialFit when it is a bare symbol).

• A user binding named like a library name keeps its spelling. With the lowercase output, let Pi = 3; f(Pi) = Pi; f(4) was written let pi = 3; f(Pi) = pi; f(4), which reads the outer binding (3 instead of 4). A library name the expression binds — by let, assignment, function name, parameter, loop index, match pattern or the variable operand of a binder — is written as is.

• A dictionary holding a list prints in Epsil. serializeEpsil read a value of the MathJSON {dict: …} form as a nested expression when it was an array and as a symbol when it was a bare string, so the JSON of a boxed dictionary holding a list (let d = {"xs" -> [1, 2]}; d in the REPL, whose value serializes as {dict: {xs: [1, 2]}}) threw a TypeError in the --epsil output mode. A {dict} value is now read as the engine reads it: an array is a list of values, a bare string is a string, a boolean is a boolean, and an expression is an expression object ({fn: …}). The JSON {dict: {z: ["Add", 2, "x"]}} therefore prints {"z" -> ["Add", 2, "x"]} (a list of a string, a number and a string), where it printed {"z" -> 2 + x} before.

• Limits has no Epsil spelling. limits is the type the engine declares for an indexing clause, so the spelling could never resolve to the operator; the Standard Library page lists Limits with no Epsil column.

Issues Resolved

• Mod of a float near the double range keeps its remainder. At machine precision Mod(2.0, 9007199254740991.0) was 1 (the formula added the divisor before the second remainder, and 2 + 9007199254740991 rounds); it is 2. Compiled JavaScript Mod(x, 9007199254740991) had the same defect (1 for x = 2). The base-10 and base-2 logarithms of a machine float now use the correctly rounded primitives on every route, so a short list and a long list agree in the last digit, and \log_{10.0}(1000) is 3.

• Compiled And, Or and Not with an absent operand follow the three-valued tables when used as a value. With M absent at run time, compiled And(M, False) gave undefined (the interpreter: False), Or(M, False) gave false (the interpreter: Missing) and Not(M) gave true, in JavaScript and in Python; the condition position of If and Which was already right. Ordinary booleans keep the plain &&/||.

• GroupBy with a character key. A key function that returns a character (GroupBy(["apple", "avocado", "banana"], s => First(s))) grouped under the quoted text ("\"a\""), because a character key was stringified with its quotes. A character is the same value as the one-character string, so it now keys the group a, as Take(s, 1) does.

• A complex numeric value whose imaginary part is outside the double range keeps it. (1+i)10^{800} under .N() was ~oo and is 1e+800 + 1e+800i; (10^{-200}(1+i))^2 was 0 and is 2e-400i; \sqrt{i\cdot10^{-600}} was 0; 2\cdot10^{-800}i evaluated to 0; a complex part below the double range serialized to LaTeX as 0. The inexact complex value now holds its imaginary part as a big decimal. See docs/plans/2026-09-27-big-decimal-imaginary-part.md.

• Exp(Ln(−2)).N() is −2. It printed −1.99…98 + 4.8e-16i: the logarithm of a negative real took a 16-digit imaginary part, whose rounding error the working-precision Exp then read as a value.

• An exact complex number whose imaginary part is too small or too large for a double keeps it. i\cdot10^{-800} evaluated to 0, (1+i)10^{-800} to 1/1e+800 with the imaginary part lost, (10^{-200}i)^2 and (10^{400}i)^2 stayed symbolic, Mean([1, 10^{400}i, 3]) read the exact datum as complex infinity, and an exact 10^{-800} + i was taken for i by the imaginary-unit recognizer: the exact value stored its imaginary part exactly but every "is this complex?" test read the cached machine double, which underflows to 0 or overflows to Infinity. Every numeric value now answers isComplex from its own representation (NumericValue.isComplex, BoxedNumber.isComplex, public on the number-literal interface), the finiteness, integrality and "is the real part zero?" tests on exact values read the exact fields, and the exact-to-double projection of a large rational is finite ((10^{400}+1)/10^{400} projected to NaN; it is 1). The double im stays available as a projection for double kernels. This is Phase 1 of docs/plans/2026-09-27-big-decimal-imaginary-part.md; the inexact route (.N() of such values, and the imaginary part's printed precision) is unchanged until Phase 2.

New Features

• ce.number({ re, im }) builds a complex number from two parts, each a JavaScript number or a BigDecimal. ce.number({ re: ce.bignum('1e-800'), im: ce.bignum('2') }) keeps both parts at full precision; a zero imaginary part gives a real number. ce.complex(a, b) is unchanged: it returns a machine-precision Complex (its documentation now says so and points at the lossless routes, this overload and the MathJSON Complex node).

0.139.0 2026-09-27

New Features

• Epsil comprehensions. A for clause inside a list or brace literal builds the collection from an iteration. The brackets select the kind: [x^2 for x in 1..10 if x % 2 == 1] is a list, {x % 3 for x in 1..10} a set, and {s -> length(s) for s in ["ab", "cde"]} a dictionary. Clauses are separated by commas, and a later clause can use an earlier binding ([(x, y) for x in 1..3, y in 1..x]). A binding can be a tuple pattern ([p + q for (p, q) in pairs]), and each clause takes an optional if guard. The set literal {1, 2, 3} is unchanged.
• Guards on Comprehension and Loop clauses. ["Element", x, xs, cond] (the form Sum and Product already accept) visits only the elements for which cond is True. This works in the interpreter and in compiled JavaScript and Python (a guarded Loop is not compiled to Python).
• SVD of a complex matrix. SVD([[2, 1+i], [1-i, 3]]).N() returns a complex U, a real diagonal Σ and a complex V. It stayed unevaluated before. A complex matrix with exact entries is decomposed only with .N().

Behavior Changes

• Exactness is decided by the route a number arrives on. A literal with a fraction part is a float on every route: 1.0, 0.0, -1.0, 1.00 and the MathJSON {num: "1.0"} were the exact interned constants (1.0x evaluated to x, \frac{1.0}{3} to the exact 1/3, \sqrt{1.0} to the exact 1, \sin(1.0) stayed \sin(1)) while 2.0 was a float. They now behave as 2.0 does: 1.0x is 1.0·x, \frac{1.0}{3} is 0.333…, \sin(1.0) is 0.841…. On the API, ce.number(bigDecimal) with an integer-valued big decimal is exact at any magnitude (ce.number(1) was exact and ce.number(ce.bignum('10000000')) was a float; both are exact, a bigint above the safe integers). A float ±1 produced by a computation is kept as a coefficient (0.5·2x evaluates to 1x, it was x), a float 0 or ±1 is no identity in a power, a quotient or a sum either (x^{1.0} stays x^{1.0}, 1.0^x stays, 2/1.0 and 2.0^2 are floats, 0.0 + x stays x + 0.0, 0.5x + 0.5x is 1.0·x), while a float 0 still absorbs a product (0.0x is the float 0). GCD, LCM and Factorial2 of big integers are exact at the default precision (they were floats). An exponent literal with no fraction part (1e3) is still exact, as Mathematica's 1*^3 is. See "Exact and Inexact Numbers: the Spelling Decides" in the numerical evaluation guide.

• Derivatives serialize with the differentiand in the numerator. D(x^2, x) now serializes as \frac{\mathrm{d}x^2}{\mathrm{d}x} instead of \frac{\mathrm{d}}{\mathrm{d}x}x^2. The old form did not round-trip when a term followed it: D(x, x) + 1 re-parsed as D(x + 1, x). The old form is still accepted as input.

Resolved Issues

• An exact coefficient raised to a rational power stays exact. (8x+8)^{2/3}·y evaluated to 3.99999999999999944548·y·(x+1)^(2/3) (an exact input gave a float with a wrong last digit) and \frac13(2x^2+2)^{-2/3} to 0.2099…/(x²+1)^(2/3): a rational exponent p/q with p ≠ 1 went through the big-number route. p/q is now the q-th root followed by the integer power p, exact when the root is (8^{2/3} = 4, (1/8)^{-2/3} = 4), and a root that is not exact is kept as a symbolic factor ((2x+2)^{2/3}·y is 2^{2/3}·(x+1)^{2/3}·y). On the way, (-8)^{2/3} gave 8^{2/3}·i; it is 4, the real root, as Root gives.

• A compiled comparison chain with an absent operand keeps its decided links. Less(x, 0, Missing) with x = 1 is False in the interpreter (the first link decides), but compiled to undefined, so an If over it took no arm; a chain now combines its pairwise links with Kleene "and". A two-operand relation with an absent operand and an operand with effects still evaluates that operand (Less(Missing, t + Random()) consumes its draw), and a chain whose shared operand has effects is refused rather than evaluated twice.

• A complex quotient whose numerator products underflow is scaled. 2^{-1000} / (2^{-100} + 2^{-100}i) gave 0 on the interpreter's numeric values (the products 2^{-1000}·2^{-100} round to 0, so the numerator read as an exact zero); it is 2^{-901} − 2^{-901}i, on both the interpreter and the compiled JavaScript.

• The interpreter fallback of a compiled function spells an absent value as compiled code does. When a compiled function falls back to the interpreter, every Missing became NaN, while compiled JavaScript spells an absent point, list, tuple, colour, boolean or string undefined and only an absent number NaN. The fallback now spells an absent result, and an absent cell inside a list or tuple, by the type of its position (user decision 2026-09-27); the interval-js fallback spells it {kind: 'empty'} as its compiled code does (it gave {lo: NaN, hi: NaN}). Also, a compiled comparison with a written Missing or Undefined operand answered false (NotEqual true) and If/Which over it took an arm; the six relations now compile to the undecided value (undefined, Python None), a branch over them takes no arm and answers NaN, and And/Or keep the Kleene rules. A restricted number is still NaN with its IEEE comparison.

• A lazy collection whose element type carries a nan arm is accepted at a list<…> parameter. With u and s declared (list<real>) -> unknown and u assigned an up-sampling comprehension, s(u(L)) evaluated to incompatible-type (u(L) is a lazy comprehension typed indexed_collection<integer | nan>, the nan arm coming from an element read) while the compiled JavaScript returned values. The argument check now accepts a lazy collection at a list, collection or indexed-collection parameter when its element type without the nan arm fits (user decision 2026-09-27); the list<real> contract is unchanged, an eager list that holds NaN and a mismatched element type are still rejected.

• Map with a bare-symbol callback is typed from the callback's signature. Map(\sin, [-1, 2]) was typed vector<integer^2> (the source type was copied because a bare symbol callback reads as unknown); it is vector<number^2>, from Sin's signature, and a declared or assigned function's result type is used the same way (user decision 2026-09-27). The source type is still copied when the callback is an undeclared symbol. Also, .N() of such a Map did not numericize ([sin(-1), sin(2)] stayed symbolic while the lambda form gave floats); it does now.

• A list-valued integrand distributes over the integral. \int_0^1\int_0^{G} xy\,dx\,dy with G = [1, 2, 3] stayed unevaluated (the outer integral had a list-valued integrand); it is [1/4, 1, 9/4], one integral per element, as a list BOUND already evaluates (user decision 2026-09-27), under evaluate() and .N(), on the parse, box and function routes. Two list bounds of equal length pair element by element; different lengths are the incompatible-dimensions error under evaluate(). Also, \int_0^1 (x, 2x)\,dx parsed to a two-statement Block integrand and gave 1/2 under evaluate() and 1 under .N(); it parses to the Tuple integrand, as (x, 2x) does on its own, and stays whole in both modes.

• A float multiple of π that is a special angle is exact in every angular unit. e^{0.25i\pi} was the exact √2/2 + √2/2·i in degree mode and a float in radian and turn mode, while \sin(0.5\pi) in radian mode was already the exact 1. A float coefficient within one unit in the last place of a fraction p/q with q in the recognizer's table (1, 2, 3, 4, 5, 6, 8, 10, 12) is now read as that fraction on the e^{iθ} route and in Sin, Cos and Tan (user decision 2026-09-27): radian e^{0.25i\pi} is √2/2 + √2/2·i, \sin(0.3\pi) is 1/4 + √5/4. A float that is not a special angle stays a float (e^{0.35i\pi}), and .N() is unchanged. Found on the way: in degree and gradian mode e^{0.35i\pi} came out as the symbolic cos(63) + i·sin(63) for a float input; it is now a float.

• Xor and Equivalent accept an absent operand. Xor(True, Missing) and Equivalent(True, Missing) were incompatible-type errors, and Xor(True, Undefined) stayed as ¬Undefined, while And, Or, Not, Implies, Nand and Nor answer Missing for an absent operand. The two now do the same (user decision 2026-09-27); a decided result is unchanged (Xor(True, False) is True, Xor(True, B) is ¬B).

• Complex division in the interpreter no longer overflows or underflows. 1/(10^{308}+10^{308}i) evaluated numerically to 0 and 1/(10^{-200}+10^{-200}i) to ~oo; Inverse([[1+i, 10^{308}+10^{308}i], [0, 1]]) gave -oo for the entry that is -10^{308}. The interpreter divided with the textbook formula, whose intermediate products overflow or underflow the double range. Every complex quotient the interpreter forms (the numeric-value div and inv, the complex matrix field behind Inverse, Determinant and LUDecomposition, the reciprocal trigonometric functions, Rational with complex operands, polynomial root finding) now uses the scaled division the compiled JavaScript route already had (Smith's algorithm with power-of-two scaling), so the two routes share one algorithm. Ordinary quotients are unchanged bit for bit; a zero, infinite or NaN operand keeps its previous answer. ComplexRoots(10^{200}+10^{200}i, 2) gave NaN from an overflowing modulus and is now correct.

• An operator applied to an error-typed operand has the type error. With E = Sin(Tuple(A, B)) for A, B: list<real>, Add(1, E) had the type error | integer and Multiply(2, E) had the type number. The arithmetic operators, the elementary functions, Abs, Hypot, Norm, Max, Min, Supremum and Infimum now have the type error.

• Lazy collection operators over an unevaluated operator evaluate. Sum(Join([3], Sort([2, 1]))) stayed unevaluated, while Sum(Join([3], [2, 1])) was 6. This also applies to Append, Reverse, Drop, Take, Filter, Zip and the other lazy collection operators.

• A comprehension clause sees only the indices of the clauses before it. In Comprehension(x, Element(x, Range(1, 3), x < y), Element(y, Range(1, 2))) the y in the guard is a free variable, as it is for Sum and Product. Compiled code returned [] for every y.

• The Epsil serializer prints a set-builder as a comprehension. ["Set", body, ["Element", v, domain]] prints as {body for v in domain}. It printed as {body, v in domain}, which reads as a set of two elements.

• The real part of a complex number prints at the working precision. .N() of √2 + √2i printed a 25-digit real part (1.414213562373095048801689 + 1.4142135623730951i).

• A binder's free variables follow the clause order. In Comprehension(x, Element(x, Range(1, 3), x < y), Element(y, Range(1, 2))) the guard's y is the enclosing variable — a clause sees the indices of the clauses before it, not after — and the node now reports y free, as Sum and Product do for the same shape. Before, every index was subtracted from the whole node, so the compiled routes folded such a node with y unbound and answered [] for every y. The Python route declines a guard that names a later clause's index (a Python comprehension is one scope). The Epsil serializer prints the engine's set-builder, ["Set", body, ["Element", v, domain]], as the comprehension {body for v in domain} instead of the literal {body, v in domain}, which read as a two-element set.

0.138.0 2026-09-27

New Features

• Zeta evaluates at complex s and takes a second operand, and HurwitzZeta is new (#340, contributed by enumeratio). Zeta(0.5 + 14i).N() evaluates. Zeta(s, a) is the same as HurwitzZeta(s, a) for Re(a) > 0. ζ(−n, a) at a rational a is exact, Zeta(2, 1/2) is π²/2, and Zeta(+oo, 2) and Zeta(2, +oo) are 0. The two-operand and complex values are computed at machine precision only, even when precision is higher. Both functions compile to JavaScript, GLSL and WGSL for real operands.

Resolved Issues

• The hyperbolic functions evaluate at 0, as the circular functions do (#341, contributed by enumeratio): Sinh(0) is 0, Cosh(0) is 1, Arcosh(1) is 0, and Coth(0) and Csch(0) are ComplexInfinity. Coth(0).N(), Csch(0).N() and Arcsch(0).N() are now ComplexInfinity (they were PositiveInfinity). simplify() gives the same values.
• Number theory on large integers (#339, contributed by enumeratio). With p a large prime, FactorInteger(p^2) (and so Totient, DivisorSigma and related functions) and Divisors(p^2) now evaluate, and MultiplicativeOrder(2, p) is faster. ModularInverse accepts a negative modulus: ModularInverse(3, -7) is -2.
• EllipticE(m) is correct at complex m (#346, contributed by enumeratio). EllipticE(0.57 + 0.23i) was 1.3175 − 0.1205i and is 1.3248 − 0.1197i. EllipticE(φ, m) for φ outside [−π/2, π/2] or m very close to 1 is also fixed, and EllipticE(10³⁰⁰) now evaluates to 10¹⁵⁰i.
• The base of a Power is bracketed when necessary in LaTeX (#345, contributed by enumeratio). ["Power", ["Complex", 1, 1], 2] serialized as 1+\imaginaryI^2 and is now (1+\imaginaryI)^2. The same rule fixes other serializations:
  • A Sum or Product raised to a power: (\sum_{k=1}^{n}k)^2, not \sum_{k=1}^{n}k^2. Also (A\cup B)^2, not A\cup B^2.
  • With fractionStyle: 'inline-solidus', ["Divide", "a", ["Divide", "b", "c"]] is a/(b/c), not a/b/c.
  • The derivative of a fraction, Sum or Product puts the function in the numerator, \frac{\mathrm{d}(\frac{x}{y})}{\mathrm{d}x}, so it reads back correctly inside a larger expression.
  • A factorial raised to a power is written (n!)^2 instead of n!^2.
• Three incorrect Chebyshev identities are corrected (#343, contributed by enumeratio). With the identities loaded by loadIdentities, simplify() rewrote T_n(x)² + (x²−1)·U_{n−1}(x)² to 1, U_n(cos x)·sin x to sin(n·x), and T_n(2x²−1) + U_{n−1}(2x²−1) to U_{2n}(x).

0.137.3 2026-09-26

Resolved Issues

• The result type of a function follows changes to the functions it calls. With f := x ↦ g(x) and g defined in another scope, a declaration that changed the signature of g did not update the signature of f.

0.137.2 2026-09-26

Resolved Issues

• Functions with an unknown parameter no longer slow each other down. With k: (unknown, T) -> unknown and w: (T, unknown) -> unknown, the type of sin(cos(k(x,y) + w(x,y))) took 10 s to compute; it now takes a few milliseconds.

0.137.1 2026-09-26

Resolved Issues

• Declaring several functions that call each other no longer takes minutes. With three nested functions declared (T, …) -> unknown, an assignment that did not finish in 100 s now takes 20 ms.
• A function declared (T, …) -> unknown no longer keeps an outdated result type. Assigning x ↦ a to f: (integer) -> unknown with a: integer kept the signature -> integer after a was retyped real, and in complex mode a function with a real-declared parameter refused a complex argument.
• A recursive function declared (T, …) -> unknown has a signature. The signature of f := n ↦ n + f(n − 1) was never computed.

0.137.0 2026-09-26

Behavior Changes

Types

• Operations on extended reals have more precise types. With y: real | signed_infinity, a: real | signed_infinity | nan, L: list<real>, K: list<integer> and C: list<real | signed_infinity | nan>:

  • Two types were wrong: y² was real<0..> and e^y was real<0<..>, although ∞² = +∞. They are now real<0..> | signed_infinity.
  • y³ is real | signed_infinity, tanh(y) is real, sin(y), cos(y), tan(y) and sec(y) are nan | real, arsinh(y) is real | signed_infinity, max(y, r) is real | signed_infinity, Max(K) is integer | nan, Sum(L) is real and Mean(K) is nan | rational. All were number.
  • A literal list keeps an infinite or NaN element in its element type: [1, ∞] is list<integer | signed_infinity^2> (it was vector<2>).
  • A value that may be complex stays number, or is complex | nan: arccos(y) and arcsin(y) are complex | nan. A compiler that cannot produce complex values now refuses such an expression.

  Code that compared these types with number or vector<n> now sees the more precise type.

• A power that may be 0^−k or 0^0 includes this in its type. x^k with x: real, k: integer was real and is now infinity | nan | real; x^n with n: integer<0..> is nan | real. A base or exponent that is provably non-zero keeps the old type (2^n is integer).

• A product or sum that may be infinite is typed on the extended real line. With y: real | signed_infinity and r: real, 2y was real and is now real | signed_infinity; r·y is nan | real | signed_infinity, because 0 · ∞ is NaN. y + 1 is real | signed_infinity and ∞ + y is nan | signed_infinity. Lists, matrices and tuples get the same element types: y·L is list<nan | real | signed_infinity>. As a result, Round(4Q) with Q: real | signed_infinity is integer | signed_infinity, not integer.

• Clamp, ElementMax and ElementMin are typed by their bounds. Clamp(x, −1, 1) is real<−1..1> (it was real), with nan when x may be NaN, also when x: number. ElementMax(x, 0) is real<0..>. So Arccos(Clamp(x, −1, 1)) is real (it was complex).

• Abs, Real, Imaginary and Arg have more precise types. |q| with q: real | nan was real<0..> | signed_infinity and is nan | real<0..>, so type.matches('real<0..>') on |x| is now false when x may be NaN. Re(z), Im(z) and Arg(z) with z: complex | nan are nan | real (they were number).

• Sum and Product are typed from their body. Sum(k², k, 1, 10) is integer and Product(1/k, k, 1, 4) is rational (they were number). With an infinite or unknown upper bound, the type also includes nan and signed_infinity. The index over integer limits is typed by its range: in Sum(body, Limits(i, 1, 40)), i is integer<1..40>.

• An operation on a scalar-or-list union is typed from the element type. With v: real | list<real>, 2v and sin(v) are list<real> | real (they were list<number> | number).

• A function declared with an unknown result reports the type of its body. (real | signed_infinity | nan) -> unknown assigned t ↦ t + 1 reports -> nan | real | signed_infinity (it was -> number).

• A call with a tuple argument for a scalar parameter is typed any. h((1, 2)) with h: (real) -> real and h := x ↦ 2x evaluates to (2, 4); its type was real.

• A symbol whose value is a pole is typed accordingly. Tan(z) with z := π/2 was real and evaluates to ~oo; its type is now number.

• The sign of a function is read from a ranged type. With t: real<0.9985..0.9999>, 1 − t is positive. So |π − 4| simplifies to 4 − π.

Functions and Pipes

• A function literal applied to a collection maps over it, as a named function does. Apply(x ↦ (x, x), [1, 2]) is [(1, 1), (2, 2)] (it was ([1,2], [1,2])). A function literal whose parameter is a collection (x ↦ Length(x)) or generic still takes the whole argument, and a tuple or string argument is not mapped. Over an infinite collection the result is an infinite list: Apply(x ↦ (x, x), Range(1, ∞)) is [(1, 1), (2, 2), …].
• Mapping a user function keeps a tuple argument whole and rejects lists of different lengths. For f := (x, y) ↦ (x, y), f([1, 2, 3], (10, 20)) now gives three results, each with the whole point (it gave [(1, 10), (2, 20)]). For f := (x, y) ↦ x + y, f([1, 2], [1, 2, 3]) was [2, 4] and is now an incompatible-dimensions error. To truncate, use Zip. Built-in operators are unchanged.
• A pipe stage behaves exactly as a call: xs |> f is f(xs). [[1,2],[3,4]] |> x ↦ (x, x) maps at every depth and is [[(1,1),(2,2)],[(3,3),(4,4)]]. A stage whose parameter is a collection takes the whole value: [[1],[2,3]] |> l ↦ Length(l) is 2 (it was [1,2]); write xs |> ((a, b)) ↦ a ∧ b to map over a list of pairs. A parenthesized stage now maps as an unparenthesized one does.

Absent Values and Collections

• A number read from an absent collection is NaN. With t := -1, First((1,2){0<t}) gave NaN or Missing depending on how it was evaluated; it is now NaN everywhere, typed integer | nan. The same applies to Second, Third, Last and At. A row, point or string read from an absent collection is still Missing.
• A search finds NaN, Missing and Undefined where the same value is. IndexOf([1, NaN], NaN) is 2, Contains([1, Missing], Missing) is True and Count([NaN, NaN], NaN) is 2. Before, Contains(L, Missing) was Missing and IndexOf(L, Missing) was NaN. NaN = NaN is still False.
• A predicate that returns Missing does not select the element. Filter([1, Missing, 3], x ↦ x > 0) is [1, 3] and Count of the same is 2 (Count threw an error). Any and All combine Missing as Or and And do: Any([1, Missing], x ↦ x > 2) is Missing, while Any([1, Missing, 3], x ↦ x > 2) is True.
• Element stays unevaluated when an unknown could make a value a member. Element(x, [1, 2]) with x free was False.
• An operator over an absent collection gives Missing. Insert, ReplaceAt, Append, Union, Intersection and SetMinus over Missing were incompatible-type errors, Join(Missing, [1]) was [Missing, 1], and Subset(Missing, {2,3}) was False. Append([1], Missing) is now [1, Missing] (it was an error).
• Types of collection operations that may give an absent value are corrected. Reduce([Missing], Max) is missing | number, and 2y with y: integer<1..3> | missing is number (it was integer<2..6>). Map(a ↦ 0, [[1,2],[3,4]]) is typed as a list of two results, not a matrix.
• Range membership accepts a value within rounding of either end. Element(1000, Range(0, 999.9999999999, 0.1)) was False and is True.

Numerics

• At machine precision, .N() of an integer too large for a double is ±oo. \frac{10^{400}}{10^{-400}} gave 1e+800 and is +oo, as 10^{800} already was. evaluate() is unchanged.

Resolved Issues

Arithmetic and Numerics

• A complex power, root or exponential keeps a small result. (10^{-10}i)^2 was 0 and is -10^{-20}. At machine precision, ∛(8i) was 0 and is √3 + i, and (10^{300}+10^{300}i)^{0.3} was NaN.
• .N() of an exact value beyond the double range is correct. \frac{10^{300}+1}{10^{400}} was 0 and is 1e-100; \ln(10^{400}) was +oo; 10^{400}-10^{400}+1 was NaN. An exact integer beyond 2^53 no longer compares equal to its neighbor with .is().
• Exact values: \log_2(2^{100}) is 100, and ((2·10^{30}+1)/10^{30})·2^x no longer simplifies to 2^{x+1}.
• .N() of an infinite series with no limit stays unevaluated. Σ_{k≥1} (−1)^k gave −1 and Π 2^((−1)^k) gave 0.5.
• Min and Max of a range. Min(Range(5, 1)) was 5 and is 1; Min(Range(1, −∞)) is −∞. The extremum of an empty range is NaN: Max(Range(1, 5, −1)) was 2.
• A multiple integral evaluates a scalar bound once, so a Random() bound gives a single value. Range(0, 1, +∞) is [0] (it was [NaN]).

Linear Algebra

• Norm, SingularValues and Eigenvalues are correct for matrices with very small or very large entries. Eigenvectors returns a basis for a repeated eigenvalue: [[2,0],[0,2]] gave [[1,0],[1,0]] and now gives [[1,0],[0,1]]. Norm(Linspace(a, 1, 3)) with a symbolic a stays symbolic (it was 0). Mean, Median and Variance accept exact values beyond the double range.

Absent Values and Collections

• Sum([Missing]), Product([Missing]) and Sum(Join([3], Take(Missing, 1))) are NaN (the last was 3). Sort by a key puts absent values last.
• A Join with an operand that depends on an undecided If stays unevaluated.
• A rebuilt Sum, Product, Loop or Comprehension keeps the type of its index. [10i for i in [1, 2, 3]] could be typed indexed_collection<number> instead of indexed_collection<integer>.

Compilation

• JavaScript: a Sum with a square root no longer gives NaN when the square root is real by its type.
• Complex division in compiled JavaScript no longer overflows or underflows. A recursive function that returns complex matrices compiles.
• Python: Degrees parenthesizes its operand, Norm handles NaN and infinite entries, and Dot and Cross refuse an operand that can be absent.
• GLSL and WGSL: color functions reject infinite channels, and Gamma has a pole at every negative integer. interval-js no longer treats a very small nonzero angle as the pole at 0.

LaTeX Parsing and Serialization

• A Tuple parsed in an explicit scope keeps the \operatorname{Tuple} spelling when serialized outside that scope.

0.136.2 2026-09-25

Resolved Issues

• A call with a nan | real argument is typed from the function body. With d := (l, b) ↦ l + b·Length(l), L: list<real> and a: nan | real, d(L, a) was typed collection | number; it is now list<nan | real>. A union argument that may hold a collection still keeps the declared result.
• The product of a nan | real factor and a list keeps nan in its element type. b·L, with b a function parameter, was typed list<real>; it is now list<nan | real>, as for a declared symbol. A factor that may also be infinite gives list<infinity | nan | real>.

0.136.1 2026-09-25

Behavior Changes

Absent Values

• The arithmetic methods read an absent operand (Missing or Undefined) as NaN, as the Add, Multiply and Divide operators already did. ce.box('Missing').add(ce.box(1)) is NaN (it was "Missing" + 1), and ce.box('Missing').mul(0) is NaN (it was 0). Beside a point the whole point is absent: ce.box('Missing').mul(ce.box(['Tuple', 1, 2])) is Missing. The .neg() and .inv() methods are unchanged.
• Canonicalization no longer removes an absent operand from arithmetic. Multiply(Missing, 1), Add(Missing, 0) and Divide(Missing, 1) were Missing, Divide(Missing, Missing) was 1 and Multiply(0, 2, Missing) was 0. They are now NaN, and so is Divide(Missing, 2). Multiply(2, Missing) and Add(Missing, x) keep their form and evaluate to NaN.
• An out-of-range read of a numeric collection with an absent element is NaN. Third((1, Missing)) and At([1, Missing], 5) are NaN (they were Missing), as Third((1, 2)) is. A collection with only absent or non-numeric elements still gives Missing.
• Trace(Missing) and Trace(Undefined) are NaN, as Norm(Missing) is. They were incompatible-type errors.
• Sort and Ordering put an absent element last. Sort([3, Missing, 1]) is [1, 3, Missing] and Ordering([3, Missing, 1]) is [3, 1, 2]; both stayed unevaluated. An absent element sorts after NaN, also with a comparator or a key.

Types

• Arithmetic over a Tuple with a list coordinate is typed error. Its value is already an incompatible-type error, but Add(Tuple(1, 1), Tuple(A, B)) (with A, B lists) was typed tuple<list<real>, list<real>>. It, and the functions that apply to each coordinate of such a tuple, are now typed error.
• An operand typed error is kept as it is inside a sum, product, power or negation. Before, it was wrapped in a second incompatible-type error that hid the real one. (1 − t)D − tD, with D the tuple above, now evaluates to one error that names the tuple.

Linear Algebra

• SVD returns the singular values in descending order. The Σ of SVD([[1e200, 0], [0, 2e200]]) was [[1e200, 0], [0, 2e200]]; it is now [[2e200, 0], [0, 1e200]]. The sign (or phase) of each pair of singular vectors is fixed so that the largest entry of each column of V is real and positive.
• SingularValues and SVD accept entries whose magnitudes span up to 10^290 (it was 10^150). SingularValues([[1e200, 3e-80], [2e-80, 1e-85]]) was unevaluated; it is now [1e200, 1e-85].

Arithmetic and Numerics

• The fourth root of a negative number evaluates to its exact principal value when it has one. ∜(−1) and (−1)^{1/4} are (√2/2)(1 + i), and ∜(−4) is 1 + i; they stayed a Root. ∜(−2) still stays a Root, and an odd root of a negative number keeps its real value (∛(−8) = −2).

Resolved Issues

Compilation

• A restricted list, point or list of points compiles on the interval-js target, as on javascript. [1,2]\{0<t\}, (A\{0<t\}, B) and A\{0<t\} + 1 failed with "no lowering for List". When the condition fails, the result is empty.
• Compiled JavaScript gives the shifted points for (A\{0<t\}, B) + (1, 1) and (A, B)\{0<t\} + (1, 1), as the interpreter does. They failed to compile. When the condition fails, compiled code gives NaN where the interpreter gives Missing.
• A compiled out-of-range First, Second or Third of a numeric collection is NaN, as in the interpreter. Third((1, x)) was undefined.
• Sort and Ordering of a list with an absent element compile to JavaScript, with the absent element last: Sort([3, Missing, 1]) gives [1, 3, undefined]. It gave [1, 3, NaN].

Linear Algebra

• The small singular values of a matrix with graded entries were 0. SingularValues([[1e20, 0.5], [0.5, 2.5]]) was [1e20, 0]; it is now [1e20, 2.5]. The same applies to SVD and to complex matrices.

Arithmetic and Numerics

• An even root of a negative number no longer drops the sign. \sqrt[4]{-16} simplified to 2 and ce.number(-16).root(4) gave the same; they now give √2 + √2·i. (-x).root(4) stays ∜(−x).
• ce.number(-8).root(-3) is −1/2: a negative odd index now keeps the real root. At high precision, the square root of a negative real is imaginary (2i for −4) instead of NaN.

Absent Values

• A point minus an absent value is Missing, as the sum is. (1, 2) - Missing was an incompatible-type error. The same applies to Undefined, to 2·Missing + (1, 2) and to .sub(). 3 - Missing is still NaN.
• Collection operators that reorder or select elements keep an absent element in the type. Sort([3, Missing, 1]) was typed list<unknown>; it is now list<integer | missing>. This applies to Reverse, Take, Drop, Unique, Filter and similar operators.

0.136.0 2026-09-25

Improvements

• A call of a user function that takes a whole list is typed and compiled from its argument. When the parameter is broader than the argument (a bare function declaration, or a list, collection or unknown parameter), the call is typed from the body and compiles to JavaScript. With L: list<real>, a chain such as d(s(u(L)), 1) was typed list<unknown> and failed to compile; it is now typed list<nan | real> and compiles.
• A seeded list draw compiles on the interval-js target. WithRandomSeed(seed, RandomChoice(domain, k)), with a literal seed and a literal Interval, Range or list domain, compiles to the same values as evaluate() and the javascript target. Before, it failed to compile.

Behavior Changes

Restrictions and Absent Values

• A restriction over a number masks to NaN. x\{c\} with c false is NaN when x is a number; it was Missing. 1\{c\} is typed integer | nan, and [10,20,30]\{[1,2,3] > 2\} is [NaN, NaN, 30]. A point, list, string or non-numeric value still masks to Missing. A scalar restriction beside a list or point goes into each element: [1,2,3] + 2\{c\} is [3\{c\}, 4\{c\}, 5\{c\}].
• A restriction over a list stays one restriction while its condition is undecided. [1,2,3]\{c\} evaluates to itself instead of a list of restricted elements, so re-evaluating it once c is false gives Missing, as a fresh evaluation and compiled code do. Its type is missing | vector<integer^3>.
• A collection operator over an absent collection gives the absent value of its result. Reverse(Missing), Sort(Missing), Map(f, Missing) and Filter(Missing, p) are Missing, and Length(Missing) is NaN. They were incompatible-type errors. Reverse([1,2]\{c\}) is Reverse([1,2])\{c\}.
• An Undefined element of a list, tuple or set is typed as absent. [1, Undefined] is typed list<integer | missing> (it was vector<2>), and (2·[1, Undefined]).N() is [2, NaN] (it was [2, 2·"Undefined"]). Arithmetic on a point with an absent coordinate gives NaN there: (1, Missing) + (1, 1) is (2, NaN).
• Dot([1, Missing], [1,1]) is NaN, as Norm((1, Missing)) is, instead of an error.

LaTeX Parsing and Serialization

• A bracketed list in the parentheses of a function call keeps the parentheses in the raw and structural forms. With A a function, A([1]) parses to ["A", ["Delimiter", ["List", 1]]]; it parsed the same as A[1]. The canonical form is unchanged: both are ["A", ["List", 1]].

Arithmetic and Numerics

• √i evaluates to (√2/2)(1 + i), and √(−i) to (√2/2)(1 − i). They stayed a symbolic Sqrt.
• e^{a + bπi} with an exact real part evaluates to an exact product. e^{1 + iπ} is -e and e^{ln 2 + iπ} is -2; they stayed unevaluated. e^{1 + 0.5iπ} is 2.718281828459045i with no rounding residue.
• The sign of a trigonometric function of an exact rational multiple of π is known. Cos(π/3).isPositive and Cos(2π/3).isNegative are true; they were undefined.
• Csc and Cot of a tiny nonzero angle give a signed infinity. Csc(5e-324) is +oo and Csc(-5e-324) is -oo; both were ~oo. Compiled JavaScript and Python give the same result.
• A Range whose end point is on the step grid keeps its last element. Range(0, 0.3, 0.1) had 3 elements because 0.3/0.1 is 2.9999999999999996; it now has 4. The element count allows a very small tolerance, so Range(0, 2.5, 1) still has 3 elements. A unit-step range one rounding error short of an integer span gains its last element: Range(1.3, 2.3) is [1.3, 2.3] (it was [1.3]). The interpreter and every compile target use the same rule.

Collections

• Map with its arguments in the wrong order is always an error. Map(P, fn) is an incompatible-type error also when P is a symbol; it stayed unevaluated.

Compilation

• Last compiles on the GLSL and WGSL targets for a point or a list of 2 to 4 numbers, as First does.

Resolved Issues

Compilation

• On the interval-js target, a point within rounding error of a pole of Tan, Cot, Sec or Csc is singular, as the interpreter gives ~oo. It gave a very large finite interval, which plotted as a wrong segment. Also, above 2⁴⁰ every point was singular (tan(10²²)).
• A bare function over a list compiled to wrong values. s(L) returned { re: NaN, im: NaN } objects instead of numbers.
• A compiled Range with a compound step or start has the right elements. Range(0, 10, d - 498) at d = 500 gave [-498, 2, 502, …]; it now gives [0, 2, 4, 6, 8, 10], as the interpreter does.
• A compiled count, bound, position or index that is not provably real is read through its real part. At K = 4, Take(L, \sqrt{K}) was [], Range(1, \sqrt{K}) was [], and RandomChoice(L, \sqrt{K}) threw.
• A compiled Range whose step points away from its end (Range(5, 1, 1)) threw a RangeError, and a zero step gave an infinite length. Both give [].
• A mutually recursive function pair that uses an operator with no lowering reports that operator. Compiling h(2) reported a misleading type error instead of "Could not compile Simplify".
• PointX(Missing) and other component reads of an absent value compile to NaN on GLSL and WGSL. The WGSL output was invalid.

Types

• Multiply keeps a nan member. 2·x with x: nan | real was typed number; it is now nan | real, as Add already was.
• An unknown symbol in a sum or quotient is accepted where a matrix is expected. With P declared unknown, Histogram(P + 0.5, 1/100) was an incompatible-type error; it is now valid and types P as matrix, as Histogram(2P, 1/100) did. Determinant(A + 1) works the same way.
• An Undefined operand of arithmetic is typed as absent. Multiply(Undefined, [1, 2, 3]) was typed vector<integer^3>; it is now vector<3>, the same as with Missing.
• At with two indices through a row that may be absent is typed. At([[1,2]\{0<t\}, [3,4]], 1, 2) is typed integer | missing | nan (it was unknown).
• A union type no longer lists a member twice. C + 1 with C: list<real | signed_infinity | nan> was typed list<infinity | infinity | nan | real>.

Calculus

• The normal CDF written as an integral evaluates over a list. With E(x) = \frac{1}{\sqrt{2\pi}s}\int_{-\infty}^{x}\exp(-\frac{1}{2}\frac{Z^2}{s^2})\,\mathrm{d}Z, s = 0.577 and G = [-1, 0, 1], E(G).evaluate() was a list of symbolic integrals; it is now [0.0415…, 0.5, 0.9585…]. An integral with a list bound gives one integral per element under .N(), and PolynomialDegree(x²/a, x) is 2.
• .N() of a number times a list of Measurement values evaluates each element. 0.5·[2 ± 0.1, 3 ± 0.1] is [1 ± 0.05, 1.5 ± 0.05].

LaTeX Parsing and Serialization

• A number before a comprehension or a range in brackets is a product. 0.01\left[u+1 \operatorname{for} u = [1,2,3]\right] and 4[1...3] were unexpected-operator errors; they now parse to Multiply, so the serialized form of such a product parses back. A bracket after a symbol is still an index, so x times a comprehension now serializes with \times.

Arithmetic and Numerics

• At machine precision, a rational is the correctly rounded double. Divide(Pi, 6).N() with precision: 'machine' was 0.5235987755982998, and (1/6).N() printed 0.166666666666667. Machine-precision values that involve a non-integer rational can change in the last digits.
• At 21 digits, \sin(\pi/6), \cos(\pi/3) and \sin(30) in degree mode are exactly 0.5. They were 0.500…001.
• Trigonometric functions of a large exact integer, or a large multiple of π, are correct under .N(). Sin(12345678901234567890123).N() is −0.4205828591… (it was 0.9918…) and \sin(10^{30}\pi).N() is 0 (it was 0.8838…). In degree mode, \sin(10^{30}\pi).N() is 0.350160229920896711994 (it was 0). \cot(5\pi/2) and \tan(\pi) at machine precision are 0.
• An imaginary literal with an integer coefficient is exact. 4i, -2i and ["Complex", 4] were inexact, so \sqrt{4i} gave 1.414… + 1.414…i; it is now \sqrt2(1 + i), and (4i)^2 is -16. 1.5i stays inexact.
• A large double prints in exponent form at machine precision. ce.box(1e300).toString() printed 301 digits, and 1e23 printed as 99999999999999991611392.
• ComplexInfinity has one MathJSON form at every precision. At machine precision, Add(1.5, 2.5, ComplexInfinity).N() gave ["Complex", 1, "PositiveInfinity"].
• A number literal follows the engine's current precision. √2 created at machine precision still gave a double from .N() after ce.precision = 50. Because of this, Max, Min and Sort also ordered √(2 + 10⁻³⁰) below √2 at machine precision.
• A perfect-power root of a large integer is extracted. ∛(10^60) is 10^20 and ∛(10^61) is 10^20·∛10; they stayed a root.
• Simplifying \sin(10^{30}\pi) no longer fails an internal assertion.
• A constant that contains a special function can be ordered. Max(Γ(1/3), 3) is 3 (it stayed unevaluated), Min(erf(1), 1/2) is 1/2 and Max(ζ(3), 1) is ζ(3).
• Artanh(10⁻³⁰).N() is correct at 21 digits. It was 0.
• Zeta(1 + 10⁻³⁰).evaluate() is no longer the pole ~oo. The exact rational argument was read as the integer 1.

Absent Values and Points

• Distance and Dot of a point with an absent coordinate are NaN. Distance((1, Missing), (0, 0)) was an expected-value error and Dot((1, Missing), (1, 1)) stayed unevaluated. Compiled JavaScript also gives NaN instead of throwing. PointY((1, Missing)) is typed missing.
• A function applied to a list of points gives Missing in an absent element. Sin([Missing, (3,4)]) gave NaN there, as did Negate and Sqrt. A numeric list keeps NaN (Sin([Missing, 1]) is [NaN, sin 1]).
• An absent point inside a matrix of points stays Missing when a restricted point is added whose condition fails.
• Norm of a matrix with an absent element is NaN. Norm([[1, 2], [3, Missing]]) stayed unevaluated. A text element is an incompatible-type error.

Other

• A seeded draw from an open interval is cached. WithRandomSeed(s, RandomChoice(Interval(0, Open(1)), n)) was treated as impure, so each read of one element drew the whole list again. Open and Closed are now defined operators, and ce.box(['Open', 1]).isPure is true.
• A Delimiter nested thousands of levels deep boxes to its operand. Past about 1000 levels, the result was a non-canonical Delimiter and an error was logged to the console.

0.135.0 2026-09-25

Behavior Changes

• A parenthesized LaTeX list with a list coordinate is a list of points. With A = [1, 2, 3] and B = [10, 20, 30], (A, B) canonicalizes to PointList(A, B) and evaluates to [(1, 10), (2, 20), (3, 30)]. Arithmetic, Distance and functions apply point by point; a scalar coordinate is repeated ((A, 0)), and lists of unequal lengths stop at the shortest. This applies only to parentheses, and only when every coordinate is a number or a list of numbers. (A, B) serializes as \operatorname{PointList}(A, B). Compiled javascript and interval-js code gives the same points; GLSL and WGSL do not compile it.
  • A MathJSON ["Tuple", …] built by code (for example the result of Tally or SVD) stays data. Arithmetic or a function such as Sin applied to such a tuple with a list coordinate is now an incompatible-type error: Sin(Tuple([1, 2], 3)) was ([sin(1), sin(2)], sin(3)). Use PointList(A, B) for a list of points. PointX, Norm, Dot, Length and Abs of such a tuple are unchanged.
  • A user function whose parameter is a point (tuple<real, real>, …) or has no declared type maps over a list of points: with x = [1, 2] and y = 3, k((x, y)) is [k((1, 3)), k((2, 3))]. A parameter declared as a list still takes the list whole.
  • PointList with a source that may be absent (PointList(A\{0<t\}, B)) evaluates to Missing when the source is absent (it repeated Missing in every point).
• Dot and Norm accept an absent operand. Dot(Missing, (1, 1)) was an incompatible-type error; it is now Missing (NaN when the result is a number). Norm(Missing) and Norm([1, Missing]) give NaN, as Abs does.
• In strict mode, At rejects a colon index. [1,2,3][1:2] parses to At(List(1,2,3), Colon(1,2)), which is now an incompatible-type error (it stayed unevaluated). Use the range notation: [1,2,3][1...2] is [1,2].
• Map over a range or an indexed collection is typed list<T>, not indexed_collection<T>: Map(f, Range(1, 3)) is list<…>.
• A product of rational numbers is typed rational, not real: 2x with x declared rational, and k/2 for an integer k.
• Sin, Power and a scalar factor of a restricted point whose condition is false answer Missing, not NaN (\sin((0,1)\{t<0\}) at t = 1). Code that tested the result with isNaN must test for Missing.
• Products and quotients with a restricted point are rejected when the expression is created, as for a point without a condition. With P = (0,1)\{0<t\}, P·Q, t/P and (2,3)/P are now invalid (no-product-between-points, no-division-by-point). P/t, t·P, -P, P + Q and Dot(P, Q) do not change.
• A point plus a list of numbers is an incompatible-type error: (0,1) + [10,20,30] was valid and evaluated to a list of errors. A point plus a list of points still adds the point to each point.
• The same rules apply to a list of points L, with or without a condition. L·(1,1), L·L, 2/L, L/(1,1), L + 1 and L + [1,2] are now invalid when created (before, most evaluated to a list of errors, and L/L was [1, 1]). L·2, L/2, -L, L + (1,1), L + L, L·[1,2] and [1,2,3]·(0,1) do not change.
• On glsl and wgsl, a number added to a point (t + \operatorname{PointList}(t, 1)) no longer compiles, as the interpreter gives incompatible-type.

Improvements

• New operator-definition flag threadsConditionals. When an operator sets it, a conditional value whose condition is not decided (P\{c\}, that is ["When", P, c], or a Which) moves out of the application: f(When(v, c)) evaluates to When(f(v), c). The value is true (every operand) or a list of 0-based operand positions. Dot, Cross, Norm, Distance, At, First, Last, PointX and similar accessors set it. A custom operator that reads a point or a vector whole can set it too.
• simplify() applies more of the loaded Fungrim identities. Identities loaded with loadIdentities() now run before the built-in rules, are also tried on the expression before its operands are simplified, and are used when their result is strictly cheaper. LambertW(-π/2) simplifies to iπ/2, Hypergeometric0F1(3/2, -z²/4) to Sinc(z), and RisingFactorial(1, n) to n!. When a later rule makes a result more expensive, simplify() now keeps the cheapest value reached: 3x/(x+1)² + 5/(x+1)² simplifies to (3x+5)/(x+1)².

Epsil

• The Epsil MCP server accepts LaTeX. The evaluate and parse tools accept format: "latex" (\int_0^1 x^2\,dx), and serialize can write LaTeX. Every evaluate result includes a latex form of the value.
• The epsil command accepts LaTeX. --from latex reads a LaTeX expression and --latex writes the result as LaTeX: epsil --from latex --latex -e '\frac{1}{2}+\frac{1}{3}' prints \frac{5}{6}.
• A Compute Engine guide for AI agents, a condensed reference for writing JavaScript or TypeScript with the library. The Epsil MCP server serves it as the resource epsil://docs/compute-engine-api.
• The Epsil MCP server has a compile tool. It shows the code generated for an Epsil program or a LaTeX expression: to is javascript (default), glsl, wgsl, python or interval-js; declarations gives the types of free symbols. The result has the code and freeSymbolTypes, or an error and a diagnostic.

Resolved Issues

LaTeX Parsing and Serialization

• A juxtaposition that was not canonicalized (InvisibleOperator, from a raw parse) now serializes with the parentheses it needs, so it parses back to the same expression: (x+1)(x+2), not x+1x+2; (2x)^2, not 2x^2.
• A number before a bracketed list is a product: 4[1,2] evaluates to [4, 8]; it was an unexpected-operator error. a[1,2] is still an index.
• A space before an index bracket is ignored: a [1,2] parses like a[1,2]. A visual space (\,, \quad) before a list makes a product: a\,[1,2] evaluates to [a, 2a].
• A bracketed list after a function name is its argument: \sin[a,b] and \Gamma[a] parse as Sin(List(a, b)) and Gamma(List(a)), and broadcast. They were an incompatible-type error.
• A Tuple with a list-of-numbers coordinate round-trips. It serializes as \operatorname{Tuple}(…), since (A,B) parses as PointList(A, B).
• Empty sets and lists serialize with no space: \lbrace\rbrace, not \lbrace \rbrace.

Points and Restricted Values

• Erf, Erfc, Erfi and ErfInv apply to each element of a list: Erf([0.5, 0.25]) is [0.5205…, 0.2763…]. It was an incompatible-type error.
• Distance, Norm, Dot and Cross accept a restricted point or a restricted list of points. With t free, Distance((3,4)\{0<t\}, (0,0)) is 5\{0<t\} (it was an error), and Dot([(1,2),(3,4)]\{0<t\}, (1,1)) is [3\{0<t\}, 7\{0<t\}]. A list that holds an absent point gives one value per point: Norm([(3,4), Missing]) is [5, NaN].
• The point accessors read restricted points. With t free, PointX([(1,2),(3,4)]\{0<t\}) is [1\{0<t\}, 3\{0<t\}] (it was the first point), and First((1,2)\{0<t\}) is 1\{0<t\} (it was an error). PointX([Missing, (1,2)]) is [Missing, 1] (it was Missing).
• The types of accessors and functions of restricted values include the absent case. First((1,2)\{0<t\}) was typed integer; PointY of a list with a restricted point is typed list<missing | number>; \sin((0,1)\{0<t\}) is typed missing | tuple<number, number> (it was number). PointX(Missing) is typed number, as its value is NaN.
• Accessors of a symbol declared as a restricted point with no value stay unevaluated; they were an incompatible-type error.
• A numeric function of a list restricted by a list of conditions works cell by cell: \sin([10,20,30]\{[1,2,3]>2\}) is [NaN, NaN, sin(30)]; it was a 3×3 matrix.
• An absent scalar times a list gives NaN in every cell: Missing · [1, 2, 3] was [Missing, NaN, NaN].
• An absent point cell of a list answers Missing, as its type says: Map and 2·[P{c}, (2, 3)] gave NaN, or a mix of NaN and Missing.
• Dot of a restricted list of points and a point of another width is one incompatible-dimensions error, not a list of errors.

Compilation

• Compiled products of a list and a point match the interpreter. [1,2,3]\cdot\operatorname{PointList}(t,1) ran to NaN in JavaScript; it now gives the list of points. On GLSL and WGSL, a point times or divided by a list now fails to compile instead of giving one vector.
• Compiled arithmetic over a restricted point is correct. t\operatorname{PointList}(0,1)\{0<1\} ran to null in JavaScript. The product of two points, a division by a point, a point plus a number and an ordering of points now fail to compile on every target, where they gave a wrong value.
• Compiled Norm and Distance of restricted points match the interpreter. Norm([(3,4),(6,8)]\{0<t\}) gave 11.18 (the Frobenius norm); it is now [5, 10]. Distance, PointX, First, Second and Third of an absent point no longer throw a TypeError. Python refuses to compile Norm, Dot and Cross of an operand that can be absent.
• freeSymbols and freeSymbolTypes of a compiled block with a local declaration are correct: the symbols read by let y = x + 1 were missing.
• Compilation error messages are clearer. They start with what could not be compiled, then give the reason: Could not compile `SinIntegral`: the operator is known to the engine but target 'glsl' has no lowering for it. Code that matched the old text must match the new one; diagnostic.code is unchanged.
• A declined JavaScript compilation names the reason of a user-function call that could not be compiled (for example "argument 2 is a point with a complex-valued coordinate").

Other Fixes

• An element-wise result over more than 100 elements has the type of its expression. Assigning PointList(0, k) + PointList(cos k, sin k) with k = [1...101] to a symbol declared with that type threw a TypeCompatibilityError.
• The type of a numeric function of a value that can be absent keeps missing: PointX(V)^2 is typed list<number> | missing | number.
• PolynomialGCD and simplify no longer compute over long floats. A coefficient with more than 15 significant digits gives the GCD 1; a short decimal such as 1.5 is read as exact (PolynomialGCD(x² − 1.5x + 0.5, x − 1) is x − 1). With the integration rules, ∫ 1/(2+3x⁴)² dx takes 0.1 s instead of reaching the 30 s limit.
• Assigning an invalid function to a symbol declared with a signature no longer says the initializer "is not a function".
• The documentation of isEqual() said an identity such as (x+1)^2 vs x^2+2x+1 is true; it is undefined. Use isIdenticallyEqual().

Performance

• Repeated Apply(Derivative(f, n), x).N() at machine precision computes the derivative once: 1,000 points take 0.16 s (they took 1.94 s).
• Interpreted evaluation of scalar expressions is about 14% faster.
• Max, Min, Clamp and Sort of constants that are very close are faster: Sort of 200 such values is about 3× faster.

0.134.0 2026-09-24

Behavior Changes

Ordering

• Ordering functions order exactly. Max, Min, Clamp, Sort, ArgMax and the other ordering functions no longer treat two numbers closer than the tolerance as a tie: Max(1e-12, 2e-12) is 2e-12. Relational operators keep the tolerance, so 1e-12 < 2e-12 is still False. Real(1 + √2 i) now evaluates to 1.
• Sort without a key stays unevaluated when two elements cannot be ordered. Sort([2, y, 1, 3, x]) was [2, y, 1, 3, x]. Ordering follows Sort.
• Sort and Ordering put NaN last. Sort([3, NaN, 1]) is [1, 3, NaN], in the interpreter and in compiled JavaScript.

Colors

• Rgb colours are extended sRGB. A channel outside [0, 1] is kept: AsRgb(Rgb(2, 0, 0)) is Rgb(2, 0, 0) on every target, and conversions among Rgb, Oklab and Oklch no longer clip.
• Hsv and Hsl clamp saturation, value and lightness, also when they are infinite: AsRgb(Hsv(30, ∞, 3.14)) is Rgb(1, 0.5, 0) (it was an error). The hue is reduced modulo 360.
• Colour output is gamut-mapped, not clipped. ColorToString and the conversions to Hsv/Hsl map into the gamut with the CSS Color 4 algorithm: ColorToString(Oklch(0.7, 0.4, 30)) is #ff5843 (it was #ff0000). New operator GamutMap(color, gamut?), with gamut "srgb" (default) or "display-p3". ColorToString(c, "display-p3") gives color(display-p3 r g b). On a shader, wrap a final colour in _gpu_gamut_map_oklch() instead of _gpu_oklch_to_srgb().

Arithmetic and Numerics

• Only a safe integer is boxed as an exact integer. ce.number(2.0) is the exact 2, but ce.number(1e200) is the float 1e+200. For an exact large integer, use a bigint or a string ({num: "1e200"}). A literal with a decimal point (12345678.0) is a float. IsPrime(7.0) is still True.
• A float argument of a trigonometric function is never a special value. Sin(3.141592653589793) is 2.38e-16 (it was 0) and Arcsin(0.5) is a float (it was π/6); write Sin(π) or Arcsin(1/2) for an exact value. This fixes wrong exact answers such as Sin(π − 10^-30) = 0. e^{iπ} in degree mode is now −1, and Cot(10^-25).N() is 1e25 (it was ~oo).
• .N() of a Gaussian integer is a float: (1 + i).N() is no longer exact. .N() of a sum of quantities gives a float magnitude: (1/3 m + 1/3 m).N() is 0.6666… m.
• Tangent-like poles are detected relative to the argument. A value is a pole of Tan, Cot, Sec or Csc when the argument is within about 100 ulps of a pole: Cot(1e-13) is 1e13 and Tan(1.5707954) is 1078987.38… (both were ~oo); cot(1000π) is ~oo. Compiled code gives Infinity (JavaScript) or np.inf (Python) at a pole.

Linear Algebra

• Norm(A, 2) of a matrix is the spectral norm (the largest singular value); it was the Frobenius norm. Norm(A) and Norm(A, "Frobenius") are still the Frobenius norm. The result is exact for small or simple matrices; otherwise it stays Norm(A, 2) under evaluate() and gives a number under .N().
• Compiled JavaScript linear algebra selects real or complex computation from the operand types. A matrix<number> or vector<number> operand is read as real, except under mode: 'complex'. The compiled function checks at run time that such an input has no complex entry and throws an error that names it; entryChecks: false turns the check off. Before, Determinant(M) + 1 gave NaN for a complex entry. A function declared with a real result whose body is complex now fails to compile on JavaScript, as on GLSL and WGSL.

Integration and Time Limits

• Symbolic integration gives up at the same point on every machine. The integration rules and the compiler stop after a number of steps (300,000 per integral; new option stepBudget of loadIntegrationRules), not after a time. timeLimitMs now defaults to 30 s (it was 10 s) and guards only against a hang.
• A caller's deadline propagates out of compile(), integration and derivatives. When a ce.withTimeLimit span expires, the CancellationError is thrown: compile() of an Integrate throws instead of returning success: false, and NIntegrate throws instead of returning a partial result.

Improvements

• On glsl and wgsl, user functions accept complex arguments. A function whose parameter type does not say complex gets a second, complex definition when a call passes a complex value; \arg(f(x+iy)) now compiles. A real argument to a complex parameter is converted to vec2(x, 0.0).
• Every compile result has freeSymbolTypes: for each free symbol, its type, the type the target reads it as (float, vec2, complex, …) and whether it was declared or inferred.
• New compile option strictTypes: true: a compilation with an undeclared free symbol fails and names the symbol.
• On interval-js, PointX/PointY of a PointList of list symbols compile, and so does the sum of two such PointLists.

Resolved Issues

Arithmetic and Numerics

• Exact complex numbers stay exact. √2·(1 + i) is √2 + √2 i (it was 1.414… + 1.414…i), (√2(1 + i))² is 4i, |√2(1000 + 1000i)| is 2000, Abs(π i) is π and Conjugate(1 + √2 i) is 1 − √2 i. Erfc of a complex argument evaluates.
• Values very close to an exact number are no longer rounded to it: √(1 + 10^-30) − 1 evaluated to an exact 0 and 2^{10^-30} to an exact 1; they now stay symbolic.
• Exact numbers outside the float64 range keep their value. Norm([10^400, 1]) was +oo, Max(10^{-400}, 2·10^{-400}) was 10^-400, and √(10^30 + 1) was 10^15.
• Ordering is sound. An order is decided only when the rounding error allows it, so a value near 0 no longer gets a sign from rounding noise. Max(3.141592653589793, π) is π, Max(∞, π) is +∞, and complex values are never ordered. When the precision cannot decide, the engine tries a proof of equality (Max(sin²1 + cos²1, 1) is 1), then a higher precision. Sorting symbolic constants is about 8× faster.
• Abs of a real constant evaluates: |1 − π| is π − 1. Conjugate distributes over sums, products and powers. Distance((π, 0), (0, 0)) was an error.
• Arsinh and Arcosh of a big decimal are accurate near 0.
• A degree literal such as \sin(30^\circ) evaluates correctly in grad and turn mode: it gave 0.0082 in grad mode; it is 1/2.
• isIdenticallyEqual no longer answers undefined for some true identities such as (x+y)² ≡ x²+2xy+y².
• The hash of a dictionary does not depend on the order of its keys, so like terms that hold equal dictionaries combine.
• Writing valueDefinition.type or operator.signature updates the types already computed for expressions that use them.
• A use that narrows a symbol's type updates expressions boxed before: List(x) becomes vector<real^1> after a use makes x real.
• Timing returns the time first, in microseconds, then the value. Its description and signature now say so.

Statistics

• Mode breaks ties by the smallest value also under .N(): Mode([3.5, 2.5, 1.5]) is 1.5, not 3.5.
• The statistics operators accept real constants. Median([π, 3, e]) is 3 and Mean([π, 3]) is (π + 3)/2.

Linear Algebra

• Matrix functions keep the imaginary part of complex entries. SingularValues([[1, i], [0, 1]]) was [1, 1], Eigenvalues([[0, i], [i, 0]]) was [0, 0] (it is now ±i), and CholeskyDecomposition gave a real L. Decompositions of a complex matrix that are not supported now stay unevaluated.
• Exact complex matrix entries stay exact: Norm([[1, 1+2i], [0, 1]], 1) is 1 + √5 (it was 3.236…), and Inverse of a Gaussian-integer matrix is exact.
• SVD and SingularValues of matrices with very large or very small entries no longer give NaN or 0. When entry magnitudes span more than about 10^150, they stay unevaluated; Norm(A, 2) still answers.
• Norm of a lazy collection is computed: Norm(Sin(L)) stayed unevaluated for a list of a thousand numbers.
• determinant() of the numerics entry point threw a TypeError for a matrix of size 3 or more.

Compilation

• Compiled JavaScript: Sort/Ordering refuse element types they cannot compare; Unique accepts list<real | signed_infinity | nan>; Norm of a list with a complex entry is correct; Dot, Determinant, Inverse and the other linear-algebra functions fail to compile on a complex entry instead of giving NaN; Float64Array matrix rows are accepted; a nested function such as Trace(MatrixPower(A, 2)) compiles over matrix<real>.
• Compiled user functions read the right global when a Sum index or a local has the same name: \sum_{k=1}^{3} h(x) with h(s) := s + k returned the string "[object Object]…"; it is now 6 + 3i.
• Compiled Cot, Coth, Round, Fract, Haversine and odd roots compute their operand once, so Cot(Random()) uses one random draw.
• GLSL and WGSL: real-only functions such as Gamma, Erf and BesselJ fail to compile on a complex operand instead of giving invalid code; Root of a complex radicand is the principal root and its compound degree is parenthesized; an If with one complex and one real value is valid code; a complex value in a point component fails to compile; a complex result declared over a real body is converted; 0^w is 0; a hue outside [0, 360) is correct. Mandelbrot and Julia take complex points, passed as vec2.
• Python: GammaLn of a negative real uses scipy.special.gammaln (it gave nan); Norm of an operand of unknown rank checks the rank at run time; Artanh, Arsinh, Sign, Mean, Variance and Trace compute complex values; 2(x + z) compiled to 2 * x + z.
• interval-js: a free symbol declared complex fails to compile (it was read as real); degree literals compile.
• Colour conversions no longer round each channel to 8 bits: AsRgb(Hsv(30, 1, 1)) compiled had a green channel of 0.498.

Integration

• Numeric integration of an integrand that is NaN everywhere returns NaN quickly; it returned 0 after about 32,000 evaluations. A compiled nested integral that used its whole budget returns NaN.

Performance

• evaluate() of a Sum of symbolic terms is no longer quadratic: Sum(sin(i), i, 1, 1000).evaluate() takes about 80 ms (it took about 770 ms).
• Like terms in a sum are found faster: Expand((x+y+z+1)^32) is about 2.7× faster.
• .N() of Add and Multiply approximates each operand once: (1 + Σ_{i=1}^{1000} sin i).N() takes about 12 ms (it took about 0.9 s).
• More element-wise operations over large lists of machine numbers run on doubles, such as L / 3 and Exp(L) under evaluate().
• Monte Carlo integration is faster: 10⁷ samples take about 0.4 s (about 3.6 s before).
• Repeated derivatives of the same function are cached.

0.133.0 2026-09-22

Breaking Changes

• StringFrom(value) without a format reads integers as Unicode code points. A finite non-negative integer, or a list of such integers, is now decoded: StringFrom(65) is "A" and StringFrom([127467, 127479]) is "🇫🇷"; they were "65" and "[127467,127479]". Other values (NaN, non-integers, negative numbers, tuples such as (65, 66), strings, symbols) are still printed. To get the decimal text of a number, use StringFrom(128287, "default"). The empty list is now "" instead of "[]".
• Equal and NotEqual of two collections are undecided when a pair of elements is undecided. Equal([1, Missing], [1, Missing]) was True and Equal([1, Missing], [1, 2]) stayed unevaluated; both now answer Missing (NaN in compiled code), and NotEqual answers the same. One undecided pair is enough: Equal([1, Missing, 3], [2, Missing, 3]) is Missing, not False. A length mismatch is still False, and [x] = [y] with free variables still stays unevaluated. .isEqual() and IdenticallyEqual still answer true for two such lists.

Behavior Changes

• Undefined is an absent value, the same as Missing. In a numeric argument it answers NaN: Cos(Undefined), 2 · Undefined and Sqrt(Undefined) are NaN, where they stayed unevaluated. The comparison and logical operators and the statistics operators read it as Missing: Equal(Undefined, 1) and And(Undefined, True) answer Missing instead of False and Undefined; Mean([1, Undefined, 3]) is NaN; IsMissing(Undefined) is True; Coalesce(Undefined, 2) is 2; First(Undefined) is Missing instead of an incompatible-type error. A bare Undefined still evaluates to itself, with type unknown.
• A function declared with scalar parameters broadcasts over collection elements in callbacks. With ce.declare('f', '(number) -> number') and f := x ↦ 2x, Map(f, [[1, 2], [3, 4]]) now answers [[2, 4], [6, 8]], as Map(x ↦ 2x, …) already did; it was an incompatible-type error. This applies to Map, FlatMap, Reduce, Fold, Scan, MaxBy, MinBy, ArgMax, ArgMin, Sort, Ordering, GroupBy and ChunkBy, and to compiled JavaScript. Predicates that must answer a single boolean (Filter, Any, All, CountIf, TakeWhile, DropWhile) keep the error, and tuple and string parameters do not broadcast.
• A symbol value keeps the meaning it had where it was assigned, in compiled code too. With a := 3t + 1 and g := t ↦ t + a, the t in a is the global t, not the parameter of g. The interpreter already gave g(2) = 3t + 3; compiled code now agrees (it substituted a into g and answered 9). The same applies inside Sum and other big operators. The glsl and wgsl targets, and a compiled lambda that shadows the name, now fall back to the interpreter for such values.
• A loop index no longer changes a global read inside a called function. With w declared real without a value and W(x) := [w·x, x], [W(w)[1] for w in [1, 2, 3]] is now [w, 2w, 3w] and Σ_{w=1}^{3} W(w)[1] is 6w, the same as with another index name. They were [1, 4, 9] and 14.
• An element-wise operation on a single empty list answers the empty list. Sin([]), -[], Not([]), Abs([]) and [] < 3 answered Nothing; they now answer [], as 2·[] already did. Sin([]) + [1, 2] is now an incompatible-dimensions error instead of [1, 2]. PointList and Zip still pair up to the shorter input.
• RandomChoice of a string answers a string. RandomChoice("abc", 5) is now a five-character string instead of a list of characters, and RandomChoice("abc", 0) is "". To get the old result, use Characters(RandomChoice("abc", 5)) or draw from Characters(s). Random(s) still answers one character. Compiled JavaScript now supports a string source.

Resolved Issues

Types

• A function declared with unknown slots follows its current body. With ce.declare('phi', '(unknown) -> unknown'), phi := x ↦ 1 + c(x), and c defined later as c := i ↦ 2i, phi now reports (unknown) -> number; it kept -> broadcastable<number>, and compiling 2\cos(\phi(x)) failed. A re-assignment also updates the reported signature.
• A restricted value can be assigned wherever its unrestricted value can. With v > 3 assumed, ce.assign('v', ce.parse('5\{a>0\}')) threw an error about the assumptions; it now binds, as 5 does. 1\{a>0\} is still refused, and x := 7\{a>0\} is now refused under x = 5 as x := 7 is. A bare Missing can now be assigned to a symbol with an assumption.

Arithmetic and Numerics

• A deep expression no longer overflows the stack. ce.parse('1-2-3-…-N') threw RangeError: Maximum call stack size exceeded past about 500 terms. A 5,000-term chain and a 2,000-deep Sin nest now parse, canonicalize and evaluate.
• Round rounds a half away from zero at every precision. At machine precision, Round(-0.5) was 0 and Round(-1.5) was -1; they are now -1 and -2, as at the default precision and in compiled code.
• Exp(x).N() and Exp(x).evaluate() agree to the last digit at machine precision. Exp(1.1) gave 3.0041660239464334 or 3.004166023946433 depending on the method, and .N() could vary with the Node version. Exp(π).N() is now 23.140692632779267 (it was 23.140692632779263).
• Linspace between endpoints whose difference overflows a double. Linspace(-1e308, 1e308, 3) was NaN, +oo, +oo; it is now -1e308, 0, 1e308.
• A huge float product with a vector stays a float. 1e308 · [0.9, …] answered 9e307 as an exact 308-digit integer; it is now a float.
• FindFit and FindRoot report a time limit reached during setup as the "setup" phase again, instead of "solve".

Collections

• Dot broadcasts over a list of points. With L: list<number>, Dot(PointList(1, L), PointList(3, 4)) was an incompatible-type error; it now answers a list<number>, one inner product per point, as Dot((1, L), (3, 4)) already did. Two point lists are paired element by element; different lengths or widths give incompatible-dimensions. This also works in compiled JavaScript. The Python target now rejects a point-list operand instead of returning a wrong result.
• Ordering stays unevaluated when the order cannot be decided. Ordering([[5], [1], [3]], x \mapsto 2x) answered []; it now stays unevaluated, as Sort does.
• Max and Min fixes. Max([[1, Missing], 3]) is now NaN (it was Max(3, Missing)), and Max(Linspace(1, 5, 0), 1) is now 1 (it stayed unevaluated). Max(Reverse(Map(f, xs))) no longer calls f twice per element.
• simplify() keeps Max, Min, Supremum and Infimum of one collection. Min(Map(Sin, RealNumbers)).simplify() returned the Map itself; with the Fungrim identities loaded it is now -1. Max().simplify() is NaN.

Compilation

• Missing and Undefined compile to a value, not a free variable. Compiled IsMissing(Missing) answered false, Coalesce(Missing, 2) answered undefined and Median([1, Missing, 3]) answered 3; they now match the interpreter (True, 2, NaN). In JavaScript they compile to undefined, in GLSL, WGSL, interval and Python code to NaN. IsMissing on a variable not given to the compiled function now answers true.

LaTeX Parsing and Serialization

• A call whose argument uses the same function is read as a call. f(yf(x))(x+y) was the product Multiply(f, y, x + y, f(x)); it is now f applied to y·f(x), times x+y. q(2q) with an undeclared q is still a product.
• A sequence with a second ellipsis after its last term. a_1 a_2 \dots a_k \dots parsed to a Range with an incompatible-type error; it now parses to the sequence a_1·a_2, …, a_k, ….

Epsil

• A sum type can conform to a protocol under its own name. With type shape = circle(r: number) | square(s: number), type shape is Area { … } was rejected with protocol-conformance-target-invalid; it now applies the implementation to each variant, including variants added in a later program. Generic sums and ce.declareProtocolImplementation() are unchanged.

Performance

• Arithmetic and common functions over large lists of machine numbers are computed on doubles. At machine precision, 2·L, L + M, L·M, −L, Sin(L), Sqrt(L), Ln(L), L^2 and similar expressions over lists of more than a hundred elements, and Sum, Max, Min and PointX/PointY/PointZ of such lists, are now much faster: over ten thousand elements, Sum(Sin(L)) went from 58 to 2 ms and Min(1, 2 − 2·L) from 81.5 to 4.0 ms. The result is a List of numbers where it was a lazy Map(f, L), so Norm(2·L) is now a number. Some values can change in the last digit.
• Reading a symbol that holds a large list of numbers or points is faster. A written-out list of numbers now evaluates to itself, and coordinates are read directly: over ten thousand points, Length(C) went from 1.2 to 0.03 ms and PointZ(C) is about 14 times faster.
• Factoring a deeply nested quotient is much faster. The third derivative of √(x+√(x+√(x+√(x+x)))) takes half the time it took before.

Benchmarks

Numeric performance (200-digit precision)

Median time per call, in microseconds — lower is better. — means the tool returned no usable result at that precision.

Expression	CE (current)	CE 0.132.3	SymPy	math.js	Mathematica
\pi^2	12	11	173	66	3.9
\sin 1	22	22	217	190	5.2
\cos 1	20	22	219	253	7.0
\ln 2	16	18	340	4,784	3.8
e^{\pi}	17	17	215	5,055	4.7
\zeta(3)	1,397	1,426	275	—	51
\Gamma(\tfrac13)	736	729	355	—	203
\psi(\tfrac13)	649	647	2,812	—	167

Symbolic capability & performance

Each cell is how many times faster than Mathematica that engine is on the case (higher is better). — means the engine can't do the case. The CE + R/F column loads the optional Rubi integrator and Fungrim identities (loadIntegrationRules / loadIdentities).

Operation	CE (current)	CE + R/F	CE 0.132.3	SymPy	math.js	Mathematica
Antiderivatives
\int\frac{1}{\sqrt x}\,dx	3.4×	2.1×	2.5×	0.6×	—	1×
\int\frac{x}{\sqrt{1-x^2}}\,dx	5.7×	1.1×	4.7×	0.09×	—	1×
\int\frac{1}{x^3+1}\,dx	3.8×	0.6×	3.4×	0.3×	—	1×
\int\frac{\sqrt x}{1+x}\,dx	—	1.1×	—	0.1×	—	1×
\int\frac{x}{(1+x)^{1/3}}\,dx	—	0.8×	—	0.01×	—	1×
\int\frac{x^2}{(1+x)^{1/3}}\,dx	—	0.8×	—	0.005×	—	1×
Derivatives
\tfrac{d}{dx}\sqrt{1-x^2}	0.02×	0.02×	0.02×	0.001×	0.004×	1×
Simplification
\sqrt{3+2\sqrt2}	41×	25×	24×	—	—	1×
\sqrt6\,x+\sqrt2\,x	290×	279×	234×	12×	90×	1×
Evaluation
\lim_{x\to0}\tfrac{\sin x}{x}	27×	14×	24×	2.9×	—	1×
\lim_{x\to\infty}(1+\tfrac1x)^x	4.3×	4.1×	4.1×	2.1×	—	1×
\int_1^2\tfrac1x\,dx	2313×	3076×	1921×	90×	—	1×
\int_{-\infty}^{\infty} e^{-x^2}\,dx	182×	73×	157×	2.2×	—	1×
Solving
x^4+x^2-1=0	0.2×	0.2×	0.2×	0.06×	—	1×
x^3-x-1=0	1.4×	1.4×	1.2×	0.04×	—	1×

Across the cases both solve, Compute Engine is a median 3.8× faster than Mathematica (up to 2313×).

_{Measured 2026-09-22 · SymPy 1.14.0 · math.js 15.2.0 · Mathematica 15.0.0 · Node v26.9.0. Correctness is verified numerically against an independent mpmath reference.}

0.132.3 2026-09-20

Improvements

• Max and Min of a scalar and a collection compile on the interval-js target. Min(1, L), Max(0, Min(1, L)) and Min(1, [x, 2, 3]) used to fail to compile there. An empty collection beside a scalar contributes nothing (Min(1, []) is 1). Min(L, M) also compiles.
• Faster coordinate access on a large written-out list of points. PointX, PointY and PointZ over a List of more than 100 points now read the coordinates directly. On ten thousand points, PointZ(C) went from 165 ms to 79 ms.
• A symbolic integration that timed out during compilation is not searched again for a target that declares wider types for the same symbols. For example, a search that timed out with k: real<0..>, x: real is not repeated with k: real, x: number. The emitted code does not change.

Resolved Issues

• Integrating xⁿ·eˣ with a symbolic exponent no longer hangs. ce.parse('\int x^n e^x dx').evaluate() did not return. The same happened with xⁿ·sin x and (2x+1)ⁿ·eˣ. These integrals have no elementary closed form and now come back unevaluated at once. Integrals with a positive integer exponent, such as x³·eˣ, are unchanged.
• The record of a timed-out symbolic integration is found again when the host declares its symbols again for each compilation. In 0.132.2, a host that made the same declarations in a new scope for each compilation repeated the two-second search every time.

0.132.2 2026-09-19

Improvements

• Faster compiled point arithmetic over a list of points. PointX(0.3(t, t) + 4[(x₁, y₁), …]) now compiles to 0.3t + 4[x₁, …], without building the points. On a list of 7,225 points, a call went from 2.55 ms to 0.14 ms. On the glsl and wgsl targets, a list of points whose coordinates are computed at run time now compiles to a float[N] array of coordinates.
• A symbolic integration that timed out is not repeated by the next compilation of the same integral. Compiling an Integrate searches for a closed form for up to two seconds, then uses numeric integration. The next compilation of the same integral (for example, for another target) now goes to numeric integration at once. An assignment, an assumption, or a change of the angular unit or the precision clears this record. The emitted code does not change.
• Functions over a list that mixes complex and real elements compile to JavaScript. For example, Abs, Divide, Power, Real, Exp and Sign over Which(0 < L, i·L, True, 0) used to fail to compile. Before, only Add, Subtract, Multiply and Negate compiled.

Resolved Issues

• Compiled complex arithmetic in JavaScript gives the same results at a pole as the interpreter. 1 / (0·i) is ~oo, its absolute value is +∞ and its reciprocal is 0; the compiled code gave NaN for all three.
  • A complex power of zero is 0 for an exponent with a positive real part, ~oo for a negative real exponent, and NaN otherwise (0⁻² was −∞, 0⁰ was 1).
  • At a complex zero, Cot and Csc are ~oo, and Coth, Csch and Arsech are +∞. Arg(~oo) is NaN; it was π/4.
• Fixes to compiled coordinate access over a wide list of points.
  • A point that is a sum of points of different sizes, (a, b) + (1, 2, 3), now compiles to NaN, as the interpreter reports an incompatible-type error. It gave a + 1.
  • A vars entry given as JavaScript source is no longer removed when its coordinate is not used.
  • An operators or functions override of Add, Negate or Multiply inside a point now receives the points, not their coordinates.

0.132.1 2026-09-19

Resolved Issues

• resolveApplication callbacks preserve range precedence. A callback could change f(x)+1...n into f(x)+(1...n), even when it returned undefined. This also affected offsets inside range bounds, such as L[I(l,r,i)+1...n], and explicitly parenthesized ranges.

0.132.0 2026-09-19

New Features

• New resolveApplication(context) parse hook. It receives the head, the parsed arguments, the LaTeX source offsets and the enclosing structure of a parenthesized occurrence such as f(x). Return apply, multiply, or undefined to choose how that occurrence is read. The decision is kept in the raw MathJSON, so it applies even when canonicalization happens later. Explicit declarations, function parameters and resolveSymbol facts take precedence.

Behavior Changes

• Explicit declarations take precedence over resolveSymbol. The handler is consulted only when a name has no explicit binding. An explicit unknown declaration also takes precedence; a type only inferred from earlier use can be overridden. The facts the handler supplies are kept with the expression, not installed in the caller's scope. To choose the reading of each occurrence, use resolveApplication.

Resolved Issues

• In non-strict mode, a letter run before a parenthesis is one name. ce.parse("foo(x)", { strict: false }) was f·o·o(x); it is now ["foo", "x"], and myfn(2, 3) is ["myfn", 2, 3]. Without a parenthesis, foo is still f·o·o, and a run that contains a Greek constant keeps its split (pix(3) is π·x·3).

0.131.3 2026-09-18

Resolved Issues

• The static type of PointList includes the width of a matrix component. PointList(M, V) with M a matrix<3x2> and V a vector<3> now has the type list<tuple<vector<2>, number>^3>. The width was missing before.

0.131.2 2026-09-18

Resolved Issues

• The resolveSymbol handler is also used outside ce.parse(), and takes precedence over a function declaration the engine inferred. A raw or structural result canonicalized later now also uses the engine-wide latexOptions.resolveSymbol. Before, b(\cos(NX)-1), with a handler that says b is a value, could be read as an application of b. A declaration made by the host, or one that holds a value, still takes precedence.
• The static type of PointList over list components includes the list width. PointList([0.1, 0.4, 0.8], [0.2, 0.7, 0.3]) now has the type list<tuple<real, real>^3>, so PointX of it has the type vector<3>. When the components have different widths, the smallest one is used. On the shader targets, 2 · PointX(PointList(-6, v)) over a vector<3> now compiles to 2.0 * vec3(-6.0).
• ce.appliedNonFunctions() reports a head that is a function only because of an earlier application. After the first parse of g(x), the call returned [] for g. Only a host declaration or an assigned body now makes g a function for this call.

0.131.1 2026-09-18

Breaking Changes

• \overline{z} is the complex conjugate. \overline{…} now parses as Conjugate (it was OverBar), and Conjugate serializes as \overline{z}. It used to serialize as z^\star, which parses as ConjugateTranspose, so a conjugate did not round-trip through LaTeX. OverBar built as MathJSON still serializes as \overline{…}. The repeating decimal 0.\overline{3} and the extended sets \overline\C, \overline\R, \overline\Z and \overline\Q are unchanged. z^\star still parses as ConjugateTranspose.

Resolved Issues

• The resolveSymbol handler decides between a product and an application, as a declaration does. With a handler that answers { type: "number" } for s, ce.parse("s(x+1)") was an application of s, which then made s + 1 an error. It is now a product. A function type gives an application, and a head the handler does not resolve uses the scope. ce.parse(latex, { speculative: true }) now reads a head declared unknown as a normal parse does.
• A head declared unknown by the host is read as a product. In 0.131.0, k(1 - w) was an application whenever k was declared unknown. An explicit unknown declaration says k is a value whose type is not known yet, so k(1 - w) is now k·(1 - w), as for a declared number. Only an undeclared head, or one with a type inferred from use, is applied.
• ConjugateTranspose over a vector of more than 100 elements evaluates. z^\star over a 200-element vector stayed unevaluated, while Conjugate of the same vector evaluated.

0.131.0 2026-09-18

Breaking Changes

• A symbol with no type information followed by a parenthesized argument is an application. f(x) with f undeclared, or declared with an unknown type, now parses as ["f", "x"], not as the product f·x. The product reading gave silent wrong results: \int f(x) g(x) dx took g out of the integral as a constant. A wrong application stays visible (f(2) evaluates to itself), and a later value for f overrides the inferred declaration.
  • 2f(x) is 2·f(x), and f(3)^2 is f(3)².
  • The product reading remains for a head that is a value: declared with a concrete type (a: real), assigned a value, a constant (\pi(x+1)), a parameter of the enclosing function ((x, N) \mapsto x(N+1)), or a head that its own argument refers to (x(x+1)).
  • A function defined before the head gets a value is read again: with g(t) := 2x(t+1) then x := 5, g(3) is 40.
  • Multiply(s, x+1) now serializes as s\times(x+1), so it round-trips.
  • With strict: false, foo(x) is still f·o·o(x).

New Features

• Host capabilities: ce.effects and ce.withEffects(). The application can replace or deny the handlers the engine uses to reach the host, one engine at a time. The console handler is used by Print and Input (Epsil print and input):

  const lines: string[] = [];
  ce.effects = {
    console: { log: (line) => lines.push(line), readLine: () => "Ada" },
  };

  readLine(prompt) returns the line read, null at end of input (Input evaluates to Nothing), or undefined when there is no interactive input (Input stays unevaluated). The defaults are the real console in Node and the prompt() dialog in a browser. ce.effects = {} restores the defaults.

  A handler set to null denies the capability: the operator evaluates to Error("capability-denied", "console"). ce.withEffects(overrides, fn) applies a change for one callback (also an async one). Use it to evaluate an expression that is not trusted:

  const result = ce.withEffects({ console: null }, () => expr.evaluate());

  The entropy handler, { random(): number }, is the unseeded source of randomness (Math.random by default). RandomExpression and the random operators use it outside a WithRandomSeed frame. A constant handler makes them reproducible in tests; entropy: null makes them evaluate to Error("capability-denied", "entropy"), and a compiled function throws CapabilityDeniedError.

  A change made while an evaluation runs does not affect that evaluation. Operator handlers receive the handlers as options.effects; a custom operator may use console or entropy if its signature declares that effect. EvaluateHandlerOptions has a new required member, effects: code that builds its own options object to call a handler directly must add it.

Resolved Issues

• \operatorname{conj}(z) parses as Conjugate(z). It was the product conj·z. Over a list, it was an unknown function that did not broadcast. Conjugate still serializes as z^\star.
• The square of a base that ends with a superscript serializes with braces. Power(Transpose(A), 2) serialized as A^T^2, which is invalid TeX. It is now {A^T}^2, and Power(Conjugate(z), 2) is {z^\star}^2.
• At with an index that is not an integer returns the missing-element value. L[2.5], L[3/2] and L[5 + √17] stayed unevaluated. They now return NaN for a numeric collection and Missing otherwise, as for an out-of-range index and as compiled code already did. An index whose value cannot be decided, such as a symbol with no value, stays unevaluated.
• More expressions compile on the interval-js target:
  • Missing values: a restriction over a list or a relation, a Which whose branches are lists, and IsMissing and Coalesce, which gave wrong results. A type that is not numeric, such as a string, still does not compile.
  • PointX, PointY and PointZ over a list of points, and over a value that is either a point or a list of points. A helper function that reads a coordinate of its parameter can be called with one point or a list of points.
  • Point arithmetic: Add, Subtract, Negate, Multiply and Divide with point operands, for example V.x·(cos a, sin a) + V.y·(−sin a, cos a). Operations the interpreter rejects, such as point + number, still do not compile.
  • A Which whose branches return a list in one case and a number in another.
• sqrt, abs, ln and gamma on the interval-js target keep a missing value missing. They returned an interval such as [0, ∞) for it.

Epsil

• The MCP server captures print and input with a console handler. It no longer replaces global functions such as console.log during an evaluation. The output of the evaluate tool is unchanged.
• A denied print or input is a capability-denied runtime error (see epsil doc capability-denied). The program continues after it.
• The VS Code extension supports type names and named-argument labels. Go to Definition, Find All References and Rename now work on a type name used in an annotation such as let p: point. Renaming a parameter also renames its labels in calls such as g(a: 2). A rename is refused when not every use can be resolved.
• epsil check checks calls to a function literal bound with let, const or :=. let k = (n: integer) => n + 1 followed by k(1.5) now reports the same diagnostic as a declared signature. A named call such as k(n: 2) no longer reports a false argument-names-unavailable.
• A constant bound to a function literal can no longer be redefined. const k = (n) => n + 1 followed by k = (s) => s, or by k(x) = x * 2, replaced k without an error. It is now refused, as const c = 5 then c = 6 is. From the host, ce.assign('Pi', (x) => 2x) is also refused; it replaced Pi on that engine.

0.130.0 2026-09-17

New Features

• New expr.digest property: a 128-bit digest of the expression's structure, usable as an in-memory cache key. Unlike expr.hash (a 32-bit bucket that needs an isSame() check on a hit), two expressions have the same digest exactly when their MathJSON is the same tree. It is much faster than JSON.stringify(expr.json) on expressions that share sub-expressions. It is not an isSame key: 1/2 and 0.5 digest differently. The value is 32 lowercase hexadecimal characters, stable only within one release, and not cryptographic. Do not persist it.

Resolved Issues

• expr.hash now agrees with isSame on number literals. The exact rational 1/2 and the float 0.5 hashed differently, so Unique, Tally and set membership could fail to match them.
• Compiled callbacks compute repeated sub-expressions and loop invariants once. This applies to the callback of a numeric derivative and to the integrand of a compiled Integrate (for example Γ(k/2)·√2^k in a chi-square tail is no longer computed at every sample). One measured case went from 84 µs to 4 µs per call.
• On the interval-js target, a constant list is built once, and a constant index selects its element at compile time. A sum over a 400-element assigned list no longer builds a new array at each read.

0.129.2 2026-09-17

Resolved Issues

• ce.rebind() now gives the same result as ce.expr(expr.json, …) for an expression that already contains an Error. Type errors below an existing error were not reported again.

0.129.0 2026-09-17

New Features

• New ce.rebind(expr, { form, scope }) rebuilds a boxed expression as if it was boxed from its MathJSON, without producing the MathJSON. ce.expr(expr, { scope }) keeps the existing bindings of a boxed expression. To read it under other declarations you had to serialize it and box it again, which can exhaust memory for an expression with many shared sub-expressions. rebind copies each distinct node once. See "Rebinding a Boxed Expression in Another Scope" in the structural-forms guide.

Resolved Issues

• A sum, a product or a literal collection with a few hundred thousand operands no longer overflows the call stack (RangeError: Maximum call stack size exceeded).
• Compiling an expression with many shared sub-expressions no longer runs out of memory. The analysis of free symbols and unsupported operators visited shared nodes once per path. A custom compile handler is also no longer called twice for each node.
• Lazy collections and assigned symbols evaluate faster. [l[i] for i = 1…Length(l)] over a lazy list l recomputed from the first element at each step (quadratic time). A symbol assigned an unevaluated expression now keeps its evaluated value until a dependency changes. One measured chain of list helpers went from 16.7 s to 0.24 s.
• A symbol that holds a lazy Map or Range value compiles on the interval-js target. R[k] and length(R) with R = mod(10⁴ sin(10⁴ · [0...100]), 1) failed with "Map: no lowering".

0.128.13 2026-09-16

Resolved Issues

• Collection values compile on the interval-js target. A list is compiled as an array of intervals. Previously, a list outside a comprehension failed with "List: no lowering". The target now:

  • applies scalar functions element by element to an operand whose type is a list of numbers (sin(L), L + 1, sin(L) · M);
  • passes lists into and out of user functions, and maps a scalar-parameter function over a list argument (f([1, 2, 3]) with f(x) := x²). This call used to return an incorrect result with success: true;
  • supports a Range with literal or symbolic bounds;
  • supports Sum, Product, Max and Min over a list computed at run time;
  • returns a list of numbers, a range or a Map at the root as an array.

  Also fixed: Σ_{k=1}^{x} k with x ∈ [2.5, 3.5] returned [6, 6], which is not an enclosure. A bound that is not constant over the interval now returns entire.

• A restriction with a list of conditions compiles on the JavaScript target. u\{[1, 2, 3] v < 2\} compiles element by element ([u, NaN, NaN] for v = 1). In the interpreter, a point under a list of conditions is now kept whole at each position: (1, 2)\{[1, 2, 3] < 2\} is [(1, 2), Missing, Missing] (it was [1, Missing]). The shader and interval targets still do not compile a list-valued condition.

0.128.12 2026-09-15

Resolved Issues

• A comprehension or a sum evaluated in a child scope uses that scope's values. With n declared in the root scope and n = 4 in a child scope, [k/n for k in 1..n] and \sum_{k=1}^{n} k stayed unevaluated, while 90n returned 360.
• Compiled JavaScript: a point with a complex coordinate passed to a user function no longer gives NaN. The call is inlined when possible, or the compilation fails with an error message. 1 + g((x, i x)) with g(P) = P.y returned the string "[object Object]1".
• A coordinate of a written-out point keeps the type of that component. PointX((1, \sqrt{1 - e^2})) is typed integer | nan instead of complex | nan, so a division by it compiles correctly.
• A coordinate accessor over a union of numbers, points and lists gets a precise type. PointX(P) typed collection<number>, so arithmetic on it did not compile to JavaScript. It is now number | tuple<number, number> | list<number>.
• Juxtaposition with a union-typed operand is multiplication. 2\mathrm{PointX}(P) with a number | list<number> coordinate parsed as the tuple (2, PointX(P)). A union with a non-numeric member (string | number) still parses as a Tuple.

0.128.11 2026-09-14

Resolved Issues

• A function that builds a list by recursion compiles to a loop on the JavaScript target. For example F(n) = \{n = D - 1: [P(n)], [P(n)].join(F(n + 1))\} overflowed the stack near 5,000 levels. It now builds a 10,000-point list in about 35 ms. The loop obeys ce.iterationLimit (default 1024), like the interpreter: raise the limit for longer lists. The shader targets still do not compile recursion.

0.128.10 2026-09-14

Behavior Changes

• The cross product of two points is a point. Cross((1, 2, 3), (4, 5, 6)) returned the list [-3, 6, -3], so Cross(A, B) + P failed. It now returns a point typed tuple<number, number, number>. A list operand still gives a list.
• Map over a tuple returns a list. The result is typed list<R>. It was typed as the tuple or as collection, and could become a Set that removed duplicates: Map(x ↦ 0, (1, 2)) had one element.

Resolved Issues

Arithmetic and Evaluation

• A function that builds a list by recursion evaluates iteratively. It no longer grows the JavaScript stack. It obeys ce.iterationLimit: a missing base case throws a CancellationError with cause iteration-limit-exceeded. Raise the limit for larger builds, such as a 10,000-point list.
• The numeric derivative of a point-valued function has a value. .N() of the derivative of F_0(t) = f'(t)/|f'(t)| for a space curve f returned an incomplete symbolic result; compiled JavaScript returned NaN. Both now differentiate each component and return a point. ND also supports functions that return points or lists.
• The norm of a point with an empty list coordinate is the empty list. Norm(([], 3)), Abs(([], 3)) and Hypot(([], 3), 4) now evaluate to []. They returned √10 and √(16 + []²).

LaTeX Parsing

• A list method with arguments parses as a function call with the list as first argument. L.\operatorname{join}\left(M\right) parses as Join(L, M), and [1].join(2, 3) as Join([1], 2, 3). Before, it did not parse. Methods without arguments (.x, .total) are unchanged.

Types

• A coordinate accessor over a point-or-point-list union has a precise type. PointX over list<tuple<…>> | tuple<…> is typed number | list<number> instead of collection<number>, so arithmetic on it compiles to JavaScript.

Compilation

• More point and list expressions compile to JavaScript:
  • a point multiplied by a symbol bound to a list of numbers (for example R × (cos t, sin t) with R built from a Range);
  • a call to a function with a list argument next to a point argument;
  • a value that can be a point or a list of points, added to a list of points. It returned NaN for every element with success: true. When the declared type cannot tell the two apart, the compilation now fails with a message;
  • Norm, Abs and Hypot of a point with a list coordinate return one norm per element: |([1, 2], 3) − (4, 0)| is [4.24, 3.61]. Before, the result was a single wrong number;
  • Which([T, F, F], (1, 2)) returns [(1, 2), NaN, NaN] instead of NaN.
• A literal index into a list expression computes only the selected element. This applies to element-wise Which/If, relations, Mod, list arithmetic, a literal Range and a PointList built from columns, on every target. For example (0.1·[a, b, c] + 0.4)[2] compiles to 0.1·b + 0.4, and [a, b][1] compiles to a. Likewise PointX of a PointList built from columns compiles to the first column.
• More list expressions compile on the interval-js target. Lists of any width and helpers that return lists are written out, so F(x, y)[1] and Total(f(x + 1, y) + f(x − 1, y)) compile.
• More expressions compile on the GLSL and WGSL targets:
  • a comprehension over literal ranges or lists, up to 64 elements, compiles to an array of vectors;
  • a helper function that returns a list;
  • a Which, If or When whose branch contains a loop-form Sum or a block. The loop runs only when its branch is selected;
  • a block with local variables used as a Sum term or an operand.
• Fixed a shader miscompile of a reduction assigned to a block local. On GLSL and WGSL, a := Σ… in a function or Loop body produced invalid code that was reported as a success.
• Shader code is smaller. Repeated sub-expressions in a conditional branch are computed once. A deeply nested Max(e, e) no longer expands into megabytes of code.
• Shader helper names are valid. They no longer contain two consecutive underscores or dollar signs.
• Color helpers in shaders keep a three-channel return type.

Performance

• Compiled code reuses more calculations. A user function called from each term of a Sum with the same argument computes the parts that depend only on that argument once, on all compile targets. The interval-js target also reuses small repeated operations, such as a reciprocal.
• Nested element-wise operations on a list run in one loop in compiled JavaScript. One measured case went from 21 µs to 3.6 µs per sample.

Benchmarks

Median time per call at 200-digit precision, in microseconds (lower is better):

Expression	CE (current)	CE 0.128.9	SymPy	math.js	Mathematica
\pi^2	11	13	179	143	3.9
\sin 1	23	23	221	449	5.1
\ln 2	18	18	357	4,460	3.7
\zeta(3)	1,584	1,588	281	—	50
\Gamma(\tfrac13)	841	848	368	—	204

On the symbolic cases that both solve (antiderivatives, simplification, limits, definite integrals, equation solving), Compute Engine is a median 3.1× faster than Mathematica.

0.128.9 2026-09-11

Performance

• Compiled JavaScript skips redundant shape checks on constructed vectors and point lists. Helpers with a fixed-size collection return type, and PointList over constructed sources, no longer emit shape checks and broadcast fallbacks that the compiler can prove unnecessary. Runtime inputs keep their validation.

0.128.8 2026-09-11

Resolved Issues

• Compiling nested point helpers to JavaScript no longer repeats the same analysis. This removes a compile-time slowdown in composed point helpers.
• List-valued point coordinates are kept in compiled JavaScript helpers. Dot broadcasts across a list in any coordinate of a runtime tuple input.
• Compiled products over a one-element list are correct. The element is emitted directly and evaluated once, and negative zero is preserved. An empty product is still 1.

0.128.7 2026-09-11

Resolved Issues

• Colors in GLSL and WGSL keep their vector representation in locals, helper parameters, return values and shared temporaries. Converted colors are normalized to OKLCh, so the shader types are valid and the colors are kept.
• A caller parameter with the same name as a global no longer changes how a compiled helper result is classified. List-valued coordinates keep broadcasting when called through such a helper.

Performance

• Compiled JavaScript uses direct arithmetic for vectors and points of known width. Scalar lists returned by helpers or held in local variables, and points passed to specialized helpers, skip redundant shape checks, broadcast fallbacks and small .reduce calls. Dot of two points with proven scalar coordinates is a direct multiply-add. Runtime inputs keep their checks.

0.128.6 2026-09-10

Resolved Issues

• Compiling nested shared scalar functions to JavaScript no longer takes excessive time.

Performance

• Compiled JavaScript sums and products use direct arithmetic when their terms are proven scalar, even when the declared result of a block permits collections. Collection terms still broadcast.

0.128.5 2026-09-10

Performance

• Faster compiled JavaScript for reductions and helper chains.
  • Sums and products use real square roots and logarithms when a known lower bound proves the argument non-negative.
  • Sum(Map(f, xs)) and Product(Map(f, xs)) run in one pass, without an intermediate array, when the source and callback are pure.
  • Nested helper calls avoid unnecessary broadcasting when local variables and loop counters are proven scalar.

0.128.4 2026-09-10

Improvements

• The interval-js target compiles applications of small closed-form derivatives, such as Apply(Derivative(f, n), x).

Resolved Issues

• Compiled derivatives of odd roots stay real at negative inputs. For f(x) := ∛x, compiled f''(-1.2) now returns approximately 0.16399052 instead of a complex value, as the interpreter does. This applies to all compile targets.
• Compiled closed-form derivatives handle complex arguments. f''(z) for f(x) := x^3 now uses complex arithmetic when z is complex, instead of returning NaN.
• Compiled Dot broadcasts list-valued point coordinates. For points declared as tuple<broadcastable<number>, broadcastable<number>>, Dot(a, b) with a = [1, [1, 2]] and b = [3, 4] now returns [7, 11] instead of NaN. Complex point coordinates are not compiled.

Performance

• Repeated Sum or Product over the same range slice P[a...b] computes the result once in compiled code.

0.128.3 2026-09-10

Improvements

• Dot of points whose coordinates may be lists returns the broadcast inner product. With L := [1, 2], Dot((1, L), (3, 4)) is [7, 11]; it stayed symbolic before. Dot(c, d) for two tuple<real, real> symbols is now typed real instead of number.

Resolved Issues

Compilation

• Scalar calls keep their types through user-function chains and local variables. A function that accepts points or lists can compile a scalar point call to shared JavaScript, GLSL and WGSL helpers without losing its list inputs.
• A point built from a symbol that holds a number no longer compiles to code that throws. (⌊x⌋, ⌊y⌋, s) with s assigned 2.41 emitted […, ...2.41], and each call raised "2.41 is not iterable".
• A point plus a scalar is no longer compiled. (x, y) + 3 returned [3.3, 3.7] at x = 0.3, y = 0.7, while the interpreter reports incompatible-type. The expression is now left to the interpreter.
• A compiled element-wise comparison marks an absent element with NaN instead of false. Equal([1, Missing], 1) compiles to [true, NaN], as the interpreter returns [True, Missing]. A piecewise over such a condition gives an absent result at that position instead of taking a branch. A NaN operand still equals nothing, and the scalar comparison still returns a boolean.

Types

• Dot no longer reads the operands of any tuple-typed expression as coordinates. Dot(P + (0, 0), Q) was typed tuple<real, real> and Dot(2P, Q) a union with a tuple, although both values are numbers.

Performance

• Point arithmetic and Dot compile to per-component code on the javascript target when the width of the points is known. 2 · (1, 2) compiles to [(2 * 1), (2 * 2)].
• Fewer runtime broadcasts in compiled user functions. Calls that take points or use a bounded Sum can now be proven scalar. On one noise kernel, the time per sample went from 15.5 to 2.5 microseconds.

0.128.2 2026-09-10

Behavior Changes

• An absent value in a function with an undeclared result is Missing, not NaN. With g(t) := \begin{cases} t & t > 5\end{cases}, g(3) + 1 is now Missing. Declare the function, for example g: (number) -> number, to get NaN.

Resolved Issues

• Point accessors read a list whose element type is a union of tuple types as a list of points. For Q = [(a, 1), (2, b)] with a and b declared number, PointY(Q) is typed list<number>. Before, PointY(Q) + 1 had the wrong type and compiled JavaScript returned the string "1,71".
• A scalar cannot be added to a point that can be absent. For a point with a restriction, such as h(t) := (t, t+1)\{t > 0\}, h(1) + 2 was accepted, and compiled code returned the string "21,2". It is now rejected as for a plain tuple<number, number>. A scalar multiple of such a point is still accepted.
• An absent point stays absent instead of becoming NaN. With P declared list<tuple<number, number, number>>, 2 \cdot P[0] returned NaN, and 2 \cdot P[0] + (1, 1, 1) then failed with incompatible-type. It is now Missing. Add(Missing, (1, 1, 1)) and Multiply(Missing, (1, 1, 1)) are also Missing. An operation typed as a number still returns NaN: 2 \cdot Missing and \sin(Missing) are unchanged.

0.128.1 2026-09-10

Breaking Changes

• Compiled Equal and NotEqual are exact. A compiled comparison of two numbers uses the IEEE 754 test (=== in javascript, == in python), with no tolerance. The interpreter still compares within engine.tolerance, so 0.1 + 0.2 = 0.3 evaluates to True but compiles to false. This also applies to KroneckerDelta and to comparisons of collections. NotEqual is still true for NaN. For tolerant equality, evaluate with the interpreter.

Resolved Issues

LaTeX Parsing and Serialization

• Nested absolute values round-trip. An Abs whose body contains a vertical bar serialized as \vert\vert x\vert-0.5\vert, which did not parse. It now serializes with \left\vert … \right\vert. The parser also accepts nested bars such as \vert\vert x\vert-0.5\vert, |x-|y||, |\frac{|x|}{2}| and |x^{|y|}|.
• \operatorname{real}, \operatorname{imag}, \operatorname{erf} and \operatorname{erfc} parse as Real, Imaginary, Erf and Erfc.

Compilation

• An integer exponent over a possibly negative base is correct in GLSL and WGSL. (-1)^n with a runtime n returned NaN on most drivers, because pow(x, y) is undefined for a negative x. The sign is now kept for an odd exponent.
• A compiled point accessor treats a row with non-numeric cells as not a point. PointX(L) over [[1, "a"], [3, "b"]] returned [1, 3]; it now returns [1, "a"], as the interpreter does.

Types and Assignment

• A value with a restriction can again be assigned to a declared symbol. In 0.128.0 this threw a TypeCompatibilityError:

  ce.declare("P", "tuple<number, number>");
  ce.assign("P", ce.parse("(1,2)\\left\\{a>0\\right\\}"));

  It works again, with ce.assign, ce.declare, Assign, := and destructuring declarations. P.type stays tuple<number, number>. With a := -1, P evaluates to Missing. missing <: number is still false, so the type of an undeclared symbol or of an expression still includes missing.

Collections

• A point accessor over an empty list of points returns the empty list. PointX([]), and PointY, PointZ over any empty collection, returned Missing, and 2 · PointX([]) returned NaN. PointX([]) is now [], typed list<never>. The empty case follows the element type:

  • a point element type (list<tuple<number, number>>), an unknown element type, or the literal [] returns [];
  • a numeric element type (list<number>) is one point, so an empty one still returns Missing (NaN in compiled JavaScript);
  • any other element type (list<string>) returns Missing.

  Compiled JavaScript returns the same values. PointX(s) for s declared string now compiles to its first character instead of NaN.

Performance

• Higher-order derivatives of user functions (f''(x), f'''(x)) compile much faster in JavaScript. They use automatic differentiation: the third derivative of a nested radical went from 28.7 s and 392 KB of code to 1 ms and 174 B. This also fixed (x-e)·f'(e), which compiled to NaN when the body has a complex value.
• Faster compiled complex arithmetic in JavaScript. Complex operations no longer allocate a closure each. i · b compiles to { re: 0, im: b }.
• More constant folding in compiled code. \sum_{k=1}^{3}(-1)^{k}x compiles to (-x + x + -x) instead of three pow calls. In GLSL and WGSL this also fixed a wrong value: a literal pow(-1.0, 1.0) is undefined in both languages.
• Better common-subexpression elimination in compiled code. (sin(u)+1)^2 + 1/(sin(u)+1) called Math.sin twice and now calls it once. Values moved out of a loop also share their subexpressions.
• Element-wise arithmetic over lists of known width compiles to per-component code in JavaScript, instead of a runtime broadcast. A reduction over a real vector no longer uses complex arithmetic.
• A reduction over a range slice compiles to a loop. Sum, Product, Mean, Max, Min and Length over P[a...b] no longer build the slice. A source that is not an array, or a bound that is not an integer, now returns NaN instead of throwing a TypeError.
• interval-js kernels no longer rebuild constant intervals on each call. A plotted cases expression is 25–40% faster per sample.

0.128.0 2026-09-09

Breaking Changes

• A false restriction evaluates to Missing, not Undefined. When c is false, When(e, c) (the e\{c\} restriction) now evaluates to Missing, the same value as a Which with no selected clause or an If with no else branch. The element-wise form masks each element: [10,20,30]\{[1,2,3] > 2\} is [Missing, Missing, 30]. Compiled output is unchanged. Code that tested for "Undefined" must test for "Missing" (or use IsMissing, which now returns True for a masked value).

  The type of a restriction now includes the absent case: When(5, x > 0) is integer | missing, and a list with a masked element is list<integer | missing>. Only a restriction whose condition is the literal True keeps the bare type. Code that checks type.matches("number") must remove the missing member first.

• On the JavaScript target, a compiled color is an object, not an array. compile() now returns a color as { space, c0, c1, c2, alpha }, where space is 'oklch', 'rgb', 'hsv', 'hsl' or 'oklab', the channels are in that space, and alpha is undefined when there is none. Before, a color was a flat [L, C, H] (or [L, C, H, alpha]) OKLCh array, and a conversion such as AsRgb returned a flat array in its own space, with no way to tell them apart. Read raw.space and raw.c0…raw.c2 instead of raw[0]…raw[2]. The color helpers throw a TypeError for a numeric array or an unknown space. A list of colors is an array of these objects; a color string is still a string. ColorToColorspace is not affected: it returns components, and its compiled value is still a plain array.

  Nested conversions are now correct: AsRgb(AsRgb(c)) equals AsRgb(c) and ColorDelta(AsRgb(a), b) equals ColorDelta(a, b). The interpreter fallback returns the same object and accepts one as a vars value. The GLSL and WGSL targets are unchanged: a color is still a vec3 in OKLCh.

• ColorFromColorspace returns a color. Its signature is now (color | tuple, string) -> color. ColorFromColorspace((0.5, 0.1, 20), "oklch") is Oklch(0.5, 0.1, 20); it was the sRGB components tuple (0.58, 0.29, 0.29), while the compiled code returned a color. To get the old components, use ColorToColorspace(color, "rgb"). Passing the result to another color operator needs no change. Indexing the result (At(result, 1)) is now an incompatible-type error, because a color is not a collection. On the JavaScript target the compiled color keeps the space it was built in; on GLSL and WGSL, a shader whose whole expression is ColorFromColorspace and that expects OKLCh must wrap it in AsOklch.

New Features

• Arg compiles on the interval-js target when the complex value is built in the expression (a + ib), for example \arg((x - 0.3127) + i(y - 0.329)). A box that crosses the branch cut on the negative real axis reports the jump. Other complex operands, such as Sin(z), still fall back to the interpreter.

• More fixed-width list expressions compile to straight-line code. Map(h, [e1, …, eN]) with a user function h becomes [h(e1), …, h(eN)], so Min(Map(h, list)) now compiles on interval-js. Product([a, …]) and a Reduce over Add, Multiply, Min or Max compile to an n-ary operation: the JavaScript target emits _.a * _.b * … instead of a reduce over an array.

• On the JavaScript target, the loop-invariant part of a Reduce/Scan combiner is computed once. In Reduce(list, (acc, x) ↦ acc + x·Sin(u)·Cos(v), 0), Sin(u)·Cos(v) was computed again for each element.

• A point compiles on interval-js when it is the whole expression. Tuple(a, b) and an all-scalar PointList(a, b) compile, and run() returns one {lo, hi} interval per coordinate. A point inside a larger expression (2·(a, b)) still does not compile on this target.

Resolved Issues

Compilation

• Compiled Equal and NotEqual are correct for NaN and for two infinities of the same sign (JavaScript and Python). NotEqual(NaN, 3) compiled to false, for example r \ne 3 when r was not given to run(); it is now true, as in the interpreter. Equal(oo, oo) compiled to false; it is now true. An impure operand (the Random family) is evaluated once.

• An else-less If compiles. \mathrm{If}(x > 0, x + 1) was refused by every target. It now compiles on all targets and returns NaN (for a number) when the condition is false. An If whose branch is a statement is unchanged.

• A selection with no selected value compiles to the same absent value on a target. A Which with no matching clause, an If with no else branch and a false restriction return NaN when the value is a number or of unknown type, and undefined (JavaScript) or None (Python) only when the value is provably not a number, such as a string. Before, Which and If could return undefined for a number, and Python always returned float('nan'), so a compiled IsMissing over a string selection disagreed with the interpreter.

• A point passed to a user function whose parameter has no declared type compiles correctly. With r(P) := 2·P, compiled r((a, b)) returned NaN on JavaScript instead of (2a, 2b). Such a call now compiles correctly on every target. When that is not possible (an impure argument, a recursive callee, or a body the interpreter rejects for a point), compilation fails and fallback: true uses the interpreter; before, it could return a wrong list.

• A user function whose body uses an inner lambda parameter with the same name as a variable of the caller now compiles inline. For example, a body that contains Map(p ↦ …, list) called from a definition that binds p.

• A Sum or Product over a collection (Sum(2n, Element(n, [3, 5, 7]))) fails to compile with a clear message instead of an internal error about Nothing. It still falls back to the interpreter.

• An unrolled Sum or Product no longer uses complex arithmetic for a square root whose argument is a non-negative constant in each term. For example \sum_{i=1}^{6} (1 - j(1 - \sqrt{1 - 0.025^2(i - 0.5)^2}))\cdot s compiled to six complex square roots.

• A Range with a non-integer upper bound has integer elements. Range(1, 2.5) and Range(1, n) with n declared real were typed indexed_collection<real>; they are now indexed_collection<integer>. An equality against such an index now compiles to k === 9543 instead of a tolerance test.

• An equality with an operand of type integer | nan (such as an element read P[i]) compiles to ===/!==, which gives the interpreter's result for NaN.

• Shader float literals use the shortest single-precision spelling. 2π was emitted as 6.2831854820251465; it is now 6.2831855.

• Constant factors inside parentheses are folded. \frac{2\pi s}{P} with P assigned 100 emitted (2 * Math.PI * s) / 100; it now emits (6.283185307179586 * s) / 100.

Interval Arithmetic

• Interval sign and heaviside are tighter. sign([0, 0.5]) returned [-1, 1]; it now returns [0, 1]. heaviside (with H(0) = 1/2) returns [0.5, 1] for an input with no negative part and [0, 0.5] for an input with no positive part. Only an input that spans zero keeps the full range.

• Interval atan2 reports a jump along its x operand. atan2([0, 0], [-1, 1]) returned the bounded interval [0, π]; it is now singular with the same enclosure. A singular result has a new optional atOperand field: the index of the operand that the at location refers to (absent for the first operand). -0 is read as zero: atan2([0, 0], [-1, -1]) is +π.

Colors

• A tuple variable is no longer read as a color by compiled code. With v declared tuple<number, number, number>, ColorMix(v, Rgb(0, 0, 1), 0.5) compiled but threw Not a color at run time on JavaScript, and gave a wrong color on the shader targets. Operators that take a color (ColorMix, ColorDelta, ColorContrast, ContrastingColor, ColorToString) now do not compile such an operand and fall back to the interpreter; the message suggests AsRgb((r, g, b)). The As* conversions and ColorToColorspace read it as 0-1 sRGB components, as the interpreter does.

• Color conversions of a possible list of colors compile again on JavaScript. AsRgb(w) with w declared broadcastable<color> did not compile in 0.127.0, and before that converted one color channel by channel. AsRgb("red") compiles again too.

• Nested color conversions give the interpreter's color in compiled code. AsRgb(AsRgb(Hsv(0.3, 0.5, 0.5))) compiled to [0.7137, 0, 0.3686] instead of Rgb(0.5, 0.2513, 0.25). This applies to all color operators. On GLSL and WGSL, a Which or If whose branches return colors in different spaces does not compile.

• Compiled ContrastingColor returns an OKLCh color on JavaScript. Before, AsOklch(ContrastingColor(bg, AsRgb(a), AsRgb(b))) read sRGB channels as OKLCh.

• Compiled AsHsv and AsHsl of a non-finite color return NaN for all channels. They returned a hue of 0 (red).

Absent Values and Types

• Add and Multiply return NaN for an operand that evaluates to Missing. With g(3) evaluating to Missing, g(3) + 1 stayed Add(Missing, 1); it is now NaN. A list operand gives NaN per element (Add(Missing, [3, 4]) is [NaN, NaN]).

• A juxtaposition with a maybe-absent operand is a product. With g(t) := \begin{cases} 0.5 & t < 1 \end{cases}, 2g(0) parsed as the tuple (2, g(0)); it is now 2 \cdot g(0). The same applies to 2x\{x>0\}, 2\,\mathrm{If}(c, x) and t P\{0 \le t \le 1\}. A non-numeric operand, such as a string, still gives a Tuple.

• A maybe-absent Sum/Product bound is accepted. \sum_{x=g(0)}^{3} x was an incompatible-type error; an absent value is now handled at evaluation. A non-numeric bound ("lo") is still an error.

• A bound written as the literal NaN is an error, like one written as Missing: Sum(x, (x, NaN, 3)) now reports incompatible-type.

• Partition accepts a maybe-absent chunk size, like Take, Drop and Chunk. Partition([1,2,3,4], k(0)) with k(0) typed integer | missing was an incompatible-type error; it now returns [[1,2],[3,4]].

• A function with a missing | T result keeps its result type when it is assigned to a symbol declared function. After ce.declare('l', 'function'), assigning a piecewise function without a default made l(t) of type unknown, so \sin(l(t)) was typed broadcastable<number> instead of number.

Performance

• Faster compiled JavaScript. A repeated Min/Max call is computed once; Mod(a, 1) uses a shorter formula; a factor of 1 is removed from a product; PointX(PointList(a, b)) emits a; and √a³ and a^(3/2) emit a * Math.sqrt(a) instead of Math.pow (also more accurate). Loop indexes and unrolled sums that cannot be NaN skip their run-time NaN tests. A product with one complex factor uses real multiplication for the real factors.

• Faster compiled interval-js code. Multiplying or dividing by a constant uses a cheaper operation; a constant such as π/50 in 2πs/100 is computed at compile time; and a constant interval in a function or loop body is created once per call instead of once per use.

0.127.0 2026-09-09

New Features

• Fixed-width collections compile to scalar code on every target. A collection whose width is known at compile time (a literal List, possibly exposed by inlining a user function) compiled to runtime array code on the JavaScript target and did not compile on glsl, wgsl and interval-js. The compiler now unrolls Map(f, List(…)), PointX/PointY/PointZ over a list of points, Min/Max/Sum/Product over a literal list, and scalar-list arithmetic. On JavaScript, a loop-invariant part of a Map, Filter, CountIf or Find callback is computed once instead of once per element. A nine-point Voronoi row written as user functions went from 58 to 2 µs per sample on JavaScript and now compiles on glsl, wgsl and interval-js.
• Texture-backed At on the WGSL target. With the storage compile option ({ storage: { S: 'sampler2D' } }), At(S, k) now compiles on WGSL. On GLSL and WGSL, a literal index and a gather of several indexes over a texture also compile (an out-of-range literal index gives NaN). These cases did not compile before.
• Rest parameters on function literals. The last parameter can collect all remaining arguments: (a, ...rest) => … in Epsil, ["Function", body, "a", ["Spread", "rest"]] in MathJSON. rest is a Tuple of the trailing arguments (empty when there are none), and the function types as (unknown, any*) -> R. Spreading it back into a call passes the arguments on: (...args) => Conjugate(g(...args)) applied to (1, 2) gives Conjugate(g(1, 2)). The rest parameter cannot have a type annotation. compile() refuses a function with a rest parameter. Named Epsil definitions accept one too: h(a, ...rest) = Length(rest) and function h(a, ...rest) { Length(rest) }.
• Primes, the set of all prime numbers. A lazy, infinite set like Integers: Element(7, Primes) is True, iteration yields 2, 3, 5, …, and \sum_{p \in \mathbb{P}} p^{-2} stays symbolic with an integer index. \mathbb{P} parses to Primes and serializes back.

Resolved Issues

Compilation

• User functions passed as values compile correctly. A user function used as the callback of Map, Filter, Reduce, CountIf, Find, Tabulate and similar operators:
  • if its definition could not be compiled, the output compiled with success: true and then threw TypeError: _f is not a function. It is now refused at compile time with a message that names the function, and the default fallback: true uses the interpreter.
  • now broadcasts over an element that is itself a collection, as the interpreter does. For f(x) := 2x, the compiled Map(f, [[1, 2], [3, 4]]) gave [NaN, NaN] and now gives [[2, 4], [6, 8]]. A multi-clause function gave strings such as ["3,61", "91"]. A function with a collection parameter still receives the element whole, and a function whose parameters are declared scalar gives NaN for a collection element.
• Built-in operators and inline function literals used as callbacks broadcast over a collection element. Over [[1, 2], [], [3]], Map(Sin, xs) now gives [[sin 1, sin 2], NaN, [sin 3]] and Map((x) ↦ 2x, xs) gives [[2, 4], [], [6]]; they gave NaN before.
• A function parameter with the same name as a user function no longer compiles as a call of that user function. With gsh(t) := sin t + t² and psh(gsh, t) := gsh(t) + 1, psh(u, u) compiled to a call of the global gsh and ignored its first argument. It is now refused.
• angularUnit = 'deg' applies inside inlined user functions. A Sin in a user function body inlined at a call site compiled without the degree conversion.
• The compile targets respect broadcastExemptions: ['tuples']. A tuple argument written at the call site was mapped element by element even for an operator that takes a tuple whole, so AsRgb((1, 0, 0)) did not compile on JavaScript.
• Infix operands of identity operations keep their parentheses. Abs of a non-negative value, Floor/Round of an integer, Re of a real, Identity and similar operations inserted their operand without parentheses: 3·Re(x + 1) compiled to 3 * _.x + 1. Fixed on the JavaScript, GLSL, WGSL and Python targets.
• WGSL emitted a two-argument atan, which WGSL does not have. Arctan2 and the argument of a complex value now emit atan2.
• A compiled Mod with a non-literal divisor evaluated the divisor three times (2 mod sin(0.2θ), or Random()). It is now evaluated once.
• Division by an exact rational compiles as a division. x/49 compiled to 0.02040816326530612 * x, which is not correctly rounded: floor(x/49) gave 0 at x = 49. The JavaScript, GLSL, WGSL, Python and interval-js targets now emit x / 49. A power-of-two denominator (x/2) still compiles to 0.5 * x.

Interval Arithmetic

• The interval-js target rounds every result outward. Results were computed in round-to-nearest, so a product, a quotient or a transcendental function could give an interval that did not contain the true value. Inexact endpoints are now widened outward; exact results (an integer, 2 · 0.5, 49/49, cos = 1) stay points. Irrational constants (\pi, e), rationals such as 1/3 and radicals such as \sqrt{2} compile to two-ulp enclosures instead of a single point. atan2 across its branch cut now contains π. A 40-term sum went from 32 to 72 µs per call.
• Rational powers and roots on interval-js are correct enclosures. A rational power did not contain the true value in about 5% of random cases, and Root of a large operand failed in most cases. Exact results stay points: 8^(2/3) is 4, 27^(2/3) is 9 (it was 8.999999999999998), Root(64, 3) is 4 and Root(1000, 3) is 10.
• Round(x, n) on interval-js with a negative n scales exactly, so an exact multiple of the step stays a point.

Types

• Rest parameters type as variadic. f := (a, ...rest) => Length(rest) was typed (unknown, unknown) -> integer; it is now (unknown, any*) -> integer, so f(1) and f(1, 2, 3) are valid. A literal with a rest parameter matches a variadic declaration such as (integer, any*) -> integer. An Epsil definition with a type parameter, function h<T>(x: T, ...rest) -> T { … }, is typed (x: T, any*) -> T where T; h(1) and h(1, 2, 3) were errors.
• f(L) := Sum(L) treats L as a collection, so f([1, 2, 3]) is 6. It gave [1, 2, 3], because L had type unknown and the call was mapped over the list. A Sum(L) or Product(L) with no index over a symbol with no other type information now types the symbol as a collection.
• The index of a Sum/Product over an Element clause has the element type of the collection. It was always integer, so Sum(2x, Element(x, [0.5, 1.5])) threw, and a sum over a set of functions gave an expected-function error.
• Generic signatures narrow their operands. For f: (T) -> T where T: number | function, f(u) + 1 now types u and the sum as number (the sum was function | number). An operator declared (T) -> T where T: number with a canonical handler now types an undeclared operand as number.
• LinearRegression(xs, ys, x) and PolynomialFit(xs, ys, n, x) are typed number. LinearRegression([1,2,3],[2,4,6],x) was typed tuple<number, number> while it evaluates to 2x.
• The LaTeX pipe shorthand has a type. [1,2,3] |> \_^2 was typed unknown and is now list<integer<0..>^3>; [1,2,3] |> \mathrm{Length}(\_) + 1 is integer. A type handler's context.derive now also maps a broadcastable operator over a collection operand: derive('Power', [d, 2]) with d typed list<integer^3> gave number.
• A second pipe on the same engine no longer throws. After xs |> \_^2, xs |> Take(\_, 2) threw "Function body must be a scoped Block expression", and [1,2,3] |> Sum(\_) gave the list itself.

Arithmetic and Numerics

• Assumed inequality bounds are exact. A bound was rounded to a JavaScript number: after assume(v > 1 - 10^{-30}), v > 1 was "proven" and assume(v < 1) was rejected as a contradiction. Bounds are now compared exactly, and ce.ask(['Greater', 'x', '_k']) gives the exact bound (1/3, not 0.333…).
• Covariance and PopulationCovariance no longer overflow on large data. Covariance([1e200, -1e200, 0], [1e200, 1e200, -2e200]) gave NaN; it is 0. A covariance larger than the largest double is +oo.
• Conjugate accepts a function. Conjugate(f) is (x) ↦ Conjugate(f(x)).

Colors

• A bare tuple as a color argument is 0–1 sRGB everywhere. The compiled targets read it as OKLCh: ColorToString((1, 0, 0)) gave #ff0000 in the interpreter and #ffffff when compiled. A list of three numbers is not a color. AsRgb, AsHsv, AsHsl, AsOklab and AsOklch now accept a color string and a tuple (AsRgb("#ff0000") was incompatible-type).
• Color strings are validated the same way everywhere. A string that names no color is refused by every operator (ColorMix("bogus", "#ffffff") gave a half-transparent grey). A four-digit hex color (#f00f) is read as CSS reads it. Malformed strings such as rgb(255, 0, 0 and #gg0000 are refused. #00000000 and rgba(0, 0, 0, 0) are transparent black instead of an error. A color tuple must have three or four finite numeric components.
• Color operators give the same results in the interpreter and in compiled code. On GLSL and WGSL, ColorContrast(Rgb(0,0,0), Rgb(1,1,1)) gave about −114 instead of about −1.079, so ContrastingColor chose black for a 0.6 grey. ColorFromColorspace(Rgb(1,0,0), "rgb") was not red when compiled. "hsv" is accepted by ColorToColorspace/ColorFromColorspace everywhere. Colormap samples the same color as ColorMix. ColorToString(c, "oklch") keeps out-of-gamut chroma when compiled.
• ContrastingColor returns the chosen candidate in its own color space. ContrastingColor(Oklch(0.98, 0.02, 90), Oklch(0.2, 0.1, 29), Oklch(0.95, 0.2, 264)) now gives Oklch(0.2, 0.1, 29) instead of an Rgb value.

Performance

• Compiled JavaScript caches the last call of a pure user function that is called from several places inside other user functions. A repeated call with the same arguments (and the same variable values) returns the cached result. A nine-point Voronoi row went from 3.4 to 2.7 µs per sample.
• Fewer runtime shape checks in compiled user functions. Inside a user function body, a call that passes on a scalar parameter of that function is a direct call. An untyped chain f(x) := g(x) + 1, g(x) := 2h(x), h(x) := x² went from 155 to 77 ns per call.
• interval-js compiles a subtraction as one sub call instead of an addition of a negation.
• Better generated code on the JavaScript, GLSL and WGSL targets. Literal arithmetic is folded; loop-invariant parts of a Sum/Product body are computed once before the loop; user function bodies on GPU targets share common subexpressions; the GPU preamble includes only the helpers the shader uses. GLSL uses fract, exp2, dot and inline small integer powers where possible, and rounds half away from zero like the interpreter. JavaScript compiles an equality of integers to ===, and \arg(a + i b) and |a + i b| to Math.atan2 and Math.hypot.

0.126.2 2026-09-08

New Features

• New expr.isMachineNumeric property. It is true when ce.list(expr.array) reproduces the list exactly: every element is a float or an integer that a double holds. It is false when an element is an exact non-integer, such as 1/2 (expr.array reads it as 0.5, but 0.5 / 3 is a float where 1/2 ÷ 3 is 1/6). On a number, the same rule applies to the number. It is false for all other expressions.

Resolved Issues

Compilation

• Complex compile mode: a call to a user function could give a wrong result. With b(x) := If(x > 0, x, 3), b(z) + z on a complex z computed "1[object Object]". It now gives 4 + 2i. Also, a call to a function whose body already returns a complex value is no longer wrapped in an unnecessary conversion.
• The python target no longer binds a repeated square as a list comprehension. (x^2+1)/(x^2-1) compiled to [(_cse1 + 1) / (_cse1 + -1) for _cse1 in [x ** 2]][0] since 0.126.0. It is again (x ** 2 + 1) / (x ** 2 + -1).
• interval-js target: a \sum with a fixed number of terms is up to 4× faster. Constant subexpressions are computed at compile time and terms that do not depend on the index are computed once per call. Results do not change.
• interval-js target: GCD, LCM and Binomial of an integer at or above 2^53 did not return. They now return a wide enclosure.

Collections

• expr.array threw a RangeError for an exact rational outside the double range. It now returns undefined, and isMachineNumeric is false.
• A user function applied to a list-typed symbol that has no value yet stays unevaluated. With xs: list<integer> declared but not assigned, h(n: integer) = n + 1; h(xs) gave an incompatible-type error. It now evaluates to h(xs), and to [2, 3] once xs = [1, 2]. With an untyped parameter, pair(n) = (n, n); pair(xs) no longer becomes (xs, xs).

Other

• Chained set relations. A ⊂ B ⊂ C and a one-operand Subset(D) boxed with an error. Subset, SubsetEqual, Superset and SupersetEqual now accept one or more operands. A chain evaluates all adjacent pairs (ℤ ⊂ ℝ ⊂ ℚ is False; before, only the first pair was checked). The LaTeX \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} parses to Subset(Integers, RationalNumbers, RealNumbers), as a < b < c parses to Less(a, b, c).
• An operator with only an evaluateAsync handler is awaited inside Sum, Product and Block. Σ_{i=1}^{3} AsyncOnly(i) and {x = 1; AsyncOnly(x)} stayed unevaluated with evaluateAsync().

0.126.1 2026-09-07

Resolved Issues

• expr.array no longer approximates exact elements at precision: "machine". ["List", 1, ["Rational", 1, 3]] returned [1, 0.3333333333333333]. It now returns undefined, as at other precisions. An exact value is accepted only when a double holds it exactly (for example 2^70 or 3/4).
• javascript target: faster selection over a list declared as indexed_collection<number> (1.73 ms to 0.4 ms for a 200×200 Game of Life step).
• A function literal with wildcard parameters round-trips through LaTeX. () \mapsto 2 parsed as a function of one parameter named Nothing, and () \mapsto \_ + 1 applied to 3 returned _ + 1. Both now parse as ["Function", body]. Also, x_1 \mapsto x_1 + 1 serialized as ()\mapsto x_1+1, which lost the parameter.

0.126.0 2026-09-07

New Features

• ce.list(values) builds a List of numbers quickly. The input can be a number[], a Float64Array or any array-like of numbers. A 40 000-element list is created in a fraction of a millisecond instead of about 5 ms. The result is the same as ce.box(['List', ...]). A non-number element throws a TypeError.
• expr.array reads a List as plain machine numbers. It is undefined when an element is not a machine number (an exact rational such as 1/3, a radical, a complex number, a symbol, a nested list): values are never approximated. The array can be passed directly to a compiled function.

Improvements

• Generated code shares repeated squares and specializes more numeric operations on the JavaScript, GLSL and WGSL targets.
• javascript target: numeric selections over arrays run in one loop, with no intermediate arrays.
• javascript target: a list-declared input accepts a typed array. A Float64Array passed for a list<number> or vector<3> input was read as a scalar and gave NaN. It is now copied into a plain array.

Behavior Changes

• javascript target: a symbol declared with a scalar type no longer broadcasts over a list. With u declared number and f(x) = 2x + 1, run({ u: [1, 2, 3] }) for f(u) returned [3, 5, 7]; a list is now outside the declared contract. A symbol whose type was inferred from use still broadcasts.

0.125.1 2026-09-07

Improvements

• Faster generated code on the JavaScript, GLSL and WGSL targets. Repeated expressions are computed once, including in sums and products; small powers use multiplication; loops use constant bounds when known; array reads skip checks when the index is proven in range; and fewer function wrappers are emitted.
• javascript target: list pipelines are about 10× faster. A Game of Life step over a 40 000-cell board went from 13.3 ms to about 1.1 ms. Results do not change.
• javascript target: a symbol with an assigned value is computed once per compiled function, not on every call. A plot that read a 22 500-element list per pixel went from about 180 ms to 1.5 µs per sample.

Resolved Issues

• Interval compilation: a piecewise expression with no matching branch and no default returns empty instead of entire.
• simplify() treats \log_e as \ln. \log_e(x) simplifies to \ln(x), \ln(x) + \log_e(y) gives \ln(xy), and \log_e(x^3) - 3\ln(x) gives 0. Contributed by @yelliver (#332).

0.125.0 2026-09-06

New Features

• GLSL target: read a fixed-length list from a texture. The new storage compile option stores a list symbol in a sampler instead of a uniform array:

  glsl.compile(ce.box(["At", "S", "k"]), { storage: { S: "sampler2D" } });

  The host declares uniform sampler2D S; and uploads a single-channel float texture with the list in row-major order from texel (0, 0). Indices are 1-based, a negative index counts from the end, and an invalid index reads as NaN. The symbol can be read only with At and a runtime index. The option is ignored on the other targets, and WGSL does not support it.

0.124.3 2026-09-06

Breaking Changes

• Operator type handlers now return BoxedType | undefined. Return context.engine.type('real') for a fixed result, or context.engine.type(computedType) for a computed result. undefined continues to use the declared signature fallback.
• BoxedType.type is now readonly. Use engine.type(newType) to construct a boxed type with a different value.

Improvements

• Faster symbolic operations: simplification is 14–16% faster, solving 7%, integration 9%, and definite integrals 25%, compared with 0.124.2.

Benchmarks

Numeric performance (200-digit precision)

Median time per call, in microseconds — lower is better. — means the tool returned no usable result at that precision.

Expression	CE 0.124.3	CE 0.100.0	SymPy	math.js	Mathematica
\pi^2	9.1	7.9	179	106	3.9
\sin 1	22	21	227	496	5.1
\cos 1	21	21	221	536	7.1
\ln 2	16	14	347	4,428	3.7
e^{\pi}	14	13	212	4,824	4.6
\zeta(3)	1,585	1,602	268	—	49
\Gamma(\tfrac13)	863	829	362	—	213
\psi(\tfrac13)	726	712	2,835	—	171

Symbolic capability & performance

Each cell is how many times faster than Mathematica that engine is on the case (higher is better). — means the engine can't do the case. The CE + R/F column loads the optional Rubi integration rules and Fungrim identities (loadIntegrationRules / loadIdentities).

Operation	CE 0.124.3	CE + R/F	CE 0.100.0	SymPy	math.js	Mathematica
Antiderivatives
\int\frac{1}{\sqrt x}\,dx	2.9×	1.8×	3.0×	0.5×	—	1×
\int\frac{x}{\sqrt{1-x^2}}\,dx	6.4×	1.2×	6.0×	0.09×	—	1×
\int\frac{1}{x^3+1}\,dx	3.5×	0.6×	3.8×	0.3×	—	1×
\int\frac{\sqrt x}{1+x}\,dx	—	1.4×	—	0.1×	—	1×
\int\frac{x}{(1+x)^{1/3}}\,dx	—	0.8×	—	0.01×	—	1×
\int\frac{x^2}{(1+x)^{1/3}}\,dx	—	0.8×	—	0.007×	—	1×
Derivatives
\tfrac{d}{dx}\sqrt{1-x^2}	0.03×	0.03×	0.02×	0.001×	0.004×	1×
Simplification
\sqrt{3+2\sqrt2}	36×	27×	30×	—	—	1×
\sqrt6\,x+\sqrt2\,x	64×	55×	49×	3.2×	17×	1×
Evaluation
\lim_{x\to0}\tfrac{\sin x}{x}	27×	11×	32×	2.9×	—	1×
\lim_{x\to\infty}(1+\tfrac1x)^x	3.8×	3.1×	3.6×	2.1×	—	1×
\int_1^2\tfrac1x\,dx	1693×	2285×	3777×	83×	—	1×
\int_{-\infty}^{\infty} e^{-x^2}\,dx	186×	86×	230×	2.5×	—	1×
Solving
x^4+x^2-1=0	0.2×	0.2×	0.2×	0.06×	—	1×
x^3-x-1=0	1.5×	1.6×	1.4×	0.04×	—	1×

_{Measured 2026-09-06 on Node v22.13.1 with SymPy 1.14.0, math.js 15.2.0 and Mathematica 14.3.0. Correctness is verified against an independent mpmath reference.}

0.124.2 2026-09-06

Performance

• Symbolic evaluation is 5–12 % faster on simplification and solving, and about 4–10 % faster on indefinite integrals. For example, solving x⁴+x²−1=0 went from 3.17 ms to 2.86 ms, and ∫1/(x³+1)dx from 2.66 ms to 2.39 ms. Results are unchanged.

0.124.1 2026-09-06

Resolved Issues

• A coordinate accessor over a value that may be one point or a list of points now canonicalizes. With P declared list<tuple<number, number>> | tuple<number, number>, PointX(P[2]) reported an incompatible-type error. An operand with no compatible type is still refused: PointX(5) is still an error.
• A chained slice no longer makes its base a collection of collections. With Z undeclared, (Z[1..p-1])[W] = Z[p] was typed list<boolean | missing>, so a Filter with that comparison as predicate was refused. It is now typed broadcastable<boolean> and the predicate is accepted.
• glsl target: Dot over a point list whose component is itself a Dot compiles again. It reported that dot "has no array overload". Lists of complex values, booleans, strings or lists still do not compile.
• interval-js target: a power whose base reaches zero is no longer reported as clipped. A base of [0, 2] with an exponent interval such as [1.9, 2.1] returned a partial result, so a smooth field looked discontinuous. The base is now clipped only when it has a negative part or the exponent can be negative. The upper bound at such a pole is now +∞, and an exponent that spans zero no longer excludes reachable values ([0, 2]^[-1, 3] contains 0.5³).

Performance

• Symbolic evaluation is 25–45 % faster than 0.124.0 on simplification, integration, solving and construction of expressions with exact literals such as √6. For example, simplifying √6x + √2x went from 1.53 ms to 0.96 ms, and ∫1/(x³+1)dx from 4.7 ms to 3.1 ms. Results are unchanged.

0.124.0 2026-09-05

Breaking Changes

• Epsil: a bare lowercase library name refers to the library definition. Because of the new lowercase spellings (see New Features), mean, sum, count, pi and the other spellings resolve to the library when nothing in the program or the engine binds them: mean + 1 is now a type error, where it was a sum with an unknown mean. Declare the variable (let mean = 5) to shadow the library name. MathJSON is not affected (["Add", "mean", 1] still names an unknown symbol), but the engine aliases print and input are removed: use Print and Input in MathJSON.
• Epsil: a let/const cannot re-declare a name of its own scope. This includes an earlier let of the same block, a parameter of the enclosing function, and the index of the enclosing loop: function f(x) { let x = x + 1; x } is now a variable-redeclaration error, reported by epsil check. Before, the interpreter and compiled JavaScript gave different results. Shadowing a name of an outer scope in a nested block is still allowed, and the shadowing let now reads the outer name on each loop turn: let t = 1; for k in 1..3 { let t = t * 2; xs = Append(xs, t) } collects [2, 2, 2], where it collected [2, 4, 8]. To update a binding, assign to it. A top-level let in a re-run notebook cell still re-declares.

New Features

Epsil

• Lowercase spellings for the standard library. Every library function and constant can be written with an initial lowercase letter: sin(x), map(f, xs), isPrime(7), gcd(12, 18), pi, print("hi"). The MathJSON names (Sin, Map) still work. Names with a dedicated syntax (Add is +, If is if) have no lowercase spelling. A user binding shadows the spelling. Hosts that call parseEpsil and box the tree themselves must call the new resolveLibraryNames(tree, source, ce) first.
• Common mathematical Unicode notations. Double-struck letters are types (c: ℝ is c: real; also ℤ, ℚ, ℂ, ℕ, and in ce.declare('x', 'ℝ')). √3, ∛8 and ∜16 are radicals (√2x is Sqrt(2)·x). Superscripts are exponents (x², x⁻¹, xⁿ⁺¹). A subscript of letters and digits joins the name (xₙ is the symbol x_n); other subscripts give a Subscript (xₖ₊₁). ∫, ∑ and ∏ are Integrate, Sum and Product. The highlighter and the VS Code grammar support these notations.
• Unicode output. The serializer option fancySymbols (command line: epsil --epsil --fancy-symbols; MCP serialize and evaluate: fancySymbols: true) writes √x, ∛x, ∜x and x², with parentheses where needed (√(x + 1), (−x)²). The default output stays ASCII. The --epsil output now writes a square as x ^ 2 instead of Square(x).
• Dot call for protocol functions. c.area() is area(c) and c.scale(2) is scale(c, 2), so calls chain (c.scale(2).area()). Named arguments follow the receiver (c.scale(k: 2)), and c.(Shape.area)() names the protocol when two protocols declare the same member. Other functions are the new error dot-call-not-a-protocol-function (xs.Sort()). c.area without parentheses is still a field read, and a declared field still wins over a member of the same name.
• epsil check --effects reports the inferred effects of each top-level function, for example f (line 1): console or g (line 2): pure (declared). With --json the report is the effects array; the MCP check tool accepts "effects": true. The VS Code hover shows the same line.

LaTeX Parsing and Serialization

• Typed lambda parameters. (i: integer) \mapsto 2i and (f: (real) -> real) \mapsto f(1) parse to Function literals with typed parameters and serialize back as (i\colon integer)\mapsto 2i. Other uses of a colon (set-builder, piecewise, f: A \to B) are unchanged.
• Inline lambda application. (x \mapsto 2x)(3) and ((x, y) \mapsto x + y)(2, 3) evaluate to 6 and 5; they parsed as products. A power on the argument list applies to the application: (x \mapsto 2x)(3)^2 is 36, and (f)(3)^2 reads as f(3)^2.

Types

• A use of an element refines the collection's element type. xs[1] + 1 makes an undeclared xs an indexed_collection<number>, xs["a"] + 1 a dictionary<number>, and m[1][2] + 1 makes m an indexed_collection<indexed_collection<number>>. A lambda parameter shows the refinement: (v) => v[1] + 1 is typed (v: indexed_collection<number>) -> broadcastable<number>. A later assignment replaces the inferred type, and a declared type is never changed. This applies to At, First, Second, Third and Last; an operator definition can declare its own rule with the new inferOperandTypes handler.

Resolved Issues

LaTeX Parsing and Serialization

• A set-builder with several conditions now parses as a comprehension. \{ k : k \in \{1, 2, 3\}, k > 1 \} parsed as a two-element set; it now evaluates to {2, 3}. The domain can also be on the left of the colon (\{ k \in S : k > 1, k < 3 \}). A trailing comma in a set literal or a piecewise no longer adds a stray element.
• An inferred parameter type is no longer serialized. With C a list of integers, \operatorname{Filter}(C, Z\mapsto 0<|Z|) printed (Z\colon integer)\mapsto…. A written annotation still round-trips. To see inferred annotations, use toMathJson({ inferredAnnotations: true }).

Epsil

• A multi-clause function with a typed scalar parameter maps over a collection. With fib(n: integer) = … defined by clauses, 5..10 |> fib |> Sum failed with incompatible-type; it now returns 136.
• A number literal keeps all its digits. 12345678901234567890.5 parsed as 12345678901234570000, 1e400 as Infinity, and hexadecimal or binary integers above 2^53 were rounded.
• Serialized Epsil re-parses to the same expression. Power(-2, 2), Power(Power(x, 2), 3), Subtract(a, Subtract(b, c)) and Factorial(-2) now serialize as (-2) ^ 2, (x ^ 2) ^ 3, a - (b - c) and (-2)!.
• A match or if let pattern that starts with an unknown glyph reports an error. match x { ⊕ => 1 } silently dropped the case.
• A destructuring comprehension binder gets its type from the collection. In [p + q for (p, q) in pairs] with pairs: list<tuple<number, number>>, p and q are number. A pattern of the wrong arity, or a name bound twice, is now an error.
• An empty block evaluates to Nothing. if 1 < 2 { } returned the empty block {} with type unknown; it now has type nothing.
• A return-type annotation no longer gives a function the effects of the lambda it returns. function make() -> ((number) random -> number) { x => Random() * x } was typed () random -> …; it is now pure.

Types

• q + 1 with q: boolean | missing is now refused, as Sin(q) already was. unknown | missing and list<number> | missing are still accepted.
• A collection of non-numbers at a numeric element-wise operator is refused when boxed. Ln(["a", "b"]) and Add(["a"], 1) are now incompatible-type errors, as Sin(["a", "b"]) already was. A collection that could hold numbers (such as list<integer | string>) is still checked per element at evaluation.

Compilation

• A compiled function no longer returns a string when given a matrix where it expects a list of numbers. With h := (v) => v[1] + 1, the compiled h on [[1, 2], [3, 4]] returned "1,21". It now throws an error that suggests declaring the parameter as list<list<number>> or matrix.
• javascript target: a Sum with index i gives correct results. \sum_{i=1}^{3}\cos(i(|x|-\sqrt{9.81/k[i]}\,t)) returned "[object Object]…" because i was read as the imaginary unit. With mode: "complex" it also returned a wrong value.
• javascript target: Abs of a parameter that may be a point or a point list is decided at run time. With g(P) := |P - (4, 0)|, g((5, 1)) returned [1, 1] instead of √2. A point subtracted from a point list is now subtracted from each point.
• interval-js target: Arctan2 over a box that crosses its branch cut reports a jump. It returned a bounded interval; it now returns singular with the enclosure [-π, π].
• interval-js target: a domain clip is kept by the operations above it. x + \sqrt{x^2 - 0.01} over x ∈ [-0.3, 0.3] returned a bounded interval, although the function is undefined on |x| < 0.1; it now returns partial.
• python target: a for loop over a Range with a non-integer bound matches the interpreter. A symbolic bound of 2.5 raised TypeError, and a literal bound of 5.5 walked 5, 4, … instead of 5.5, 4.5, ….

Arithmetic and Numerics

• A subscript with a computed index evaluates. With k := 3, x_{k+1} evaluates to the symbol x_4, and to its value when x_4 is assigned.
• Recursive functions in a comprehension are no longer recomputed for each element. [P(i) for i in 0..n], with P(i) defined from P(i-1), called the step function n(n+1)/2 times; it now calls it n times.
• An asynchronous-only operator inside a comparison, If or Which is awaited. Less(15, AsyncOnly(2)).evaluateAsync() returned 15 < AsyncOnly(2). Inside the body of a big operator or a block, such an operator still stays unevaluated.
• GammaRegularized(a, z) evaluates for a negative non-integer a. GammaRegularized(-1/2, 2) is now −0.00849… and GammaRegularized(-1/2, 0) is −∞. A negative z still stays symbolic.
• Variance, Covariance and Correlation are accurate for data far from zero. Variance([10⁸+1, 10⁸+2, 10⁸+3]) returned 0 instead of 1, and Correlation([1.5e200, 2.5e200, 3.5e200], [1, 2, 3]) returned NaN; it is now 1. Correlation is typed real<-1..1>.
• Operations whose work grows with an argument value are bounded. Eulerian, Stirling, NPartition, StirlingS1, BernoulliB, MatrixPower, Chunk and others could run without limit or exhaust memory (Stirling(4000, 2000)). Past a limit they now stay symbolic or report iteration-limit-exceeded. Eulerian(60, 30) now returns immediately.

Benchmarks

Numeric performance (200-digit precision)

Median time per call, in microseconds — lower is better. — means the tool returned no usable result at that precision.

Expression	CE 0.124.0	CE 0.123.2	SymPy	math.js	Mathematica
\pi^2	10	12	175	103	3.9
\sin 1	22	23	219	438	5.1
\cos 1	21	23	216	522	7.1
\ln 2	15	17	342	4,364	3.7
e^{\pi}	14	16	211	4,659	4.6
\zeta(3)	1,538	1,587	270	—	49
\Gamma(\tfrac13)	855	839	339	—	213
\psi(\tfrac13)	729	723	2,795	—	171

Symbolic capability & performance

Each cell is how many times faster than Mathematica that engine is (higher is better). — means the engine can't do the case. CE + R/F is the current build with the optional integration rules and identities loaded (loadIntegrationRules / loadIdentities).

Operation	CE 0.124.0	CE + R/F	CE 0.123.2	SymPy	math.js	Mathematica
Antiderivatives
\int\frac{1}{\sqrt x}\,dx	2.2×	1.4×	1.7×	0.5×	—	1×
\int\frac{x}{\sqrt{1-x^2}}\,dx	5.0×	0.9×	3.6×	0.09×	—	1×
\int\frac{1}{x^3+1}\,dx	2.1×	0.6×	1.7×	0.3×	—	1×
\int\frac{\sqrt x}{1+x}\,dx	—	0.7×	—	0.1×	—	1×
\int\frac{x}{(1+x)^{1/3}}\,dx	—	0.5×	—	0.01×	—	1×
\int\frac{x^2}{(1+x)^{1/3}}\,dx	—	0.5×	—	0.007×	—	1×
Derivatives
\tfrac{d}{dx}\sqrt{1-x^2}	0.02×	0.02×	0.01×	0.001×	0.004×	1×
Simplification
\sqrt{3+2\sqrt2}	16×	15×	14×	—	—	1×
\sqrt6\,x+\sqrt2\,x	23×	20×	16×	3.2×	20×	1×
Evaluation
\lim_{x\to0}\tfrac{\sin x}{x}	23×	11×	19×	3.0×	—	1×
\lim_{x\to\infty}(1+\tfrac1x)^x	3.1×	3.0×	3.0×	2.1×	—	1×
\int_1^2\tfrac1x\,dx	1441×	2430×	1187×	85×	—	1×
\int_{-\infty}^{\infty} e^{-x^2}\,dx	140×	57×	118×	2.5×	—	1×
Solving
x^4+x^2-1=0	0.08×	0.08×	0.08×	0.06×	—	1×
x^3-x-1=0	1.0×	1.1×	1.0×	0.04×	—	1×

_{Measured 2026-09-05 · SymPy 1.14.0 · math.js 15.2.0 · Mathematica 14.3.0 (carried over from the 2026-08-19 run) · Node v22.13.1. Correctness is verified against an independent mpmath reference.}

0.123.2 2026-09-04

Improvements

• Typing a large list of tuples is about three times faster. Typing a list of 5,000 pairs of radicals is about ten times faster.
• Growing a list one element per loop turn is much faster. xs = Join(xs, [k]), xs = Append(xs, k) and xs = [...xs, k] in a loop took 2.9 s for 400 turns and about 30 s for 1000; they now take 89 ms and 379 ms. A Join of list literals, and an Append to a list literal, now become one list literal at canonicalization: Join([1, 2], [3]) is [1, 2, 3]. Such a result is no longer a lazy collection, and its type carries the shape: Join([1, 2, 3], 10) types vector<integer^4> (was list<integer>).

Resolved Issues

• Real-only functions over a list that may hold complex values now compile in the auto and complex modes. Min, Max, Floor, Mod, the statistics functions and the ordering comparisons over such a list (√L for L: list<real>, a list<complex> symbol) were refused on the JavaScript and Python targets. Each real element now gets the real result and every other element is NaN: ⌊√L⌋ at L = [4, -1] is [2, NaN]. This also fixes two wrong results: √L < 1 at L = [0.5, 2] returned [false, false] (now [true, false]), and min(Z) for a list<complex> symbol returned NaN. strict mode is unchanged.
• List and set literals now await asynchronous elements. [1, AsyncOnly()].evaluateAsync() returned [1, AsyncOnly()] for an operator with only an evaluateAsync handler. Elements are now awaited in order.
• evaluateAsync() now honors numericApproximation on numbers and symbols. [1/3].evaluateAsync({ numericApproximation: true }) kept the exact value. A set comprehension now also evaluates the values of a literal domain: {k : k ∈ {x, 2}} with x := 5 returned Set(x, 2).
• A tuple, set or list of inexact literals no longer carries a numeric range in its type. (√2, 1) typed tuple<real<1.4..1.5>, integer> and now types tuple<real, integer>. A list of 5,000 such points typed list<tuple<any, any>> and now types list<tuple<real, real>^5000>. A record holding ~oo now types record{x: infinity}.
• A point with a list component now compiles under Norm, Abs and Hypot. Norm(([1, 2], 3)) is [√10, √13] and Hypot(([1, 2], 3), 4) is [√26, √29]. The JavaScript target used to fall back to the interpreter.
• Hypot with a point and a list paired the point's components with the list's elements. Hypot((3, 4), [1, 2]) returned [√10, √20]; it is now [√26, √29] (the point's norm is one leg of each hypotenuse). Abs follows the same rule. Power((1, 2), [3, 4]) is still [1, 16].
• Destructured block locals compile. After (xs, n) := ([1, 2, 3], 2) in a block, Length(xs) and At(xs, n) now compile instead of failing.
• A list times a point now types as a list of points. [1, 2] · (3, 4) evaluates to [(3, 4), (6, 8)] but typed tuple<integer, integer>; it now types list<tuple<integer, integer>^2>.

0.123.1 2026-09-04

Resolved Issues

• A single-letter builtin name in an optional or variadic argument is read as a variable. Range(1, N), Max(1, N) and Min(2, N, M) returned an incompatible-type error on N, while N + 1 and Range(N) read N as a variable. For example, [1, \ldots, N] with an undeclared N failed in 0.123.0.
• Join and Append no longer drop an operand that has no value yet. With v declared list<number> and unassigned, Join([1], v) evaluated to [1] and Append(v, 2) to [2]. They now stay unevaluated until v has a value. First(Join([1], v)) still returns 1.
• A symbol declared with the bare tuple type is treated as a tuple in arithmetic. With ce.declare('w', 'tuple'), Sin(w), Sqrt(w) and w^2 typed number; they now type tuple. Abs(w) is the norm, and w · w is the no-product-between-points error.
• A list scaled by a symbolic tuple kept a unit factor. [1, 2, 3] · p evaluated to [1p, 2p, 3p]; it is now [p, 2p, 3p].

0.123.0 2026-09-04

New Features

• Epsil: match exhaustiveness warning. A match whose subject has a closed type (a sum such as type light = red | green | yellow, or boolean) now reports match-not-exhaustive when a value reaches no case, and names the missing pattern (yellow(), false). A guarded case does not count as covering. See epsil doc match-not-exhaustive.
• Typed match bindings and is tests compile. On the JavaScript target, MatchesType(v, T) (used by n: integer => …, while let v: T = … and x is T) now compiles for numbers, strings, booleans, literal types, numeric ranges, unions of these, and tagged non-generic sums. Other types, and other targets, still fail to compile.
• Destructuring loops compile. for (p, q) in pairs { … }, and the same pattern in a Comprehension, now compile on the JavaScript and Python targets, when the element type is known to be tuples of the right arity (a Zip, a literal list of tuples, a list<tuple<…>> annotation).
• The Python target compiles more loop forms. A Loop over several Element clauses, a Range with integer literal bounds (Range(10, 1, -3) → range(10, 0, -3)), and a Comprehension (as a list comprehension).
• Zip types its elements. Zip(xs, ys) now types list<tuple<X, Y>> from the element types of its sources.
• Epsil: a standard library index (published at /epsil/library/) lists every function and constant by category, with its signature, description and a tested example.
• Epsil: a style guide (published at /epsil/style/) describes idiomatic Epsil with an example for each idiom. It recommends Map/Fold, or ListFrom, instead of growing a list with Join in a loop.

Resolved Issues

Types and Validation

• Operators with custom canonical forms now validate their arguments against their signature. PoissonDistribution(s) with s := -3 was accepted, while a literal -3 was refused. It is now refused too. Related changes:
  • Sin(+oo) and a range used as a range bound are now refused when the expression is created, instead of when it is evaluated.
  • Some signatures are now wider: Apply takes any function (Apply(3, 5)), a relation takes a single operand (Less(3) is True), At takes a base of any type, and the set operators take any operand.
• Function signatures print unions correctly. (complex | infinity, (complex | infinity)+) -> number printed without the inner parentheses, and (real<0..>) | nan printed unnecessary ones. A union may now print its members in a different order (nan | real<0..>).

LaTeX Parsing and Serialization

• Norm(v, p) round-trips through LaTeX. The order was dropped, so a 1-norm re-parsed as a 2-norm. It now serializes as a subscript (\|v\|_1, \|v\|_\infty, \|v\|_F for Frobenius) and parses back from every norm notation.

Statistics

• A discrete PDF or CDF at a symbolic point now matches the numeric result. PDF(PoissonDistribution(2), x) with x := 0.5 returned 0.216, while PDF(PoissonDistribution(2), 0.5) returns 0. The symbolic forms now include the support condition, for example Which(⌊x⌋ = x ∧ x ≥ 0, …, True, 0). A point declared integer<0..> keeps the plain closed form.

Compilation

• A value that may be a scalar or a list compiles correctly. On the javascript target, an operand typed integer | vector<integer^2> made Sum(x) fail, and 2·x, x + 1 and x < 1 returned wrong values (NaN, or the string "1,21"). They now handle either shape at run time. Such a union now types 2·u as number | vector<2> (was number).
• A compiled condition with a random operand handles NaN. If(Random() > 0.5 ∧ x > 0, a, b) took the else branch at x = NaN; it now returns NaN, and Random() is called at most once.
• The Python target emitted JavaScript for a Comprehension.
• A Range whose bound is not a number compiled to an empty range. Sum(1..10..2) compiled to 0. It now fails to compile.
• Compiled Sum, Product and Comprehension loops no longer recompute loop-invariant list values on each iteration. A 10,000-element histogram took 25 s; it now takes under 0.1 s at 16,000 elements. cse: false turns this off.

Arithmetic and Numerics

• A NaN in a branch condition selects no branch. If(NaN > 0, 1, -1).evaluate() returned -1, while compiled code returned NaN. It now evaluates to Missing. Greater(NaN, 0) is still False, and If(NaN > 0 ∧ False, 1, -1) is still -1. A NaN element in an element-wise condition still takes the else value.
• Exact numbers are compared exactly. Less(10^400, 10^500) and Greater(10^500, 10^400) were both False, and 1 + 10^(-20) compared equal to 1.
• Factorial of a very large integer no longer hangs. Factorial(10^400) now stays symbolic, and its .N() is +oo. The same applies to Factorial2.
• Dividing a very small or very precise exact rational no longer rounds it. Divide(1/10^400, 3) was 0 and Divide(99999999999999999999/10^20, 1) was 1. CDF(UniformDistribution(0, 1), 10^-400) is now 10^-400.

0.122.0 2026-09-03

Breaking Changes

• A type handler in an operator definition now receives operand descriptors, not expressions. The signature is (operands: OperandDescriptor[], context: TypeHandlerContext) => Type. A descriptor has the operand's type, facts such as facts.finite, facts.sgn and facts.elementType, and a structural view from structureOf(). Use context.derive(operator, operands) to type a sub-expression. A handler can no longer declare, canonicalize or evaluate. The typeHandlerKind flag, the OperatorTypeHandlerOnExpressions type and the operandTypes option are removed. To migrate: ops[i].type.type → operands[i].type; .ops, .operator and .string → structureOf().

  Some built-in types changed: {n² : n ∈ ℤ} now has elements integer<0..> (was unknown); an Interval with a symbolic endpoint has real elements (was never); √2 ∈ [√2, 1] now types boolean.

New Features

• RuntimeError(code) creates an error value when evaluated. A function whose body contained Error("neg") was never defined. With function g(x) { if x > 0 { x } else { RuntimeError("neg") } }, g(-1) is an error value that if let v: !error = … and match can inspect. The javascript target does not compile it.
• Epsil if let. if let pattern = subject { … } else { … } binds the pattern when the subject matches, and runs else otherwise: if let v: !error = f(x) { v } else { 0 }. Without else, a failed match evaluates to Missing. New warnings: if-let-irrefutable, if-let-equal-expected.
• Epsil while let. while let pattern = subject { … } loops until the pattern fails to match: while let [h, ...t] = xs { s = s + h; xs = [t] }. New warnings: while-let-irrefutable, while-let-equal-expected. It compiles on the JavaScript target but not on the Python target.

Resolved Issues

Errors and Evaluation

• An error from evaluating an operand is now the result. Sin(Length(5)) evaluated to sin(Error(…)) and 1 + Length(5) to 1 + Error(…); both now return the error. A collection still keeps a failed element in place.
• A user function whose body evaluates to an error returns the error. len := x ↦ Length(x) then len(5) returned len(5). An Epsil function can now return an error value, such as match-no-case.
• Deeply nested tuples or lists with shared parts are fast. A value built as Tuple(t, t) twenty levels deep took seconds and printed a type with a million entries. Types are now limited in size.
• Nested list literals with shared sub-lists no longer overflow the stack. Length or Element over List(t, t) 18 or more levels deep overflowed.
• toLatex({ materialization: false }) no longer evaluates anything. Printing Join(p) with p bound to a long unevaluated chain could take minutes.

Pattern Matching

• match sees lazy lists. match Rest([1, 2, 3]) { [h, ...] => h; _ => "no" } took the wildcard case, and a while let [h, ...] = xs loop ended after one turn. A lazy list now matches a list pattern.
• A match case can break or continue in compiled code. It caused an Unexpected token 'break' syntax error.
• A compiled ...rest capture now splices into a list. [h, ...t] => [t] compiled to a list that held the tail as one element, so a while let [h, ...t] = xs { xs = [t] } loop never ended.
• A binding with a guard compiles. n if n > 3 => … failed on the JavaScript target.

Types

• The stage of a pipe keeps its element type after evaluation. xs |> p ↦ p[1] ∧ p[2] over a list of boolean pairs types list<boolean^3> before and after evaluation.
• A scalar function over a list of points keeps the point shape. With P: list<tuple<number, number>>, Sqrt(P) and Power(P, 2) typed list<number>; they now type list<tuple<number, number>>.
• A comprehension variable's type comes from the collection. In [q.x + 1 for q in L] with L: list<number>, the body changed the type of q and the expression failed only when evaluated. It is now a type error when the expression is created.
• Reshape with a non-literal size no longer throws. Reshape([1,2,3,4], (n, 2)) threw Failed to parse type; it now types list<number>.
• point + (list, list) types as a tuple of lists. G + (L, L₂) now types tuple<list<number>, list<number>>, not a union.

Compilation

• The javascript target refuses invalid point arithmetic. P·P for a point list, P + 2, P − L, and a scalar divided by a point list compiled to plausible values where evaluate() returns an error. They now fail to compile, and so does a comprehension whose variable appears in its own collection ([P.x + 1 for P in P]).
• List arithmetic with radicals and logarithms compiles. L + √L, 2·√L and L + ln L failed to compile on the javascript target.
• List arithmetic over a mix of real and complex elements compiles. Add, Subtract, Multiply and Negate over 2·[1+i, 2] or a list<complex> parameter now compile.

LaTeX Parsing and Serialization

• The serializer overflowed the stack on a product with a never-typed factor, such as f(f(b)) - f(f(a)) = (f'(c))^2 (b - a).
• An ellipsis over an indexed family is no longer read as an arithmetic progression. \{\frac{x_1}{1+x_1}, \frac{x_2}{1+x_1+x_2}, \dots, \frac{x_n}{1+\dots+x_n}\} parsed to a Range; it now stays a set, like \{x_1, x_2, \dots, x_n\}.
• A two-element Tuple in a set position serializes unambiguously. It was (a, b), which reads back as an open interval; it is now \operatorname{Tuple}(a, b).

0.121.1 2026-09-03

New Features

• Join accepts scalar operands. A scalar is appended as one element: Join([1, 2], 3) and Join(1, 2, 3) are [1, 2, 3]. This was an incompatible-type error. The javascript and python targets now also append a tuple operand as one element instead of spreading its components.
• Sign accepts complex numbers. Sign(z) is z/|z|: Sign(i) is i and Sign(3 + 4i) is 3/5 + 4i/5. This was an incompatible-type error. For real values the result is still −1, 0 or 1; Sign(~oo) is still an error and Sign(NaN) is NaN. The javascript target compiles the complex case.

Resolved Issues

• Fixed compiled PointX, PointY and PointZ reading a list of points as a single point when the operand shape was not known at compile time. For f(v) := PointX(v) + 1, f(P) over a list of points compiled to [2, 3, 4] instead of [2, 5], or threw inside a PointList.
• Fixed compiled base-10 and base-2 logarithms losing precision: log(x) / log(2) at x = 4 gave 1.9999999999999996 instead of 2.
• Fixed ["Function", ["Add", ["Negate", "q"], ["Tuple", 1, 2]], "q"] being an incompatible-type error while q + (1, 2) was accepted.
• Fixed the glsl and wgsl targets missing parentheses before a swizzle: PointX((x, y) + (1, 2)) compiled to vec2(x, y) + vec2(1.0, 2.0).x. The same applied to First, Second and Third.
• Fixed compiled color constructors (Rgb, Hsv, Hsl, Oklab, Oklch) with a non-finite channel: they now return a NaN color. Hsv(90, 1, ~oo) compiled to the same color as Hsv(90, 1, 1).
• Fixed compile() taking exponential time and memory, and eventually throwing Invalid array length, on expressions with deeply shared sub-expressions.

0.121.0 2026-09-03

Breaking Changes

• Numeric types now distinguish finite values, infinities, and NaN. integer, rational, real, and complex contain only finite values; number is now complex | infinity | nan. The new infinity type contains +oo, -oo, and ~oo; signed_infinity contains only +oo | -oo. The old name non_finite_number is a temporary alias for signed_infinity, and the finite_* names are temporary aliases for the bare types (finite_number maps to complex).

  Use BoxedType.complex, BoxedType.integer, BoxedType.real, and BoxedType.setInteger instead of their finite_* counterparts. Use isExtendedReal instead of the removed isReal property.

• Infinity and NaN have precise types and matching runtime behavior. Arithmetic, functions, assignments, collection accessors, and compiled code report or propagate infinity, signed_infinity, and nan. Finite numeric types no longer accept infinite arguments. Infinite endpoints of intervals and ranges are not members of them.

• Assumptions refine values only while they are in scope. They no longer permanently change declarations or inferred signatures. forget() and leaving a scope remove the refinement, and an assignment that contradicts an active assumption is rejected.

  The assumption API now exposes assumedValues, and ce.context.assumptions contains immutable fact arrays instead of booleans. assumptionBindings, ContextAssumptions.bindings, TypeProvenanceEntry.previousType, and the assumed provenance kind are removed.

• Numeric predicates are three-valued. isNumber, isInteger, isRational, and isExtendedReal return true when membership is proven, false when it is impossible, and undefined when it is not known.

• Functions enforce their mathematical domains. Operations that need an ordered value (Heaviside, Sign, rounding, Clamp, ElementMin, ElementMax) accept only finite reals and signed infinities; complex values and ~oo are rejected. NaN propagates through numeric evaluation.

• Elementary and special functions handle infinities consistently. They return their limits at infinities, reject arguments outside their domain, and stay symbolic when no numeric value is available. evaluate(), N(), and simplification agree more closely at exceptional points. Examples: Sqrt(~oo) = Ln(~oo) = ~oo, (+oo)^(+oo) = +oo, IEEE values for Arctan2 at infinite corners, and NaN for indeterminate forms.

• Complex accessors propagate NaN. This applies to Real, Imaginary, Argument, AbsArg, and Conjugate. The real part, imaginary part, and argument of ~oo are NaN.

• Power and Root use the same rules for infinities. ~oo is not a valid exponent. Root(x, n) follows Power(x, 1/n), and a zero root index is an error.

• Arithmetic and number-theory operators validate their arguments. Mod, Fract, Rationalize, ContinuedFraction, the parity and prime predicates, and the numerator/denominator accessors reject invalid arguments. The continued fraction of an inexact value uses its best rational approximation at the current precision. IsEven is now kept as its own operator.

• Linear-algebra operations validate numeric inputs. Norm, Trace, MatrixPower, and Determinant reject incompatible operands and have more precise result types. Frobenius norms work for vectors and higher-rank tensors, the norm of an empty collection is 0, and numeric half-integer powers of exact 2×2 positive-semidefinite matrices are supported.

• Combinatorial and regularized functions have corrected domains. Binomial, Choose, Pochhammer, GammaRegularized, and BetaRegularized handle infinite limits consistently, and several results for negative arguments and gamma poles are corrected.

• Probability distributions validate their parameters and arguments. Normal, uniform, Poisson, exponential, and binomial distributions reject invalid parameters. Uniform and exponential PDF/CDF are piecewise, discrete distributions return correct values outside their support, and N() computes binomial and Poisson quantiles.

• Statistics functions reject invalid samples. Non-numeric or nested values are an incompatible-type error. NaN and Missing propagate, quartiles of a single value are supported, and quartile tuple fields are ordered lower, mid, upper.

• Selections are lazy, and undecided conditions stay unevaluated. If, Which, And, and Or evaluate only the operands needed, so an error in a branch that is not selected does not affect the result. A condition that is provably not boolean is an incompatible-type error. A selection with no matching branch and no default returns Missing. Custom lazy operators can get the same behavior with selectsOperands.

• Compiled conditions use three-valued logic. Compiled JavaScript and Python no longer select a branch for a NaN, absent, or undecided condition. And, Or, and Not still short-circuit.

• Compiled float-only targets represent ~oo as Infinity. These targets cannot distinguish signed and complex infinity.

• Result types include possible exceptional results. Partial operations can have nan or infinity in their type, and domain failures of non-numeric operations return Missing. Division and negative powers include non-finite results when the divisor can be zero. An expression with an empty-range operand has the type never.

New Features

• Operator definitions can declare their error behavior. The new nanBehavior, partiality, definedWhen, and requires fields describe how NaN is handled, whether the operation is partial, its domain, and its preconditions. Result types include nan or missing only when needed.
• Hypot accepts infinite arguments. An infinite argument gives +oo even when another is NaN; otherwise NaN propagates. Euclidean norms and distances follow the same rule.
• Factorial, Factorial2, Gamma, and GammaLn accept signed infinities. +oo gives +oo; -oo gives NaN.
• D differentiates piecewise expressions. Each branch of If and Which is differentiated and its condition is kept.
• Componentwise functions broadcast over tuples. Sin((1, 2)) evaluates each component. Functions that treat a tuple as a point, such as Abs, are unchanged.
• The javascript target compiles more user functions: bounded generic functions, functions that return tuples, array arguments broadcast at run time, complex-valued callbacks, and element-wise selections with complex branches.
• The python target compiles statement-form conditionals. If with an else branch compiles inside blocks and loops.
• The interval-js target supports more expressions: Choose, direct application of function literals, bounded random draws, and sums over collections. A finite jump discontinuity gives a singular result with a correct value enclosure, instead of being treated as a pole.
• Predicates can have the type true or false when the operand types decide them; otherwise the type is boolean. Compiled code can remove unreachable branches.
• Numeric ranges support open endpoints. real<0<..> means x > 0, real<..<3> means x < 3, and real<0<..<3> means 0 < x < 3. An empty range is never.
• Numeric result types include computed bounds for Add, Multiply, Abs, Divide, and integer powers. Inexact literals get a small enclosing range type.
• New infinity and nan primitive types, usable in type strings, declarations, signatures, matches(), and isSubtype(). ~oo and ~∞ are the type of complex infinity.
• BigDecimal.toPrecisionToward(n, direction) rounds to n significant digits toward negative or positive infinity.

Resolved Issues

Types

• Fixed the TypeScript signature of ce.declare() rejecting an existing operator definition, for example ce.declare('Sqrt', { ...ce.expr('Sqrt').operatorDefinition!, evaluate }). The accepted type is exported as SymbolDefinitionInput.
• Fixed type inference and broadcasting through type aliases, including recursive aliases that could overflow the stack.
• Fixed assumptions leaving declarations behind after they were removed, and types that depended on temporary assumptions.
• Fixed a function parameter named i or e not shadowing the library constant.
• Fixed assigning a function signature through .type corrupting later type reads.
• Fixed unions that contain broadcastable<T> not broadcasting.
• Fixed exact literal types changing with the precision setting.

Arithmetic and Numerics

• Unexpected failures during evaluation now use the internal-error code instead of evaluation-error.
• Fixed arithmetic with infinities in scalars and collections, including zero times infinity, division of a collection by zero or infinity, and the reciprocal of infinity.
• Fixed Euclidean norms and distances with infinities and NaN, including matrix norms that ignored a NaN row or column.
• Fixed numeric integration and special functions discarding the imaginary part of complex values. Unsupported complex evaluations now stay symbolic.
• Fixed very large prime, Lucas, and Catalan computations running indefinitely.
• Fixed one-sided limits at jump discontinuities.

Collections

• Fixed Length and Count staying unevaluated for values known not to be collections.

Compilation

• Compilation output is now deterministic.
• Fixed compiled Block locals losing their declared or assigned types.
• Fixed compiled Heaviside(NaN) and Sign(NaN) not returning NaN.
• Fixed compiled JavaScript for a point added to a list of points, and for collection arguments to compiled functions.
• Faster JavaScript compilation of expressions that reference assigned values many times; self-referential assignments no longer overflow the stack.

Colors

• Fixed OKLCh conversion producing negative hue angles; hues are now in [0, 360).

0.120.0 2026-08-27

Breaking Changes

• The bivariate statistics reject complex data instead of silently using its real part. Covariance, PopulationCovariance, Correlation, LinearRegression and PolynomialFit previously projected every data point through its real part, so Covariance([1, 1+2i], [2, 3]) returned 0 — the answer for the data [1, 1]. A non-real data point now returns a structured incompatible-type error naming the operator and the offending datum. A complex literal with a zero imaginary part (Complex(2, 0)) still canonicalizes to a real and participates normally, and the complex infinity ~oo is not complex data in this sense: it is the single point at infinity rather than a sample point off the real line, so it propagates NaN. Additionally, Correlation now propagates NaN for any data it has no real value for — NaN, ±∞ and ~oo — matching Covariance, instead of misreporting it as "zero variance". That error is now raised only when a column is genuinely constant, so at ce.precision = 'machine' it also no longer appears for finite data whose sums of squares overflow the double range (values around 1e200), which now yields NaN. (At the default precision the bignum kernel handles such data without overflowing.) (Complex covariance — E[(X−μX)·conj(Y−μY)] — is not implemented; if you need it, compute it from Mean directly.)

• The one-sample statistics no longer silently use the real part of complex data: they either compute the complex answer or reject it. Every one of them used to project each datum through its real part, so Mean([1, 1+2i, 5]) answered 2.333…, the mean of [1, 1, 5]. Now:

  • Mean returns the COMPLEX mean — Mean([1, 1+2i]) is 1 + i — with no convention involved, since the mean is linear. Exact data still gives an exact answer: Mean([1, i]) is (1 + i)/2, not 0.5 + 0.5i.
  • Variance, PopulationVariance, StandardDeviation and PopulationStandardDeviation compute E[|X − μ|²] — the mean squared MAGNITUDE of the deviations, with the same sample (n − 1) or population (n) divisor as before. The result is a real, non-negative number: Variance([1+i, 1−i]) is 2 and PopulationVariance([1+i, 1−i]) is 1.
  • Median, Mode, Quartiles, InterquartileRange, Skewness, Kurtosis, Histogram, BinCounts and the empirical data form of Quantile return the same incompatible-type error the bivariate statistics use. Order statistics need a total order, and the complex plane has no canonical one; the standardized moments have only convention-laden, branch-dependent complex extensions. Histogram/BinCounts reject a complex bin EDGE for the same reason. (Quantile of a DISTRIBUTION is unaffected.)

  Real data is unaffected, including the exact results (Mean([1, 1, 5]) is still 7/3), and NaN and a real ±∞ keep the readings they have always had in these heads (Mean([1, +∞, 5]) is +∞, Median([1, +∞, 5]) is 5).

• The complex infinity ~oo now reads as NaN in the one-sample statistics, under both of its spellings. The real part ~oo reports is an artifact of how it was written — ComplexInfinity reports Infinity, Complex(1, Infinity) reports 1 — so the same value produced different statistics depending on the spelling: Mean([1, ~oo, 5]) answered +∞ one way and 2.333… the other, and Median answered 5 one way and 1 the other. Every one-sample head now projects it to NaN, which is what the bivariate heads already did, so both spellings agree: Mean, Median, Mode, the variance family, Skewness, Kurtosis and InterquartileRange return NaN, Quartiles returns (NaN, NaN, NaN), and the empirical Quantile returns NaN. A real ±∞ is unaffected.

• Mean and the variance family return NaN for complex data mixed with a non-finite value. Variance([1+2i, +∞]) returned +∞, an accident of boxed arithmetic that contradicted the real-only path (Variance([1, +∞]) is NaN). A complex sample point together with a point at infinity has no reading — +∞ is a limit along the real axis, and no direction in the plane makes it a value a complex number can be averaged with — so NaN is what these now answer. The real-only paths are unchanged.

• Histogram and BinCounts stay inert on data (or bin edges) that is not a number literal, instead of silently dropping it. A Sqrt(-2) datum has no real value, and it used to be filtered out of the sample: BinCounts([1, Sqrt(-2), 5], 2) reported the counts of [1, 5]. It now leaves the expression unevaluated, the same way the empirical Quantile does; under .N() the datum numericizes to a complex literal and is rejected with the incompatible-type error.

• A PolynomialFit degree that is not an integer is reported as a bad degree. The degree was read with a helper that takes a number's real part and rounds it, so PolynomialFit(xs, ys, Complex(1, 2)) silently fitted a degree-1 polynomial and PolynomialFit(xs, ys, 2.5) a degree-3 one, while a NaN or ±∞ degree was blamed on the argument list as invalid arguments. Every number in the degree position is now answered by the degree must be an integer in [0, 12] error unless it is exactly an integer.

• Histogram and BinCounts reject non-finite data and non-finite bin edges instead of silently dropping them. A data point with no finite real reading — NaN, a real ±∞, or the complex infinity ~oo under either spelling — used to be filtered out of the sample, so BinCounts([1, +∞, 5], 2) reported the counts of the two-point dataset [1, 5] with no hint that a value had been discarded. A non-finite explicit bin EDGE was worse: every interval comparison against it is false, so the head fabricated a row of zero counts (BinCounts([1, 2, 3], [0, NaN, 10]) returned [0, 0]). Both now return the same structured incompatible-type error the complex rejection uses, naming the finite_real constraint, the operator, and the offending value. These two heads cannot absorb a non-finite value the way Mean([1, NaN, 5]) returns NaN does: their result is a vector of COUNTS, and no count means "there was no reading". Finite real data and explicit finite edge lists are unaffected.

  A finite real too large for a machine float — 10^400, an exact integer — is refused too, but as an out-of-range error naming the machine floating-point range, because the limit belongs to the binning arithmetic and not to the value: the bin width and every interval comparison are computed in doubles, where such a datum reads as infinity. The statistics that sum their data exactly (Mean, Covariance, the least-squares fits) have no such limit and accept it.

• LinearRegression and PolynomialFit propagate NaN for non-finite data instead of misdiagnosing it. A NaN, ±∞ or ~oo value in the X column made the least-squares pivot search fail, and the heads reported unexpected-argument: "degenerate data" — a claim about the geometry of the sample that the data does not support — or, when the elimination happened to pivot on a later row, returned the half-NaN tuple (NaN, 0) whose 0 slope is not a fit of anything. Both now answer with every coefficient NaN, in the shape each head declares: (NaN, NaN) for LinearRegression, a NaN-filled list of degree + 1 coefficients for PolynomialFit, and the fitted expression with NaN coefficients when a trailing variable symbol is given. This is what both already answered when the non-finite value sat in the Y column, and it matches Covariance/Correlation. The "degenerate data" error remains for rank-deficient finite real data — LinearRegression([2, 2, 2], [1, 2, 3]) still reports it.

• Two collections of different lengths are a dimension error in the fits. LinearRegression([1, 2], [1, 2, 3]) and the corresponding PolynomialFit call fell through to the least-squares rank guard and were misdiagnosed as "degenerate data"; they now report incompatible-dimensions 2 vs 3, the error Covariance and every other pairwise head already used for a length disagreement.

• LinearRegression reports a sample with fewer than two points as such. One point determines no line whatever its value is, but the head answered (NaN, NaN) for LinearRegression([NaN], [2]) and "degenerate data" for LinearRegression([1], [2]), while PolynomialFit([NaN], [2], 1) said there were not enough data points. LinearRegression now reports not enough data points for any sample shorter than two, ahead of anything the values could say.

• Re, Im and Arg reject the argument lists their targets reject. These aliases rewrote to Real/Imaginary/Argument through a construction path that skips signature validation, so Arg(1, 2) silently dropped the second operand and answered 0 where Argument(1, 2) reported the unexpected argument. They also declared a narrower type than their targets, so Re(NaN) claimed the type real, which does not admit NaN, wherever the expression was left uncanonicalized. Each alias now validates and types exactly as the name it stands for.

New Features

• Re and Im are now defined, as aliases of Real and Imaginary. ["Re", z] and ["Im", z] used to stay inert as unknown operators; they now canonicalize to Real/Imaginary (the preferred names) and evaluate, exactness included: Re(1/3 + 2/5i) is 1/3. The \Re and \Im LaTeX commands already parsed to Real/Imaginary and are what the serializer emits, and \operatorname{Re}(z) now resolves too.

• A type handler can now be declared as a function of operand DESCRIPTORS instead of operand expressions. An operator definition that sets typeHandlerKind: 'types' receives, in place of each operand, an OperandDescriptor: the operand's handler-visible type, a deliberately minimal set of three-valued facts carrying only what the type cannot (finiteness — for the NaN literal, whose type is number; sign from pure value sources such as a held numeric value; closedness; the collection capability facets; a static shape) and an on-demand structural view — never the operand expression itself, so deriving a type cannot declare, canonicalize, or evaluate anything. Validity has no fact: an error operand's type is 'error'. The legacy expressions shape remains the default and is unchanged; the flag — never the handler's parameter count — selects the shape. Under test (and with CE_TYPE_PURITY_GUARD elsewhere) a runtime guard turns any engine-state write from a 'types' handler into an immediate error. The built-in Coalesce, Hold and ReleaseHold definitions were the first to use the new shape, with byte-identical derived types; DigitCount, Block and When have since converted the same way, each proven equivalent by a differential shadow that runs both shapes side by side across the test suite. 34 further operators (the number-theory and combinatorics constants) went one better: their constant type handler was retired outright and its result moved into the declared signature — NthPrime now declares (integer) -> finite_integer instead of pairing a wide signature with a narrowing handler, with byte-identical derived types.

• GammaRegularized and BetaRegularized no longer claim a finite real result unconditionally. Their old constant claim was unsound off the proven domain (GammaRegularized(-1, 2) evaluates to NaN). They now claim finite_real only when the domain is proven — a > 0 and z ≥ 0 for the gamma; x ∈ [0, 1], a > 0, b > 0 for the beta — and answer number otherwise, per the non-finite typing convention.

Resolved Issues

• Square roots of large exact perfect squares are exact at every precision. Sqrt(10^402) returned +oo at machine precision (the radicand was narrowed to the numeric format before rooting) and Sqrt(10^12/9) stayed unevaluated where 10^6/3 exists. Exact integer and rational radicands now reduce exactly — perfect squares fold, negative radicands give exact imaginary results — regardless of the precision setting. Sqrt(1000000/49) now evaluates to 1000/7.

• Conjugate, Real and Imaginary preserve exactness. Conjugate(1/3 + 2/5i) returned machine floats (so z·Conjugate(z) answered 0.2711… where 61/225 is available); Real(1/3 + 2/5i) returned a 21-digit approximation. All three now read the exact components: Real(1/3 + 2/5i) is 1/3, Imaginary(√2 i) is √2, and z·Conjugate(z) is 61/225.

• A fence-less Delimiter around a bracketed or braced collection serializes to valid, round-trippable LaTeX. Delimiter(List(x, y)) — the parse of ([x,y]), and the operand shape a call f([x,y]) carries — serialized as \lbrackx,y\rbrack: the [ fence maps to the \lbrack command, plain concatenation glued it onto the first operand (an unknown command, so the output did not re-parse), and the Delimiter's recorded parentheses were dropped. Fences are now joined with joinLatex, which separates a command from a following letter (this also fixes the same glue for \lbrace and custom bracket delimiters), and a fence-less Delimiter keeps its parentheses around an operand that carries its own fences: Delimiter(List(x, y)) emits (\bigl\lbrack x, y\bigr\rbrack), which re-parses to the same structure, and f([x,y]) round-trips to the call. The same rule covers the tuple family — Delimiter(Tuple(x, y)) used to emit f(x,y) under a call, which re-parsed as TWO scalar arguments where the source had one tuple argument; it now emits f((x,y)). A Sequence operand still fuses with the parentheses, since a sequence IS a bare argument list. (Tycho item 230; the Desmos importer's persisted rows were corrupted by the glued form.)

• Negate distributes over a tuple whose components are collections, and a tuple divides by a scalar under the same rule that multiplies it. A "zipped" point list — a Tuple of coordinate lists, ([1,2], [3,4]) — negated component-wise only when every component TYPE was a scalar number, so -P stayed an inert Negate, and P − P or the interpolation (1−t)·P − t·P (with t substituted later) never reduced, while 2·P and P + P folded fine. The negate arm now uses the same structural tuple test as the Add/Multiply dispatch. The tuple / scalar arm had the sibling gap plus one more: it now admits the same tuples as the Divide type handler (transparent type aliases included), and each component quotient is evaluated the way the multiply and add arms already evaluate theirs — so P / s folds once s has a value, and P / 0 answers (~oo, ~oo) with the point shape kept. Dividing BY a tuple still errors (no-division-by-point), and a non-numeric tuple still rejects with the same error the multiply path gives. (Tycho item 229 — the interpreted fallback of the declined rows painted nothing because the difference of two evaluated tuples stayed symbolic.)

• The JavaScript compile target no longer withdraws a value-correct kernel because a tuple holds an exact constant with a complex-hedged type. √(5−√5) — an exact value of cos/sin(π·rational), ordinary in Desmos documents — types the finite_complex hedge (the engine does not prove 5−√5 ≥ 0 at type time) though its value is the plain real 1.6625…. The broadcast lowering's per-element shape analysis read that element complex, the element verdicts of (√(5−√5), 0) disagreed, and every Add/Multiply over such tuples failed closed with a misleading "list-valued operand" diagnostic. The complexness oracle (isComplexValued) now answers fold-first inside a JavaScript compilation: a closed pure scalar whose memoized constant fold is a real number reports real, and the constant folder's complex-shape gate — reading the same oracle — then inlines that folded literal, so every analysis and the emitted code describe the same plain number (an indexed read of such a broadcast result computes correctly rather than reading .re off a plain number). The fold-first verdict stands down where the emission cannot fold: under constantFold: false, in a symbolDeps capture, on the shader targets, and outside a compilation — there the previous fail-closed decline is unchanged, as it is for genuinely complex or symbolic elements. (Tycho item 229; 80 of the witness document's 201 line members were declined this way, all value-correct.)

• Interpreted broadcast evaluation no longer pays the literal-type cost per element. Since a number literal's public .type became its literal value (ce.box(21).type is 21), every intermediate result of an interpreted broadcast carried a structured type, and each per-element type query walked that structure instead of comparing interned tier strings — about 2,000 subtype queries per element on the witness workload, roughly doubling the cost of evaluating and draining a large computed collection (a 7,225-element Desmos color chain went from ~5.5 s to ~12.7 s and blew through the consumer's materialization budget). Ruled: a broadcast CELL is a storage-like position and may widen. A re-entrant widening window now makes a literal report its bare tier for the duration of the interpreter's own per-cell computation — opened around each element step of a broadcast map and in the shape classifier that provably discards the precision anyway. Everything user-visible keeps the literal type: the public .type of any expression, tuple components read at classification time, and the types named in per-element error messages (a diagnostic minted inside a cell re-reads the precise type outside the window). The widening recovers the drain cost to within ~5% of the pre-literal-types baseline and costs nothing on ordinary small expressions. (Tycho item 228.)

• The shader targets lower Power over a vector base, so the min-distance-field family compiles on the GPU lane. min over squared distances to a literal point list — (x−P.x)² + (y−P.y)² after vectorization — stopped at Power: the sign-preserving integer-power helper _gpu_powi is declared over scalar floats, and pow(vecN, scalar) is invalid in both shader languages (no scalar promotion). The targets now emit a per-width helper family _gpu_powi2/3/4 (WGSL has no overloading, so the width is in the name; the sign is restored per component), select it from the operands' vector width, widen the scalar side of pow explicitly, and answer vecN(1.0) for a vector base raised to the zero power — which previously emitted a shape-wrong scalar 1.0. Found and fixed alongside, on WGSL only: the emitted-call argument counter treated the comma inside a type template (array<f32, 1>) as an argument separator, so every one-for-one operand-shape check stepped aside and an invalid emission passed as success: true. Component reduction for Min/Max over a vector already existed and is unchanged. (Tycho item 231, face (a); the symbolic-length face (b) still fails closed by design — a host-side per-values substitution makes the count static, which is the supported route.)

0.119.0 2026_08_23

Breaking Changes

• A number literal's .type is now its literal type. ce.box(42).type is 42 — a value type, a subtype of finite_integer — and ce.box(0.5).type is 0.5. An exact rational carries its tier through a singleton range (ce.parse('\frac12').type is finite_rational<0.5..0.5>), and an exact value no machine number represents carries its sign on its tier (ce.parse('\sqrt2').type is (finite_real<0..>) & !0). NaN, ±∞ and complex literals keep their tier types, as do string and boolean literals. Subtype tests are unaffected — ce.box(42).type.matches('integer') is still true — but code that compares .type.toString() against a tier name must switch to .matches(). Stored types still widen to tiers: k := 42 infers k: integer, a generic call identity(5) types finite_integer (the solver never binds a type variable to a literal type), [[1, 2]] types matrix<finite_integer^(1x2)>, and {x -> 1} types record{x: finite_integer}. Type errors now name the offending literal: "expected integer, got 2.5" instead of "got finite_real".

• Declared signatures admit by overlap; conformance is checked at evaluation. An operator with a declared signature now follows the same admission model as arithmetic: an argument is refused at boxing only when it is provably incompatible — a concrete value that fails membership (FactorInteger("abc") is still an error at boxing), or a symbolic argument whose type shares no inhabitant with the parameter (string at an integer slot). A symbolic argument whose type merely OVERLAPS the parameter is admitted provisionally: FactorInteger(n) with n: number now boxes, works when n turns out to be 12, and errors at evaluation when it does not. On a strict engine, a generic runtime conformance check at non-lazy dispatch enforces the declared parameter on each evaluated concrete operand, producing the same incompatible-type error value the static gate mints for a literal — so the literal and symbol routes now agree. Consequences to note: a wrong-kind concrete value that a handler previously absorbed silently is now an error (see Resolved Issues); a diagnostic for a call like k(x) with x merely typed incompatibly (no value yet) moves from box time to run time, and the Epsil static pre-pass no longer flags it (ROADMAP: "Epsil static evidence diagnostics lost to overlap admission"). Collection-kind and function-kind parameters, and operators with a custom canonical handler, keep their existing handler-owned admission unchanged. On a non-strict engine (strict: false) nothing changes. (R1/R8 — §4.4 of docs/plans/2026-08-22-type-handlers-on-types.md; pinned by runtime-conformance-fuzz.test.ts and runtime-conformance.test.ts.)

• Sqrt no longer evaluates a closed radicand to decide its type. √(1 − 0.2²) — a machine-float radicand, which canonicalization deliberately does not fold — now types finite_complex, the same hedge as any other real radicand of statically unknown sign, instead of finite_real: the type handler used to numericize such radicands (closedRealSign, now deleted) for the sole benefit of the compile targets, and a type derivation must not evaluate. Values are unchanged (evaluate(), .N(), solve() fold the float as before) and compiled output is byte-identical: the compile targets fold constant subtrees themselves before any lowering decision reads a type (the item-137 GLSL band is pinned byte-for-byte in type-handler-audit.test.ts). A literal radicand still types precisely — √0.96 stays finite_real, its sign being statically known. Measured blast radius: 3 pinned type assertions, all in type-handler-audit.test.ts; zero snapshot changes in the full suite. (§5.4 Sqrt row and §5.8 A5 of docs/plans/2026-08-22-type-handlers-on-types.md.)

New Features

• Ranged result types: Abs, even powers and Exp carry their sign in the type. |x| types real<0..> (each real tier keeps its non-negative range), x² types finite_real<0..>, and e^x — every Power with a provably positive real base — types (finite_real<0..>) & !0, so a consumer that reads the TYPE sees the sign the sgn handlers always knew: √(x²), √|x| and ln(2^x) now type finite_real instead of hedging complex. Negate reflects a range instead of echoing it (−|x| is real<..0>). The scope stops at these heads: Add/Multiply deliberately still join bare tiers (see the Bug Fixes note below); general interval arithmetic is tracked separately in ROADMAP.md.

• Number literals carry their value in their handler-visible type. A type handler now sees 21, 0.5 (value types), an exact rational as a singleton range (finite_rational<0.5..0.5>), and a value no machine number holds exactly as a sign-carrying range ((finite_real<0..>) & !0 for √2) — so the sign and value questions handlers used to answer from the value channel (Power's integer-vs-rational claim for 10^21, arcsin(0.5)'s domain classification, Range(2, 3)'s index-span test, an even root of a negative constant) are now answered by the type alone. The PUBLIC .type of a literal is unchanged (ce.box(21).type is still finite_integer), and a handler result is widened back to ordinary types before it is stored, so literal types never leak into an expression's type. The constants e and π now declare value-bracket ranged types (finite_real<2.718281828459045..2.718281828459046>), so their positivity is a type fact as well. (Ruling O9 first half, docs/plans/2026-08-22-type-handlers-on-types.md §4.3/§6.)

• The interval-js target sizes — or declines — dynamically nested integrals at compile time. An inner integral runs once per enclosing piece, so Integrate nodes nested inside each other's integrands or bounds multiply the enclosure's work by their piece count per level — a class the per-node sizing (which only sees one node's limits) could not bound. The outermost integral of a nest now measures its whole subtree (integrateStats: how many integral runs, how deeply they nest — a pure function of the tree, memoized per node, no clock anywhere) and picks one uniform per-level count keeping runs·count^depth within the 65 536 evaluation budget; inner lowerings inherit that count through the compile target. Below 4 pieces per level the enclosure would be as uninformative as entire at full cost, so such an integral fails closed instead and the caller falls back (in a two-lane consumer, to its scalar estimate) — restoring, for the macro-expanded Tycho item-226 witness, exactly the pre-0.118.2 decline behavior at ~zero probe cost: the witness document's member sweep runs in ~25 s against 44 s on 0.118.1 and 90 s with the runtime budget alone. Visible chains that fit are now genuinely cheaper AND honest: a three-deep chain that formerly ran 256³ evaluations into the runtime budget's entire now answers a real (coarse) enclosure at ~28 pieces per level in milliseconds. Composition through by-reference function calls shows no Integrate node in the tree and is still bounded at run time by INTERVAL_NESTED_QUADRATURE_BUDGET. A Sum/Product with literal bounds multiplies its body's integral runs by its iteration count in the sizing; with symbolic bounds the body is counted once and that shape, too, is runtime-backstopped. Both mechanisms are pinned in compile-interval-integrate.test.ts. (Tycho item 226)

Resolved Issues

• Rational(n, d) is an integer-pair constructor again: a proven non-integer argument is rejected instead of silently dividing. The two-argument form canonicalizes to Divide, which erased the declared integer contract, so Rational(3, 2.5) — and Rational(3, x) with x := 2.5 — evaluated to 1.2. The signature is now the overload ((real) -> rational) | ((integer, integer) -> rational) and a non-integer literal, or a symbol holding one, errors with incompatible-type on every construction route. A valueless symbol is still admitted and rewrites to Divide as before, and the Divide spelling itself is unchanged (Divide(3, 2.5) is still 1.2).

• Divides no longer answers a rounded question. Divides(2.5, 3) returned True — the handler converted operands with the rounding toBigint, so it actually tested 3 | 3 — and NotDivides(2.5, 3) returned False (the truth is True). Non-integer operands now stay symbolic, as the operator's documentation always stated. The same integrality gate protects FromContinuedFraction, which used to reconstruct a value from a rounded non-integer term, and NotDivides' declared signature now matches Divides' ((number, number)).

• The unary Multiply/Subtract fold no longer lets a held non-numeric value through unexamined. Multiply(s) with s := "str" folded to the lone operand — the operator vanished before any check could run — and evaluated to "str", while the literal Multiply("str") refused. A lone operand holding a concrete string, boolean, or character now errors with incompatible-type; a lone collection still folds (the broadcast identity) and a valueless symbol still folds.

• Intersection of a single collection is that collection as a set, not the empty set. Intersection([1, 2, 2]) evaluates to Set(1, 2) (symmetric with Union), and a set-shaped operand returns itself without enumeration — Intersection(Integers) is Integers. Nullary Intersection() is still EmptySet.

• Vector declares the content leniency it always had. Its signature is now (any+) -> vector, matching the tensor family (Matrix accepts non-numeric entries on every route), and its type handler claims the numeric vector<n> only when every element is provably a number.

• The Epsil linter is now deliberately stricter than engine admission for assignment evidence. let x; x = g(); k(x) with g: () -> number and k: (integer) -> integer flags a static-type-error again: the pre-pass refuses a call whose recorded evidence type does not FIT the parameter, even when it merely overlaps — the way TypeScript flags code that would run. Lint-only by construction (the evidence exists only while a pre-pass runs); the executed program is unchanged, overload verdicts stay per-arm, and an exact-rational initializer now flags too. One adjacent lint gap is recorded in ROADMAP.md: a LAMBDA-valued callee's argument errors are deferred to run time by design, so the pre-pass does not surface them yet.

• Folding an assigned value that would explode the emitted source now fails closed. A DAG-shared value tower unfolds once per reference path in generated text, so compiling a member could grind through megabytes of source; tryFoldKnownSymbol now refuses above 20,000 expanded nodes (a DAG-linear identity-memoized size sum — the compile corpus's largest legitimate fold is 4 nodes) with a message that says why. The default fallback: true route degrades to interpreted evaluation with correct values; the direct registered-target route throws. The CSE initiative remains the general fix.

• Nested quadrature is bounded through Monte-Carlo composition, and compiled multiple integrals cost what the interpreter's do. _SYS.integrateMC joins the nested-evaluation budget (a runaway by-reference composition under Monte-Carlo, previously unbounded, now answers NaN in seconds; a nested Monte-Carlo integral that previously never returned now answers NaN too), and the Integrate emitter seeds starting panels by the statically visible nesting depth using the interpreter's own sizing — a compiled triple integral dropped from 1.38·10⁷ integrand evaluations (~0.9 s) to 9.1·10⁴ (~12 ms) with accuracy equal or better on every probed closed form.

• A runaway dynamically-composed scalar integral now answers NaN instead of hanging. The scalar javascript target's _SYS.integrate carries the same per-outermost-integration evaluation budget the interval target already had (2²⁵ nested integrand evaluations — ~70× headroom over the hardest measured double integral, 2.4× over a smooth genuine triple): by-reference composition no tree walk can see is cut after a few seconds, an exhausted run cannot fall through to the Monte-Carlo fallback, and a fresh outermost integral re-arms. Choosing NaN as the exhausted result follows the ROADMAP entry's own analysis (the scalar result is an estimate, not an enclosure).

• structural of an object-holding shared tree is no longer exponential. The per-node memo cannot persist a payload containing a mutable object (cache rulings B12/B22), so such trees rebuilt once per path; a transient map scoped to the outermost read — retaining nothing, validating nothing — now preserves sharing within one read (depth 18 cost 8.5 s before; depth 30 is now depth + 1 rebuilds).

• The Epsil pre-pass flags a concrete literal initializer that cannot inhabit a call's parameter again. Overlap admission had silenced let x = 1.5; k(x) for k: (integer) -> integer (the widened number overlaps integer); the pre-pass now records the initializer's exact literal type as its assignment evidence, which provably refutes the parameter, restoring the static-type-error — while let x = 2 stays clean and a symbolic x = g() stays admitted (that half is an open product decision recorded in ROADMAP.md).

• Every irrational standard-library constant declares a value-bracket ranged type (e/ExponentialE, π, γ, Catalan's G, and φ), so sign and magnitude are type facts — √φ and ln π type finite_real off the declarations alone. GoldenRatio's value is an unevaluated expression whose static type cannot witness a bracket; such standard-library declarations are now TRUSTED (the throwing value-vs-type check is user-declare-only) and validated empirically under a development-build console.assert plus the suite pin constant-declared-brackets.test.ts. The remaining literal-value type handlers (ErfInv, PolyLog, Subscript's numeral base) read the literal's handler-visible type first.

• A compilation that exhausts the shared antiderivative pool no longer starves later compilations of closed forms. The per-compilation pool bounding symbolic antiderivative-first attempts (added with the item-226 fix) was reset only in compileRoot and the public compile() entry — but registered targets invoked directly (ce._getCompilationTarget('javascript').compile(...)) enter through compileCseRoot, which reset nothing. Once any compilation drained the pool, every later direct-target compilation skipped the symbolic attempt permanently and emitted runtime quadrature (_SYS.integrate(…)) where a closed form exists (∫₀ᵗ 2x dx compiled to a quadrature call instead of t²). The reset now lives at the one choke point every route crosses — the depth-0 boundary of BaseCompiler.compile — which also gives each auto-mode escalation attempt its own pool. Nested compilations now consume the outer pool instead of re-granting themselves a fresh one, so the aggregate stays bounded by one pool per depth-0 compilation entry (a multi-statement shader body, which compiles each statement as its own root, gets one pool per statement). Pinned in compile-antiderivative-budget.test.ts. (Surfaced by the dual review of the constant-fold-before-type-read change.)

• Histogram/BinCounts no longer round a non-integer scalar bin spec. The scalar bin spec is a bin COUNT, but it was read with a rounding conversion, so BinCounts(data, 2.5) silently answered the 3-bin question. A non-integer scalar now stays INERT — the documented contract, which lets Desmos-style bin-WIDTH spellings (histogram(L, .05)) parse and remain for the importer to translate. A string bin spec, reachable through a permissively-typed symbol, is now an incompatible-type error instead of being iterated character-by-character into NaN bin edges.

• Integer-domain operators no longer silently round a non-integer operand delivered through a permissively-typed symbol. With u: any and u := 2.5, FactorInteger(u) answered [(3, 1)], NextPrime(u) 5, IsTriangular(u) True, and Fibonacci(1 + 2i) 1 — the ~45 operators of the number-theory and combinatorics families trusted the boxing gate to never hand them a non-integer, and toBigint rounds by contract. All of them now produce an incompatible-type error value at evaluation, from the generic runtime conformance check (no handler was changed).

• An unknown rule-condition name no longer crashes. Condition(x, "nonsense") threw a raw TypeError out of evaluate() (an unguarded CONDITIONS[name] lookup); checkConditions now fails closed on an unknown condition name. More generally, a native-fault crash (TypeError/RangeError/ReferenceError) escaping a built-in non-lazy evaluate handler is converted into an evaluation-error value on the expression instead of crashing the caller; deliberate throws — cancellation, redefinition discipline, predicate contracts, user-function arity — still propagate unchanged.

• A promoted radical arm in a branch is no longer double-wrapped on the JavaScript target. Under the complex discipline (the default mode: 'auto'), If(c, √−4, √(1 − 0.2²)) promotes the unknown-sign radical arm to _SYS.csqrt(…) — already a {re, im} object — while the branch coercion classified that arm by its static type and wrapped it a second time ({re: {re, im}, im: 0}), NaN-poisoning every slot read. The JavaScript target's static complex-wrap sites (BaseCompiler.isProvablyRealValued: branch-arm coercion, the Typed ascription lift, the complex-argument delivery) now consult the shape analysis (isComplexValued) before the static type, and — where the type is imprecise — the constant fold's value (constantFoldValue, split out of tryConstantFold behind every gate except the constantFold emission opt-out). A closed constant arm beside a complex arm therefore keeps the complex convention under every mode, including mode: 'strict' with constantFold: false, where only the fold-informed analysis can prove the structurally-emitted arm real. Pinned in compile-complex.test.ts ("coercion reads the value SHAPE" block).

• The runtime effects projection honors its memo for pure applications, and an expression's structural form is memoized per node. Two independent defeats of per-node caching made every walk over a DAG-shared tree — a document function applied to its own previous result embeds that result once per parameter mention, so a few levels unfold to millions of paths — exponential, which surfaced as a consumer document evaluating at 4 GB and aborting on the heap limit (Tycho item 225). First, effectsOf's memo consultation was spelled expr._effectsOf?.() ?? applicationEffects(expr), and undefined is the memo's legitimate answer for a PURE application — so every read of every pure node fell through and recomputed its entire subtree: 24 million recomputes in one document evaluation, 1.5 million once the memo is honored. Second, BoxedFunction.structural rebuilt through every operand's structural with no per-node memo, so a shared tree was rebuilt once per path — exponential time and fresh allocation for every copy; it now caches per node (generation-guarded, like type), and the rebuilt form preserves the sharing. Both pinned on the depth-30 shared tower in dag-shared-walks.test.ts. The witness document no longer OOMs; its remaining cost is a separate compile-time issue (baking a DAG-shared symbol value into compiled source is exponential in the OUTPUT without CSE) tracked in ROADMAP.md. Found in review and fixed with it: once any mutable object exists, two more walks re-scanned a shared tree once per path — the cache commit points' payload-containment scan (containsObject), and the cross-engine ingress guard that runs on EVERY function boxing (containsForeignEngineObject — constructing a depth-n shared tree cost 2^n scans). Both now carry a per-question memo and are linear in distinct nodes; pinned by the armed-scan test in dag-shared-walks.test.ts. A shared tree that CONTAINS an object still rebuilds its structural form per read (the cache commit rule refuses object-holding payloads by ruling); tracked in ROADMAP.md. (Tycho item 225, partial)

• A dynamically nested interval integral can no longer run away. The compiler sizes an integral's piece count only for the nesting it can see — the limits of one Integrate node (256 pieces for a single or double integral, fewer per level beyond that, so one node never exceeds its 65 536 evaluation budget). But integrals also nest DYNAMICALLY: a distinct Integrate node inside another's integrand, or a compiled user function that computes an integral, runs once per enclosing piece, multiplying the work by 256 per level with no syntactic trace. The Tycho item-226 witness — a six-level Newton iteration w − k·E(w)/d_E(w) whose macro-expansion substituted each level into E's integrand — emitted 728 _IA.integrate calls nested inside each other's integrands, ~256⁶ integrand evaluations, and never returned. The interval runtime now enforces its own budget (INTERVAL_NESTED_QUADRATURE_BUDGET, 4× the per-node compile-time budget): each outermost integral entry resets it, every integrand evaluation performed by a nested integral consumes it, and a nested integral that finds it exhausted answers entire — sound, since the entire line contains the true value — so the enclosure widens instead of spinning and the caller degrades to its non-interval fallback. Integrals the compiler did size are untouched (a double consumes 65 536 of the 262 144 budget, a triple about the same), pinned alongside the regression in compile-interval-integrate.test.ts. The consumer's document that hit this completes its member sweep in ~90 s where it previously never returned. With it, the antiderivative-first symbolic attempts of one compilation now share a single 4 s wall-clock pool instead of arming a fresh 2 s span per Integrate node, so a many-hundred-node emission degrades to its numeric emitters at bounded compile-time cost. The pool resets at every compilation entry — the public compile(), compileRoot (raw custom targets), and compileCseRoot (every registered target's root) — so no compilation inherits an earlier one's depleted pool; pinned by draining the pool and compiling on each route. (Tycho item 226)

• An assume() range no longer leaks through Add and Negate. After assume(x > -1); assume(y > -1), x + y typed real<-1..> — but the sum can be as small as −2 — and -x echoed real<-1..> verbatim. Join-based result computations now strip range decorations from their inputs (a join is a set union, and a sum does not lie in the union of its terms' ranges), and Negate reflects a range about zero (-x under that assumption is real<..1>).

0.118.2 2026-08-22

New Features

• The interval-js target now lowers Integrate, At, Length and PointX/PointY/PointZ. Each compiled on the javascript target and declined on interval-js with "the operator is known to the engine but target 'interval-javascript' has no lowering for it", so every plotted band over one of them fell from the interval lane to the scalar fallback. Four of the six had a common cause: the interval target had no collection value model at all — a List/Tuple literal had no lowering, and an array handed to a compiled function was flattened into an object keyed "0", "1", …, losing its length. A collection is now a JavaScript array of intervals in the OPERAND position of an accessor, and the accessor projects it back to one interval: Length answers the count as a point interval; At applies the interpreter's 1-based, negative-from-the-end convention, and since the index is itself an interval it stands for a set of indices — the result is the hull of every element the integers in it select, partial when some of them fall outside the collection, and the numeric absence marker ({ lo: NaN, hi: NaN }) when none selects anything; PointX/PointY/ PointZ read one coordinate of a single point. A literal collection — or a symbol assigned one — folds at compile time. The gates mirror the javascript target's: a string base, a gather or mask index, a list of points under a coordinate accessor, and a collection-valued RESULT all still decline, each with a message saying why (the target's value is one interval). Integrate is a different algorithm rather than a table entry: the antiderivative-first step is now shared with the javascript target (BaseCompiler.closedFormIntegral), so ∫₀ˣ sin t dt compiles to the closed form 1 − cos x on both — except that on the interval target a DEFINITE integral's closed form is guarded at run time (_IA.integrateClosed): the symbolic step differences an antiderivative at the bounds without checking that the integrand is bounded between them (∫₋₁¹ dt/t² closes to −2), and on this target that would be a zero-width "enclosure" of a divergent integral, so a coarse interval scan of the integrand over the range withholds the closed form — the scan cannot miss a pole, since interval arithmetic encloses — and hands such an integral to the enclosure, which answers singular. An integral that does not close compiles to _IA.integrate, an enclosure — a Riemann bracket over a uniform partition, with interval-valued bounds handled by the mean-value correction [0, b₂−b₁]·f([b₁, b₂]) on each side, and the accumulation's own round-to-nearest error added back by the standard dot-product bound — rather than the scalar target's Gauss–Kronrod estimate. Its width is first order in the piece size (≈ 2.5·10⁻³ on ∫₀¹ e^{−t²x} dt at 256 pieces); nested limits share one evaluation budget of 65 536 integrand calls (256 pieces per level for a single or double integral, 40 for a triple, 16 for a quadruple), so a deeper integral gets a coarser enclosure instead of a runaway cost. A pole inside the range answers singular, an undefined integrand empty, and an infinite bound entire (no finite partition exists; the scalar target's variable transform is not interval-tight). An indefinite integral with no closed form still fails closed, as on the javascript target. Found and fixed alongside, on BOTH targets: a Function integrand's parameters were paired to the limits by position, where the interpreter pairs them by name — Function(a·x·y², y, x) under (x, 0, 1), (y, 0, 2) compiled to ∫∫ a·y·x² (2a/3) instead of ∫∫ a·x·y² (4a/3), and a spare parameter (Function(x + q, x, q) under one limit) compiled as a read of an input the integral never supplies; both now fail closed unless the parameters match the integration variables one to one. And the interval target never populated varsKeys, so a symbol the caller pinned through vars was not protected from the closed-form fold (∫₀ᵏ t dt with k mapped baked k²/2 instead of reading k at run time). Two contract points were settled with it: the interval target's result is one interval per quantity, so a collection-valued expression (a comprehension) returns an ARRAY of interval values (IntervalValue) — which comprehension roots already did, now typed and honored by the interpreter fallback too, which used to collapse any list to entire; and every scalar kernel now fails closed on a provably collection-valued operand, where Add(L, 1) used to answer { lo: NaN, hi: NaN } and Less(xs, 3) 'maybe' behind success: true. The runner's variable type is IntervalInput — a number, an interval, or an array of these for a collection-valued variable. (Tycho item 220)

Resolved Issues

• Evaluating a user function over its own previous result no longer takes exponential time. A document function applied to its own result — Tycho's detail(smooth(upSample(…))) heightmap chain, evaluated while its sliders are still valueless — produces an inert value that EMBEDS the previous level once per mention of the parameter: four levels hold about 16 000 distinct nodes that unfold to over a million. Every walk that descended each operand independently paid for the unfolded tree — the Add ordering key (revlex, a string of every symbol in the term, so the key itself grew to gigabytes), the free-variable scan (unknowns, freeVariables, symbols), the binder rewrite behind parameter substitution, symbol dereference and Sum/Comprehension canonicalization (rewriteWithBinders), closure capture, the element memo's dependency snapshot, has() and the ordering tie-breaker's leaf count — and none of them reached a deadline check, so a consumer's per-evaluation time budget never fired: one document ran for more than twenty minutes at 100 % CPU and 3 GB (Tycho item 220's sweep stall). 0.118.1 finished the same document in about 12 s only because its Add broadcast over the operand; the item-221 fix keeps that Add inert, which is right, and exposed the walks. Each walk now memoizes per node within a call, so it is linear in the number of distinct nodes, and the ordering key is bounded to the trailing 1 024 characters of the symbol sequence. The bounded key is a function of the unbounded one, so every sum whose terms' keys fit orders exactly as before; a sum whose terms carry more than 1 024 characters of symbol names AND share that whole tail now breaks the tie by the structural order rather than by the longer key — deterministic, and only reachable from expressions of that size. The heightmap document evaluates in 3.7 s (78–199 s before); the engine-only replica at three levels takes 0.2 s (more than two minutes before). Pinned in dag-shared-walks.test.ts on a depth-30 shared tower (31 distinct nodes, 2³⁰ unfolded): the free-variable and symbol scans, the bounded ordering key, the degree walks on a shared polynomial spine, the leaf-count tie-breaker, has(), a Sum binder over the tower, the application of a literal holding it, and a lazy collection's dependency snapshot over it. Two walks on the way to ASSIGNING such a literal remain exponential and are tracked in ROADMAP.md ("Assigning a function literal whose body shares operands").

• A compound factor of a vector Multiply keeps its parentheses on the glsl and wgsl targets. Multiply(Add(t, 1), Tuple(x, 0)) compiled to t + 1.0 * vec2(x, 0.0), which the shader reads as t + (1.0 * vec2(x, 0.0)) — the float broadcast into the vector rather than scaling it, so a Desmos-style lerp t·P₁ + (1−t)·P₀ rendered as t·P₁ − t + P₀. A Multiply with a collection operand is lowered by the shader target's function handler, and the compile callback such a handler receives carries no precedence context, so the factor arrived unparenthesized and was joined with a bare *. Each factor is now compiled at the binding power of *. Two more instances of the same dropped grouping are fixed with it: a function handler's emission spliced into an enclosing infix operator now obeys the same op[1] < prec rule the infix path applies to its own operands (s·(P + Q) over two points emitted s * vec2(a, b) + vec2(c, d)), and a single-statement Block used as a sub-expression is compiled at the enclosing precedence on every target (Multiply(Block(Add(t, 1)), x) emitted x * t + 1 on javascript, python, glsl and wgsl). The scalar multiply, Divide, and the javascript/python lowerings of the vector shape were already correct and are unchanged. (Tycho item 224)

• A bare Nothing fails closed on every compile target. Nothing is the erasure marker, not a value; it can still reach a compiler as a bare symbol (a malformed Which with a dangling clause canonicalizes to it). Only the javascript target refused it — glsl and wgsl emitted the undefined identifier Nothing behind success: true, python the undefined name, and interval-js a _.Nothing read that is undefined at run time. All five targets now decline with the same diagnostic ("the erasure marker is not a value … Fail closed (D6)"). A Nothing arithmetic operand is still erased at canonicalization (Add(Nothing, x) compiles to x), as before.

• An element-wise broadcast no longer captures a collection-typed but still valueless operand as a per-element scalar. With A an unknown-length view (Range(0, n)/n, n unassigned) and s declared list<number> but not yet assigned, Multiply(A, s) evaluated to Map(_ ↦ _·s, A); once s := [10,20,30] and n := 2 bound, that stored form materialized as the outer product [[0,0,0],[5,10,15],[10,20,30]] while a fresh Multiply(A, s) zipped to [0,10,30]. The known-length form [1,2,3]·s → [s, 2s, 3s] was the same splice. A collection-typed operand that is not yet a collection value now leaves the operator inert — s·[1,2,3], typed list<number>, the zip's element type — and re-evaluating it after the assignment zips. The decline reaches every route: Add/Multiply/ Divide/Mod, user lambdas and declared broadcastable<…> parameters (the application is held rather than inlined — g(a,b) = (a,b) no longer stores ([1,2], s)), the relational operators ([1,2,3] < s), When's mask, PointList components, tuple scaling ((1,2)·s no longer stores (s, 2s)), and operands typed number | list<number> or broadcastable<number>. A definite length mismatch among the resolved operands still errors ([1,2] + [3,4,5] + s → incompatible-dimensions 2 vs 3), and a symbol declared bare tuple keeps scaling a list ([1,2,3]·r → [r, 2r, 3r]). (Tycho item 221)

• PointList over unknown-length lazy views is a point VIEW, with a type that carries the point arity. PointList(−√(1−A²), A) with A = Range(0,n)/n and n unassigned evaluated to an inert PointList head — isCollection false, no count, typed list<tuple> with no component arity — while the sibling A·(1,0) already produced a lazy point view. An unknown-length indexed component (a symbolic-length Range, a lazy Map over one, a Filter) now transposes lazily, exactly as the arithmetic route does: the result is a collection, resolves to the eager list of points as soon as the length does, and the stored lazy form re-evaluates to the same points a fresh evaluation gives. A provably infinite component or a non-indexed one (a Set) still fails closed; a string component is an atomic coordinate rather than a fail-closed trigger, so PointList("ab", 3) now evaluates to the point ("ab", 3) its type already promised. The static type is list<tuple<T₁, …, Tₖ>> (each list component contributes its element type, any other component its own type) instead of the arity-less list<tuple>. Consumer-visible consequence: because a two-component point list now statically has no third coordinate, PointZ(PointList(-6, n)) is a typed incompatible-dimensions error at box time on every route, where the javascript target used to emit a kernel of NaN absence markers and the GLSL target its own arity decline. (Tycho item 222)

• A zipped Divide(list<tuple>, list<number>) folds each element. [T, U] / √(PointX([T,U])² + PointY([T,U])²) evaluated to [0.999999999999999835·(0.943…, −0.331…), (0.6, 0.8)]: the value-level div() had no tuple arm, and the broadcast zip builds each element with ce._fn('Divide', …), which bypasses the canonicalDivide fold the single-tuple form T / √(…) takes. An inexact divisor left an inert number × Tuple; an exact integer divisor ([T, U] / [2, 5]) left a tuple of unevaluated Divide components. A numeric tuple divided by a numeric scalar now scales component-wise on the value route too, and tuple.div(0) on that route answers (~oo, ~oo), as the box route always has. (Tycho item 223)

• Arithmetic admits a symbol declared through a transparent type alias of a tuple, list or vector. ce.declareType('pt', 'tuple<number, number>', { alias: true }) then ce.declare('p', 'pt'): 2·p, [1,2,3]·p, Divide(p, 2) and Negate(p) all errored incompatible-type "number" vs "pt", as did an alias of list<number> or vector<2>, while the direct spelling was accepted — the shape gates read the type's kind directly and saw an opaque reference, where isSubtype already unfolded it (so an alias of a scalar worked). Transparent aliases are now unfolded at every shape gate and in the quotient's type. A NOMINAL declaration (declareType's default) stays opaque by design and is still refused.

• A lone scalar | list<…>-typed operand keeps its union in the result type, and a big operator over a body that may be a collection stays inert. With u declared number | list<number> and not yet assigned, 2u, u + 2, −u, sin(u) and a user lambda f(u) all typed a definite list<…> — type.matches('collection') answered true — yet u := 5 evaluated 2u to the scalar 10. The result now carries the union through, finite_number | list<finite_number>, each branch wrapped back in its own collection kind (number | range → finite_number | indexed_collection<finite_number>), and a definite collection sibling ([1,2] + u) still gives list<number> — the union's list branch zips and its scalar branch lifts. Two folds that committed such a body to a scalar are closed by the same rule: Sum(2u) evaluated to 2u (and Sum(u) to u) where u := [1,2] makes the sum 6; and Sum(2 + h(x)) with h not yet defined evaluated to h(x) + 2, which re-evaluated to [5, 8] once h := x ↦ [x, 2x], where the sum is 13. Sum/Product with no index now stay inert over a body typed as a scalar-or-collection union (any branch a list, vector, set, tuple, string or broadcastable<…>), a broadcastable<…> body, or the application of an undeclared head; a bare undeclared symbol still folds (Sum(y) → y).

• A definite integral with a pole strictly inside its bounds no longer evaluates to a finite number. ∫₋₁¹ dt/t evaluated to 0 and ∫₋₁¹ dt/t² to −2: the antiderivative was differenced at the bounds with no check that the integrand is bounded between them, which the fundamental theorem of calculus requires — both integrals diverge. The Integrate evaluate handler now answers what the integral actually is: +∞ or −∞ when the integrand keeps one sign across every interior pole (∫₋₁¹ dt/t² → +∞, as ∫₀¹ dt/t → +∞ already did), and inert (unevaluated, as it already was when no antiderivative was found) when the integrand changes sign across a pole and the integral has no value at all (∫₋₁¹ dt/t, whose principal value is 0 but whose integral is undefined). .N() follows suit — ±∞, or NaN for the sign-changing case, on either bound order and in the iterated form for a dimension with constant bounds — where it used to hand back a confident quadrature Measurement (∫₀² sec t dt → 8.316585 ± 0.000016, ∫₋₁¹ dt/t → −1.4 ± 3.6): the adaptive Gauss–Kronrod error estimate is blind to a singularity it straddles. A pole is reported only when it is PROVEN: located exactly — as a real root of a polynomial denominator (1/t, 1/(t² − 1), t⁻³), a pole of tan/cot/sec/csc or csch/coth of a linear argument, or a zero of a sin/cos/tan/cot/ sinh/tanh divisor (a reciprocal such as 1/tan t is not canonicalized to cot t, so it is read as written) — and then confirmed by sampling the integrand on both sides of it to grow at least as fast as 1/|t − r|. A root is told apart from a bound by the denominator's residual there, not by a distance, so ∫₀¹ (t − 10⁻¹⁰)⁻² dt is recognized as divergent while a double root exactly at a bound stays an endpoint case. That confirmation is what keeps a cancelling denominator (∫ (t² − 1)/(t − 1)), an integrable singularity (∫₀¹ dt/√t = 2), a pole AT a bound (∫₀¹ dt/t = +∞), and every integral with a symbolic or infinite bound exactly as they were. The compile targets' antiderivative- first step inherits the fix: ∫₋₁¹ dt/t² no longer compiles to the constant −2.

• The sign of a circular function of an exact argument was off by one quadrant. sgn of cos 1, sec 1, tan 1 (first quadrant) and sin 2 (second quadrant) answered negative: the quadrant helper numbers the quadrants 1..4 and the sign table was indexed 0..3 with that number. Every consumer of the sign was affected — |sec 1| evaluated to -sec(1), so ∫₀¹ tan t dt (= ln|sec 1|) came out as ln(−sec 1), a complex number for a real integral. All six functions now report the sign of their value in every quadrant (verified against the numeric value at 90 sample points).

• ∫ cot(ax + b) dx had the wrong sign. The pattern rule answered −ln|sin(ax + b)|/a; the antiderivative of cot is +ln|sin x| (its derivative is cos x / sin x). ∫₀² cot t dt, divergent at the lower bound, evaluated to −∞ instead of +∞. The power-reduction route (∫cot³x dx) already used the correct base case and is unchanged.

• Eight more rules of the built-in antiderivative table were wrong (the table the Integrate evaluate handler uses when no integration provider is loaded; the Rubi provider was unaffected). ∫ sinh and ∫ cosh answered ln|cosh| and ln|sinh| (the antiderivatives of tanh and coth); ∫ tanh answered ln|sech|, the negative of the correct ln(cosh); ∫ sech answered ln|tanh| where it is arctan(sinh); ∫ csch kept the full argument in −ln|coth| where it must be halved; and all six inverse hyperbolic rules (arsinh, arcosh, artanh, arcoth, arsech, arcsch) answered the function's own logarithmic definition — ∫ arsinh t dt evaluated to arsinh t. Each is now the by-parts form (∫ arsinh t dt = t·arsinh t − √(t² + 1), …), and every rule of the table is pinned by differentiating its answer back to the integrand at several points of its domain.

0.118.1 2026-08-22

Resolved Issues

• The javascript and python targets now compile the multi-collection (zipWith) form of Map over symbolic sources. Map((_1,_2) ↦ _1+_2, 1..N, 2..N) with N a free input declined with "Map: multi-collection form is not compiled" — which a plotting document met through Desmos' element-wise list difference c = (1..N) − Join([0], 1..(N−1)), whose bound value is exactly that zip Map, so every compiled expression reading c declined with it. The same shape with literal bounds const-folded away before reaching the lowering, which is why it looked as if every minimal zip compiled. The form now lowers like the single-collection one: each source is materialized once, the callback receives one element from each source per position, and the result is as long as the SHORTEST source — the interpreter's count for the form, and what Zip does. A parameter annotation on the callback is checked against ITS source, position by position, with the same fail-closed rule the unary form has (an integer annotation over a number-typed source declines, naming the parameter). Three shapes stay with the interpreter, by a compile-time decline rather than a wrong value: a source whose elements are not provably real numbers (list<complex>, list<list<number>>, strings, a bare list) — the callback's parameters are compiled untyped and its body treats them as real numbers; a source with observable effects (one that draws Random) — the compiled code materializes every source in full where the interpreter stops at the shortest; and a bare Add/Subtract/Multiply/Divide symbol as the mapping over anything but exactly two sources — it compiles to a two-argument function. The effects decline now guards Zip on both targets too, which had the same gap. The constant fold's cost estimate prices a zip by its shortest resolvable source instead of its first. (Tycho item 218)

• Reading the type of a nested lazy collection view is no longer exponential in its nesting depth. A view built over a collection whose length is not statically known — L := Range(0,n)/n with n a free input, then Y_0 := Y + 0·(2L−1) — nests one Map per arithmetic step, and asking such an expression for its type cost 2^depth type-handler invocations. Sqrt(1 − Y_0²) took 3.6 s, and PointList(−√(1 − Y_0²), Y_0) did not return at all. A bound list was affected too, superlinearly in its length: parsing an rgb(…) row over a 301-element list took 12.9 s. All are now single-digit milliseconds.

  Two causes compounded, both in Map's type handler. It read each source's type twice per level, which doubles per level when the source is itself a Map. And the element-type derivation added in 0.118.0 declares stand-in symbols in a scratch scope, which advanced the engine's any cache generation — the generation BoxedFunction.type keys its memo on — so every read retired the type cache of every expression in the engine, including the sources the same walk was mid-way through reading. One PointList evaluation made 998K handler calls and 2.0M cache invalidations.

  A declaration whose target scope is one the computation itself pushed and pops now advances no cache generation, since the binding cannot outlive that computation and nothing outside can depend on it. The test is on the declaration's resolved target scope, not on whether a derivation is running, so a declaration aimed at a longer-lived scope — a function literal's block scope, a protocol member's scope — still invalidates as before. (Tycho item 219)

• A compiled Sum/Product over a COLLECTION-valued body no longer skips iterations of an effectful body. 0.118.0 made the NaN early exit depend on effects rather than on who supplied the code, but only for scalar bodies; the element-wise fold kept an unconditional exit, so Sum(Random()·[1,1], n=1..31) stopped drawing at the first NaN while its scalar twin Sum(Random()+n, n=1..31) ran every term. The exit did two jobs at once, and only one of them may skip work: projecting a length mismatch to a scalar NaN is what makes the result independent of which term came last, and is now unconditional, while STOPPING the loop is an optimization and is now taken only when the skipped terms have no observable effect. An effectful body runs every iteration and answers the same value.

0.118.0 2026-08-21

Breaking Changes

• .N() now leaves a product of sums FACTORED, like evaluate(). Since 0.117.0 2(x+1) evaluated to 2(x + 1) but .N() of the same expression still came back 2x + 2: the numeric route of the Multiply handler kept distributing. .N() is evaluate() with floats, so the two now agree on shape and differ only in the numbers — √2(x+1).N() is 1.414… * (x + 1), ((a+b)/c)·d keeps its quotient shape (d(a + b))/c on both routes, and a closed constant such as 2.5(√2+1) still folds to a single float. The same rule now reaches the tuple and tensor arms of the handler on BOTH routes: (1,2)·(x+1) is (x + 1, 2(x + 1)) and [1,2]·(x+1) is [x + 1, 2(x + 1)] where the components used to be distributed even under evaluate(). A component product also now honors the closed-inexact-constant rule a scalar product has: 0.5 · [π, 1], 0.5 · (π, 1) and the element-wise [0.5, 1] · [π, 1] all evaluate their π cell to 1.57…, as 0.5 · π does, instead of leaving 0.5π. Expand reproduces the previous output; simplify() and the internal normalization paths still expand. The values are unchanged.

• A compiled Sum/Product now decides its NaN early exit on EFFECTS rather than on who supplied the code. The exit — if (acc !== acc) return NaN; between terms, valid because NaN absorbs + and * — used to be suppressed whenever the body contained caller-supplied source, on the reasoning that such code might count its own calls or mutate shared state. That test asked the wrong question, and got two answers wrong in opposite directions:

  • An impure operator of the engine's own no longer keeps the exit. A sum containing Random() was skippable, because Random is not caller-supplied. Skipping terms draws from the generator fewer times, and a later draw observes that, so Sum(Random() + n, n=1..31) now emits no exit. This is the user-visible half of the change: such a sum evaluates every term again, as it did before the exit existed.
  • A caller compile handler no longer costs the exit by existing. Whether the handler supplies source or declines for the target at hand, the suppression fired on the handler being present. The operator definition's pure / effects now governs: a definition declaring no effects keeps the exit, pure: false refuses it. A consumer measured ~33x on a 31-term sum from a handler that had explicitly declined.

  The rule holds for a piece written directly in the body. It does NOT yet reach one used through a user-defined function: wrap(t) := s(t) + 1 summed over still refuses, because the user-function admission gate reads the body's own purity and a signature-only declaration — which is what a name implemented through functions has — reports impure. That refusal is conservative, never unsound.

  The rule is now one sentence — an emission may be skipped when nothing in it has observable effects — with one oracle per spelling: a functions entry through its declared or inferred purity, an operator with a caller compile handler through its definition, everything else through the effects model. A spelling with no oracle is refused, which is where an operators entry ([op, prec], no body and no declaration slot) and a string-valued vars symbol land.

  Note the two oracles have deliberately opposite defaults. A functions entry must EARN purity — it is inferred from the source, and anything the analysis cannot model is refused — while an operator definition is GRANTED it, since pure defaults to true. A declaration is an assertion its author made; a bare source string is not. An operator whose handler emits effectful code while its definition declares none is the one shape this cannot catch, and it is the same mis-declaration that already misleads common-subexpression elimination.

• A function type must now cover every call its target type permits. Signature subtyping asked only whether a function had enough parameters; it never asked whether it could handle every call the declared type allows. So a name declared (integer, string+) -> string accepted a (integer, string) -> string function, and the mismatch surfaced later, at a call the declaration explicitly permits:

  declare join: (integer, string+) -> string
  join := narrow          // narrow: (integer, string) -> string — was ACCEPTED
  join(1, "a", "b")       // permitted by the declaration → unexpected-argument

  The error is now raised where the mistake is, at the assignment. A declaration is a contract in both directions — it tells callers which calls are legal, and it constrains what may be stored under that name — and assigning checks against that contract rather than rewriting it. Concretely, a function is a subtype only when it accepts the target's shortest AND longest permitted call, so no fixed-arity function satisfies a * or + tail (which has no longest call): only a variadic function can.

  This affects storing and substituting a function, not calling one. + and * still mean one-or-more and zero-or-more at every call site, and passing a fixed-arity function to a variadic callback slot is unchanged — that admits when the two arities overlap, which is a different question. The engine already enforced this rule for function LITERALS ("takes 1 parameter(s), but the declared signature accepts 0 or more"); only the named-function path slipped through.

• A compiled runner's declared return type now covers everything it can return. run() was typed number | ComplexResult, but a compiled predicate returns a boolean (Greater(x, 0) runs to true, never 1), a string-valued expression returns a string, and a collection-valued one returns a (possibly nested) array. TypeScript therefore accepted result.run({ x }) * 2 on an expression that could never be a number. The default is now the new exported CompiledValue union — which also covers a function-valued expression, since Derivative(Sin) runs to the callable (x) => Math.cos(x) — so such a call is a build-time error. The interval-js target keeps its own result type (an IntervalResult, or a bare Interval for a constant) rather than being folded into that union.

  Migration: a caller doing arithmetic on the result declares the narrow type it expects — compile<'javascript', number>(expr) — and gets a number back. This is the type-level replacement for the realOnly: true option removed in the previous release, which was the only remaining way to obtain a narrow numeric result; unlike realOnly it is purely a type assertion and projects nothing at run time.

  The same call's VARIABLES widened too, and that direction only ever accepts more: run() now takes number | ComplexResult on a JavaScript target, so a complex-mode call such as run({ z: { re: 2, im: 3 } }) type-checks instead of needing a cast. It was previously typed number alone, which refused the one shape complex mode exists to accept — while the _getCompilationTarget('javascript') route had always typed it the wider way. interval-js widened by the same step, from number alone to number | Interval; there too the _getCompilationTarget('interval-js') route had always used the wider type. Neither widening requires any change from a caller — both only ever accept more than before. A complex value handed to a REAL-mode runner is still rejected at run time, with an error naming the variable, unless the call passes entryChecks: false, which disables that guard along with the others and lets the value reach real arithmetic, yielding NaN.

• ~oo (ComplexInfinity) now types number, not complex. The non-finite typing convention admits an undirected infinity at the top type only — which is how every derived pole already typed (Gamma(-2), Zeta(1), (-1)!, sqrt(-oo)) — but the constant itself was the exception, so two expressions with the same ~oo value could type differently depending on whether the constant survived canonicalization (Divide(~oo, 5) answered complex, Add(1, ~oo) answered number). expr.type.matches('complex') is now false for ~oo, exactly as it is for NaN. A value carrying an infinite IMAGINARY part is ~oo for this purpose — the set the engine renders as ~oo — while an infinite real part with a finite imaginary part (∞ + i) is unchanged.

• A pole compiles to NaN on a real-valued lane. (-1)!, 1 + (-1)! and 1 + \tilde\infty compiled to the object {re: Infinity, im: Infinity}; they now run to NaN. A pole has no real value, and NaN is how the real lane spells that — the same projection _SYS.factorial already applied at a negative integer. Previously the constant fold emitted the value's own complex shape while the surrounding code was lowered from the node's number type, so a parent added an object to a number (1 + {…} → "1[object Object]") and the folded and structural paths disagreed with each other. A pole and a pole under a parent now answer NaN on both paths, and on the shader targets too, where the vec2 this produced was also a shape mismatch wherever a float was expected. A real-valued FUNCTION of a pole still differs between the two: Re(~oo) folds to Infinity (the fold evaluates symbolically, and the interpreter answers +oo) but lowers structurally to NaN, because on the real lane the pole is already NaN by the time the function sees it.

Resolved Issues

• A call of a user function whose definition a compile target cannot emit is compiled inlined. On the glsl and interval-js targets f((x,y)) with f(P) := a·P.x² + b·P.y² declined — a shader function needs a static type for every parameter and a point-typed one has none (parameter "P" has no static GLSL type); the interval target has no PointX/PointY lowering over an opaque parameter — while the same body written out compiled. The call now compiles with the body substituted at the call site (the coordinate accessors of the literal point folded), so f((x,y)), d((x,y)) with d(P) := √(P.x²+P.y²), a two-point Q((x,y),(1,2)) and a chained e(P) := d(P) + 1 all compile on both targets; the javascript target keeps compiling such calls by reference. The body is substituted, never evaluated — an impure body, a generic or recursive callee, a symbolic point or a list argument keep the definition's own decline. (Tycho item 216.)

• Applications of a pure user function to number-literal arguments are memoized within an evaluation. A recursive definition that applies itself twice per level, R(i,x,y) = R(i-1,x,y) + 0.5·S(x,y,R(i-1,x,y)), cost the interpreter 2^i body evaluations at depth i — minutes at depth 20 where the compiled artifact, whose CSE pass binds the repeated self-call, took microseconds. The same application — same literal, same number-literal arguments, same exact/numeric route — is now answered from a memo, so depth 20 evaluates in milliseconds. An entry is valid only while the engine's state version — the axis every assignment, declaration, assumption, configuration change, non-clean scope pop and checkpoint restore advances — and a mutable-object store epoch are both unchanged, so a pure body that reads an assigned free symbol or an object field is never answered from a stale result; an impure body or a symbolic argument is never memoized. (Tycho item 217.)

• The javascript compile target lowers a point multiplied by a list. [1,2,3]·(cos a, sin a) and (3((-N)..N)+cos t)·(cos a, sin a) declined with "no list-arithmetic support" while x+[1,2,3] and 2(cos a, sin a) compiled. The interpreter broadcasts over the LIST and scales the point whole at each element — a list of points — which a flat element-wise broadcast over the two arrays would have zipped instead; the product now emits a nested broadcast (list outside, point components inside) and agrees with evaluate() on every shape probed, a list-typed or tuple-typed symbol included. Shapes the interpreter does not broadcast keep failing closed (tuple·tuple, a matrix, two lists of provably different lengths). Add and Divide of a point against a list, which the interpreter answers with a per-element incompatible-type error or leaves inert, used to compile to a plausible zip ((1,2)+[3,4] → [4, 6]); they now fail closed. (Tycho item 214.)

• [1,0] = P with P declared tuple<number, number> compiles. It declined as "a tuple participant" while [1,0]=[x,y] and (1,0)=P compiled. A list and a point are never equal in the interpreter (a point binds atomically), so the pair is a constant whatever the coordinates — the literal [1,0]=(1,0) already folded to False before compilation. The list-vs-point-symbol pair now compiles to that constant (true for NotEqual) instead of declining. (Tycho item 215.)

• A zip of two unknown-length point views keeps its element tuple-ness in the TYPE. With n unassigned, A = (Range(0,n)/n)·(1,0) is a lazy view of points, but A - B evaluated to Map((_1, _2) ↦ _1 + _2, A, B) typed indexed_collection<number> — the mapping's parameters are bare, so its body was typed with both unknown — and PointY over the view folded to a scalar absence marker on the strength of the type alone, while the pulled values were correct tuples. Map now derives a bare-parameter mapping's element type from its sources' element types (no annotation is written onto the literal, so no runtime check is added and the lambda's shape is unchanged), and the same view now reads indexed_collection<tuple<finite_number, finite_number>>. Two component tier lies on the way were fixed with it: a scalar-scaled point now widens its components by the declared scalar's tier (x·(1,0) with x: number is tuple<number, number>, and (1/n)·(…, 1) no longer claims an integer second coordinate), and a collection-times-point product scales the components by the collection's element type. PointY over a point view whose length is not yet decidable answers the lazy projection instead of an absence marker. (Tycho item 212.)

• PointX/PointY/PointZ over a list of SYMBOLIC points stay symbolic. With P declared tuple<number, number> and unassigned, PointY([P]) evaluated to [NaN] and PointY([(1,2), P]) to [2, NaN] while PointY(P) stayed symbolic: the broadcast arm read a symbolic element's missing components as an absent coordinate. A symbolic element now keeps the accessor applied to it — [PointY(P)], [2, PointY(P)], [PointY(P), PointY(2P)] — and substitutes to the numbers once the point is assigned; a coordinate a point provably lacks still takes the absence marker. (Tycho item 213.)

• .N() of a point view is a view of points, not a tuple of coordinate views. N((Range(0,n)/n)·(1,0)) came back as (Map(…), Map(…)) — and, with n assigned, as ([0, 1/3, …], [0, 0, …]) — because the numeric routes of Multiply and Add ran their tuple branch on the raw operand Range(0,n)/n, which is collection-typed but not yet a collection, before the .N() step could reveal the view. Such a co-factor now defers to the post-evaluation re-dispatch, where the collection wins exactly as on the exact route. The literal-list shape N([0, 1/3]·(1,0)) used to crash (mulN had no single-operand short-circuit, so mulTensors' tuple "scalar" recursed into a zero-operand product); it is [(0, 0), (0.333…, 0)].

• A primed name with a subscript that does not fold into the symbol is a primed variable, like the bare name. \alpha_1' parses as the primed variable Prime(alpha_1), but \alpha_{i+1}', A_{i,j}' and x_{n+1}'' parsed as Derivative(Subscript(…)) — the prime parselet read every compound base as an expression to differentiate — and canonicalization then lifted the subscript into a lambda over its own base ((alpha) ↦ alpha_{i+1}). Such a base now asks the same variable-or-function question as the bare symbol, decided by its base: Prime(Subscript(alpha, i+1)) for an unknown base, Derivative(Subscript(f, n+1)) for f declared a function. The prime-first spelling \alpha'_{i+1} agrees, the applied \alpha_{i+1}'(t) is unchanged, and a parenthesized expression (x^2)' is still differentiated.

• A symbolic derivative order is accepted and stays symbolic. f^{(n)} is documented as Derivative(f, n), but with n unassigned it canonicalized to an incompatible-type error (the order was checked against number without the inference a free symbol gets elsewhere), and had it got through, evaluation read the non-numeric order as 1 and returned the FIRST derivative. The order is now inferred number and the expression stays inert until it is assigned (n := 2 then gives f''). Also fixed on the way: Derivative(g, 0) with g a function symbol evaluated to (x) ↦ g, the constant function returning the symbol, instead of g (likewise the all-zero multi-index Derivative(g, 0, 0)), and a multi-index derivative of a symbol bound to a lambda (Derivative(g, 1, 0) with g(x, y) := x^2 y) stayed inert because that arm only recognized an inline function literal.

• A compiled Sum/Product keeps its NaN exit when a caller-supplied function is applied beneath another operator or inside a user-defined callee. A function declared by signature only and implemented through the functions compile option projects unknown effects onto every application of it, so the skippability gate — which lets a Sum stop evaluating terms once its accumulator is NaN when nothing in the body has observable effects — kept all 30 exits for \sum sq(n x) but none for \sum (sq(n x) + 1), \sum \sin(sq(n x)), \sum 2 sq(n x), or \sum wrap(n x) with wrap(t) := sq(t) + 1: the functions entry's purity oracle was consulted only when the vouched head was the whole body. The gate now re-reads the effect projection per node with each oracle's answer standing in for the head it vouches for, so a pure entry keeps the exit wherever the head sits, and an impure entry, an impure built-in beside it, or a second head with no oracle still refuse.

• A product of two infinities could come out with the wrong sign at machine precision. With ce.precision = 'machine', x · (-2) · 3.1 · (-∞) · (-∞) · (y + 1) evaluated to +oo · x · (y + 1) instead of -oo · x · (y + 1). The machine-precision numeric value answered "no sign" for ANY infinity, and the product accumulator read that as positive, so once the running coefficient was -∞ the next infinity flipped it. The big-number path already returned ±1 for ±∞; the two now agree. The pairwise fold that products without a sum go through masked the bug, which is why it surfaced only alongside a factored sum.

• Multiply gave ~oo a sign. 2·~oo evaluated to +oo and -2·~oo to -oo, which contradicted Negate(~oo) — that correctly stays ~oo, so -2·~oo and -(2·~oo) disagreed. An undirected infinity takes no sign from its factors: all three are now ~oo. One neighbouring case went with it: i·~oo answered NaN, because the general complex product computed ∞·0 − ∞·1. The indeterminate form 0·~oo is still NaN, and the signed infinities are untouched (-2·∞ = -oo).

• A real infinity turned in a non-real direction is ~oo, not NaN. ∞·i evaluated to NaN, as did ∞·(2+3i) and every other product of a real ±∞ with a factor that has a non-zero imaginary part: the general complex product computes ∞·0 for the real part and lands on the indeterminate form. Such a product is infinite with no real direction left, which is precisely what the single point at infinity represents, so it is now ~oo. The rule reads from either side: a non-real COEFFICIENT does the same to an evaluated real infinity, so i·ln(0) is ~oo rather than -oo. 0·∞ remains the genuine indeterminate form at NaN, and a real factor still keeps the signed rule (-2·∞ = -oo, 2·ln(0) = -oo).

• A sum holding ~oo alongside a real infinity kept the wrong one. ∞ + ~oo evaluated to +oo and -∞ + ~oo to -oo, discarding the ~oo term; both are ~oo, since the undirected point at infinity absorbs the sum. Add selected ~oo by asking whether a term typed complex, which stopped selecting anything once ~oo moved to number, leaving the signed-infinity counters — which track only real ±∞ — to decide. It now tests the value. Two real infinities are unaffected (∞ + -∞ is still NaN).

• Gamma at a non-positive integer returned a large finite number when compiled. Gamma(-2) compiled without constant folding ran to 6413262697001887: the reflection formula π / (sin(πz)·Γ(1−z)) cannot see the pole, because sin(-2π) computes as ≈2.4e-16 rather than 0. The kernel now answers NaN at every pole. The interpreter was unaffected — it returns ~oo before reaching the kernel — and so are non-integer negative arguments (Gamma(-0.5) = -3.5449…).

• Malformed dictionary input is reported instead of thrown — or silently accepted. Building a dictionary from a malformed expression raised a raw JavaScript exception, which ce.box() promises never to do for untrusted input, and two shapes were worse than that: a tuple with more than two elements (["Dictionary", ["Tuple", key, value, extra]]) and a non-string key both produced an EMPTY dictionary that reported itself as valid. All of these now box to an incompatible-type error naming the offending entry, the way the sibling DictionaryFrom/RecordFrom handlers already did. A malformed dictionary NESTED inside another is reported too, rather than becoming a silently empty entry.

• The two dictionary construction routes agreed on empty keys. The ["Dictionary", …] form accepted an empty-string key while the plain-data {dict: …} form rejected it, so a dictionary built the first way serialized to {dict: {"": …}} and then failed to box back — a valid expression that did not survive its own round trip. Both routes now reject it.

• Dictionary keys and record-type fields named after Object.prototype members work. A __proto__ entry could not be stored at all (the dictionary rendered as {->} and listed no keys, while At still returned the value by reading the prototype), and reading a MISSING toString or valueOf key returned the inherited JavaScript function as if it were a math value, where any other missing key yields NaN. The same flaw dropped a __proto__ field when a record{…} type was parsed back from its own spelling, so such a dictionary did not round-trip through its type.

• An argument that violates a type variable's declared bound is now reported. Calling a generic function with an operand its bound refuses — f("abc") where f: (T) -> list<T> where T: number — was accepted and left to run time. The check that decides whether an operand REFUTES a parameter reads the declared signature, and for a generic arm it read the parameter with the variable still free, which no type predicate can answer; the arm is now read with each variable standing for its declared bound, so T: number refuses a string outright. An UNBOUNDED variable is unaffected: it ranges over every value type, so nothing refutes it and such a call still defers.

• A symbol whose name matches an Object.prototype member no longer reads that member. A MathJSON symbol name is an arbitrary string, but several lookups keyed plain JavaScript objects by it, so toString, constructor, valueOf or __proto__ found an inherited value instead of missing. Three consequences, all fixed:

  • ce.parse('\\mathrm{toString} + 1') threw a TypeError. The LaTeX parser's scope chain answered "declared" for a name it had never seen and handed back the inherited function as the symbol's type.
  • A compiled expression read a MISSING symbol of such a name as the inherited member: toString + 1 returned the string "function toString() { [native code] }1", where every other missing symbol yields NaN. A caller-supplied vars map was also consulted with in rather than an own-property test, on all four targets.
  • \\mathrm{__proto__} resolved as a known unit, because the unit tables answered Object.prototype for that key and the lookup tests its result for truthiness. \\sum_{\\mathrm{__proto__}=1}^{3} \\mathrm{__proto__} silently lost its index and evaluated to 3·__proto__ instead of 6.

  Ordinary symbol names are unaffected, and the emitted code is unchanged for them — the own-property guard is emitted only for a colliding name.

Breaking Changes

• The deprecated realOnly compile option has been REMOVED. It was an OUTPUT projection applied to a compiled unit's result after the kernel had run — a {re, im} collapsed to re when the imaginary part was at roundoff scale and to NaN otherwise, a top-level boolean became NaN, and an array result was projected component-wise. It never selected a lowering, so nothing about which code is emitted changes; only what the runner hands back does.

  The result convention already carries what most callers wanted: a compiled value whose imaginary part is exactly zero comes back as a plain number, and a returned ComplexResult always has im !== 0. Test typeof v === 'number' per sample. Where the projection's specific choices mattered, reproduce them at your own value boundary — a {re, im} slot, a boolean result, and ~oo ({re: Infinity, im: Infinity}, what a compiled (-1)! now returns) all reach the caller unflattened instead of as NaN. To forbid promotion and hold the real lane — which realOnly never did — pass mode: 'strict'.

  realOnly is gone from the typed option surface, so a TypeScript caller passing it gets a compile error. An untyped JavaScript caller still gets a one-time console warning naming the removal on both compile routes rather than silently losing the projection.

Improvements

• A functions entry can now be declared pure, and a compiled Sum over it stops at the first NaN again. A compiled Sum/Product exits as soon as its accumulator becomes NaN, since NaN absorbs both + and * and no remaining term can change the answer. That exit was suppressed for any body splicing caller-supplied source — a functions/operators entry, a string-valued vars symbol, an operator with a caller compile handler — because such code may count its own calls, log, or mutate shared state. The suppression was all-or-nothing: ONE caller-supplied factor anywhere in the body dropped the exit for every term, and for the scalar loop arm too.

  Note what the skipped terms could still change. Once the accumulator is NaN the sum's VALUE is settled whatever the remaining terms do, so the only thing running them preserved was the supplied function's SIDE EFFECTS. A caller who knows there are none can now say so:

  compile(expr, { functions: { s: { source: mySpline, pure: true } } });

  The functions option accepts a { source, pure? } descriptor beside the bare source-or-function spellings it always took. A declared pure is an assertion and is believed rather than re-derived, so asserting it for a helper that draws or counts will drop calls to it; pure: false pins the conservative behavior.

  An entry that declares nothing is analysed instead, so the common arithmetic helper needs no annotation: a source that is an arrow or function expression whose body uses only its parameters, numeric literals and an allowlist of Math members is taken as pure. It calls nothing else — not a parameter ((f) => f(x) is rejected, since f is whatever the caller passed at run time), and not a Math member outside the allowlist (Math is an ordinary mutable object, so Math.audit may be anything). Also rejected: Math.random, whose two calls disagree; a closure over an outer binding ((t) => t * scale mentions a name that is not a parameter); and a bare name with no body to read, which can only be declared. The grammar is small and explicitly enumerated rather than a JavaScript semantics check, which is what keeps the rejections safe; a missed helper costs the early exit and nothing else.

  Purity governs only whether an emission may be SKIPPED. A caller-supplied implementation stays opaque to every pass that would rewrite what is inside it — common-subexpression elimination included — because the emitter receives its operands as text and may drop, repeat or defer them. Values are unchanged in every arm.

• A callback with too MANY parameters is now rejected at a user-declared arrow slot, the way one with too few already was. With myOp: ((number) -> number) -> number, the call myOp((a, b) |-> a + b) reported nothing and failed at application; it now carries the same callback-arity diagnostic the opposite mismatch gets — "myOp calls its callback with 1 argument (per the declared parameter list); (a, b) => a + b declares 2 parameters". This covers a callback written inline and one supplied through a symbol alike, whether the symbol was declared with a signature or inferred from an assigned lambda.

  The check reads the slot arm's admissible range — required, then optional, then a variadic tail's mandatory occurrences — so a slot spelled ((number, number?) -> number) still accepts both a unary and a binary callback, while a ((number+) -> number) slot now rejects a nullary one it can never satisfy. At an OVERLOAD set the verdict participates in choosing the arm, so a callback that fits one arm resolves to that arm instead of being reported against a more specific sibling it cannot satisfy. The library's own collection operators are unaffected: their canonical handlers mint the richer per-operator wording before validation runs.

• Signature subtyping now checks arity in both directions. A signature requiring MORE arguments than a fixed-arity signature supplies is no longer reported as its subtype: (number, number) -> number is not a (number) -> number. Only the too-few direction was refused before, which is what let a wrong-arity callback reach an arrow slot unnoticed. A signature whose tail is optional or variadic keeps its existing leniency.

• Sort and Ordering declare the comparator result they actually accept. The comparator arm read ((any, any) any -> number), but the evaluator has always also accepted an Elixir-style BOOLEAN comparator (True means the first argument sorts first), so the arm is now ((any, any) any -> number | boolean). The previous spelling was masked by the arity hole above — a boolean comparator was admitted through the unary sort-KEY arm instead, which also meant Sort(xs, Less) type-checked for the wrong reason. Behavior is unchanged; the declared type now matches it.

0.117.0 2026-08-20

Breaking Changes

• evaluate() now leaves a product of sums FACTORED. (a+b)(c+d) evaluates to itself rather than to ac+bc+ad+bd, and 2(x+1) stays 2(x+1). evaluate()'s contract is the most EXACT form, and a factored product is exactly as exact as the polynomial it expands to while being smaller — often dramatically so, since expanding multiplies the term count at every factor. Expand reproduces the previous output verbatim, so code that needs the expanded form should ask for it. This is what made a Product of linear factors superlinear: ∏_{k=1}^{8}(kn-1) returned a nine-term polynomial with large coefficients instead of its eight compact factors, and a plotting consumer paid that cost on every sample because a plot axis variable can never be bound.

  simplify() and .N() still expand, as does every internal normalization path — the change is confined to the Multiply and Product evaluate handlers (mulFactored() and productAccumulate() respectively; the ∏(kn-1) example above is the Product half). Several results are now reported factored where they were not: ∫x²/(2(1+x²))dx is ½(x - arctan x), d/dx LambertW(x) is W(x)/(x(W(x)+1)), and a quadratic with a symbolic coefficient solves to ½(a ± √(a²-4)). The values are unchanged.

Improvements

• Raw-form subscript folding no longer depends on the base's spelling. An undeclared subscripted name parsed with form: 'raw' folded to a joined symbol for a Latin base (a_{0} → "a_0") but stayed structural for a command-spelled Greek base (\eta_{w} → ["Subscript","eta","w"]) and for a prefixed base (\operatorname{speed}_{0}), and the prime-first spelling (\alpha'_1) diverged from the subscript-first one (\alpha_1') the same way. All spellings now fold to the joined symbol ("eta_w", "alpha_1", "speed_0"), matching what canonical form always produced, and the two prime orders agree on every spelling. Unchanged: an indexed-collection base keeps the element-access reading unless the joined name is declared, a subscriptEvaluate base (such as \gamma for the EulerGamma family) keeps its subscripts, and dictionary-claimed constants such as unbraced \mu_0 are untouched. If your code reads form: 'raw' output and pattern-matches ["Subscript", base, sub] nodes for undeclared names, re-measure it — those nodes are now plain symbols; code that folded them itself now receives the already-folded name.

Resolved Issues

• Dividing a list of points by a scalar keeps the points. A collection whose elements are numeric tuples — list<tuple<number, number>>, the shape a set of points carries — typed its quotient list<number>, dropping the element tuple-ness, while the equivalent product p·(1/q) kept it. The values were always correct; only the type was wrong, and it was wrong in a way that changed downstream readings: PointX/PointY over a list<number> take the element-INDEX reading rather than the elementwise one, so a normalized point list read as its own first coordinate. Every denominator spelling is repaired (a scalar, a broadcastable<number>, a sibling collection of scalars that divides elementwise), and a divisor that can present a tuple — a point, a point list, or a broadcastable<tuple<…>> — still has no defined quotient. The same repair applies to the product of a point list with a sibling list of scalars, which used to widen to list<number | tuple<…>>.

• Scaling a list of points widens the point's components. A scalar factor folded into a collection's cells but stopped at a tuple cell, so list<tuple<integer, integer>> scaled by a real still claimed integer components even though [(3,4)]·0.5 is [(1.5, 2)]. Each component is now widened with the scalar factors, as a non-tuple cell already was.

• The derivative of a vector-valued function no longer claims a scalar result. A head declared as a bare function and only then assigned a tuple-valued lambda typed f'(t) as number even though evaluating it returned a 3-tuple — so the declared type contradicted the value, and type-strict consumers such as Cross and Dot rejected the call with incompatible-type. Derivative now reads the codomain of an assigned function literal when the declaration itself is uninformative, and D of a tuple- or list-valued body reports the shape it actually evaluates to (D((cos t, sin 2t, t), t) is a tuple, not a number). A head that is declared function and never assigned still reports the long-standing scalar compromise.

• A literal 0 operand is no longer dropped when serializing to LaTeX. Several serializers tested an operand for truthiness where they meant to test for absence, and the MathJSON of the literal 0 is the number 0 — so a zero operand read as a missing one and was silently deleted. A domain restriction on a zero lost BOTH operands (When(0, x=x) serialized to the empty string), Return(0) printed as a bare return, a big operator lost its body AND its indexing sets (Sum(0, i=1..10) became \sum), an integral lost its integrand (Integrate(0, x) became \int), and Delimiter(0) became an empty (). Each now serializes its zero, so an expression carrying one round-trips through LaTeX again — which matters wherever serialized LaTeX is a persistence boundary, since parsing was never at fault and the loss was silent. Log(x, 0) also now renders its zero base as a base (\log_{0}(x)) rather than falling back to the argument-list form.

  The round trip is an equality of CANONICAL expressions, not of bytes or of raw parse trees. Re-serializing is free to choose a different spelling of the same value (\left[.1,.3\ ...\ 1\right] comes back as (0.1..0.3..1), \operatorname{abs} as \vert…\vert, \sqrt{m} as m^{1/2}), so a byte-stability check reports damage on a bundle where nothing is wrong. Compare with .isSame() on the canonical parse. Also beware an assertion that merely looks for a 0 in the output: Sum(0, i=1..10) now serializes \sum_{i=1}^{10}0, but the upper limit contains a 0 of its own, so "contains a zero" is satisfied by output that dropped the body — the exact defect being tested for. Assert on the reparsed expression, not on the string.

• A negative subject of a When restriction keeps its precedence. When(-1, cond) serialized as -1\left\{cond\right\}, which reads back as Negate(When(1, cond)) — the negation applied to the restriction instead of to its subject. The subject is now parenthesized whenever its own precedence is below that of the restriction. The repaired form is (-1)\left\{cond\right\}, which parses back to When(-1, cond) at canonical form; a RAW parse of it keeps the parentheses and the unfolded negation (["When", ["Delimiter", ["Negate", 1]], cond]), as it does for a parenthesized negative literal anywhere else. The invariant restored here is that the negation sits inside the restriction rather than outside it — check it against the canonical parse, not the raw one.

• A symbol named after a JavaScript object member no longer misbehaves. Names such as toString, constructor and valueOf collide with members every JavaScript object inherits, and the engine looked symbol names up in tables that carried those members. The consequences were visible three ways: ce.box("toString") returned the JavaScript function of that name instead of an expression, ce.function("toString", [1]).toString() rendered a JavaScript error message as the expression's text, and toString used as a variable was refused by compile() as though it were a built-in operator. All three now treat such a name as the ordinary symbol it is.

• Rewriting a nested Sum/Product no longer breaks its index binding. .subs(), .replace() and .map() each rebuilt a scoped node onto a FRESH scope, parented at the rewriting site rather than at the rebuilt outer node. The chain from the inner body then never reached the outer binder's index, so an outer index whose name collides with a library constant resolved to the CONSTANT: Σ_{i=1}^{2} Σ_{j=1}^{2} x·At(K, i+j) substituted at x := 1 answered 60 where the same expression built directly answers 120, and Σ_{i=1}^{2} Σ_{j=1}^{2} x·i under x → 2 answered 8i — the imaginary unit — instead of 12. Nothing looked wrong: the rewritten expression reported isCanonical and printed identically to the direct one. A scoped node is now rebuilt onto its own scope, so the chain is preserved at any nesting depth. Only the OUTER index's name mattered, so sums indexed p/q were correct all along and sums indexed i/j — what most people write first — were not.

• A partial CanonicalForm[] request no longer rewrites a binder's own index. Boxing with { form: ['Order'] } (or any other partial form) runs before a binder's scope exists, and its symbol pass resolved EVERY symbol against the ambient scope — including the bound variable at its binding site. Sum(2i, Limits(i, 1, 3)) came back as Sum(2·Complex(0,1), Limits(Complex(0,1), 1, 3)) and then failed to evaluate with incompatible-type; the Number form rewrote it a second time, through its imaginary-unit normalization. A name the node's operator declares as a binding site is now left alone, at the site and throughout the binder's body. A FREE imaginary unit still folds as before.

• An unrolled Sum/Product now stops at the first NaN, and a constant collection inside one is built once instead of once per term. When both bounds of a Sum or Product are constant and the range is small, the JavaScript target unrolls it into explicit terms rather than emitting a loop. That unrolled form evaluated every term even after the running total had gone NaN — where the loop form exits on the first one — and it re-emitted every subexpression per term, including subexpressions whose value cannot depend on the index. A 31-term sum sampling a constant 101-element list built that list 31 times, in the source and again on every call: 11 KB of emitted JavaScript with 31 Array.from constructions, all evaluated even when the first term already answered NaN. Such a sum now emits 4 KB with a single construction bound ahead of the accumulation, and returns as soon as the total is NaN. This restores the cost a plot or sampling loop over such an expression had before the typing improvement that began steering these bodies into the unrolled form (they previously took the element-wise fold loop, which already had both properties) — a plot that went blank because every sample evaluated all 31 terms paints again. Values are unchanged: NaN absorbs both + and *, so stopping early returns the same answer. From four terms on, the emitted source for such a sum is a statement sequence in an IIFE rather than a flat a + b + c chain; two- and three-term sums keep the flat chain. The SCALAR loop form — a range past the unroll limit, or a bound that is not a compile-time constant — gained the same exit, which it had only on the element-wise fold path: a symbolic-bound or 100,000-term sum no longer runs every remaining iteration after the total has gone NaN. Its shape is otherwise unchanged (one emission of the body, no per-term statements). Complex-valued sums and products keep every term in both forms, deliberately: a complex accumulator with a finite imaginary part does not absorb, so an early exit there could change the answer. In both forms the exit is omitted when a term splices caller-supplied source — a functions/operators entry, a string-valued vars symbol, an operator with a caller compile handler — since such code may count its own calls or mutate shared state, so it must run as many times as it did before. Note this when checking the change yourself: a vars entry whose value is a STRING is spliced as source, which is exactly the carve-out above, so a sum compiled with vars: { x: 'number' } emits no early exit at all and looks like the old behavior. Measure with a numeric vars value, or with no vars map — the free symbol is read from the arguments object either way.

• A protocol function member read as a field now says so. b.span for a function span(self: Self) -> number requirement reported unknown-field "span" (w, h) — true of the object's LAYOUT and useless to the author, since the name they wrote does exist, on a protocol the value conforms to. It now reports protocol-function-not-a-field, naming the protocol and the call spelling: a function member is called (span(b)), only a readonly/readwrite property is read with a dot (b.area). This is the mirror of the protocol-property-not-callable warning below. Object, record and named-tuple receivers all report it, and so does an ASSIGNMENT to such a name (b.span = 5), whose message says the member cannot be written rather than recommending a call. When several conformances answer the same name, the message names them all and asks for a qualified call, since a bare one would be protocol-call-ambiguous. A name no protocol claims still reports the layout, and so does a name whose conformance is not settled — a PENDING edge carries no implementation, so there is no call to recommend yet.

• Epsil: a protocol property called as a function is now reported. A protocol's two member kinds are spelled differently — a function member is invoked in call position (span(b)), a readonly/readwrite property is read with a dot (b.area) — and using the call form for a property resolved to no operator, so it stayed a silently inert application: a program ending in print(c, area(c)) printed area(Circle(…)) with nothing to say why. It now raises a protocol-property-not-callable warning naming the property, the protocol that declares it, and the dot spelling. The check runs over the raw statement, so it fires even when the inert call is consumed by another operator and never reaches the statement's value — the shape this was reported on. Function members in call position are correct and untouched, and a name that has a real operator definition (a user function that happens to share a property's name) is never reported.

• Epsil: the protocol-implementation-pending warning no longer covers the whole program. It was emitted with a hardcoded whole-source range, because the check walks the protocol registry — which outlives the batch — rather than the source, so an editor squiggled the entire file. Each pending edge is now anchored to the type X is P statement that declared it when that statement is in this batch; one statement naming several protocols anchors every one of its edges. An edge declared in an EARLIER batch (the declare-in-one-cell, implement-in-the-next pattern the warning exists to support) still falls back to the whole program, since this source contains no statement to point at.

• A declared type now admits a value whose type is unknown. Assigning an expression the engine cannot type statically — a call to a function with no signature, say — to a symbol with a declared type was refused: let xs: list followed by xs = f(0) raised incompatible-type list unknown, and so did the let xs: list = f(0) spelling, for every declared type (number, string, list<number>, …). A type of unknown states nothing about the value ("some value, not stated which"), so there is nothing for the declaration to refute; it is admitted and the verdict is left to run time, which is what the argument-position boundary already did — the identical value passed to a list parameter was accepted. Values the engine CAN type are still held to the contract (xs = 42 against list is still an error), and a bare-constructor declaration still refines its element slot from real evidence: an admitted unknown value leaves xs reporting list, and a later xs = [1, 2, 3] refines it to list<finite_integer>. any is unaffected — it is a stated contract, not a placeholder, and is still checked, and so is a declared function SIGNATURE ((number) pure -> number), which keeps refusing an unknown value rather than installing a definition that would be callable under a contract nothing proved. Note the admission is unchecked: nothing re-examines the value later, exactly as at a typed parameter.

• A compilation reached through a target now escalates to complex mode like the standalone compile() does. Under the default auto mode a lane mismatch — a complex-shaped value, typically a promoted unknown-sign radical, reaching a binding the compilation shaped real — is answered by recompiling under mode: 'complex'. That retry existed only in the standalone compile() export, so a caller using the target route (ce._getCompilationTarget('javascript').compile(expr, options), the route an integration takes once it needs a specific target) got a thrown LaneMismatchError instead — or, with fallback: true, a success: false result with the lane-mismatch diagnostic and an interpreter-backed run — where the standalone route returned working compiled code. The escalation now lives in each target's own compile(), so both routes behave identically: such a call returns success: true with mode: 'complex', promoted: true and the failed strict attempt's diagnostic in escalation. Callers of the target route under the default mode are affected; fallback: true callers will see success: true where they previously saw success: false with lane-mismatch. mode: 'strict' is unchanged on both routes, and the shader and interval targets, which never promote to complex, are unaffected. One consequence for a THIRD-PARTY registered target: escalation is now each target's own responsibility, so a custom target that declares auto support no longer inherits the retry from the standalone wrapper — apply the exported compileWithAutoEscalation helper inside its compile() to keep the behavior.

• The Python target's interpreter fallback no longer drops the realOnly projection. With fallback: true on the Python target route, a declined compilation handed back an interpreter-backed run that ignored realOnly: true (the standalone route forwarded it; the target route did not) — a complex result then reached a caller that had asked for the real-only projection. Same route-asymmetry family as the escalation fix above, found while moving it.

0.116.1 2026-08-19

Improvements

• Cross accepts numeric tuples, like Dot always did. Cross((1,2,3), (4,5,6)) computed nothing — the signature demanded vector, so a point spelled as a tuple was rejected with incompatible-type while the same point passed Dot — and the two classic vector products disagreed about what a vector is. A provable numeric 3-tuple (a point in ℝ³, including a PointList row) is now lowered to its component vector exactly as Dot lowers it; the result is a List, consistent with collection operators not preserving the tuple kind. A tuple that is not provably numeric stays a symbolic Cross, and a wrong-length tuple gets the incompatible-dimensions report, not a type error. Affects code that builds Cross over Tuple/point operands — previously an error, now a value.

Resolved Issues

• compile() no longer reports the boolean literals as required inputs. CompilationResult.freeSymbols lists the identifiers a caller must supply, and True/False were included even though every target either bakes them into the emitted code or drops them entirely. Since consumers derive a compiled function's variable list from freeSymbols, and True is the idiomatic fallback condition of a Which, an ordinary piecewise such as Which(x > 0, 1, True, 2) acquired a phantom variable named True.

• A boolean literal compiled to invalid code on the shader and interval targets. The glsl and wgsl targets emitted the undeclared identifier True — a shader that fails to compile, reported behind a successful result — and the interval target emitted a lookup for a caller-supplied variable named True that threw at run time. Each target now emits its own spelling of the literal.

• Trigonometric operators now report a non-numeric operand instead of staying inert. The trigonometric and hyperbolic functions, Degrees and DMS validated only the NUMBER of arguments, not their types, so an operand that turned out not to be a number was absorbed rather than reported: Sin(At(["a", 2], 1)) stayed as sin("a") and Degrees(At(["a", 2], 1)) answered a bare NaN. All of them now answer incompatible-type, matching the arithmetic operators. This completes the numeric-operand checking introduced in 0.116.0, which had covered arithmetic but not these operators.

• DMS no longer mis-folds a component that is not a plain real number. Its degrees, minutes and seconds components were read in a way that silently discarded anything but a real number literal, so DMS(1, At([30, 2], 1)) answered NaN even though the minutes component resolves to 30, and DMS(1, i) answered exactly what DMS(1, 0) does. Such a call now evaluates its components first and, when one of them cannot be folded, is left unevaluated instead of producing a wrong angle.

• The root compile() export regained its precise result types. Its declared return type collapsed the target generic, so run was typed optional even for executable targets such as javascript, and realOnly: true no longer narrowed run's values to plain number — a type-precision regression against the internal target route (ce._getCompilationTarget(...).compile(...)) that 0.116.0 told consumers to migrate away from. The export's overloads now mirror the underlying compile function's, so the target name flows through: compile(expr).run is non-optional, compile(expr, { realOnly: true }).run(...) is a number, and calling run unguarded on a source-only target (python) is a type error. Runtime behavior was never affected. Reported by a consumer at 0.116.0 adoption.

0.116.0 2026-08-19

Breaking Changes

• Type now returns a type value instead of text. Use StringFrom(Type(x)) if you need the type name as a string, or use type-aware checks such as x is integer and Subtype(Type(x), TypeFrom("number")). Code such as Type(3) == "integer" now evaluates to False.
• Callback type declarations now use regular arrow types. The old callback<...> spelling no longer parses. Declare the callback slot directly, for example (T) any -> boolean for a predicate. Calls that could only fail, such as filtering a list<string> with a number-only predicate, are now rejected before evaluation.
• The minimum supported Node.js version is now 22.3.0. Older Node 21 releases are no longer supported.
• About now returns a dictionary. Read fields such as About(Pi)["type"] or About(f)["signature"] instead of parsing a display string.
• Bare collection types now mean values-only collections. list, set and similar bare types are synonyms for their <unknown> form, not <any>. Use collection<any> when absence markers such as Missing should be admitted.
• Dictionaries now print in dictionary literal form. toString() and the Epsil REPL print dictionaries as {"key" -> value}, with {->} for an empty dictionary. MathJSON serialization is unchanged.

Types And Pattern Matching

• Added first-class type values with TypeFrom(...), Subtype(...) and the primitive type. Example: Subtype("integer", "number") evaluates to True.
• is tests and match type patterns now accept full type expressions, including unions, negations and parameterized collections. Example: x is list<integer> and x is number | string.
• Protocol tests can be written with is. Example: x is Hashable & Comparable.
• Collection declarations with a bare element type can refine from assignments. Example: after let xs: list; xs = [1, 2, 3], xs is typed as a list of integers.
• Uses of assigned symbols are checked against their assigned type instead of narrowing the symbol silently. This reports incompatible uses earlier, before evaluation.

Epsil And VS Code Extension

• Added console I/O with print(...) and input(prompt?). print writes to the host console and returns Nothing; input reads one line of text where the host supports it.
• The CLI and REPL no longer echo a final Nothing result, so a program ending with print("done") only displays done.
• Very deeply nested Epsil expressions now produce an expression-nesting-limit diagnostic instead of overflowing the JavaScript stack.
• Lexing and parsing now respect the evaluation time budget, so an oversized program can time out before execution begins.
• The VS Code extension gained Go to Definition, Find All References, Rename Symbol, richer hovers and diagnostic links to the error reference.

Strings And Regular Expressions

• Added regular expressions with the regexp type and RegExp(pattern, flags?). New operations include IsMatch, StringMatch, StringMatchAll, and regular-expression forms of StringSplit and StringReplace.

  IsMatch("abc123", RegExp("[0-9]+"))
  StringReplace("a1b22c", RegExp("[0-9]+"), "#")

• Regular expressions use the host JavaScript regexp dialect. The g and y flags are rejected because their mutable position would make evaluation order visible.

• StringMatch returns match text, ranges, numbered groups and named groups. Ranges are character-based, matching the rest of the string library.

Symbolic

• And and Or once again compare and match symbolically without depending on operand order. Evaluation still runs left to right and still short-circuits.
• Applying a value that is known not to be a function now reports expected-function instead of staying inert. Example: Pi(2) is now an error.

Engine State

• Added engine checkpoint APIs: ce.checkpoint(), ce.restore(cp) and ce.discard(cp). Notebook-like clients can checkpoint before a cell, restore to that point after an edit, then replay later cells.
• Checkpoints preserve expression validity and cache identity where possible. Restoring replays state changes, but effects still run again: Random() draws again and Print(...) prints again.

Compilation And Numeric

• Numeric operators now accept operands that may be numeric and report a typed error only when evaluation proves the value is non-numeric. Example: a + 1 is allowed for a: value, but a = "hello" later produces an incompatible-type error.
• Fixed several compiled complex-number cases that could return NaN or corrupt values while reporting success, including color functions, list broadcasts, complex-declared symbols with real values, and promoted radicals inside broadcasts.
• CompilationResult.mode now reports the actual arithmetic mode used by the emitted code. A default compile that promotes to complex reports mode: "complex".
• Deprecation warnings for realOnly and complexPromotion now appear through target-level compile entry points as well as the standalone compile() export.
• Consumers that discover the engine through the global registration object can now access the supported compile() helper there too.

Collections

• Sequence now splices into List, Set and Tuple literals. Example: List(1, Sequence(2, 3), 4) becomes [1, 2, 3, 4].
• Lazy set operations now deduplicate consistently. Join(Set(5, 2), Set(2, 3)) now has three elements, not four with a duplicate 2.
• Map over a set now returns the distinct image of the set. Example: Map(x => x^2, Set(-1, 1, 2)) is Set(1, 4).
• Joining or appending dictionaries and records now merges keys with the last value winning. Example: Join({"a" -> 1}, {"a" -> 2}) yields {"a" -> 2}.
• Truncated set previews now show an ellipsis when more elements remain.

Type Fixes

• Fixed bare collection and type-variable inference so bare collection operands stay in the values-only family.
• Fixed intersections, never, negated types, collection meets and function-signature intersections so type reductions are more consistent.
• Fixed several type edge cases, including matching bare names against intersections, preserving vector and matrix spellings, and reporting arity errors for malformed Type(...) calls.

Documentation

• Removed documentation for the non-existent Extract and Exclude operators. Use At, Slice, Reverse or DeleteAt instead.

Benchmarks

Numeric performance (200-digit precision)

Median time per call, in microseconds — lower is better. — means the tool returned no usable result at that precision.

Expression	CE 0.116.0	CE 0.100.0	SymPy	math.js	Mathematica
\pi^2	8.4	7.5	177	1,316	3.9
\sin 1	21	21	220	440	5.1
\cos 1	21	21	220	618	7.1
\ln 2	15	14	350	4,344	3.7
e^{\pi}	14	13	213	5,308	4.6
\zeta(3)	1,533	1,551	267	—	49
\Gamma(\tfrac13)	829	821	342	—	213
\psi(\tfrac13)	715	716	2,786	—	171

Symbolic capability & performance

Each cell is how many times faster than Mathematica that engine is on the case (Mathematica ÷ engine, so higher is better; Mathematica itself is 1×). — means the engine can't do the case. Compare the CE 0.116.0 and CE 0.100.0 columns to see what is new since the last published benchmark (a — under 0.100.0 next to a number under the current build). The CE + R/F column is the current build with the opt-in Rubi integrator + Fungrim identities loaded (loadIntegrationRules / loadIdentities), on the same minified bundle.

Operation	CE 0.116.0	CE + R/F	CE 0.100.0	SymPy	math.js	Mathematica
Antiderivatives
\int\frac{1}{\sqrt x}\,dx	3.7×	2.0×	2.8×	0.5×	—	1×
\int\frac{x}{\sqrt{1-x^2}}\,dx	7.7×	1.4×	5.8×	0.08×	—	1×
\int\frac{1}{x^3+1}\,dx	4.2×	0.8×	3.7×	0.3×	—	1×
\int\frac{\sqrt x}{1+x}\,dx	—	1.7×	—	0.1×	—	1×
\int\frac{x}{(1+x)^{1/3}}\,dx	—	1.0×	—	0.01×	—	1×
\int\frac{x^2}{(1+x)^{1/3}}\,dx	—	1.0×	—	0.007×	—	1×
Derivatives
\tfrac{d}{dx}\sqrt{1-x^2}	0.03×	0.03×	0.03×	0.001×	0.003×	1×
Simplification
\sqrt{3+2\sqrt2}	41×	30×	29×	—	—	1×
\sqrt6\,x+\sqrt2\,x	63×	55×	49×	3.2×	16×	1×
Evaluation
\lim_{x\to0}\tfrac{\sin x}{x}	33×	12×	33×	3.0×	—	1×
\lim_{x\to\infty}(1+\tfrac1x)^x	3.8×	3.5×	4.4×	2.1×	—	1×
\int_1^2\tfrac1x\,dx	3672×	4204×	4152×	80×	—	1×
\int_{-\infty}^{\infty} e^{-x^2}\,dx	262×	100×	237×	2.6×	—	1×
Solving
x^4+x^2-1=0	0.2×	0.2×	0.3×	0.06×	—	1×
x^3-x-1=0	1.5×	1.6×	1.4×	0.04×	—	1×

Across the cases both solve, Compute Engine is a median 3.8× faster than Mathematica (up to 3672×) — in the browser, not a proprietary kernel.

_{Measured 2026-08-19 · Compute Engine 0.116.0 (current build @ c5f1f3be) · published 0.100.0 · SymPy 1.14.0 · math.js 15.2.0 · Mathematica 14.3.0 for Mac OS X ARM · Node v22.13.1. Correctness is verified numerically against an independent mpmath reference, never another tool. Reproduce with npm run build production && ./venv/bin/python3 benchmarks/gen_cases.py && node benchmarks/report.mjs && node benchmarks/report_changelog.mjs.}

0.115.0 2026-08-17

Breaking Changes

• The default JavaScript and Python compile mode is now auto. Expressions such as Sqrt(x), Ln(x) and non-integer powers may promote to complex arithmetic when the sign is unknown. For example, compiled Sqrt(x) now returns {re: 0, im: 1} at x = -1 and 2 at x = 4. Use mode: "strict" for the previous real-only compile behavior.
• complexPromotion and realOnly are deprecated. Use mode: "complex" or mode: "strict" instead. realOnly still works for this release but now warns.
• Compiled JavaScript returns plain numbers for exactly real results. A complex result object {re, im} is now returned only when im !== 0. Use typeof value === "number" to test whether a sample is real.
• StringJoin now joins one collection, with an optional separator. The old variadic concatenation form changed meaning for two strings. StringJoin("ab", "cd") is now "acdb" because "cd" is the separator between the characters of "ab". Use Join("ab", "cd") or interpolation for string concatenation.
• Several string collection operators now preserve string results. RandomShuffle("abc") and DeleteAt("abcdef", 2) return strings, not lists of characters. Chunking and combinatorics over strings now return lists of strings, for example SlidingWindow("abcd", 2) returns ["ab", "bc", "cd"].

Compilation

• Added explicit compile modes: "strict", "complex" and "auto". Unsupported modes now return a clear diagnostic instead of silently changing behavior.
• CompilationResult now includes mode, promoted, escalation and structured diagnostic data so callers do not need to parse error messages.
• Strict mode now rejects complex values where the generated code expects real numbers, with a lane-mismatch diagnostic.
• Complex mode now treats broad or unannotated numeric inputs as complex-capable and promotes unknown-sign radicals consistently.
• JavaScript runners now validate incoming values against the shapes used during compilation. For example, a complex object passed to a real-analyzed binding throws a clear TypeError.
• Fixed compiled complex behavior for multi-clause functions, protocol methods, fallback evaluation, real-only special functions and promoted radicals.
• (Recorded after release.) A compiled Reduce or Scan whose accumulator becomes complex mid-fold through an unknown-sign radical — for example Reduce([4, -4], (a, x) -> a + sqrt(x), 0) — returned success: true with value: null in 0.114.0. The complex-mode migration in this release fixed it (the radical now promotes and the fold computes {re: 2, im: 2}, matching the interpreter). Documented here retroactively after a consumer traced the fix to this release; the change landed in commit 7cfa95b4.

Epsil

• Added definition attributes for user-defined functions: bind parameters, algebraic properties such as commutative and associative, and doc comments.

  /// Square a value.
  function square(x) -> number { x^2 }
  About(square)["description"]

• Fixed pipe placement when a stage has a missing trailing argument. Examples such as xs |> Fold(f, 10) now put xs in the collection slot instead of displacing arguments incorrectly.

• Multi-clause functions now map over collections the same way function literals do. Example: fib(5..10) evaluates element-wise.

• Which and If now hold or broadcast correctly when a comparison's broadcast shape cannot be decided statically.

• Sum and Product now substitute loop indexes correctly inside held If and Which terms.

• A symbol's inferred type is updated when a later value proves the earlier guess was too narrow.

Strings

• Join now concatenates strings. Example: Join("ab", "cd", "ef") returns "abcdef".
• Added contiguous sequence search: RangeOf, ContainsSequence, StartsWith and EndsWith. Example: ContainsSequence("abc", "ab") is True, while Contains("abc", "ab") is False.
• Slice accepts Nothing, so Slice(xs, RangeOf(xs, needle)) returns the match or Nothing without an extra branch.
• Added StringReplace, Trim, TrimStart, TrimEnd, StringRepeat, PadStart and PadEnd.
• Added case operations: ToUpperCase, ToLowerCase and CaseFold. Use CaseFold for case-insensitive comparison.
• Added StringCompare(a, b) for deterministic code-point ordering.
• Added NumberFrom(s, base?) for parsing numbers from strings. Failures return an error value, not NaN.

0.114.0 2026-08-16

Breaking Changes

• Strings are now indexed collections of characters. Length("shop") is 4, "abc"[2] is "b", and generic collection code can now accept strings. If a function should accept collections but not strings, use a type such as collection & !string.
• string is no longer a scalar. Declare string, or string | number, when strings are intended.
• Added a distinct character type. Characters are single user-perceived characters and are not one-character strings, although they compare equal by value where appropriate. Characters("ab") now returns list<character>.
• String-preserving collection operators now return strings for string input. Examples: Reverse("abc") returns "cba" and Take("abc", 2) returns "ab". Transforming operators such as Map still return lists.
• Materializers and set operators read strings as characters. ListFrom("abc") is ["a", "b", "c"], and [..."ab"] is ["a", "b"].
• Max, Min, GCD and LCM no longer expand strings. Strings are treated as non-numeric values for these operators.
• Invalid UTF-16 surrogate halves are normalized when strings enter the engine. They are replaced with U+FFFD so all string operations work on well-formed Unicode.
• Compiled string collection operations are grapheme-correct in JavaScript and fail closed on targets that cannot support them. Python and shader targets no longer emit approximate string collection code.

Strings

• Added CharacterFrom(s) and host API support for character values. String(c) converts a character back to a string.
• Strings work with collection operations over grapheme clusters. Example: Tally("mississippi") counts characters, and "7" in "0123456789" is True.
• String(collection) now joins a finite collection into one string. Example: String([1, 2]) returns "12"; use Map(String, [1, 2]) to convert each element separately.

Epsil And Types

• Added hold functions, whose arguments are bound as written and evaluated only when used in the function body. Example: hold f(e) = Head(e) lets f(a + 1) inspect the expression structure instead of the evaluated value.
• Re-declaring an object type now re-checks protocol conformances that depend on its stored fields. This makes notebook-style re-runs update field-backed protocol behavior correctly.
• Protocol member effect annotations in implementation blocks are now honored and checked.
• Writes through mutable objects consistently carry the state effect.
• Head and Tail now resolve a symbol's assigned expression during evaluation while staying structural for unbound symbols.

Numeric And Symbolic Fixes

• Reduce and Scan now handle complex accumulators correctly in both interpreted and compiled numeric paths.
• Abs over a negation now returns the absolute value rather than the negated expression.
• The derivative of a vector norm now uses the vector norm rule instead of the scalar absolute-value rule.
• A declared subscripted symbol with a Greek base, such as \eta_w, now remains the declared joined symbol instead of being captured as an index when \eta becomes a collection.
• A compiled user function with a complex parameter now receives and returns correctly shaped values.

0.113.0 2026-08-16

Breaking Changes

• Protocols that can mutate state now require object types. A protocol with a readwrite property or a member declared with state effects can only be implemented by an object{...} type. Use an object type when writable protocol properties are needed.
• Assigning to a protocol property now stores into the object. p.name = "Ada" mutates the object referred to by p and returns the assigned value. It no longer calls a setter that rebuilds and rebinds p.
• Collection operators now report result types by collection kind. Operators such as Reverse, RotateLeft, RotateRight, Rest, Most and Filter no longer promise to preserve every indexed collection kind. For example, reversing a tuple now reports a list type.

Objects And Protocols

• Stored object fields can now satisfy protocol property requirements with the same name. Example: type Person = object{name: string} is Nameable can satisfy a readonly name property without writing a getter.
• Field-backed writable properties store into the object in place.
• A stored field and an explicit accessor for the same protocol property are now reported as a conflict.

Collections

• Slice now accepts an index span. Example: Slice(xs, 2..4) is equivalent to Slice(xs, 2, 4).
• Bare collection element types now consistently read as unknown rather than any.
• Assigning a value to a symbol declared unknown now refines the symbol from the assigned value. Declare any when the type should remain fully permissive.
• Speculative parsing no longer narrows symbols declared unknown in the surrounding engine.

Numeric, Compilation And Parsing

• Fixed division by a scaled vector norm, which could incorrectly collapse to 0.
• Fixed compiled calls where a complex value is passed to a user function with a wide parameter.
• Lowercase LaTeX operator names such as \operatorname{unique}, \operatorname{sort}, \operatorname{reverse} and \operatorname{total} now parse to their intended operators.
• Overload diagnostics now display the declared type shape instead of confusing ...<unknown> instantiations.

0.112.0 2026-08-15

Breaking Changes

• Epsil lambdas now use =>. The old |-> spelling reports a diagnostic with a fix-it. Function types still use ->, and dictionary entries still use ->.

  let square = x => x^2
  [1, 2, 3] |> Map(_ => _^2, _)

• record and object field types now use braces. Write record{x: integer, y: integer} and object{name: string} instead of the old angle-bracket form.

• And, Or and related boolean operators now short-circuit and preserve operand order. This can change serialized order for expressions that were previously sorted during canonicalization.

Epsil

• Lambda parameters and for loop bindings can destructure tuple elements. Example: pairs |> Map(((x, y)) => x + y, _).
• Callback arity mismatches are now reported statically. Example: Map((x, y) => x + y, [1, 2, 3]) reports that Map calls its callback with one argument.
• A pipe stage that cannot accept exactly one piped value now reports pipe-stage-arity instead of returning a partially applied function.
• Duplicate lambda parameter names are now parse diagnostics.
• for and while loops now serialize back to Epsil loop syntax instead of generic Loop(...) calls when possible.

Types And Collections

• Added the range type for ascending, contiguous 1-based index spans. It is still an indexed collection of integers, but can also be required by span-consuming operators.
• A lazy collection callback now runs once per element per consumption, avoiding duplicate side effects from probing.
• Max, Min and statistics over Linspace now handle descending, single-sample and symbolic cases correctly.
• Pipes that implicitly map over a collection now report the mapped collection type.
• Comparisons that may broadcast now report a broadcastable<...> type, allowing Which and Sum to type their results more accurately.
• Scaling or shifting collections now widens element types correctly. Example: (1..4)/2 reports rational elements, not integers.
• Loop and comprehension binders can widen inferred element types when the iterated collection under-declares its elements.

Compilation And Numeric

• Shader compilation now emits valid integer conversions for symbolic Sum and Product bounds.
• Statistics functions such as Mean(x) now compile correctly for a single value.
• Differentiating point-valued functions installed through assign is componentwise again.
• Primes before subscripts now attach to the subscripted function name. Example: F'_{0}(t) parses as the derivative of F_0 applied to t.
• Element-wise arithmetic over collections of complex values now compiles where a single real-or-complex convention fits.
• Real-only numeric operations such as Floor, Max and Mod now decline complex operands instead of compiling to NaN.

0.111.0 2026-08-15

Breaking Changes

• RecordFrom has been removed. Use DictionaryFrom. Given identifier keys, DictionaryFrom([("a", 1), ("b", 2)]) already produces a value typed as a record.

Objects

• Mutable object fields can now be assigned. Example: p.age = p.age + 1 stores into the object in place, and all aliases see the update.
• Assigning to fields of immutable values now reports immutable-value-assignment with guidance to use an object type or build an updated copy.

Epsil And Performance

• Epsil programs now stop when the active time budget expires, instead of continuing after a statement times out.
• Parsing long Epsil programs is now linear in program length.
• Canonicalization now checks withTimeLimit deadlines while it runs.

Collections And Numeric

• Operators now stay symbolic for collection-typed symbols that have no value yet, instead of guessing scalar or empty behavior. Example: Mean(L) stays symbolic for unassigned L: list<number>.
• Indexed reads from collections with complex elements now compile correctly or decline when no single result shape fits every possible index.
• Collection comparisons no longer overflow the stack for declared-but-unassigned collection operands.
• Subset and superset operators now use the documented operand order and handle ranges, intervals and strictness more accurately.
• Closed degenerate intervals, such as Interval(1, 1), are now treated as non-empty.
• Declared fixed-size collection types now expose their count without requiring a value. Example: a vector<2> symbol has count 2.

0.110.0 2026-08-15

Breaking Changes

• Functions that write to outer variables must declare the scope effect. Unannotated functions now guarantee they do not mutate outer bindings. Use function bump(n) scope { total = total + n } when mutation is intended.

Compilation

• Added complexPromotion as an opt-in compile option for complex results from Sqrt, Ln and Log when their operand sign is unknown.
• Compile-time constant folding now includes small constant indexed collections. Example: At(Map(_ => _^2, 1..6), k) can bake the mapped list into emitted code.
• Computed Range bounds now use their runtime length when compiled, instead of being mistaken for numeric literals.
• Compile-time folding is now deterministic and based on expression cost, not wall-clock timing.
• Local function definitions now compile and stay local to their block instead of leaking into the global scope.

Epsil And Types

• Writes to a function's own parameters are treated as call-local and can remain pure.
• Property writes are judged by the base variable, so writes to local objects do not incorrectly require scope.
• Declared placeholder signatures using unknown now refine from the function body. Use any for a true accept-anything contract.
• Declared function clauses can omit parameter annotations when a declared signature already supplies them.
• Assigning to a declared subscripted name now prefers the joined symbol, and ambiguous undeclared function-family assignments now report an explicit error.

Numeric And Collections

• Adding three or more collections now handles lazy views correctly instead of nesting results.
• Arithmetic on vector-valued calls with not-yet-defined arguments now remains valid when the eventual operation is element-wise.
• Dividing a point or vector now widens component numeric types honestly. Example: [6, 2] / 4 has rational components.
• Equal and NotEqual between a collection-typed application and a list now produce one whole-collection boolean.
• Nested-list broadcasts no longer retype inner collection-valued symbols.
• Large counted Take and Drop operations in compiled code now honor the requested count instead of throwing at the iteration limit.
• Filter and Unique now apply the iteration cap to unproductive pulls, so productive infinite filters can be consumed safely with Take.

Performance

• Long flat sums, products and related chains now parse in linear time.
• Deep expression trees can be boxed more reliably before hitting the host stack limit.
• Large documents involving comprehensions and broadcasted indexed reads now parse much faster.

0.109.0 2026-08-14

Breaking Changes

• Defining the same function clause twice in one Epsil program is now an error. Clauses with different parameter domains still accumulate for multi-clause functions, and re-running a later program still replaces definitions notebook-style.

Epsil

• Spreading declared-but-unassigned dictionaries and records now stays symbolic instead of erroring. Example: {->, ...d, "k" -> 3} can be built before d has a value.
• Dictionary conversion errors now return error expressions instead of throwing uncaught JavaScript exceptions.

Compilation

• Degree-mode compiled angular functions with constant arguments now agree with interpretation. Example: compiled sin(90) in degree mode now returns 1.
• Epsil programs that define a local function and call it can now compile to JavaScript, including recursive calls and use as callbacks where supported.
• Lambda parameters named _ no longer shadow the generated JavaScript vars object, fixing compiled callbacks that refer to free variables.
• Drop with a negative count now reports the same element count it actually yields.
• Compiled counted Take and Drop over infinite streams now work when the requested count is larger than ce.iterationLimit.

0.108.0 2026-08-14

Breaking Changes

• Map now takes the function first: Map(f, xs). The legacy Map(xs, f) order now reports an incompatible-type error. Pipeline calls should use xs |> Map(f, _) or simply xs |> Map(f).

Epsil

• A type, protocol or sum name may only be declared once in a single Epsil program. Re-running a later program can still replace the declaration.
• Added spread syntax in list and set literals and dictionary merges. Examples: [...xs, 3], {1, ...s} and {->, ...defaults, "verbose" -> true}.

Compilation And Numeric

• Compiled code now folds pure constant subexpressions to literals. Example: x + Sum(Map(_ => _^2, 1..5)) can compile as x + 55. Use constantFold: false to inspect structural output.
• Difficult symbolic derivatives now fall back to numeric differentiation for N() and compiled evaluation when a closed form is too large. Exact evaluate() still stays symbolic.
• At(xs, indexes) now preserves known selection counts, improving Zip and Map over gathered elements.
• expr.unknowns no longer reports typed lambda parameters as free variables.
• Numeric approximation of divergent infinite sums and products no longer returns silently truncated partial results.
• 170! no longer overflows to Infinity in machine-precision paths.
• Python-compiled counts for Take, Drop, Tabulate and Fill now round like the interpreter.
• Numeric quadrature now honors active time limits.

Strings

• IntegerString now preserves the minus sign. Example: IntegerString(-42) returns "-42".
• StringSplit(s, "") now splits into grapheme clusters instead of UTF-16 code units, avoiding corrupt non-BMP characters.

Performance

• Large documents with symbolic range bounds now avoid major slowdowns and memory pressure.

0.107.0 2026-08-13

Epsil

• Pipelines gained concise stage forms. A missing argument is filled by the piped value, inline lambdas can appear directly after |>, and a one-parameter lambda stage over a collection maps each element.

  1..oo |> Take(10) |> _^2 |> Sum

• otherwise is now accepted as the wildcard case in match.

• Named calls to a function assigned earlier in the same program no longer produce false static diagnostics.

• Diagnostics inside reordered named calls now underline the argument that actually failed.

• Inline function literals now accept named arguments. Example: ((x, y) => x - y)(y: 2, x: 10) returns 8.

Compilation

• Take-bounded infinite pipelines now compile to JavaScript safely. Unbounded infinite pipelines fail at compile time instead of throwing a runtime range error.
• Fractional Take and Drop counts in compiled JavaScript now round like the interpreter.

Performance And Static Checking

• Canonicalizing or typing expressions that reference comprehension-bound names is much faster.
• Static Epsil checking no longer mutates engine state. Inferred types, forward references and declarations from a check are discarded afterward.

Effects

• Added the state effect label for object creation and mutation. It is available in type strings and effect contracts, preparing for mutable object support.

0.106.1 2026-08-13

Issues Resolved

• Epsil no longer silently reads the equivalence glyphs as the wrong equality tier: ≡, ≢, and ≣ are now rejected outright (unexpected-symbol). Previously a typed ≡ (IDENTICAL TO) silently weakened to == — arithmetic Equal, not the identity prover the glyph means in mathematics — ≣ quietly parsed as === (Same), and ≢ was a broken mapping that errored confusingly. Rejecting all three is deliberate (rather than assigning them tiers): the glyphs' bar counts cross the =-run lengths (≡ has three bars, ≣ four, while === has three characters), so a visual transliteration in either direction lands on a different comparison that silently answers a different question. The equality tiers in Epsil are ASCII-only — == (arithmetic, tolerant, may stay inert as a condition), === (Same, structural, total) — and the identity-prover tier is spelled as a call, IdenticallyEqual(a, b). LaTeX is unaffected: \equiv still parses as IdenticallyEqual.

• .toString() no longer prints arithmetic equality in Same's spelling: Equal now serializes as == (was ===) and NotEqual as != (was !==). The old JS-flavored spellings predate Epsil, where === parses as the structural Same tier — so the string view printed an Equal in the exact spelling that reads back as the other operator. Same now also serializes in operator form (===) instead of function-call form; IdenticallyEqual stays in call form (the same no-equivalence-glyphs ruling as the Epsil entry above). Compiled JavaScript is unchanged (Equal still compiles to JS ===, which is correct there).

• An exact integer beyond the safe-integer range now has one storage form too, completing the 0.106.0 normalization (the item-178(b) residual, filed by the Tycho team at their 0.106.0 adoption; witness 8e19 from their seed corpus, minimal witness 1e16). The 0.106.0 fix normalized a bigint WITHIN the safe range to machine storage but left an integer-valued machine number BEYOND it un-normalized, so ce.box(1e16) and the parse of its own serialization were isSame with byte-equal MathJSON yet hashed differently — the same isSame ⇒ equal hash break, past the boundary (2^53 itself was clean). The constructor now stores integer-valued machine numbers outside the safe range as bigints — the storage the parse route produces — so each side of 2^53 has one canonical form (machine below, bigint above); the conversion is exact for every integer-valued float64. Serialization follows the storage, so such literals now print in the compact exponent form (ce.box(1e16).toString() is "1e+16", no longer "10000000000000000"); MathJSON output is unchanged.

• Assigning to a symbol auto-declared with type unknown now applies the same literal-type promotion as a fresh declaration, so the settled type no longer depends on whether a boxing auto-declared the symbol before the assignment (observed by the Tycho team's order-matrix probe at their 0.106.0 adoption). Boxing an expression can auto-declare a mentioned symbol at type unknown when its uses record no type evidence (a bare List sibling, a binder-body occurrence); assigning 5 to such a symbol then adopted the value's raw type (finite_integer), while the same assignment on a fresh symbol declares the promoted integer. The unknown-incumbent case now takes the declaration promotion, so box-first and assign-first orderings settle on the same type (integer). The 0.106.0 assignment-narrowing guards are unchanged: declared types are never touched, assignment-derived incumbents stay widen-only, and the D11 incompatible-guess adoption keeps the value's raw type.

New Features

• Protocol dispatcher effects are now derived from the registered conformers; a requirement's effect specifier is an opt-in ceiling. A protocol function requirement with a bare specifier no longer acts as a pure ceiling: implementations may carry any effects, and the dispatcher's effect set is the live union of the inferred effects of the registered conforming implementations — pure while every conformer is pure (calls through it stay cacheable and compile-eligible), widened the moment a mutating or drawing conformance registers, with dependent caches recomputing (a call boxed before the registration reports the new effects, and an already-inferred user function whose body calls the member re-derives). The dispatcher's serialized signature snapshots the union as of serialization. A requirement with a specifier (including the explicit pure) is a durable ceiling: a conformance whose implementation declares or whose body infers effects beyond it is rejected (protocol-signature-mismatch names the exceeded labels and points at the ceiling as the fix site). Registering a conformance that would widen a bare-requirement union past a standing declared effect contract — a function annotated pure whose body dispatches through the member — is itself rejected (conformance-widens-declared-contract, naming every violated dependent and the exceeding labels) and leaves the registry unchanged. Behavior change: an implementation declaring effects against a bare requirement was previously rejected; it is now accepted, by design (docs/TYPE_SYSTEM_ROADMAP.md Appendix B, "Changing a field is an effect" — the bare-means-pure ceiling was explicitly rejected there).

0.106.0 2026-08-13

Breaking Changes

• The Error JSON shape gains an operand: argument-validation errors now carry the FAULTED OPERAND as a site operand, before the trace. An incompatible-type error minted while validating an argument reports which operand faulted:

  before   f("str")            ["Error", EC]
           Sqrt("s").evaluate  ["Error", EC, ["ErrorTrace", …]]
  
  now      f("str")            ["Error", EC, "'str'"]
           Sqrt("s").evaluate  ["Error", EC, "'s'", ["ErrorTrace", …]]

  ErrorTrace stays LAST and is head-identified, as always — code that finds the trace by scanning for the ErrorTrace head is unaffected. Code that reads the trace positionally as operand 2 now gets the faulted operand instead — a plausible-looking value rather than an error node, so that failure mode is a silently wrong read, not a crash: grep for positional Error operand reads before adopting. Rendered messages now name the site ("… at p"). Two contract repairs ride with the change: match's Error(c) patterns ignore the site the way they already ignore the trace (pattern arity decides — Error(c, w) binds the site), and static-diagnostic deduplication keys on a SITE-LESS description so one problem does not split into per-site diagnostics. Any stored/expected MathJSON containing Error nodes (snapshots, seed corpora) will change shape even where nothing is wrong — re-baseline those deliberately rather than reading the churn as a regression.

Epsil

• Named-argument calls. An Epsil call can pass arguments by parameter name: interest(principal: 1000, rate: 0.05). Named arguments may be given in any order, may follow positional arguments (never precede them), and are matched against the parameter names of the declaration the call resolves through — including overloaded and multi-clause functions (each overload is matched with its own parameter order; a named argument is also a branch selector — a clause that does not declare the written names is never chosen, so with clauses (z: 0) and (n: integer), f(n: 0) runs the general n clause while f(0) runs the base clause) and protocol members — bare (tag(prefix: "→", self: s)) and qualified (Tagged.tag(prefix: "→", self: s)) alike, dispatching on self wherever it is written. A call that uses any name is a complete call: omitted optional parameters are fine, but it never curries, and it cannot fill a variadic tail. New diagnostics: argument-name-unknown (with a "did you mean"), argument-order-invalid, argument-name-duplicate, argument-names-unavailable, and argument-optional-skipped, all covered by epsil doc <code>. Parameters without a declared name remain positional-only, as do unannotated function literals. Design record: docs/plans/2026-08-12-named-arguments-design.md (spec: docs/TYPE_SYSTEM_ROADMAP.md Appendix C, rulings C1–C6).

• The VS Code extension shows what the engine makes of your file. Epsil: Show Representation (the {} button in the editor title bar) opens a read-only pane beside an Epsil file showing one of five representations: the MathJSON the parser produced (before canonicalization), the canonical form of each top-level statement, or the program as compiled by one of the engine's code-generation targets — JavaScript, Python (NumPy), or GLSL. The pane tracks the buffer as you type — unsaved and untitled buffers included — and nothing is evaluated, so rendering a view never runs the program. A program a target cannot compile (for now that includes any program defining a function, since DefineFunction has no lowering in any target) reports the compiler's explanation in the pane instead.

New Features

• A definition-level compile handler on Which is now honored (Tycho item 180). The per-operator compilation extension point excluded every control-flow head; Which is now the carve-out, because it has no binding structure — its operands are plain condition/value pairs a handler can compile through the callback it is given (unlike a Sum index or a Function parameter list, which remain non-overridable). The handler has the same contract as every other operator compile handler: it takes precedence over the built-in lowering, and returning undefined falls back to the stock Which compilation, complex-branch coercion and frame-protocol wrapping included. The override is per-engine — attach it to the engine's own definition, which keeps the stock evaluation semantics:

  const def = ce.lookupDefinition('Which');
  if (def && 'operator' in def) def.operator.compile = myWhichHandler;

  (Do not ce.declare('Which', {...}) for this: a re-declaration replaces the stock evaluate/canonical handlers, so the operator would compile with the custom handler but no longer evaluate correctly.)

• Speculative parsing: ce.parse(latex, { speculative: true }) leaves no trace in the engine's type state (Tycho item 179). A normal parse has two persistent side effects: an undeclared symbol is auto-declared into the current scope, and a narrowing use moves the inferred type of an already-declared symbol (Fibonacci(u) narrows an inferred u: number to integer) — and the second one is not confined by parsing inside a child scope, since the write lands on the resolved definition wherever it lives. With speculative: true the parse runs in a transient scope that is discarded on the way out, and every mentioned symbol whose ambient type is inferred is shadowed in that scope with its current type, so narrowing refines the discarded shadow instead of the ambient symbol. Symbols with a declared (non-inferred) type, constants, and operators are not shadowed, so the string parses exactly as it would without the option, and the result's type and MathJSON are identical to a normal parse's. Intended for derive-style parses that only read the result (its type, structure, or serialization): the result's bindings refer to the discarded scope, so it should not be retained, evaluated, or compared against later expressions. Mutually exclusive with the scope option, which has the opposite contract (the scope receives the writes, to be read back).

Issues Resolved

• A qualified protocol member now works as a collection callback. Map([1, 2, 3], Negatable.negated) used to answer ["Missing", "Missing", "Missing"]: the callback arrives at canonicalization as the field expression Field(Negatable, "negated"), and the shorthand-lambda lift read it as a lambda body, turning the protocol name into the parameter — so every element was bound to the protocol-name slot and read as the base of a field access. A qualified protocol member is now recognized as a function value (the same contract that already kept Apply(InverseFunction(f), 2) from beta-reducing) and the callback dispatches per element. A protocol name shadowed by a valued binding is not a protocol reference and reports the ordinary not-a-function error instead of silently dispatching.

• A protocol dispatcher call inside a shorthand lambda body no longer errors. Map([1, 2, 3], negated(_) * 10) reported incompatible-type number/unknown on the dispatcher call, while the identical shape through a plain declared function worked. Cause: at an undecided receiver, the requirement's Self-typed result substituted as a type reference named "unknown" — printing exactly like the primitive but defeating every gate that admits a primitive unknown operand as "defer and infer later". The reference is now reduced to the primitive, so the shorthand evaluates ([-10, -20, -30]) and an undecided receiver defers symbolically as before.

• An assignment can now refine a use-inferred symbol type, so the inferred type no longer depends on whether the first use or the assignment came first. Previously the inference direction rules were strictly one-way: uses narrow, assignments widen. A symbol whose numeric use was seen before its assignment therefore settled on a wider type than the same symbol assigned first (v used in x·v then assigned 5 stayed number, while assign-first gave integer) — a divergence between orderings, not a wrong type under the rules. Now, when a symbol's current type came from USE inference (not from a previous assignment or a declaration), an assignment refines it to the assigned value's type, with three guards: the promoted type installs only when it is a subtype of the current one; a symbol whose latest type evidence is itself assignment-derived stays widen-only, so alternating assignments (v := 2.5 then v := 5) do not oscillate the type; and the union-adoption path for declared unions is unchanged. Declared types are never affected. If you probe type stability across orderings (box-first vs assign-first), expect both orderings to converge on the assignment's type now — a deliberate change, not drift.

• A parsed integral now binds its integrand's free symbols the same way as boxing its canonical MathJSON does (the residual surface of Tycho item 178(a), distinct from the first-boxing divergence below — it was order-independent, and pre-declaring the symbol did not remove it). Integrate's canonical handler derived a shorthand integrand's Function literal by inferring a parameter for each free body symbol — declaring those parameters in the body scope, with the body's occurrences bound to them — and then swapped the parameter list for the integration variables, leaving the body's occurrences bound to the discarded parameters. A parsed \int_{-x}^{x} \cos(x)\,dn therefore carried a body x bound to a discarded parameter while the bounds' x bound the engine's definition, and compared isSame false against ce.box() of its own .json (and against the box route generally) with byte-equal MathJSON. The handler now passes the integration variables to canonicalFunctionLiteral as the intended parameter list, so the body canonicalizes with exactly those parameters declared — the same binding structure the explicit-Function route produces. A knock-on repair: a parenthesized explicit Function integrand (Delimiter-wrapped) now keeps its user-supplied parameters instead of having them overwritten by the integration variables. Values were never affected — evaluation resolved identically on both routes — and serialization is unchanged (zero snapshot churn).

• Boxing the same MathJSON twice now produces isSame expressions — including the first-ever boxing of a shape (Tycho items 178(a) and 178(c), two surfaces of one defect). A not-yet-declared symbol occurring both inside a binder's scope and outside it (e.g. ["List", ["Integrate", ["Function", … y_r …], …], "y_r"]) made the first boxing bind the two occurrences differently: the occurrence inside the function body auto-declared into a scope local to that one canonicalization, while the sibling outside declared into the surrounding scope. Every later boxing found the surviving surrounding binding first and resolved both occurrences to it — so on a fresh engine ce.box(J).isSame(ce.box(J)) was false (and a box and a parse of the same integral disagreed), breaking the documented contract that isSame is safe as a dedup/matching key. The boxing machinery now detects the conflict — one construction both auto-declaring a name into a scope it created and then declaring the same name into a scope that outlives it — and rebuilds the construction once, so the first result is the same one every later boxing produces. No resolution rule changed and nothing new is declared: a function body's free symbols remain invisible to the caller, and capture through assigned document variables is untouched.

• .json no longer reports a different operand order than the expression it serializes (Tycho item 178(d)). .json goes through the structural form, which re-sorted a commutative operator's operands — but a CANONICAL expression has already had its operands ordered, and that order is the one .ops, .toString(), isSame and hash all read, so re-sorting could only disagree with it. It did, because order() breaks a tie between two same-operator applications on leaf count, and a complex literal is one atom in canonical form (i) but a two-operand application in structural form (Complex(0, 1)): the operands of m(M_1, i, n)·m(M_2, n, j) counted 6-vs-4 structurally where they counted 4-vs-4 canonically, so the tie broke the other way and one object reported two different operand orders depending on which accessor you asked. A canonical expression now keeps its own order; a non-canonical one is still sorted, which is what puts it in structural form at all. Four arithmetic snapshots record the corrected order — in each, .json now matches .ops (1.1 + 2 + 5 + 5/7 + 7/9 + √2 + π serialized √2 before 1.1 while the tree held the reverse).

• An exact integer now has one storage form, so it hashes the same however it was built (Tycho item 178(b)). ExactNumericValue accepted an integer as either a machine number or a bigint and kept whichever it was handed, but numberToString() formats the two differently — a bigint with more than five trailing zeros compacts to an exponent (1e+11) while the identical machine integer prints in full (100000000000) — and BoxedNumber.hash hashes that string. So ce.box(100000000000) and its own reparsed serialization were isSame with byte-equal MathJSON yet hashed differently, breaking the documented isSame ⇒ equal hash contract; any integer with more than five trailing zeros was affected, 1000000 included. The constructor now applies the same safe-integer normalization reducedRational() already used. Magnitudes beyond the safe-integer range are untouched, so bignum results keep their compact exponent form (√(1234567e+19) is unchanged).

0.105.0 2026-08-12

Breaking Changes

• The prefix forall quantifier was replaced by a trailing where clause. A generic signature states its type variables after the signature instead of before it: forall T. (T) -> T is written (T) -> T where T, forall T: indexed_collection. (T) -> T is (T) -> T where T: indexed_collection, and forall T, U. (list<T>, (T) any -> U) -> list<U> is (list<T>, (T) any -> U) -> list<U> where T, U. The clause is always last — after the effect specifier and after the return type — and an omitted bound means any, so where T is shorthand for where T: any. Semantics, solving and bounds are unchanged. The prefix spelling is removed, not deprecated.

  Consequences:

  • where is now a reserved type name and forall is not: ce.declareType("where", …) is a reserved-type-name error, and ce.declareType("forall", …) now succeeds. A nominal type named where must be renamed.
  • A type string beginning with forall reports a targeted migration diagnostic rather than a cascading parse error.
  • .toString() emits the trailing shorthand ((T) -> T where T, never where T: any), so a type has one canonical string. Variable names round-trip as written.
  • Per-arm quantification of an overload set now needs parentheses, since where binds looser than &: ((list<T>) -> T where T) & ((set<T>) -> boolean where T). The unparenthesized form is rejected with a message naming the fix.

New Features

Epsil: language

• Protocols. A protocol declares a set of functions and properties a type implements to conform; conformance drives dynamic dispatch:

  protocol Comparable {
    function compare(self, other: Self) -> "<" | "=" | ">"
  }
  type string is Comparable {
    function compare(self: Self, other: string) -> "<" | "=" | ">" {
      if (self < other) { "<" } else if (self > other) { ">" } else { "=" }
    }
  }
  compare("a", "b")            // dispatches on the first argument -> "<"
  Comparable.compare("a", "b") // qualified form

  Includes readonly/readwrite properties with getter/setter implementations (person.name, and person.name = v as rebinding sugar over the immutable value model); protocol constraints in where clauses (function sort(xs: list<T>) -> list<T> where T is Comparable); conditional conformance (type list<T> is Comparable where T is Comparable { … }, recursive — list<list<string>> conforms when string does); qualified property disambiguation person.(Protocol.name); and the host API (ce.declareProtocol, ce.declareProtocolImplementation). Protocols are engine-global, statement re-runs replace them (host redeclaration throws), and conformance is monotone: a declared-but-unimplemented conformance warns at the end of each batch until fulfilled. Protocol names are not types — using one in type position steers you to a constrained type variable. A second implementation block for the same (type, protocol) pair within one batch is a protocol-implementation-duplicate error; the same statement re-run in a later batch still replaces (the notebook pattern).

• Sum-type declaration sugar. One statement declares a tagged sum — the variants and the union together:

  type TrafficLight = red | green | yellow
  type node = lit(num: number) | plus(op1: node, op2: node) | times(op1: node, op2: node)
  type tree<T> = leaf | node(value: T, children: list<tree<T>>)

  This is exactly N nominal variant declarations plus one transparent type alias union. The sum's own name is usable in variant payloads without a type forward-reference marker, a generic sum distributes its type parameters to each variant by usage, and a variant name colliding with an existing type, a reserved word, or a builtin rejects the whole declaration atomically. Existing spellings are unchanged: type X = A | B over known types stays the opaque nominal, type alias X = A | B stays the transparent union.

Epsil: compilation

• Protocol dispatch compiles to JavaScript. Bare (compare(x, y)), qualified (Comparable.compare(x, y)) and property GET/SET (p.name, p.name = v) calls all compile: a statically resolved call becomes a direct call, a dynamic one a most-specific-first guard chain on the receiver. A receiver no conformance covers throws protocol-implementation-missing at runtime (the interpreter yields the corresponding error value). Anything unprovable — conditional conformances, host implementations, ambiguity-capable conformance sets — declines compilation and falls back to the interpreter; Python and GPU targets keep failing closed.

• Sum types compile to JavaScript. For sugar-declared sums, match constructor patterns and variant constructors now compile (they previously failed closed). Per sum, the compiler picks a representation: when every variant has a distinct JavaScript shape (jnull | jbool(boolean) | jnum(number) | jstr(string) | jarr(list<…>)) values stay erased and dispatch uses typeof/Array.isArray; otherwise values carry their tag. Tagged values do not cross the compiled↔engine boundary (a unit whose result is a tagged sum declines); Python, GPU and interval targets keep failing closed.

Epsil: tooling

• Calls to a function your program defines are now checked statically. epsil check (and the VS Code extension) reported wrong calls to library functions but let wrong calls to your own functions through — so function foo(x: string, n: integer) { x } followed by foo("hello") checked clean and only failed at run time. Wrong arity, wrong argument type and extra arguments are now all reported before anything runs, with the signature note and the definition site. Multi-clause definitions accumulate clause by clause, and a definition inside a block stays scoped to that block. A generic definition (function id<T>(x: T) -> T) is still checked only when it runs.

• Signature errors explain the signature. A call that does not match its callee used to report only what went wrong ("a required argument is missing"). The report now carries the callee's signature and the faulted position, plus a second annotated block pointing at the definition when the program defines it:

  error: Runtime error: a required argument is missing
   --> example.epsil:3:1
    |
  3 | foo("hello")
    | ^^^^^^^^^^^^
    = note: `foo` has signature `(x: string, n: integer) -> string`; argument 2
            (`n: integer`) was not supplied
  
  note: `foo` is defined here
   --> example.epsil:1:10
    |
  1 | function foo(x: string, n: integer) { x }
    |          ^^^
    = note: `epsil doc missing` explains this error

  The same notes reach epsil check, the REPL, and — machine-readable, with the definition's location — --diagnostics json and check --json, where a diagnostic may now carry a notes array. In the VS Code extension the signature appears in the diagnostic's hover, and a note pointing at a second place in the file is published as related information.

• The VS Code extension answers hovers. Hovering a library name shows its signature (or, for a symbol, its type and value) with its description — the same entry epsil doc <name> prints, so the editor and the CLI cannot drift. Hovering a name the file itself declares shows that declaration as written: a function's header without its body, a let/const's statement. A name inside a string or a comment is not a hover target.

Collections

• isEnumerableCollection now answers for arithmetic broadcasts. A broadcast node (x + [1,2], Sin(1..99)) used to answer undefined for the whole class; it now answers from its participants: true when they agree on a length, evaluation is draw-free, and every collection-typed participant is itself enumerable; false when a participant is definitively unwalkable — including an application of an unbound head (x + Total([1,2]) with Total undeclared); undefined when participant lengths disagree or an impure participant (RandomShuffle(xs) + 1) makes per-index coherence unpromisable. true is a delivery promise: each() and at() will serve the elements.

• at() now works on impure broadcasts whose randomness is confined to a lifted scalar. ([1,2] + RandomInteger(1,10)).at(1) answered undefined; since such a broadcast distributes structurally without consuming randomness, at() now serves the same unevaluated element each() yields (1 + RandomInteger(1,10)), at any nesting depth. An impure participant (RandomShuffle(xs) + 1) still declines, since per-index reads could mix draw sets.

• Operators can declare an elementCount handler (companion to canEnumerate) to report their length without evaluating.

Resolved Issues

Epsil

• A static diagnostic no longer underlines a whole function definition. epsil check and the VS Code extension anchored every canonicalization-time error on the enclosing statement, so one wrong argument four lines inside a definition flagged the entire definition. IndexOf(digits, cs[i], 23) now underlines the 23, and the message quotes the failing call. An ambiguous match (two IndexOf calls in one statement) still falls back to the statement.

• A statement that evaluates to an error now stops the enclosing block or loop. A non-final statement's value was discarded — including a refusal that is the statement's value, such as a mistyped write or a rejected indexed assignment — so execution continued past the fault. An Error result now propagates like Return: it becomes the block's value, and a loop stops iterating and surfaces it.

• A protocol-property write in a loop body no longer silently no-ops. p.name = 10 * i was refused because the unevaluated right-hand side types wider than the property's declared type; the check now runs on the evaluated value. Genuinely mistyped writes are still refused.

• A loop index over an integer range is now integer-typed. The index of for i in 1..3 was bound as number, so 10 * i typed finite_number and tripped integer-typed checks. The binding is now typed from the collection's element type (Range(1, 3) → integer, a float list → real), including through a type alias (type ints = list<integer>).

• A fully-known argument now always dispatches a multi-clause function. With clauses a(t: integer) and a(t: real), a(0.3) stayed inert under evaluate() while compiled code correctly selected the real clause — two different answers for one input. Concrete values are now tested against each clause for both admission and refutation, so interpreted and compiled dispatch agree. Symbolic arguments keep their deliberate inertness.

• A cyclic type alias no longer overflows the stack. A non-progressing structural alias cycle (type a = a | 0) crashed with RangeError when a concrete value was tested against it, from multi-clause dispatch and from typeAcceptsValue. Genuinely recursive values against recursive aliases still admit; a non-progressing back edge now answers definitively.

Compilation

• A function declaration contradicted by its own body no longer compiles to wrong results. With a declared (number) -> number but assigned a list-valued body (a := t ↦ [cos t, sin t]), compiled code trusted the declaration wherever the result was consumed as a scalar: Σᵢ a(h(i)) and a(u) + 1 returned strings, 2·a(u) and sin(a(u)) returned NaN, comparisons returned a wrong boolean, If took the wrong branch — all behind success: true — and the GPU targets emitted shader source that does not compile. When the body provably constructs a collection while the declared result type says scalar, compilation now fails closed with a message naming both sides of the contradiction, and the interpreter answers instead. Deliberately unchanged: a bare a(u) on JavaScript still compiles (it returns the correct array today), open declarations (-> unknown) keep compiling element-wise, and boolean / broadcastable<T> big-op bodies are untouched.

• A Sum/Product whose body applies a list-valued user function now compiles correctly. With a(t) ≔ [cos t, sin t] under an open (unknown) -> unknown signature, Σ_i a(h(i)) compiled success: true and returned a JavaScript string. It now compiles to the element-wise broadcast fold and returns the correct array. On the non-JavaScript targets the same shape emitted shader source that does not even compile; GLSL, WGSL, Python and interval now fail closed with an actionable message.

• A block-local introduced by bare assignment now compiles correctly. A multi-statement body binding a local with a plain assignment — t ↦ (w ≔ 2t; w + 1) — compiled behind success: true but ran to NaN (on GLSL the local was simply undeclared). Compiled results now match the interpreter.

• Compiled writes to an outer binding now reach the same place reads look. A block assigning a name the enclosing scope already binds — including the library-predeclared single letters e, i, m, s — wrote a stray global and read undefined: (s ≔ 0; s + 1) compiled to NaN where the interpreter answers 1, and a function body accumulating into s through a loop returned undefined behind success: true. Where reads bake a folded value or a constant, so no coherent write target exists, compilation declines.

• Complex-valued locals compile correctly. A loop-body local bound to a complex value miscompiled on JavaScript (NaN behind success: true; the interpreter answered 3√10), and a GPU function whose body merely contained a complex local while returning a real produced invalid shader source. Loop bodies now share the block compiler's complex inference, and the GPU return type is derived from the body's value rather than from interior locals.

• The expression-only compilation routes now fail closed on statement bodies. compileToSource() (GPU and Python) and compileShader() promise a single expression, but a statement block escaped into them — compileToSource(Block(s ≔ x; Return s)) returned "s = x\nreturn return s" — behind a successful return. Both now throw, naming the expression-only contract and pointing statement bodies at the statement-capable sibling (compile() on GPU, compileFunction() on Python). Also gated per language semantics: an assignment body on WGSL and Python (still allowed on GLSL, where assignment is an expression), and a bare declaration body, which additionally dropped its initializer silently. The function-emission siblings decline the same shapes: Python compileFunction no longer emits def f(x): return s = x, and compileLambda catches single-line statement bodies. Expression bodies are byte-identical on every route.

• A Return in a GPU function body now emits valid source or declines. A final-statement Return emitted return return … and a Return inside a conditional emitted return in expression position — neither is GLSL or WGSL — both behind success: true. Two gates now hold: a shape gate (a Return whose value shape disagrees with the function's signature declines, naming both shapes) and a placement scan on the finished body. Early Returns of matching shape in plain statement position compile unchanged, as does every JavaScript path.

• Compiling an assignment to a non-variable target now fails closed. A sequence definition (L_0 := 5) inside a compiled body emitted the silent no-op _ = 5 behind success: true (and, in sloppy mode, wrote a stray global). Such an assignment now declines and falls back to the interpreter.

• A nominal-typed argument no longer broadcasts at compiled call sites. With type bag = list<number> and function size(b: bag) -> number, compiled size(bag([1,2,3])) answered [42, 42, 42] where the interpreter answers 42. Nominal values are atomic, like tuples: the whole value binds. A broadcastable<bag> slot now also accepts a bag argument directly.

• Typed locals now compile inside loop bodies. A let q: Person = … declared in a loop body rides a different path than a block local, so a protocol property GET/SET on it failed to compile — and an ordinary named-tuple field read (q.y) on a typed local failed on every path. Both now resolve, nested chains (q.inner.y) included.

• Contradiction gates hardened. Multi-statement lambda bodies are judged by their final statement (a body ending in a list no longer slips past the declaration-contradiction gates), and -> boolean declarations over collection-constructing bodies now decline everywhere a condition or logical operand consumes them — an If no longer takes a branch off array truthiness.

LaTeX serialization and round-tripping

• Block expressions now round-trip. Four shapes changed value on a round trip and are fixed:

  • A single-statement block (Block(s ≔ 2)) serialized as bare s\coloneq2, which reparses as a plain assignment — the block-local leaked into the enclosing scope. It now serializes with a trailing semicolon (s\coloneq2;), and the parser no longer produces the vestigial Nothing statement a trailing ; used to add.
  • A multi-statement block with no assignment (Block(s+1, s+2)) serialized as s+1; s+2, which reparses as a 2-tuple. It now serializes as a one-column \begin{cases}…\end{cases} (core amsmath), and such an environment parses back as a Block. Real piecewise (Which) always serializes two columns and is byte-identical.
  • A block of bare equations (Block(x=1, y=2)) collided with the system-of-equations reading and came back as a List; it now serializes as explicit \operatorname{Block}(x=1, y=2).
  • A Block used as an argument — of any function call, a Loop body, or a comprehension body — emitted a bare ;-statement list, which binds looser than the argument comma and the for clause, so the block swallowed what followed and the value changed. Block operands are now fenced.

• A two-element list no longer turns into an interval — in any set-operator position. Element(i, List(1,2)) ("i ranges over the two-point list") serialized as i\in\lbrack1, 2\rbrack, which reparses as Element(i, Interval(1,2)) ("i is any real between 1 and 2"). The same conversion fired on both operands of Union, Intersection, SetMinus and the Subset/Superset families, including nested. A 2-element-list operand is now spelled \operatorname{List}(1, 2) everywhere (membership, both sides of every set operator, and the big-op subscript), and the parser applies the interval reading only to operands actually written with brackets — an explicitly named List head is never overridden. Authored bracket spellings keep their interval meaning, including \mathopen\lbrack a,b\mathclose\rbrack and \mleft…\mright. Other list lengths are unchanged.

• Miscellaneous parsing fixes. An authored trailing \mathrm{Nothing} statement survives parsing, a nested Block(Block(…)) operand no longer defeats the cases-environment fence, and SymmetricDifference reads and writes its operands with the same interval/list conventions as the other set operators.

Evaluation

• count no longer invents a length for reshaping operators. The broadcast count read participants' lengths without checking that the operator actually broadcasts, so Chunk([1,2,3], 2).count answered 3 (true count: 2), and a dozen other reshaping operators (BinCounts, Histogram, Tally, Shape, …) reported their operand's length. Sort, Ordering and RandomShuffle now report their source's length (with zero draws), Chunk(xs, k) reports k for a literal k, and operators whose length is not cheaply knowable answer undefined instead of a wrong number.

• A parenthesized juxtaposition on a collection-valued symbol is a product regardless of the argument's type. With A a list and B a list, A(B) and A(t-B) canonicalized to function applications of a non-function — evaluating to Error("incompatible-type", …) — while A(m) with scalar m correctly multiplied. The decision now keys on the head: a head whose value is multiplicative (a number, matrix, list, collection, broadcastable, or numeric tuple) always reads as a product. Undeclared and function-typed heads still read as applications.

• A non-multiplicative value head applied to a scalar is now an illegal application, not a product. With t := "hello", t(2) canonicalized to Multiply and the type error blamed multiplication; string-, set- and boolean-valued heads now take the application route, so the error blames the actual mistake.

• A negated product no longer produces an unflattened Multiply. (b·-a)c built a node whose .json serialized flat (ordered abc) while isSame and hash saw the nesting, violating the documented isSame ⇒ equal hash contract; a negated rational coefficient with a Power factor hit the same problem. .json, isSame and hash agree again.

• A PointList body is now differentiated componentwise. With a lambda stored as a symbol's value — p := (t) \mapsto \operatorname{PointList}(\sin t, \cos t, t) — p'(0.25) returned a sum of inert Apply(Derivative("PointList", …), …) terms, because the differentiator applied the generic chain rule to the PointList OPERATOR. It now differentiates each component and keeps the PointList head, so p'(0.25) is the point (0.9689…, -0.2474…, 1), matching what the same curve already gave when defined as g(t) \coloneq …. This holds for components that are themselves lists (PointList(t\cdot[1,2], t^2), which zips to a list of points): differentiation and the zip commute.

0.104.1 2026-08-11

Issues Resolved

• A compiled big op whose bound is a function parameter no longer folds against a same-named library constant (Tycho item 176). Compiling a Sum/Product folds its bounds at compile time when both are constant, but that fold read the bound's value through the ENGINE symbol, ignoring that the name might be bound in the compilation context. A user function's parameter named after a library constant therefore produced silently wrong code behind success: true: in F(i) = \sum_{m=1}^{i} m the bound i resolved to the imaginary unit, whose real part is 0, so lower 1 > upper 0 read as an empty range and the entire body folded to the identity — const _fn_F = (i) => 0, against an interpreted 6. A parameter named e picked up Euler's number and summed two terms (3). The same shape emitted float _fn_F(float i) { return 0.0; } on the GPU targets. A bound that mentions any compile-bound name — a function parameter, an enclosing binder's index, a broadcast element — is now treated as symbolic and emitted as a loop over the parameter, matching the interpreter on every parameter name. This affected the javascript, GLSL, WGSL and interval-js targets; Python was already immune. A bound naming a constant that is NOT shadowed still folds as before.

• The derivative of a tuple-valued function differentiates componentwise (Tycho item 174). A Tuple is a container, not a function of its elements, but it had no case in the differentiation rules, so it fell through to the generic chain rule and differentiated the Tuple operator: with g(t) = (\cos 2\pi t, \sin 2\pi t, t), both g'(t) and D(g(t), t) left inert Apply(Derivative("Tuple", 0, 1, 0), …) nodes. Tuple now takes the same elementwise branch List already had, preserving the container head and nesting — d/dt (fₓ, f_y, f_z) = (fₓ', f_y', f_z'), so g'(0.25) is (-2\pi, 0, 1). This closes the Frenet-frame idiom (F_0 = f'/|f'|, F_1 = F_0'/|F_0'|), which never closed before because f' stayed inert. Unregistered heads are unaffected: Point types unknown, so it stays an opaque application rather than acquiring invented componentwise semantics.

• Equal/NotEqual against a list no longer overflows the stack when the other operand is opaque. ce.parse('A(t) = [t]').evaluate() — one line on a bare engine — threw RangeError: Maximum call stack size exceeded, as did q(2) = [1, 2] with q declared (number) -> unknown. These operators broadcast element-wise only in the list-vs-scalar case, and that decision was taken twice with two different rules: before evaluation the engine skips broadcasting when two or more operands are collections or possibly-collection typed, then defers to the evaluate handler — which counted only actual collections. A top-typed application is possibly-collection typed and stays so after evaluation, so the handler kept rebuilding the identical node. The two rules now agree. Such a comparison also stays inert instead of answering False: an opaque operand may still be that collection, so a structural mismatch is not something the engine can claim.

• .count no longer outruns each() on a broadcast over an unbound head. An undeclared call binds vacuously and its result type lifts over a collection argument, so Total([1, 2]) types list<unknown^2> and looked countable — but nothing produces those elements and the walk yields none. count answered 2 for a collection that can never yield an element, and it propagated: x + Total([1, 2]) counted 2 while walking 0. An unbound head now reports undefined, the same honesty Add([1,2], [1,2,3]) already gets. Broadcasts over declared operators are unchanged.

• Quartiles and InterquartileRange no longer throw on thin or symbolic data. Quartiles(2.5) — a single inexact datum — threw a TypeError (median of an empty quartile half), as did any symbolic datum reaching the numeric path (Quartiles(z) for an unassigned z). The empty median is now NaN, matching the machine-float path, and symbolic data leaves both operators inert instead of baking a (…, NaN, …) tuple that a later assignment contradicts: Quartiles([1, 2], z) stays symbolic and answers (1, 2, 3) once z := 3. A NaN literal datum still takes the documented absence path.

• ListFrom/SetFrom/TupleFrom no longer wrap an unresolved collection operand as a scalar. For a valueless list-typed symbol xs, ListFrom(xs) answered ["xs"] — a one-element list containing the symbol — and ListFrom(Divisors(n)) similarly wrapped the unevaluated producer. A collection-typed operand that is not a collection in the current state is unresolved, not a datum: the conversions now stay inert (and report isEnumerableCollection: false without evaluating). Genuine scalars still contribute themselves: ListFrom(5, [1, 2]) is [5, 1, 2].

• Dot's isEnumerableCollection no longer misreports a matrix product. The result type of Dot is the wide value, which the facet read as "not a collection" even when Dot(m1, m2) evaluates to a matrix. Dot now answers per shape: a provably scalar inner product is false, a matrix-ish product is decided by its operands (Quartiles also gained a decline-only canEnumerate now that its symbolic case is inert).

• Stacked restrictions no longer depend on the order they were written. When(When(e, c1), c2) merges to When(e, And(c1, c2)), but the merged conjunction was built without going through canonical And, so it kept the authored operand order. Since .json serializes through the structural form, which DOES order a commutative operator's operands, the two spellings of one restriction set disagreed depending on which accessor you asked: x\{|z|<5\}\{0<y\} and x\{0<y\}\{|z|<5\} produced byte-identical MathJSON while isSame() answered false and their hashes differed, .json and .toString() of the SAME expression listed the conjuncts in opposite orders, and ce.box(expr.json) was not isSame to expr. Reported by Tycho as the 11 differs-json-equal rows of the item-153 corpus seed.

• Indexed wrappers over an eager collection producer no longer read it as empty (design: docs/plans/2026-08-11-eager-collection-enumerability.md). An eager producer (Divisors, Characters, Eigenvalues, … — an operator whose collection exists only as its evaluate() result) could be walked through each() but not indexed through at(), so every wrapper that reads its source by index walked it as empty — wrong values on fully-ground input: Filter(Take(Divisors(12), 3), _ > 1) answered [] (now [2, 3]), Any(Reverse(Divisors(12)), _ > 1) answered False (now True), and Sum(Take(Divisors(12), 3)) stayed inert (now 6). at() now has the same materialize-on-demand fallback each() always had — pure sources only, and the evaluated form is computed once per instance, not once per index.

  With it, eager producers can now declare a canEnumerate handler — the decline test their evaluate handler already starts with — so isEnumerableCollection answers without evaluating: Divisors(n) for a free n reports false (wrappers over it stay inert instead of answering []/False/0), Divisors(12) reports true, and an operator whose success is not cheaply decidable (Solve) stays undefined and resolves by evaluating, as before. Adopted in this release — 45 of the 73 eager producers: Characters, GraphemeClusters, UnicodeScalars, Utf8, Utf16, StringSplit, Divisors, PrimeFactors, FactorInteger, IntegerDigits, Shape, Keys, Values, AbsArg, ComplexRoots, ExtendedGCD, PlusMinus, and (decline-only) Sort, Ordering, Unique, Tally, Eigenvalues, Eigenvectors, Eigen, SingularValues, SVD, LUDecomposition, QRDecomposition, Cross, MatrixMultiply, HadamardProduct, Flatten, Chunk, GroupBy, ListFrom, SetFrom, TupleFrom, DictionaryFrom, RecordFrom, BinCounts, Histogram, ContinuedFraction, RandomShuffle, RandomChoice, RandomSample (the impure trio answers from its domain operand's facets alone — consulting the predicate consumes no random draws). The rest are deliberate skips: solver-class operators stay in the undefined tier by design, and a handful (Pair, Vector, Tail, …) never exist as canonical leaves at all.

• A type alias naming a union of nominal types now confers membership, so a sum type is usable through its own name. Given

  type lit = tuple<num: number>
  type plus = tuple<op1: type expr, op2: type expr>
  type alias expr = lit | plus

  every declaration was accepted and then every use failed: lit <: expr answered false, so passing a lit where an expr was expected was an incompatible-type error. Recursive shapes failed at the first nested construction — total(node(1, [node(2, [])])) reported expected list<tree<finite_integer>>, got list<node<finite_integer>^1> even though node<integer> <: tree<integer> held when asked directly.

  Three defects in the unfolding of a structural alias standing on the RIGHT of a subtype check: isSubtype short-circuited to a name comparison whenever both sides were references, so a reference on the left never reached the unfold at all (which is why the relation was asymmetric — expr <: lit held while lit <: expr did not); an applied reference SNAPSHOTTED its alias flag while delegating def to the declaration record, so a forward reference captured inside a variant's payload kept the placeholder's alias: false after a type alias fulfilled it; and the unfold compared against the alias's open body without substituting the application's arguments, so node<integer> <: tree<integer> asked node<integer> <: leaf | node<T> and failed on the type variable.

  Nominal opacity is unchanged: type X = A | B still declares a new opaque type that neither A nor B inhabits — only type alias X = A | B is a sum.

• A type whose body is nothing now has a nullary constructor, so a payload-free variant can be built. type leaf = nothing minted (nothing) -> leaf, which no call could satisfy: nothing is the unit type and its sole inhabitant Nothing elides as an operand, so leaf() was a missing argument and leaf(Nothing) collapsed to the same call. The constructor is now () -> leaf, the shape an empty tuple body (type unit = tuple<>) already had.

• A ground union arm binds its type variables to never even when the union is reached through an alias. Rule U gives an operand accepted by a ground arm no say over the variable, so it contributes never and a constraining operand wins outright — but the rule lives in the solver's union case, which a parameter still spelled as a forward-reference alias never reached. The union was hidden behind the reference, no arm contributed, and the variable fell through to the unknown default: plus(lit(5), lit(2)) typed plus<unknown> and was then rejected by an expr<number> parameter, where plus<never> is accepted.

0.104.0 2026-08-11

Breaking Changes

• Types are now engine-global; type statements are top-level only. Type declarations (and the value constructors they mint) now live in one engine-level registry rather than the lexical scope chain: a declared type name means the same thing everywhere on the engine, for the engine's lifetime, and a duplicate name is an error. Consequently a type statement inside a do block, function body, if branch or loop body is now a hard error — type-declaration-not-top-level at parse time in Epsil, an invalid-type-declaration error value on the MathJSON ["DeclareType"] route — instead of declaring a block-local type; there is no hoisting, and the former type-shadow warning is gone (shadowing is impossible). The top-level statement-replace flow (re-running an edited notebook cell) is unchanged. This closes the class of bugs where a value of a block-local nominal type escaped its block and re-bound to a different same-named type (or typed as unknown), makes any serialize→reparse of a type sound anywhere in the engine, and is the prerequisite for protocols. Host note: ce.declareType() under a pushed scope now targets the engine registry — popScope() no longer removes a type. See docs/plans/2026-08-10-global-type-registry.md.

New Features

• Annotation-bound parameters in Epsil typed declarations (the "lambda lift"). A parameter name now binds wherever it appears: when a declaration's annotation is a literal function type with named parameters and the initializer is not a lambda, the annotation's names become the parameters of an automatically built lambda — const f : (x: number) -> number = x^2 + 2x - 1now means= (x) |-> x^2 + 2x + 1. When the initializer is an explicit lambda, the two parameter lists must agree positionally; a disagreement is the new parameter-name-mismatchdiagnostic, with a fixit that renames the annotation to the lambda's names. Alias annotations stay opaque (names bind only where they are written), only the outermost arrow of a nested signature lifts, and zero-parameter, generic, effectful, optional/variadic, and partially named signatures never lift. Seedocs/plans/2026-08-08-annotation-lambda-lift.md.

• The ->-for-|-> typo is now diagnosed and recovered in Epsil. A KeyValuePair arrow whose left side is shaped like a parameter list — a typed parameter ((x: number) -> x^2), a parameter tuple ((x, y) -> x + y), an empty (), or a bare symbol right after ( or = (f = x -> x + 1) — is a function written with the wrong arrow: none of those shapes is a valid dictionary key. The parser reports the new mapsto-arrow-expected diagnostic with a fixit on the arrow and recovers as the intended lambda, so the program still runs. It also no longer emits a spurious unexpected-symbol ":" for typed parameters before a ->. Legitimate dictionary spellings ({one -> 1}, "a" -> 1, {->}) are untouched.

Issues Resolved

• D no longer swallows what follows it when serialized (Tycho item 166). The Leibniz spelling put the differentiand undelimited and trailing the fraction, where the parser binds it greedily — so any D with a right neighbour re-read as a different expression: ["Add", ["D", ["Subtract","x","d_t"], "y"], 1] serialized to \frac{\mathrm{d}}{\mathrm{d}y}x-d_{t}+1 and re-parsed as D(x - d_t + 1, y). The isolated case round-tripped only because that greed happened to re-absorb exactly what it emitted.

  Delimiting the body is not sufficient — the differentiand is parsed at addition precedence, so the parser consumes past a closing delimiter and \frac{d}{dy}(x-d_t)+1 still re-read as D((x-d_t)+1, y). A COMPOUND differentiand now folds into the numerator instead (\frac{\mathrm{d}(x-d_t)}{\mathrm{d}y}), a spelling the parser already accepts at every order and which is self-delimiting by construction.

  The parser's greed is deliberately unchanged, so documents relying on the current trailing binding are unaffected: a TIGHT differentiand (a symbol, a number, a function application, a power) serializes exactly as before.

• .count answers for an un-evaluated arithmetic broadcast (Tycho item 167). [1,2,3]+1 reported no count even though its type says vector<finite_integer^3>, and 2(1..99)/99-1 reported none even though the Range inside it reports 99 — because the broadcasting arithmetic operators carry no collection handlers (broadcasting is a property of how they evaluate, not a collection operator), so count had nothing to delegate to. A caller wanting to prove finiteness BEFORE deciding whether to evaluate then had no way to do so short of the eager walk it was trying to avoid.

  count now reads the operands when no count handler is declared. That is exact because the length rule for a lifted operator is agreement, not zip-to-shortest (docs/BROADCAST-MODEL.md): a scalar operand is a lift and never participates, and participants of differing lengths are incompatible-dimensions rather than a shorter result — so mismatched or unknown-length participants report undefined rather than a guess. A declared count handler still owns its own answer, and isCollection/isFiniteCollection are deliberately unchanged: this answers the length question only.

• toLatex() and .latex are total: formatting no longer throws (Tycho item 168). Serializing a lazy collection materializes it first, and materialization EVALUATES — so a comprehension over an unresolvable binding raised Condition must evaluate to "True" or "False" out of a formatting call, taking down whatever asked for it. An unresolvable binding is not a formatting error: the tree is perfectly printable, and prints identically when nothing is bound at all. The materialization is now guarded and falls through to that symbolic spelling. Successful output is unchanged — the guard only covers what previously threw.

  A CancellationError still propagates: deadlines are installed only by an enclosing ce.withTimeLimit() span, and a caller who set a budget must see it expire rather than receive a silently degraded spelling.

  Callers who want the un-materialized form by contract rather than by fallback should pass toLatex({ materialization: false }), which never evaluates. That is the supported opt-out and predates this fix.

• A broadcastable<T> parameter now admits exactly what T admits (Tycho item 157(4), generalized). The filed case — (broadcastable<value>) accepting a function-typed argument that (value) rejects — was one instance of a wholesale hole: (broadcastable<number>) also accepted a string and a boolean.

  Neither the admission path nor isSubtype was at fault; both were already correct (isSubtype(function, broadcastable<value>) is false). The gap was in provablyDisjoint: a broadcastable<T> spans two category buckets (T and indexed_collection<T>), so the category test found no bucket and fell through to the conservative "may overlap". That is safe for the predicate in isolation, but argument checking only KEEPS a type error when every candidate parameter is provably disjoint from the operand — an un-rejection that exists so a bare symbol with a provisional type is not eagerly refused. A parameter kind that is never provably disjoint therefore admitted every operand with a free variable. provablyDisjoint now distributes over broadcastable<T> exactly as over a union, using the same T | indexed_collection<T> expansion isSubtype uses.

  What broadcastable is for is unaffected: scalar and collection arguments are admitted as before, and broadcastable<value> still accepts string and boolean because value does.

• Divide no longer drops the tuple type of a PointList numerator (Tycho item 165). canonicalDivide scales a tuple / scalar component-wise only when the numerator's components are ACCESSIBLE — a Tuple/Pair/Triple/ Single head. Any other tuple-TYPED numerator, notably the PointList head importers emit, stays an inert Divide, and the type handler had no tuple branch, so that inert form collapsed to number. PointList(x, y) / n typed number while PointList(x, y) · n and (x, y) / n both kept the tuple — Divide was the sole outlier among the arithmetic operators (Multiply, Add, Subtract and Negate were already correct).

  The collapse cascaded: a list of such quotients typed vector<n> — a list of NUMBERS — so PointX/PointY over it took the element-INDEX reading instead of the elementwise one, and a mixed p + p/n failed outright with incompatible-type. Downstream, a consumer proving collection-ness by type saw a scalar and failed closed, so dependent rows never registered and a plot silently drew nothing (while getting faster, because the work was skipped).

  The fix mirrors the Multiply handler, which also stays inert on this input yet reports the tuple type. It is type-only: no canonical form changes, so the PointList head — which carries a consumer compile contract — is preserved, and the accessor narrowing that surfaced the bug is untouched (PointX over a genuinely flat numeric list remains the element-index read).

Improvements

• Declared-type mismatches on let/const are now caught statically. The Epsil static pass (epsil check, editor diagnostics, executeEpsil's pre-run check) never evaluates, and the declared-type check lived only in Declare's evaluate path — so let s: string = 42 produced no diagnostic until the program ran, and none at all in the editor. The pass now checks each annotated declaration's initializer against its annotation without evaluating, reporting only provable mismatches so there are no false positives: disjoint types (let s: string = 42, and the unnamed-signature near-miss const f : (number) -> number = x^2 + 1, which carries the same explanation as the runtime error), plus closed literals that fail the full covariant check (let n: integer = 1.5). Unknown-typed initializers, overlapping types, and cross-statement bindings remain the run phase's job — incomplete rather than unsound.

• The declared-function-type near-miss now explains itself. const f : (number) -> number = x^2 + 1 used to fail with a bare incompatible-type (expected "(number) -> number", got "finite_number") — cryptic enough that readers mentally auto-insert the missing parameter name and cannot spot the mistake. When a declared type is a function signature and the initializer is not a function, the error (on both the host ce.declare/ce.assign throw and the Assign/Declare error-value routes) now spells out the near-miss: a signature's parameters bind only when they are named, so nothing in (number) -> number binds and x^2 + 1 is an expression in the unknown x, not a function of x — and names the exact rewrite when the initializer's unknowns pair off with the unnamed parameters ("(x: number) -> number", or "(x) |-> x^2 + 1").

• New expr.isEnumerableCollection: telling an empty collection from one that cannot be walked. each() yields nothing for two unrelated reasons — the collection is empty, or its elements have no computable value (Range(a, b) over free variables, Linspace(a, 1, 3), Repeat(3, n), a declared-but-unassigned symbol, or any wrapper over one of those). The walk alone cannot tell them apart, and the facets that could (count, isEmptyCollection) are undefined in both cases, so the library decided by EVALUATING the source. The new predicate answers structurally, without evaluating: true (an empty walk means empty), false (an empty walk means nothing), or undefined for the one undecidable case — an eager collection operator such as Characters(s), which has no collection handlers until it is evaluated. Custom collection operators declare it with the optional isEnumerable collection handler; a wrapper propagates it from its source (default: true).

  This closes a family of wrong answers over wrapped unknown sources: for a valueless xs, Filter(Take(xs, 2), p) answered [], Any(Reverse(xs), p) False, CountIf(Take(xs, 2), p) 0, Position(Take(xs, 2), p) [], Ordering(Take(xs, 2)) [], Count(Take(xs, 2), 3) 0, and Length(Filter(Take(xs, 2), p)) 0 — answers a later xs := [1, 5] contradicts. All of them now stay inert, as Length, Total, Sort and Map always did over the same source spelled directly. The same applies to a symbolic-bound source: Filter(Range(a, b), p) no longer reports itself empty, and the multi-source Map now consults every source rather than only the first.

  One shape is knowingly left: an eager collection leaf under a wrapper (Filter(Take(Characters(s), 2), p) for a valueless s) still reads as empty, because an eager operator has no collection handlers to answer from and the wrapper's evaluated form is still a wrapper. Tracked in ROADMAP.md; the fix is to give those operators lazy collection handlers.

• Engine construction is ~3× faster, and the gradual registration accretion is recovered (ROADMAP P-BOX, residual half). Every new ComputeEngine() re-parsed the standard library's type strings from scratch — 1,953 parseType() calls per construction, all resolver-aware and therefore ineligible for the resolver-less memo cache, despite only ~308 distinct strings. As the library grew each release, construction and fresh-engine registration accreted a few percent per release. parseType() now uses two-step resolution: the cached resolver-less parse runs first — sound because a user type name can never shadow built-in type syntax — and only strings that actually name a user-declared type (or use the type X forward-reference spelling, which the parser now tracks via sawForwardRef, since it parses resolver-less into an unresolved placeholder and registers a forward reference as a resolver side effect) fall through to the uncached resolver-aware parse. Full parses per construction drop from 1,953 to 343 on the first engine and 48 on subsequent ones; a 120-function registration drops from 2,278 to 0. Construction: ~2.7 ms vs 7.6–8.2 ms for all published 0.10x versions; re-registration is now at or below the 0.100.1 baseline.

0.103.3 2026-08-10

Issues Resolved

• Indexed-then-accessed chains compose again (Tycho item 164): the 0.103.2 narrowing of PointX/PointY/PointZ to (collection | tuple) -> any rejected At's optional result — an in-range unprovable access types missing | tuple<…>, so every S[n].x chain errored incompatible-type at canonicalization. The accessors (and First/Second/Third/Last and Distance, which had the same composition failure) now declare a missingBehavior, admitting the missing arm through strip-before-validate (§3.B of the missing-value design) while keeping the narrowed parameter evidence. At run time an absent point's coordinate is NaN (numeric-slot marker), an absent element access propagates Missing (mirroring At), and Distance of an absent point is NaN.

Improvements

• Wildcard dimensions no longer display as negative lengths: a list type with open (sentinel -1) dimensions and a non-numeric element printed as list<T^(-1x-1)>, which neither parses back nor means anything. The serializer now expresses open rank by nesting (list<list<T>>, list<list<T>^2> for a bounded outer dimension), which round-trips.

• Deadline cancellations during canonicalization name their budget: the error canonicalizing console line printed a bare Timeout exceeded, indistinguishable from an engine-imposed deadline (no such deadline exists — deadlines are only ever installed by an enclosing ce.withTimeLimit() span). The line now appends the expiring span's owner label and span chain (Tycho item 163).

• The 0.103.0 per-box slowdown is fixed (ROADMAP P-BOX): generic boxing had regressed ~1.25× in 0.103.0 through an interaction between the R-D5 ground-display cache — which runs on every symbol .type read and is keyed on BoxedType identity — and type inference, which rewrote an already-inferred symbol's type with a freshly allocated BoxedType on every use, defeating that cache twice per boxing call and tripling GC pressure. Three changes, each independently useful:

  • primitive types bypass the display-projection cache entirely (a primitive carries no callback<S>, so the projection is the identity);
  • a re-inference that lands on the type already recorded no longer writes — which also stops the spurious per-use _writeVersion bump on value definitions and the engine-wide _semanticVersion/_worldVersion bumps on operator definitions;
  • ce.type() now interns primitive type names per engine, so ce.type('number') is identity-stable.

  The box-microloop canary in benchmarks/effects-registration.ts returns to its 0.102.0 baseline (0.0091 vs 0.0105–0.0108 ms/iter regressed).

0.103.2 2026-08-10

Breaking Changes

• The prose ellipsis no longer binds inside an inline /. 1/2...5 is Range(1/2, 5), where it was Divide(1, Range(2, 5)). Division was the one operator that captured the ellipsis into its right operand: +, *, ^, the implicit product and \frac (a primary) all took the whole preceding element as the range anchor, so the SAME expression parsed two ways depending on how its division was spelled — [1+\frac{8}{d}...5] anchored on 1+8/d, while [1+8/d...5] anchored on d alone and produced 1 + 8/Range(d, 5). [1/2, 1/3...0] was [1/2, 1/Range(3, 0)].

  A PARENTHESIZED range still divides — 1/(2...5) is unchanged, and pinned. That form was the stated reason the ellipsis sat above the / right-operand floor, but parentheses are not reachable by precedence, so the constraint cost the anchor without buying anything.

• A two-sample range fuses over a SYMBOLIC first anchor. [1+\frac{4}{d}, 1+\frac{8}{d}...5] with d := 500 is a 500-element Range, where it was a 2-element List — the gate required the first anchor to be numerically KNOWN, so any anchor built from a document constant fell through. Nothing about the symbol's value is consulted (the parser cannot read one, and a value read there would be frozen into the document's parse): both anchors and the step are emitted as expressions, so re-assigning the constant re-evaluates the range.

  Two consequences for shapes that used to stay a placeholder List. A symbolic OFFSET is now a step — [m+n, m+n+x, ..., m+n+60] steps by x, matching the symbolic steps this pass has emitted over a numeric anchor since 0.100.0. And where the second anchor is the first with terms appended, the step is those terms rather than s1 - s0, so it reads x instead of m - m + n - n + x.

  What still stays a List: bare symbols ([x_1, x_2, ..., x_n]), a function application at ANY depth ([f(1), f(2), ...] and [1+f(1), 1+f(2), ...] — previously this fell out of the numeric reduction's recursion and is now a stated rule), and an anchor pair that is not one family, i.e. where the first anchor mentions a symbol the second does not ([m+n, m+k+15, ..., m+n+60], whose "step" k+15-n is fabricated from two unrelated bases).

Issues Resolved

• A declared signature's PARAMETER types now reach the function it is assigned to. Reconciliation ascribed a declaration's result onto the stored literal but dropped its parameters, so declare L : (list<real>) -> list<real> with L(a) := a + 1 stored a literal typing (unknown) -> list<real> and the parameter fell back to usage inference, which read the body as scalar arithmetic. The two halves of a compiled call then disagreed: the call site consulted the DECLARED type and passed the list whole, while the emitted body was scalar code, so L([3, 4]) ran [3,4] + 1 and returned the string "3,41" behind success: true — including under realOnly: true, which promises a number. L(a) := |a| degraded to NaN the same way, making the declared spelling worse than the undeclared one.

  Two knock-on gains: Length(a), Map(a, …) and Sum(Map(a, …)) over a declared list parameter now compile on the JavaScript target, where they previously declined on every target; and a declared list parameter no longer needs the vector<n> spelling to compile at all.

  Scope is deliberately narrow. Only NON-SCALAR parameters are ascribed — that is precisely the disagreement, since only a parameter that binds its argument whole can be handed a value scalar-compiled code cannot read. A scalar parameter is left alone (ascribing one re-canonicalizes the body against a narrower type and changed how a tuple argument broadcasts), as is broadcastable<T>, which is a declaration contract with its own enforcement, and any parameter the author annotated or whose type mentions a quantified variable. The multi-clause route is unchanged: a clause is checked as an arm of the declared signature, and stamping the general parameter types onto it would make that check vacuous.

• A function parameter passed to PointX/PointY/PointZ (or the .x/ .y/.z spelling) now infers as non-scalar, so a list argument is applied instead of broadcast. g(a) := PointX(a) answered [g(3), g(4)] for g([3,4]) — per-element nonsense that compiled to [null, null] behind success: true — where it now answers 3. Parameter evidence comes from the callee's declared parameter type, and these accessors declared (any) -> any, which contributes none; they now declare (collection | tuple) -> any. That is what they already accepted: a scalar or a string was rejected at run time before, so the rejection has only moved earlier, to canonicalization. Distance was narrowed the same way, and for the same reason, in 0.100.1.

  Norm is deliberately unchanged: Norm(-5) is 5, so a signature excluding scalars would be false and its parameter stays unlifted.

• A point accessor over a flat point compiles on the shader targets. PointX([3, 4]) declined with "a list of points has no GPU lowering" while PointX((3, 4)) compiled to vec2(3.0, 4.0).x — the GPU lowering read every indexed collection as a list of points, so the flat list<number> spelling a data import produces could not reach a shader. A numeric list of width 2–4 IS a vecN there, so it now swizzles exactly like the tuple spelling, and gains that spelling's static arity check (PointZ([1, 2]) is a typed error rather than a decline). A list too wide for a vecN lowers to an array and still declines, as does a genuine list of points, in either spelling.

• A point accessor over a flat list types the scalar it returns. PointX([3, 4]) is the coordinate 3 but typed vector<2>: the result type read broadcast-vs-index off the static type alone, while evaluation peeks the first element to decide, so the two disagreed on every indexed collection of scalars. Values were already correct — the JS target's runtime shape dispatch absorbed the mismatch — but it emitted a needless broadcast wrapper, and a non-indexed source (a Set of points) had the mirror-image defect, typing an element access while returning a list.

• An indexed Sum/Product no longer discards its indexing set when the body is a collection. \sum_{k=0}^{2}[k, 2] canonicalized to Reduce([k, 2], Add, 0) and answered k + 2 — the range gone and the bound index leaking out free. The collection-reduce rewrite is the meaning of the NO-INDEX form (Sum([1, 2, 3]) is still 6); with an indexing set the operator iterates and accumulates element-wise, so that expression is now [3, 6], matching the list-valued CALL spelling (\sum_{k=0}^{2}a(k)), which always took the index loop. Compiled and interpreted results agree.

• A Sum/Product body that cannot accumulate numerically now fails the compile instead of emitting a bare +. \sum_{i=0}^{2}\text{ab} compiled to ("ab") + ("ab") + ("ab") and ran to the string "ababab" behind success: true — including under realOnly: true, whose overload is typed to return a number. Add/Multiply reject such an operand at box time, but a big-op body stays raw, so nothing re-ran that check before the emitters saw it. Both lowerings (unrolled and looped), Product, and every target now decline. The decline needs positive evidence, so wide and unknown bodies are unaffected; boolean bodies keep compiling (the counting idiom \sum_k (x_k > 0), where + over booleans is numerically faithful), as do broadcastable<T> bodies, which are routinely scalar at run time.

• A Range step built over an assigned symbol no longer freezes its value. [2a, 3a...9a] with a := 2 canonicalized to Range(2a, 9a, 2) — the step folded to a literal while the bounds stayed symbolic. Re-assigning a := 3 then produced an 11-element range of spacing 2 instead of the 8-element one of spacing 3: the start moved with a and the step did not, so the result was silently wrong rather than merely stale. Range's canonical handler folds its step operand so a decimal progression stays exact (1.016 - 1.008 must become 0.008, not 0.008000000000000007); it now folds only when every symbol in the step is a CONSTANT. Pi/4 folds as before, and a step over a document variable stays unevaluated and re-reads its binding per use (Range has supported symbolic bounds and steps since 0.100.0).

• A nested Map that closes over the enclosing lambda's parameter now substitutes it. Min(Map([1,2], k ↦ Max(Map([1,3], j ↦ j·k)))) evaluated to Min(Max(k, 3k), Max(k, 3k)) — the inner map reduced, but k stayed free — where it is 3; with a +x in the outer body the result went non-finite instead of 3+x.

  The drain-time Map fusion bypasses makeLambda and evaluates the level itself, so it pushes a scope to resolve operands the lambda closed over. It pushed the mapping function's BODY scope; the chain those operands must resolve in starts one level OUT. Canonicalizing a nested literal auto-declares the free names of its body — including one the enclosing lambda binds — into that body scope, VALUELESS, so pushing it let the valueless shadow win over the binding that holds the value. The element then came back symbolic, and the outer application's binding-keyed substitution correctly declined to touch it, because it genuinely is a different binding. The drain now pushes the parent, read at drain time (a body scope is re-parented for the duration of each call, so a parent captured when the level was lowered would be a stale link into a frame that has since returned). The general route never had the bug: makeLambda pushes the call's fresh scope and reaches the closure chain through its parent, so a body scope is not in its lookup path for a non-parameter name either.

0.103.1 2026-08-10

Breaking Changes

• A point inner product is now typed as the sum it is, instead of a flat number. Dot((1, 2), (3, 4)) reports finite_integer — exactly what 1·3 + 2·4 reports — where it previously reported number for every provably numeric pair. The component types carry through: real components give finite_real, a complex one gives a complex result. The answer is taken from the arithmetic handlers rather than restated, so it cannot drift from the written-out form.

  The old claim was not merely loose. number includes complex, so a point inner product was rejected by every real-declared slot in the library: Hypot(Dot(p, p), Dot(q, q)) — real by construction — reported incompatible-type('real', 'number'), as did Degrees, Haversine and PrimePi over the same operand. Those compose now, and a genuinely complex inner product is still refused there, loudly.

  Nothing else about Dot moves: an operand whose components cannot be read (a symbol declared tuple<number, number> or vector<3>) keeps the wide number; a tuple that is not provably a fixed numeric point keeps value (the rule against claiming a numeric result on retractable evidence); unequal fixed lengths still report incompatible-dimensions; and the matrix product is still value. Update any pin asserting the literal string number on a fixed numeric Dot.

• A norm and a distance are now typed as the reals they are. Norm((3, 4)), Abs((3, 4)), Norm([3, 4]) and Distance((0, 0), (3, 4)) report finite_real where they reported number, and real when a component's finiteness is not provable. A norm is √(Σ|xᵢ|²), which is real whatever the components are — ‖(3+4i, 0)‖ is 5 — so the number claim (which admits complex) was refused by the same real-declared slots as the Dot claim above: Hypot(‖p‖, ‖q‖) did not typecheck.

  The demotions follow the convention Abs already uses for the same question one operand down: a provably NaN component gives number (only a literal proves NaN, so a merely-unknown component does not demote), and so does a provably non-finite one. No narrower tier than finite_real is claimed — unlike |·| of a scalar, a norm does not preserve the integer or rational tier, since ‖(1, 1)‖ is √2.

  Unchanged: a point whose component carries a collection still reports list<number> (it zips into one norm per element), a list of points still broadcasts to list<number>, Distance over an undecidable operand still reports number | list<number>, and an operand with no readable components (a symbol declared tuple<number, number>, a matrix, whose elements are rows) keeps the wide number.

Issues Resolved

• A point whose component is itself an inner product now compiles on the GPU targets. Dot((Dot((x, y, z), (1, 2, 3)), 0), (1, 2)) — a 2-D point product one of whose components is a 3-D one, the shape a gradient-noise field has — emitted correct GLSL and was then rejected by the shape gate that had just produced it: dot(vec2(dot(vec3(x, y, z), vec3(1.0, 2.0, 3.0)), 0.0), vec2(1.0, 2.0)) declined with "no room for the aggregate vec3 value standing in each slot". The gate scanned a vector constructor's argument text for an aggregate constructor anywhere in it, so it could not tell a vec3 standing in a slot from one consumed by an enclosing dot(). What stands in a slot is now judged after the scalar-reducing builtins (dot, length, distance, determinant, any, all — each returns a scalar whatever its argument shapes) are taken out of the source. The composition nests to any depth, works with the reduced component under arithmetic (vec2(dot(…) + 1.0, 0.0)), and holds on both GLSL and WGSL. An aggregate genuinely standing in a slot — Hypot over two point operands, emitting length(vec2(vec2(x, y), vec2(1.0, 2.0))) — still fails closed with the same diagnostic.

0.103.0 2026-08-09

Breaking Changes

• A collection operator whose source has no value now stays inert, instead of answering as if the source were empty. Filter and TakeWhile answered an empty collection, Find answered Nothing, IndexWhere 0, Any False and All True when handed a source they could not see into — a symbol declared but never assigned, an undeclared symbol, or an application of an unknown operator. Those answers were not conservative: with xs declared list<integer> and unassigned, Any(xs, x ↦ x > 2) answered False, and assigning xs := [1, 5] afterwards makes the same expression True. The walk that produced them cannot tell "this collection is empty" from "there was nothing here to walk". Every one of these now behaves the way Length, Total, Sort, Map, CountIf and Position always have on the same input, and answers normally as soon as the source has a value. A genuinely empty collection still gets the definite answer.

  This covers the source operand itself. A WRAPPER around a valueless source — Filter(Take(xs, 2), p), Any(Reverse(xs), p) — still answers as if the collection were empty, because the wrapper does have collection handlers while its own walk yields nothing. That case is unchanged by this release and tracked in ROADMAP.md; it needs an O(1) propagating enumerability facet on the collection handlers.

• A parameterless operand at a callback slot is now rejected across the whole collection family. Map(xs, 5) answered [5, 5, 5] and Any(xs, 5) carried a constant thunk, while the identical Sort(xs, 5) and CountIf(xs, 5) reported incompatible-type function/finite_integer. The split was an artifact: the lazy operators routed their operand through the shorthand path, which LIFTS a value with no wildcard and no free unknown into the constant () ↦ 5, while the eager ones validated against the declared slot. All of them now report the declared slot's error — for a plain value, a string, and a symbol whose declared type is provably not a function (Map(xs, k) with k := 5). What a function slot admits is unchanged: named operators, function-typed symbols, wildcard shorthands (Map(xs, _)), free-unknown shorthands (Map(xs, q + 1)), an explicitly written nullary literal (["Function", 42]), and any symbol whose type is not yet known — the forward reference — all still pass. Apply(3, 5) is unaffected: it is not a callback slot, and applying a constant is its documented shorthand.

• The Cortex language has been renamed Epsil. The experimental scripting language previously called Cortex is now Epsil, and every public surface follows: the CLI binary is epsil (was cortex), the conventional source file extension is .epsil (was .cortex / .cx), the package subpath is @cortex-js/compute-engine/epsil (was …/cortex), and the API entry points are parseEpsil(), serializeEpsil() and executeEpsil() (with the corresponding ExecuteEpsilOptions/ExecuteEpsilResult types). The npm package name and @cortex-js scope are unchanged. There are no compatibility aliases: update imports and scripts to the new names. A ./cli package export exposes the CLI entry point so the standalone epsil launcher package (and other tools) can forward to it.

• The structural: true boolean is no longer part of ce.function()'s typed signature — use { form: 'structural' }. The form option has been the documented spelling for the creation modes since the structural tier was introduced, and ce.expr(), ce.box() and ce.parse() had already dropped the boolean from their public types; ce.function() was the last one still carrying it, which made the surface inconsistent and advertised a spelling the guide tells you not to use. The boolean is still accepted at runtime — optionsToInternal() maps { structural: true } and { form: 'structural' } to the identical internal form — so only code type-checked against ComputeEngine/IComputeEngine is affected, and the fix is a rename at the call site. { canonical: false } (equivalently { form: 'raw' }) is in the same position and was already untyped.

• Cortex: = is now positional — it assigns only as a whole statement, and compares everywhere else. := always assigns and == always compares; a bare = means Assign when it is the top-level operator of a statement whose left side is a binding target (a name, or a field/index path rooted at one), and Equal in every other position. The canonical trap simply works: Solve(x^2 = 4, x) is the equation and returns [2, -2], where it used to assign and report no solutions. So do if a = true { … }, while x = 5 { } and [a = 1], each of which previously assigned silently — the C footgun no longer exists in Cortex.

  The decision is purely syntactic: it never depends on evaluating a value, on scope, or on which definitions are installed. As a comparison = binds at the relational tier, so if x = 5 && y groups as (x = 5) && y; as an assignment it binds loosest and takes the whole right-hand side.

  Two consequences. A statement whose left side is not a binding target now compares — x^2 = 4 on its own line is the equation, not an assignment to a power. And a = b = 5 would assign a the boolean b == 5, which is never what a chained assignment means, so it reports chained-assignment; write a := b := 5 to chain, or a = (b = 5) if the comparison was meant.

  := is unconditional, so it still reaches a condition where a bare = no longer can. if flag := true { … } assigns and then uses the assigned value as the test — with no type error to catch it — so it now reports assign-in-condition, a warning (the spelling is deliberate). It fires only where a value is consumed as a boolean, not for f(a := 1).

  The assign-in-argument diagnostic is removed: f(x = 4) is now an ordinary comparison, so there is nothing left to diagnose. Serialized output always uses the explicit := and ==, never a bare =, so a round-trip is exact regardless of position — which means the formatter rewrites an authored = to := where it assigns.

New Features

• Dot accepts numeric Tuple/PointList operands. Dot((1, 2), (3, 4)) — and the equivalent PointList spelling — is now valid, typed number, and evaluates to the inner product, matching the already-supported List vectors; mixed tuple · list products work too. Dot is the sanctioned representation of a point inner product (tuple · tuple has no implicit product and remains an error), so the explicit operator now accepts the operands the implicit one rejects. Point operands lower to the native dot() builtin on the GLSL/WGSL compile targets (with the usual operand-shape fail-closed guards: matching widths, vec2–vec4 only) and to np.dot on Python. Unequal fixed lengths report incompatible-dimensions; a tuple operand that is not provably a fixed numeric point (a symbolic point, a point list with a collection component) stays symbolic.

• Inline callback lambdas now infer their parameter type from the call site. An unannotated function literal passed directly as an argument is re-derived with its parameter typed, exactly as if you had annotated it by hand, in two situations: when the callee — user-defined functions included — declares a concrete function-typed parameter (function apply2(f: (number) -> number, x) { f(x) } types the n of apply2(n |-> n + 1, 3) as number), and when the callback of Map or Filter is applied to a collection whose element type is a provable composite (a tuple or a nested collection). The headline win: point-list predicates now compile without annotation — Filter(points, pt |-> pt == (0, 0)) with points: list<tuple<number, number>> lowers to the same element-wise code as the hand-annotated spelling, and the literal's signature reflects the element type everywhere it is read.

  The inference is per-application and behaves exactly like a hand-written annotation, loud type errors included. It never touches a shared callback (a lambda bound to a name keeps its own typing at every call site), never overrides an explicit annotation, and does not fire for union element types of Map/Filter — heterogeneous "errors are values" programs keep their per-element behavior, and the vectorization default for evidence-free lambdas is unchanged. Optional and variadic callback positions are covered. The whole mechanism is driven by the callee's declared signature, and a generic function you declare yourself gets the same inference — provided the callback slot is spelled callback<(T) -> …> rather than as a plain generic arrow (T) -> …. The callback<…> spelling is what asks for the inference, and it is also the permissive one: the slot still admits any function, exactly as a bare function parameter does, and a callback that does not match is applied and judged per element as before. A plain generic arrow does neither — it declines the inference and rejects a callback whose declared type does not fit the slot. A callback that combines several collections at once (Map(xs, ys, f)) is admitted and applied dynamically, as it always was, rather than annotated.

• Annotated lambda parameters no longer disable the Map fast paths. The Map-fusion and exact-compile gates previously required bare (unannotated) parameters, so the hand-annotated spelling of a mapping function paid an interpretation penalty. The gates now accept an annotated parameter whenever the source collection's element type provably satisfies the annotation — the per-element type check is a no-op in that case, so the fused drain is unobservable — and still fall back to the enforcing path (loud error included) when the annotation is narrower than the elements or the source type is unknown. With this, the element-type inference above extends to scalar element types too: Map(Range(1, 200), x |-> Mod(x, 7)) infers x: integer, keeps fusion, and still compiles through the exact tier. A fused result is also re-validated when an inferred source type changes, so a collection that retracts to a wider element type raises the annotation error instead of reusing a stale fused pipeline.

  One consequence of the annotation contract to be aware of: an expression you retain (a lazy Map/Filter you hold and re-evaluate) keeps the parameter type it inferred when it was created. If the source collection is later reassigned to elements of a wider type (an inferred list<integer> becomes floats), re-evaluating the retained expression reports a per-element incompatible-type error — exactly as the hand-annotated spelling would — rather than silently adapting. Boxing the expression afresh after the reassignment infers the new element type and evaluates normally.

• JavaScript target: string equality now compiles. Scalar Equal/NotEqual with a provably string participant — and no provably numeric one — lowers to a strict ===/!==, the interpreter's own string semantics, instead of failing closed. All-string collection equality keeps the _SYS.eq/_SYS.neq dispatch, whose scalar leaf gained a string branch, so Equal(["a","b"], ["a","b"]) is True and Equal(["a","b"], "a") is the element-wise [True, False]. A mixed provably-string/provably-number equality and the chained (n-ary) form still fail closed. The Python target mirrors the scalar rule with a structural ==/!=. With this, character-scanner programs (skipWs(cs, i) = skipWs(cs, i+1) if i <= Length(cs) && isWs(cs[i]) else i over Characters(text)) compile end to end.

• JavaScript target: Characters, GraphemeClusters and StringJoin now compile. Characters segments UAX #29 grapheme clusters through the same Intl.Segmenter the interpreter uses, so a combining sequence, a ZWJ emoji or a flag is one element — matching interpretation element for element (neither [...s] nor split('') is faithful; both are pinned as counter-examples). StringJoin compiles the variadic all-string form and the single string-collection form; shapes the interpreter leaves inert fail closed. Known accepted divergence: compiled comparisons do not Unicode-normalize a raw non-NFC string bound to a compiled parameter (the interpreter stores every string in NFC; literals and Characters/StringJoin results are normalized) — normalize at the host boundary if you feed unnormalized user input into compiled predicates.

• broadcastable<T> as a parameter declaration is now an elementwise contract. A parameter declared broadcastable<T> (e.g. ce.declare('f', '(broadcastable<number>) -> unknown'), or an Epsil function f(x: broadcastable<number>) { … }) maps an indexed-collection argument element-wise — even when T would admit the collection whole (broadcastable<value>: the collection arm wins) — and the map descends exactly one rank: each element binds to the parameter whole, even a nested collection, unlike the unannotated default, which descends to the scalar leaves. T is checked per element (a violating element produces a loud per-element error carrying the broadcast context, and its siblings still evaluate), and a scalar argument binds directly with no list wrapper. Applications now also type like the inferred path — a definite collection argument gives list<R>, a possibly-collection argument broadcastable<R>, a scalar R — where the declared spelling previously typed the strictly weaker bare declared result (an explicitly more precise declaration yielded strictly less type information). A collection- or tuple-declared sibling slot still binds its argument whole, and the strict length-mismatch policy applies across the mapped slots only. The unannotated vectorization default is unchanged. Design record: docs/plans/2026-08-08-broadcastable-param-semantics.md.

• Compiling an application of a function with a declared broadcastable<T> parameter over a possibly-collection argument fails closed with a diagnostic (interpreted evaluation handles it) instead of emitting scalar code that returned garbage on arrays — the compiled broadcast helper recurses into nested arrays, which is exactly the leaf descent the one-rank contract rules out. The decline is per slot: an argument at a slot the declaration binds whole, an atomic tuple the element type admits, and a provably scalar argument all still compile. A multi-clause function with one broadcastable arm declines conservatively for the whole set.

• Epsil debugging support in the engine. Two additions that let a debugger (such as the VS Code Epsil extension's DAP adapter) pause and inspect Epsil programs at statement granularity:

  • Source positions survive canonicalization. The sourceOffsets metadata the Epsil parser attaches to every node is now preserved through canonical boxing — custom canonical-handler results and the numeric fast-path constructors re-attach it, the .canonical getters (function and symbol) thread it, closure capture keeps it on rebuilt bodies, and the recursion knot-tying re-box serializes it. Positions are advisory metadata: default JSON serialization does not emit them (opt-in via metadata: ['sourceOffsets']), and interned singletons are never stamped.
  • Debug statement hooks. src/common/debug-hook.ts exposes a synchronous, module-global pre-statement hook and post-statement result hook, fired by the statement sequencer (Block bodies, lambda bodies, if branches) for source-mapped statements only. One comparison per statement when unset; not part of the public engine API.
  • Function-application scopes are now pushed with the context name 'call' (previously an anonymous placeholder name), so ce.trace and debuggers can delimit activation frames.

• Destructuring assignment — (a, b) := (b, a). A tuple pattern may now appear on the left of a Cortex assignment, writing bindings that already exist instead of declaring new ones. The pattern grammar is the destructuring let's — at least two elements, each a bare symbol, a _ skipping that position, or a nested tuple pattern — and a shape mismatch is the same incompatible-type error value.

  The right-hand side is evaluated once, in full, before any target is written, which is what makes a swap mean what it reads: (a, b) := (b, a) exchanges the two values rather than assigning b to both. The same holds for a rotation ((a, b, c) := (c, a, b)) and for the pair-carrying loop step that is the usual reason to want this — (a, b) := (b, a + b) is an entire Fibonacci iteration, and (a, b) := (b, a % b) an entire Euclid step, with no temporary.

  Unlike a destructuring let, the targets keep their identity and their declared type: a value that does not fit a target's type is an error value, and assigning to a const fails. Those two are found only by attempting the write, so they are not atomic — targets earlier in the pattern stay written. A shape mismatch is atomic and writes nothing, including when it is nested under a position that would have bound; the destructuring let gained the same guarantee, which it did not previously have. Lowers to the Assign primitive with a Tuple pattern in the target position, held raw (canonicalizing it would fold a single-letter target such as i into the constant of that name), and accepted on all routes.

  In compiled code it lowers to per-leaf temporaries followed by per-leaf writes — (a, b) := (b, a + b) becomes let _tv1 = b; let _tv2 = a + b; a = _tv1; b = _tv2 — which is what keeps the compiled form honest: the targets already exist, so the naive a = b; b = a would read the a it just clobbered. Temporaries never capture a name the program already uses. Every target — JavaScript, Python, GLSL and WGSL — compiles it in any statement position, including a loop body, so the Fibonacci and Euclid steps above compile. Value position (a block's last statement, whose value is the block's) and a non-literal tuple value fail closed (D6) and the interpreter takes over; a destructuring let in value position is now refused the same way, explicitly, rather than by emitting source that happens not to parse. (This also fixes a silent divergence in the same family as the destructuring-declare one: a tuple target previously compiled as _ = …, leaving every target at its old value behind success: true.)

  ce.box(['Assign', ['Tuple', 'a', 'b'], ['Tuple', 'b', 'a']]).evaluate();

• Cortex diagnoses a tuple pattern written with a bare =. A parenthesized left side is not a binding target, so (a, b) = (b, a) resolves — correctly, under the positional-= rule — to a comparison of two tuples whose result is discarded: the swap it looks like silently does nothing. That shape is almost always a typo for the destructuring assignment above, so it now reports destructuring-bare-equal; write (a, b) := (b, a) to destructure, or == if the comparison was meant. The node is unchanged — the diagnostic reports, it does not reinterpret.

  The check is deliberately narrow: it fires only statement-leading, and only when the left side is shaped exactly like a destructuring pattern (bare names, _, nested tuples), so a genuine tuple equation with computed components — (x + 1, y) = t — stays silent.

• Parameterized nominal types: type tree<T> = tuple<value: T, children: list<tree<T>>>. A nominal type declaration now takes the same type-parameter clause a generic alias takes, in Cortex as above and from the host with ce.declareType('tree', '…', { typeParams: [{ name: 'T', variance: 'out' }] }). Unlike an alias, an application is opaque — tree<integer> is never expanded — which is precisely what lets the definition refer to itself, so a recursive parametric container (a rose tree, a JSON with a payload, a zipper) is expressible for the first time. The arity, bound and unused-parameter rules are the alias's, shared and generalized; self-reference, which an alias forbids, is the point here.

  A parameter carries a variance marker — out (covariant), in (contravariant) or inout (invariant) — saying how two applications relate: under out, a tree<integer> is usable where a tree<number> is expected. No marker means out, declared rather than inferred and verified against the definition like a written one: values are immutable, so covariance is sound and is what a payload container wants, and only the consuming minority pays an annotation. Because the default is declared, a definition that uses its parameter in an input position does not quietly change the type's subtyping contract — it is a variance-violation at the declaration, naming the violated variance and where it came from, the offending occurrences by path (notify.(arg 1)), and exactly the markers that would verify. inout verifies against any definition. Variance and bounds do not interact.

  A tuple definition mints a quantified constructor (tree: forall T. (T, list<tree<T>>) -> tree<T>), so tree(1, []) solves T = finite_integer from its arguments; a record definition is still inhabited by a constructor function, whose own clause is independent of the type's. Field access reads the definition instantiated at the application's arguments — with t: tree<number>, t.value is a number. match is a binding of values, not a projection of the annotation: each capture takes the matched value's own type, usually narrower — on a t built as tree(1, []), match t { tree(v, cs) => … } binds v: integer and cs: list<never>, not number and list<tree<number>>. Compilation erases the tag at the instantiated definition, as it already did for an unparameterized nominal type: tree<integer> compiles like the equivalent tuple, and declines identically where that would. One documented limitation: a construction solves its parameters from its arguments alone and an annotation does not widen them, so an explicitly inout/in parameterized type can only be constructed at exactly its argument type. See the new "Parameterized Nominal Types" section of the types guide.

• A type variable may now appear in one arm of a union: type opt<T> = T | missing and forall T. (T | missing) -> list<T> are accepted, and at a call the argument takes exactly one arm — the open arm binds the variable (refutation included), a ground arm binds never, the narrowest member of the family. At most one arm of a union may mention a variable (T | U is unsolvable by construction). Intersections and negations remain rejected wherever the declaration mints a constructor — the minted signature is what is checked, so a record body or a mint: false declaration goes unchecked — and the intersection diagnostic now steers to the spelling that replaces it, a bound (forall T: number.).

• A Cortex type statement re-declaration (a notebook re-run, or an edited definition) now UPDATES the existing type record in place instead of installing a new one. Types that mention the name — and applied references such as box<integer> already built — follow the new definition, so a node parsed before the re-run and one parsed after can no longer give different subtyping answers for the same pair of types, and a mutually recursive set converges on the second run rather than the third. A re-declaration that breaks a type depending on it now fails on the run that introduces it, rather than silently leaving that type reading a stale definition: an edit that changes the type-parameter count while a dependent still applies the old arity is a generic-alias-arity error, and one that makes a dependent's declared variance unsound is a variance-violation. Both are attributed to the dependent and name the re-declaration as the trigger, and both roll the statement back completely — definition, type-parameter clause, verified variance and minted constructor all restored. Re-declaring a type through the host ce.declareType() API still throws, unchanged.

• Cortex: break and continue. They leave, or skip to the next iteration of, the innermost enclosing while/for loop, and lower to the engine's existing Break()/Continue() primitives. Valid anywhere in a loop body, including inside an if, a match case, or a do block. The loop context resets at every function and lambda boundary — a break written inside a lambda defined in a loop body does not target that loop, and is a control-outside-loop diagnostic — because the engine's Block short-circuits on Break/Continue structurally, so a laxer rule would permit non-local control flow. Value-carrying break value remains unspelled, pending the ruling on a general return.

• Cortex: ?? for absence coalescing. a ?? b is Coalesce(a, b): the value of a unless it is absent (Missing or NaN). It discharges absence; it does not rescue an Error. Right-associative, at precedence 18 — looser than |> so xs |> f ?? 0 defaults the pipeline's result, tighter than |-> so x |-> x.a ?? 0 defaults inside the body.

• Cortex: is for dynamic type tests. x is integer lowers to Element(x, integer) — the same test a match type pattern performs. The right operand is a type name, so a typo (x is intger) is a parse-time diagnostic rather than a comparison against an undeclared symbol. Simple named types only for now: a compound type (!error, integer | string, list<integer>) reports type-pattern-unsupported, as the equivalent typed pattern already does. is is a contextual word — let is = 5 stays legal.

Improvements

• Signature(op) works on the ce.box and ce.parse routes. The operator holds its operand and had no canonical handler, so the name arrived unbound and Signature answered Nothing for every operator unless the caller went through ce.function(), which boxes its arguments first. The name is now resolved by lookup — which also keeps the route read-only, where canonicalizing the operand would have DECLARED an unknown name as a side effect of asking about it.

• A malformed predicate is reported by the operator that consumed it. CountIf, Find, IndexWhere and Position all threw an error whose message named Filter, an operator the user never wrote — Filter's message had been copied verbatim into each of them.

• The signature-driven callback stamp declines a wrong-arity literal instead of half-annotating it. A user signature declaring a concrete arrow parameter ((cb: (integer) -> boolean) -> boolean) annotates an inline literal passed there by pairing parameters positionally; given (a, b) ↦ a > b it annotated a and left b bare. The whole stamp now declines, as the contextual-callback<S> route already did. Evaluation is unchanged either way — the arity error dominates — but a declined application no longer carries a half-written contract. Optional and variadic declared parameters widen the admissible arity accordingly.

• src/math-json/OPERATORS.json regenerated, picking up the contextual callback<S> signature strings, the variadic Append, and three operators missing from it entirely (Adjoin, IdenticallyEqual, QuotientRing). Two defects in the generator are fixed along the way: a forall-quantified signature reported its arity as "unknown", and the arity count split on every comma — including those inside a parameter's own type, which made Fold 4-ary.

• Element-wise (broadcast) failures are now self-diagnosing. A user function with scalar parameters is automatically applied element-wise over an indexed-collection argument; when that fired unexpectedly, the resulting per-element failures gave no hint a broadcast was in flight — the motivating symptom read Condition must evaluate to "True" or "False" with nothing tying it to a mapping. An error produced inside a broadcast element now carries an ["ErrorBroadcast", name, index, length] entry in its ErrorTrace breadcrumb, rendered as (while applying 'skipWs' element-wise over 4 elements (element 2)); an error thrown out of an element (a non-boolean if condition, say) has the same context appended to its message. Every failing element is annotated with its own index. Errors that never went through a broadcast are byte-identical to before, successful broadcasts pay nothing, and Epsil runtime-error diagnostics report the context too.

• The Epsil debugger's variables pane marks broadcast behavior on function signatures: (n: number) -> number [elementwise] versus (xs: list<number>) -> integer [binds whole], so the derived vectorization behavior is visible before anything runs. A function-typed (callback) parameter does not suppress the marker, matching the eligibility rule.

• Membership in a value collection now types the tested function parameter. Element(c, digits) — Epsil c in digits — inside a function body narrows a not-yet-typed parameter to the collection's element type (digits: list<string> ⇒ c: string), the membership counterpart of the collection evidence Length(cs) and cs[i] already contribute. The evidence lands on the parameter's binding only: the function's arrow still reports a scalar parameter slot as unknown, so the lambda auto-broadcast default is unchanged (isDigit(["5", "x"]) still maps elementwise). Two deliberate exclusions: a global symbol is never retyped — membership is a predicate (x in [1, 2, 3] on a string-valued x is False, not a type error), and a Solve domain spec such as Element(x, Range(1, 9)) constrains its unknown without narrowing it — and membership in a set (x ∈ ℤ, x ∈ {1, 2, 3}) stays with the assume machinery, which applies such refinements scoped.

• Epsil debugger: function signatures in the Variables panel show inferred parameter evidence. The engine's arrow deliberately hides evidence that does not rule out broadcasting, so a function like skipWs(cs, i) = … cs[i] … displayed as (dictionary | indexed_collection, unknown) -> …. The debugger now reads the parameter bindings instead and shows names alongside everything inference recorded: (cs: dictionary | indexed_collection, i: boolean | indexed_collection | number | string) -> …, and isDigit shows (c: string) -> boolean. Display-only; the engine's types are untouched.

• Cortex: most reserved words are now ordinary identifiers. Only the words the grammar actually consumes are reserved: the literals (true, false, Infinity, oo, NaN) and the active keywords and word operators (break, const, continue, do, else, for, function, if, in, match, while). The other 76 words in the documented list — set, with, label, where, to, each, and so on — can name a binding, be assigned to, be a |-> parameter, and be called. Previously a binding name accepted them but a bare assignment target, a mapsto parameter, and a call's callee did not, so label(6) = 1 was accepted while label(6) was an error.

  Relatedly, assigning to a literal word is no longer silently accepted: true = 5 and NaN = 1 now report reserved-word like every other binding position.

Resolved Issues

• tuple · tuple and x / tuple now reject consistently, regardless of how precisely the element types are known. The canonicalization guards only counted provably numeric tuples, so a product (or a division) with a tuple<broadcastable<number>, …> operand was accepted (typed number) and the identical incompatible-type rejection surfaced only at evaluation — making validity appear to depend on the order in which operand types were refined (a more precise type turned an "accepted" expression into an error). Both guards now count operands by tuple-ness, exactly like the evaluation path: a tuple product with another tuple, or a tuple divisor, never becomes valid under any element refinement. The LaTeX rendering of these rejections now suggests Dot — the operator that does accept points — instead of only reporting that a tuple is not a number. (scalar + tuple with unproven elements deliberately stays symbolic with an honest union type: per the retractable-evidence ruling, Add bakes its error only when both the tuple and the scalar are proven.)

• A seedless Reduce now seeds with the collection's first element. Without an initial value, Reduce folded from the Nothing sentinel, which only looked right for a reducer that splices it away: Reduce([1, 2, 3], (a, b) |-> a - b) answered -6 — ((nothing - 1) - 2) - 3 — where the corresponding Scan ends at -4, and a reducer that does not splice leaked the sentinel into the result (Reduce([2, 3, 2], Power) produced a Nothing-valued power instead of 64). The seedless fold now starts at the first element and folds from the second, matching Scan and the compiled fast path; an empty seedless fold still answers Nothing. Folds given an explicit initial value are unchanged.

• Any and All now surface an element's type error instead of silently discarding it. When a predicate fails on an element — for instance a callback whose inferred parameter type a retracted element no longer satisfies — the quantifiers previously treated the error as "undetermined" and returned nothing, and Any could even short-circuit True straight past the failing element. They now return the element's error, in enumeration order: a definite answer found before the failing element still short-circuits (matching the family's laziness), and an error encountered before any decision is the result.

• Sum, Reduce, Total and other finite-only operations now work over FlatMap. FlatMap never reported whether its result was finite, so every consumer that requires a finite collection stayed inert over it. It now reports finite when its source is finite and the mapping function's result is provably finite (a scalar, or a finite collection); anything unprovable is left undetermined, as before.

• A partially annotated function literal no longer rejects arguments at its unannotated parameters. Annotating one parameter of a literal activated per-application validation for every parameter, and a bare (unannotated) parameter was checked against its inference residue — which, among other things, made a seedless Reduce with an annotated element parameter reject its own Nothing seed with incompatible-type unknown nothing. A bare parameter now imposes no constraint; explicitly annotated parameters are enforced exactly as before.

• Membership queries on derived collections no longer answer a definite False when the underlying membership is undecidable. Nine collection contains handlers (Filter, Join, Append, Reverse, RotateLeft, RotateRight, Cycle, CartesianProduct, PowerSet) collapsed an undecided sub-query into false — so Element((1, x), {1, 2} × ys) with ys a symbolic set evaluated to "False" instead of staying symbolic. All nine are now three-valued: a definite refutation still answers False immediately, and only a genuinely undecidable membership is left undecided. In the same pass, a Filter whose predicate returns a non-boolean now reports the malformed predicate from a membership query exactly as it does from iteration, counting, and emptiness — it was the one silent facet.

• An element that fails a callback's declared parameter type is now loud everywhere. Previously it could surface as a nonsensical Filter predicate must return "True" or "False". Unknown symbol … message (which spell-checked the lambda's own parameter), be silently swallowed by TakeWhile/DropWhile (an Error treated as an ordinary "stop"), or be laundered into NaN by a fused Map pipeline doing arithmetic on the Error value. All three paths now surface the element's own incompatible-type error: stream operators emit it in the element's place, scalar operators (CountIf, Find, Position, IndexWhere, Partition) return it as their result, and fused pipelines propagate it exactly like the unfused evaluator.

• Compiled collection callbacks no longer ignore parameter type annotations. On both the JavaScript and Python targets, a callback such as (n: integer) |-> n > 0 compiled to a plain function with the annotation dropped, so compiled Filter/Map/TakeWhile/… could silently produce different values than the interpreter on elements that violate the annotation. Compilation now admits an annotated callback only when the collection's element type provably satisfies the annotation — the same rule the evaluator's fast paths use — and declines to compile otherwise, so the interpreter's per-element errors are preserved. Applies to inline and named callbacks across all collection operators, including Reduce/Scan and Tabulate/Fill.

• An ordering such as j <= Length(cs) no longer declines to compile when the same local is also used as an index (cs[j]). At's index slot is typed boolean | indexed_collection | number | string (a gather index may be a collection, a dictionary key a string), which permanently widened the local, and the string arm of that union was read as positive string evidence by the mixed-string ordering gate. A type that admits a string and a number and a boolean and an indexed collection — that exact four-armed shape — says no more than the top type does, and is no longer treated as evidence, so ordinary scanner loops (while j <= Length(cs) && isWs(cs[j]) { … }) compile again, on both the JavaScript and Python targets. A deliberately written union of scalar sorts (string | number | boolean) is still evidence, and genuinely mixed shapes (Less("a", 1), a number | string participant, Less("a", [1, 2])) still fail closed.

• An else-less if statement in a function body compiles (JavaScript target). if cs[j] == "-" { sign = -1; j = j + 1 } in statement position lowers to a bare if (…) { … } instead of failing with If: wrong number of arguments. Loop bodies already handled this; a plain block's statements did not. An else-less if in value position still fails closed — it has no value. The Python, interval-JavaScript and GPU targets still decline the shape in a plain function body.

• Iterating over a function parameter now counts as collection evidence. A parameter whose only use was for c in cs stayed untyped, so the lambda auto-broadcast mapped the function over the argument's elements instead of binding the collection whole: a function that accumulated t = t + c over for c in cs and was applied to [1, 2, 3] returned [0, 0, 0] instead of 6, and its inferred signature read (unknown) -> …. The iterated operand of a Loop/Comprehension Element clause now narrows an as-yet-untyped symbol to collection, matching what Length(cs) and cs[i] already did. Scalar parameters are unaffected — f(x) = x * 2 still broadcasts over a list.

• A function forward-declared with the bare function wildcard (ce.declare('clean', 'function')) now participates in parameter collection-inference. Previously a caller such as proc(cs) = clean(cs) + 1 learned nothing from clean in either definition order: the wildcard declaration carries no parameter types, and assigning a function literal deliberately leaves the declared type alone (so re-assignment at a different arity stays legal), leaving no signature to narrow from. The application site now reads the assigned literal's own signature — for argument narrowing and for the application's result type, which used to be broadcast-typed (list<unknown^3>) where the value was a scalar — and a caller canonicalized while the callee is still a bare wildcard registers for re-derivation, so the later assignment repairs it: proc([10, 20, 30]) answers 11 instead of broadcasting into [proc(10), proc(20), proc(30)]. The wildcard remains a widening: re-assignment at a different arity or parameter type still works, and applications are still not validated against the assigned signature.

• Compiled string comparisons fail closed instead of returning wrong values. The JavaScript compile target is numeric at heart: Equal and NotEqual lower to a tolerance test (Math.abs(a - b) <= tol), which for string operands is NaN <= tol — so s == "a" compiled to false behind success: true, and IndexOf over a list of strings returned 0 for the same reason. Both now fail closed (D6) when an operand is provably a string (a string literal, or statically string-typed — an unknown-typed symbol never gates, so inferred-parameter plot equalities compile byte-identically), and the interpreter fallback returns the correct value. Orderings (Less, Greater, …) are gated more narrowly, on the mixed case only ("a" < 1 — inert in the interpreter, false compiled): an all-string comparison compares strings exactly as the interpreter does (raw code-unit order) and keeps compiling, with parity pinned. match on string constants was never affected — it emits a real === — and is now pinned too. The Python target needed no gate: its wrong shapes raise a loud runtime TypeError rather than a silent value, and its IndexOf is genuinely correct.

• The broadcast route no longer miscompiles string comparisons, and whole-array string equality fails closed. Two stragglers of the string-comparison class above reached the emitter through different doors: a mixed ordering over a collection (Less("a", [1, 2])) broadcast to [false, false] via _SYS.bcast before any gate ran, and whole-array equality (Equal(["a","b"], ["a","b"])) compiled to false because _SYS.eq's per-element tolerance test makes equal strings unequal — the gate tested operands, and neither operand is a string scalar. The string gates are now element-aware ("participants": scalar operands and the provable element types of collection operands), and the string-evidence test is recursive: nested string lists ([["a"]] == [["a"]]), heterogeneous literals (["a", 1] == ["a", 1]), a symbol typed broadcastable<string> or list<string> in an ordering, and — via the same walk — whole-value equality over dictionary/record/tuple-typed symbols (whose element types reach string) all previously compiled to wrong booleans and now fail closed. Admission is deliberately narrower than decline: only flat all-string orderings keep compiling (their interpreter parity is pinned); nested all-string shapes decline. Numeric shapes keep byte-identical codegen. (The Python target has the same broadcast-route defect — np.less("a", [1, 2]) — recorded as a known open hole, not yet fixed.)

• The Python compile target now fails closed on the string and aggregate comparisons it miscompiled. Two shapes returned wrong values behind success: true: an ordering that mixes strings with numbers, where NumPy coerces the number to a string and compares as text (Less(["a", 10], ["b", 9]) ran to [True, True], while the interpreter answers ["True", "False"] — 10 < 9 is False), and equality with a tuple participant, where the tuple is looked inside instead of binding atomically (Equal(Tuple(1, 2), List(1, 2)) ran to True against the interpreter's False). Both now decline (D6) and the interpreter fallback answers correctly, as do dictionary/record comparisons (no positional lowering at all) and scalar string equality (abs("a" - "a") raises). Gating is at COMPILE time on static type evidence, not on the emitted code raising: several mixed shapes are loud on NumPy 2.x but historically returned a scalar False with a FutureWarning, and emitted code runs on the user's NumPy. As on the JavaScript target, an unknown-typed operand is never string evidence, so numeric and plot comparisons compile byte-identically. What keeps compiling, each with executed-parity pins: all-string orderings (scalar, chained, list-vs-list, list-vs-scalar), tuple-vs-tuple equality, and — unlike the JavaScript target, whose kernels are numeric there — IndexOf in every shape (including a tuple needle in a point list) and all-string collection equality, both of which lower to Python's own structural comparison and are faithful.

• Compiled IndexOf's element test is now EXACT and bool-aware, like the interpreter. IndexOf([1, 2], True) ran to 1 on both the JavaScript and Python targets, where the interpreter's structural element test answers 0 (and symmetrically, IndexOf([True], 1) ran to 1): Python's True == 1 is True, and Math.abs(true - 1) is 0. Both targets now compare bool-ness first — Python through a new _ce_indexof/_ce_same runtime adapter, JavaScript through a strict === in the emitted element test. This is a runtime adapter, not a compile-time gate, so boolean IndexOf keeps compiling instead of declining. Both element tests also dropped their numeric TOLERANCE leaf: the interpreter's number .isSame() is exact, so IndexOf([0], 5e-11) and IndexOf([0.30000000000000004], 0.3) answer 0, where the compiled forms answered 1. (The tolerance leaf came from a probe artifact — IndexOf([0.3], Add(0.1, 0.2)) agrees only because Add(0.1, 0.2) evaluates to exactly 0.3 by exact decimal folding, so the element test never saw a near-miss float. The residual: a needle computed to a near-miss f64 at runtime is not found, which is the ordinary exactness loss of compiling to f64 arithmetic.) The adapters also FIND a NaN needle, as the interpreter's structural element test does: IndexOf([NaN, 3], NaN) ran to 0 on both targets (nan == nan is False in Python, and Math.abs(NaN - NaN) <= tol is false) and now answers 1. Otherwise unchanged: numbers still match across int/float, a tuple needle is still found in a point list, and a missing needle is still 0. On the Python target an np.ndarray on either side is now normalized to a nested list, so a caller-supplied (n, 2) point matrix works instead of raising an ambiguous-truth-value error.

• Compiled Python orderings over two collections now reject a length mismatch. The interpreter answers Error("incompatible-dimensions", …) for Less([1, 2, 3], [1, 2]) and for Less([1], [1, 2]). NumPy only half agreed: it raises on 3-vs-2, but silently BROADCASTS 1-vs-n (np.less([1], [1, 2]) → [False, True]) — a wrong answer behind a success: true. The ordering ufuncs are now emitted through a _ce_ord runtime shape guard that raises on any list-like-vs-list-like length mismatch, the 1-vs-n case included. Scalar-vs-collection broadcasting is unaffected.

• compileLambda (Python target) no longer emits references to undefined runtime helpers. It guarded only _ce_bcast; an IndexOf body, a collection/tuple Equal body, or a collection ordering produced a lambda referencing _ce_indexof/_ce_eqcoll/_ce_ord, which a bare lambda has no place to define — a NameError at call time. All four now fail closed with a message pointing at compileFunction.

• A destructuring Declare with a positional initial value now binds (tuple patterns). ["Declare", ["Tuple", "x", "y"], "unknown", ["Tuple", 3, 4]] — the positional-value spelling the Declare contract documents and the scalar path already honors — silently declared nothing on the tuple path, which only read the trailing-attributes dictionary. The two forms now share one value resolution. A positional type on a tuple pattern, previously a silent no-op on this dead path, is now applied per bound name and surfaces an incompatible-type error value when it doesn't fit — loud over silent, and atomic: every leaf is validated against the type before any binding is installed, so a failure on the second leaf no longer leaves the first one declared. (No surface route emits either spelling: the Epsil parser uses the dictionary form.)

• A destructuring assignment is now atomic too. (x, y) := (7, 4.5) with both targets declared integer used to write x and then fail on y, leaving the tuple half-assigned; the same happened when a later target was a constant. Every leaf is now validated against its target's existing binding — declared type, constness — before the first write, using the very check the write itself performs, so the diagnostic is unchanged and a rejected pattern leaves every target at its OLD value. (Failures raised deeper inside the install machinery — function-literal reconciliation, effect contracts — stay sequential.)

• A destructuring declare or assign whose right-hand side is a tuple-valued expression now compiles (JavaScript target). let (v, j) = parseValue(cs, i) and the state-threading idiom (v, j) := step(j) previously failed closed unless the right-hand side was a literal tuple. When the pattern is flat and the right-hand side's static type pins the tuple arity (-> tuple<T1, T2>), the compiler now binds the whole result to one temporary and reads components positionally — let _tv1; _tv1 = step(k); let v = _SYS.at(_tv1, 1); … — preserving the interpreter's evaluate-once-then-write order (so swaps and _ positions behave identically). A tuple-typed symbol right-hand side rides the same path. Nested patterns, statically-unknown arity, and every non-JavaScript target (GLSL, WGSL, Python, interval) keep the fail-closed refusal.

• A function no longer broadcasts over a collection argument its body consumes whole. A user function's unannotated parameters default to scalar, and a scalar-parameter function maps over an indexed-collection argument (the vectorization convention: f(x) = 2x applied to [1, 2, 3] is [2, 4, 6]). But the collection evidence a body provides was being lost in three ways, so functions that plainly consume a collection whole were broadcast too — the body then saw a single element, and conditions inside it failed (Condition must evaluate to "True" or "False") or loops never terminated. All three are fixed, and each writes its evidence onto the parameter so the inferred signature reflects the use:

  • a parameter referenced only from a nested block scope (an if branch, a while body — while cs[j] != "z") auto-declared a throwaway per-scope shadow binding that swallowed the inference; bare parameters now share one cached binding across the whole body, which the literal's parameter declaration then adopts;
  • a function that merely forwards its parameter (g(xs) = f(xs)) learned nothing, because calls to inferred-signature functions skip argument validation — and with it, its narrowing side-channel; the collection-only parameter types of the callee now narrow unknown symbol arguments even on that route;
  • Length(x) contributed nothing because its parameter is deliberately any (Length(5) stays symbolic); it now treats a not-yet-typed symbol operand as collection evidence, like an indexed read does.

• Collection evidence now survives definition order: a function defined before its callee is re-derived when the callee arrives. The narrowing described above — a callee's collection parameter teaching the caller's parameter — could only fire when the callee was already defined. Write the caller first (process(cs) = clean(cs) + 1 before clean(v: list<number>) = v[1]) and process was canonicalized while clean was still unknown: cs learned nothing, so process([10, 20, 30]) broadcast elementwise, handing clean one number at a time — each call then failed against the declared list<number> parameter, and the errored elements surfaced as inert [process(10), process(20), process(30)]. Definition order changed semantics, which the engine already refuses to accept for juxtaposition (g(t) := 2a(t) written before a is defined re-reads as an application when a arrives): the same repair machinery now covers forward-referenced calls. While a function literal's body is canonicalized, a call whose callee is undefined — or known only by a guessed signature — registers the literal as a dependent of that name; when the name later gains a definition (assigned a body, or merely declared with a signature), the literal is re-derived from its raw operands and the parameter picks up the evidence, so the caller binds the collection whole and answers 11. Helpers can now be written below their callers, and mutually recursive pairs work in either order. A scalar callee defined later leaves the caller vectorizing, exactly as if it had been defined first, and a self-recursive function skips the pointless re-derivation of itself. Chains repair transitively: re-deriving a forwarder is that forwarder's own name gaining a definition, so A → B → C written in that order all bind correctly once C arrives — including diamonds, where the apex is rebuilt once, after every forwarder below it is current — and mutually recursive forward references terminate instead of chasing each other.

• A while loop inside a zero-argument function now terminates. function f() { let j = 1; while j < 3 { j = j + 1 }; j } hit the iteration limit: the nullary apply path skipped the sweep of stale canonicalization bookkeeping that the parameterized path performs, so the loop condition read a hoisted valueless binding forever. The nullary path now hides those bindings for the duration of the call, exactly like the parameterized path.

• And/Or/Not accept a possibly-absent condition, Kleene-style. A comparison on an indexed read — cs[j] == "a", honestly typed boolean | missing since the index may be out of range — was rejected at canonicalization by the logic operators' boolean parameters, so the guarded loop condition j <= Length(cs) && cs[j] == "a" errored with incompatible-type. The three operators now declare the handle missing-value behavior and evaluate Kleene over absence: False dominates And, True dominates Or, Not(Missing) is Missing, and a surviving absent condition still surfaces through If's absent-condition error.

• Epsil: a pinned match case after a result line is a new case. The case body 1 => "one" followed by a line starting == lim => … fused into the comparison "one" == lim (leading-operator line continuation), and the => then diagnosed. At the top level of a case body a linebreak now ends the body; parenthesized subexpressions keep the ordinary continuation. Two diagnostics were also sharpened: comma-separated cases get a targeted match-case-separator (with a fix-it to ;, and parsing recovers instead of dropping the remaining cases), and a conditional tail accidentally placed at the start of a line (x + 1 / if x > 0 else 0) reports conditional-if-line-start instead of the misleading opening bracket expected.

• A shader loop body no longer emits a return inside the loop. On the GLSL and WGSL targets, a for body with more than one statement compiled as a value — its last statement became return <statement>, so the shader returned on the first iteration while reporting success: true. Two plain scalar assignments (a := a + k; b := b * 2) were enough to hit it. A loop body is now compiled as a statement list, which is also what lets a destructuring assignment lower on those targets.

• An even root of an even power reduces, and no longer does so for complex values. \sqrt[4]{x^2} now returns \sqrt{|x|}. It did not before, because the result is structurally larger and the cost check rejected it — which was quietly masking a soundness bug: the rewrite had no real-domain guard, so for a value declared complex it would have produced \sqrt{|z|}, where the principal value of \sqrt[4]{z^2} at z=i is e^{i\pi/4}, not 1. The guard is now in place and the reduction is kept for being the reduced real-domain form rather than the smaller one. A complex-declared base is left alone.

• A closure returned from a function now resolves captured variables from inside a nested block. k ↦ (x ↦ if x > 1 { k } else { 0 }) applied at k = 100 returned the symbol k instead of 100, while the same body without the branch blocks — or a plain do { k } — returned 100. captureClosures rebinds a returned function literal so its body block closes over the call's frame, but it reused the body's operands verbatim, so a scoped block nested inside that body kept its canonicalization-time parent chain and reached the stale copies of the same lexical levels. The walk now re-roots nested blocks onto the captured chain, keeping their own locals. Held operands are what introduce such a block — If branches are the common case, and Cortex compiles every if branch to a block, so any if inside an escaping lambda was affected, including one drained later from a lazy Map.

• An annotated function parameter read from inside a nested block now resolves. In Cortex, function s(k: number) { if 1 > 0 { k } else { 0 } } returned the symbol k rather than the argument — while the same function with a bare k returned it correctly. evaluateBlock sweeps stale canonicalization bookkeeping from the block's scope, and its keep-test was "is this binding's type inferred". An auto-declared shadow inherits the DECLARED type of the outer binding it shadows, so an annotated parameter left an explicitly-typed valueless shadow that survived the sweep and hid the call value in the lambda's fresh scope. The keep-test is now "was this created by a Declare statement", which is what the sweep meant to ask; genuine block-locals still survive, including across a re-entered block. Cortex wraps each if branch in a block, which is why the conditional shape surfaced it.

• A lazy Map returned from a function no longer loses the variables its mapping function closed over. f(k) = Map([1,2], x ↦ x + k) drained by the caller produced [k + 1, k + 2] instead of [101, 102] — a silently wrong value, with no error. Drain-time Map fusion serves each element with a direct operator application in the ambient scope, bypassing makeLambda and so its scope push, and the shape gate treated every parameter-free operand as a "closed" value. A free symbol is not closed: it resolves by binding lookup, and once the defining call has returned that binding is no longer ambient. The closure chain was always intact — captureClosures rebinds the literal to the call's frame — so the fix is for the drain to evaluate inside it. A level whose operands are all literals, the shape the fusion was built for (1 + Mod(Range(0,899) + 29, 900)), records no scope and keeps the original zero-scope-work path; the 3×900 witness is unchanged.

• Annotating a callback parameter no longer switches off broadcasting for the whole function. function map(f: (A) -> B, t) stopped broadcasting over every parameter the moment f was annotated, so a recursive map(f, t.children) over a tree went inert while the identical function with a bare f worked. Broadcast eligibility (paramsAreScalar) is all-or-nothing across the parameter list and a function type is not a scalar type, so one callback annotation vetoed the rest. A function-typed parameter receives a function, never a collection, so it can never be broadcast over and now abstains instead of vetoing — which is the position the inference path already took, so a declared (A) -> B parameter and an inferred one of the same shape no longer disagree. A collection-typed parameter still suppresses broadcasting for the function: it consumes a whole collection, and that suppression is what keeps a nested collection argument from being descended into elementwise.

• Reading a field through a recursive type's own recursive field no longer fails with Converting circular structure to JSON. With type tree = tuple<value: any, children: list<tree>>, the expression t.children[1].value produced that error rather than the field's value. Joining the element types of a list<tree> reaches unionTypes, whose de-duplication key was JSON.stringify(type) — and a recursive type reference reaches itself through its resolved def. The key now omits def, which is both cycle-safe and lossless: a reference is identified by its name, and def is the only edge by which a type cycle can close.

• A recursive type alias no longer overflows the stack when a value fails to match it. type alias json = missing | boolean | finite_real | string | list<json> | dictionary<json> accepted every JSON shape correctly, but any rejected value — a function, a complex number, NaN — produced Maximum call stack size exceeded instead of an incompatible-type error. hasValueComponent unfolded structural alias references with no cycle guard, unlike isSubtype, which has had one at each of its own unfold sites. Cutting the back edge is exact rather than merely conservative here: every component reachable around the cycle is already reachable on the first unfold.

• A forward type reference can now be fulfilled by a later declaration, so a mutually recursive set of types is writable. The documented spelling — ce.declareType("json", "… | type json_array") followed by ce.declareType("json_array", "list<json>") — failed with The type "json_array" is already defined in the current scope: the forward reference installed a type record, and the declaration meant to complete it read that record as a redeclaration conflict. The declaration now completes the record in place, so the types that captured the reference resolve through to the definition (a fresh record would leave them pointing at the empty one, and the recursion could never close). Only an unfulfilled reference is completable; a name that already has a definition is still a redeclaration error. This applies to the Cortex type statement equally.

• A parameter typed by an alias of a collection now binds its argument whole instead of broadcasting over it. With type alias u = list<number>, a function (u) -> … applied to [1, 2] was mapped over the list and each element then failed the parameter check — while the inline (list<number>) -> … spelling bound correctly. isScalarType did not unfold alias references, so an alias of a collection read as a scalar. Nominal types are unaffected and still broadcast: their values are tagged applications, never collections, so a list of them is a genuine elementwise call.

• simplify() now reduces inside \int, \sum, \prod and \frac{d}{dx}. The body of those operators was never simplified, so \int(\sin^2x+\cos^2x)dx stayed as written rather than becoming \int 1\,dx, and \sum(n+n) never reached its closed form. That was deliberate: the closed-form rules match on the body's shape, and simplifying first rewrites the shape out from under them — \sum k(k+1) becomes \sum(k^2+k) and the sum-of-products rule stops recognising it. The body is now simplified once at the fixpoint, after those rules have had every chance and none has fired, so both work: the integrand above collapses AND \sum k(k+1) still returns its closed form.

  This is new work on expressions that contain a binder — roughly 3x on binder-heavy input in exchange for simplification that previously did not happen at all. Expressions without a binder are unaffected.

• Coalesce no longer evaluates its tail past an undecided operand. When an operand still carried free variables, the handler evaluated every remaining operand before returning the partially evaluated expression — so a later operand's effects ran, and its errors surfaced, on a path that a decided first operand would never have taken. The tail is now left unevaluated, which also makes the nested form Coalesce(a, Coalesce(b, c)) and the flat Coalesce(a, b, c) observationally equal.

• The cost function no longer prices the same expression differently depending on which form it is handed. Square(x) cost 6 unevaluated but 1 canonical, and Exp(x) cost 10 versus 1 — yet both canonicalize to a Power. Since simplify()'s cost gate compares an incoming expression against a rule's result, and the two need not be in the same form, that made some comparisons off by up to 2×. Both now price through the same helper as Power, so the cost is a property of the expression rather than of its representation.

  Relatedly, a negation is now priced by what it is applied to: a sign on a term costs 1, while negating a whole sum keeps the higher cost of 4, since that forces delimiters. Previously both cost 4, which said -a - b - c is nearly twice as complicated as -(a + b + c) when it reads more simply — and made Subtract(a, b) cost less than the Add(a, Negate(b)) it canonicalizes to. One visible consequence: \int\sqrt{x^2-1}dx now returns \frac12(x\sqrt{x^2-1} - \operatorname{arcosh} x), matching the factored form its two sibling trig-substitution integrals already returned.

• A coefficient no longer jumps in cost at an arbitrary size. The cost function treated a numeral times something as cheap only for an integer up to 10 (but any rational), and discarded the coefficient's own size entirely — so 10x cost 4 while 11x cost 10, a 2.5× step for the same shape. The coefficient's cost is now counted, which prices size continuously (integer literals are already priced by digit count), and the magnitude test is gone: 11x costs 5, 1000x costs 7. 2x still costs less than x + x, which is what the discount exists for. One visible consequence: \sum n^2 returns \frac16(2b^3+3b^2+b) rather than \frac13 b^3+\frac12 b^2+\frac16 b, matching the shape \sum n already returned.

• Raising something to a power now accounts for what is being raised. The cost function priced a power by its exponent alone, so (a+b+c+d)^{20} — which expands to 1,771 terms — scored 2, the same as x^{20} and barely above x^2. The base is now counted. 2q^2 is still cheaper than the repeated multiplication it replaces, which is what that rule exists for. One visible consequence: (\sqrt2+\sqrt3)^2 now reaches its closed form 5+2\sqrt6 instead of staying unexpanded.

  Two long-standing shortcuts came out with it. A negated power was priced without its base, so -\sin^2x scored 4 where \sin^2x scored 12 — the same subexpression valued three-fold apart on nothing but a leading sign. And \sqrt{} carried hidden surcharges for a perfect-square or odd-power argument, added to push factoring rewrites like \sqrt{x^2y}\to|x|\sqrt y past the cost check. Those rewrites introduce an absolute value, so they genuinely do grow the expression; they are kept because they are correct on the reals, not because they are smaller, and they now say so directly rather than relying on a surcharge to disguise their size. Same for distributing an exponent over a product. Behavior is unchanged in every case.

• A radical is no longer priced as if it were an ordinary decimal. The cost function reduced an exact value to its floating-point magnitude, so \sqrt3, 2\sqrt3 and \sqrt{17} all scored the same as the plain decimal 0.5 — the radical was invisible to every comparison. An exact value is now priced as rational × \sqrt{radical} with the radical counted, calibrated so a radical literal costs about what the equivalent expression costs (\sqrt3 and \sqrt y both score 6). Plain integers, decimals and fractions are unchanged.

  Because a radical now carries weight, keeping one factored out beats spreading it across several terms, so a number of antiderivatives come back in a tidier form: \int\frac{1}{x^2+x+1}dx returns \frac{2\sqrt3}{3}\arctan\left(\frac{\sqrt3}{3}(2x+1)\right) rather than distributing the \sqrt3 over both terms of the argument. Collapsing nested radicals (\sqrt{\sqrt{12}} \to \sqrt[4]{12}) also no longer needs a cost-gate exemption — it now wins on its own merits.

• An unevaluated integral now weighs heavily against a closed form. Integrate was priced like any unrecognized operator, but an antiderivative can be much larger than its integrand — \int\sec^3x\,dx scored 49 against a closed form of 78, and \int\frac{1}{x^4+1}dx 62 against 145 — so a rewrite that resolved such an integral could be rejected for being "more complicated". Integrals now carry a large flat premium. It is a weight rather than an absolute rule — a closed form vastly larger than its integral could still lose. It is flat rather than proportional so that comparing two expressions which each contain one integral still turns on the integrands, and so that an expression with fewer integrals still wins.

• ce.parse() now accepts a scope option in its public type signature. The implementation has always honored it — the whole parse runs with the supplied scope as the current lexical scope, so name resolution walks scope → parents and every auto-declare and inference lands rooted there — but the option was missing from IComputeEngine.parse(), which is the type the public ComputeEngine resolves to. Passing it was therefore a compile error (TS2769) even though it worked at runtime. ce.expr(), ce.box(), ce.function() and ce._fn() all already declared it; parse() was the lone omission. No runtime change.

• Pochhammer(), Degrees() and DMS() no longer numericize an exact irrational argument. Found by auditing for the Beta() bug below, which turned out to be one instance of a small class. (\sqrt2)_2 returned 3.41421356… instead of the exact 2 + \sqrt2, and \mathrm{Degrees}(\sqrt2) returned 0.0246826… instead of \sqrt2\pi/180. Two different causes, one symptom: Pochhammer built its rising-factorial terms with the .add() method, which folds two exact literals to a machine float (the same slip as Beta, and its own symbolic branch alongside already did it correctly); Degrees fell back to ce.number(arg.re) for a non-rational argument, and .re is a machine float. Exact rationals, integers, floats, poles and symbolic arguments are unchanged in all three.

  The audit also found the nightly exactness grid was covering only 104 of the engine's numeric operators — which is why these went unnoticed. It now covers 28 more. Mandelbrot/Julia are deliberately excluded (a float is the answer for an escape-time sampler), as are Rational/Rationalize (they exist to turn a float into an exact value).

• Beta() no longer numericizes an exact irrational argument. \mathrm{B}(\sqrt2, 2) evaluated to 0.2928932188… instead of the exact 1/(\sqrt2(1+\sqrt2)), breaking the contract that evaluate() returns the most exact form and only N() produces a float. The closed form \mathrm{B}(a, m) = (m-1)!/(a(a+1)\cdots(a+m-1)) was being built with the .add()/.mul() methods, which fold two exact literals to a machine float — so \sqrt2 + 1 collapsed on the very first factor and the whole result went inexact. Integer and rational arguments were unaffected (they fold exactly), which is why only an irrational argument showed it. Poles (\mathrm{B}(-1, 2) = \tilde\infty), the finite negative cases (\mathrm{B}(-2, 2) = 1/2) and float arguments are unchanged.

• A collection rebound to a Map over itself no longer overflows the stack. Assigning xs the value Map(xs, f) — the shape an accumulator loop produces — made any later query throw a raw RangeError: Maximum call stack size exceeded out of evaluate(), rather than being absorbed as an Error value; a host calling evaluate() directly saw a hard crash. The self-referential-binding guard was in place and firing (xs.value reads undefined, as designed), but Map is the one lazy collection operator that answers isFinite/isEmpty from its source rather than its own iterator, and that path reached the stored value through evaluate() instead of the guarded value. Each turn — isFinite → source resolution → evaluate() → isFiniteCollection → isFinite — completed its dereference before the next began, so the existing cycle guard, which is released when the dereference returns, never observed the re-entry. Such a source is now left unresolved, putting Map on the same symbolic residual that Filter and every other lazy operator already produced. Eager and broadcast sources (Map(X - 1, f)) still resolve exactly as before.

0.102.0 2026-08-05

New Features

• Two ring constructions, Adjoin and QuotientRing, with their standard notations. \mathbb{Z}[\sqrt2], \mathbb{Z}[\sqrt2,\sqrt3], \mathbb{Z}[i] and \mathbb{Z}[x] now parse as ring adjunction (["Adjoin", "Integers", …]), and both \mathbb{Z}_n and \mathbb{Z}/n\mathbb{Z} as the quotient ring (["QuotientRing", "Integers", "n"]), which serializes back to the subscript form. The blackboard-bold ring and field constants — \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C} — are accepted as bases. Both operators are inert in this version: they stay symbolic, carry no membership test and no arithmetic in the constructed ring, but they do report an honest type — \mathbb{Z}[\sqrt2] is a set<finite_real>, \mathbb{Z}[i] a set<finite_complex>, \mathbb{Z}_n a set<finite_integer>. Note that \mathbb{Z}_p is read as the integers modulo p — the quotient reading — not as the alternative number-theoretic reading the same notation carries in some texts, and that field adjunction written with parentheses (\mathbb{Q}(\sqrt2)) is not parsed. Sign-restricted spellings are unaffected: \mathbb{Z}_+, \mathbb{R}_- and \mathbb{Z}_{\ge0} still name PositiveIntegers, NegativeNumbers and NonNegativeIntegers.

• Quantifiers accept an undelimited parenthesized body. \forall x > 0 (x^2 > 0) — a condition followed by a parenthesized body, with no comma between them — now parses as ["ForAll", <x > 0>, <x^2 > 0>] for all five quantifiers (\forall, \exists, \exists! and the negated forms). Previously the group was absorbed into the condition as an implicit product. The split is accepted only when the group reads as a proposition — a relation, a logic connective, a membership, or a predicate application such as (P(x)) — so \forall x > 2 (\sin y) still reads the group as a factor of the condition.

• More spellings of the quotient ring and the sign-restricted number sets parse. \frac{\Z}{n\Z} (and \dfrac, \tfrac) now reads as ["QuotientRing", "Integers", "n"], like the inline \mathbb{Z}/n\mathbb{Z}, and serializes back to \Z_{n}. The terse blackboard-bold family also accepts the short comparison commands — \R_{\ge 0}, \R_{\gt0}, \N_{\ge1} and their siblings — which previously required the spelled-out \geq/\geqslant forms.

• Cortex: the conditional expression a if c else b. When both branches are single expressions, the braces of the block form are noise, and the conditional spells the same ["If", c, a, b] without them: let y = 10 if x > 3 else 20. It is the same operator the block form builds — only the branches differ, plain expressions instead of Blocks, so the conditional introduces no scope and no statement can appear in a branch. The else is mandatory (it is what ends the condition; a missing branch would leave the false case with no value to name), and 1 if c reports the new conditional-else-expected diagnostic — use the block form if c { 1 } when there is nothing to return. Chains nest to the right ("zero" if n == 0 else "negative" if n < 0 else "positive"), so there is no else if spelling to learn. The conditional binds looser than every operator that computes — as in Python, looser than || — but tighter than the four that bind or pair, =, |->, |> and ->, so the whole conditional is the right-hand side of an assignment, the body of a function, the piped value, or the value of a dictionary entry. Used as an operand it needs parentheses: 1 if c else 2 + 3 reads as 1 if c else (2 + 3). One layout rule: the if must be on the same line as the value before it — a line break separates statements, so an if that starts a line always begins a new if-statement. The block form is unchanged, and a match-case guard (n if n > 0 => …) is unaffected: patterns have their own grammar, which has no conditional.

Improvements

• Cortex: If now serializes as if, not as a function call. The if-expression syntax has always parsed, but the serializer had no rule to emit it, so a program written if c { 1 } else { 2 } came back from the formatter as If(c, do {1}, do {2}). It now round-trips to the form it was written in. The spelling is chosen by the shape of the branches, which is exactly what the parser distinguishes: Block branches give the block form (chaining to else if when the alternative is itself a block-form If), plain expression branches give the conditional form a if c else b. A shape with neither spelling — mixed branches, or an If with no else and a non-Block consequent — keeps the generic If(c, …) call form, which also re-parses faithfully. An If in operand position is parenthesized according to the conditional's precedence, so Add(If(c, 1, 2), 3) serializes (1 if c else 2) + 3 rather than the differently-parsing 1 if c else 2 + 3.

• simplify() no longer returns a result more complicated than its input. The cost gate that decides whether to keep a rewrite used to tolerate growth of up to 30% — a proportion, so the bigger the expression, the more growth it allowed. It is now strict: a rewrite is kept only if it does not increase the cost.

  Two things were wrong with the tolerance. It let large expressions run away: instrumenting the gate across the test suite caught a single rewrite adding 1,693 cost units, and a chain walking one expression from cost 7,098 to 10,133 — every step recorded as a "simplification". And 90% of what the tolerance actually bought was the generic expansion rule, which was making results worse by blowing factored closed forms apart.

  Removing it restores them. \sum_{n=0}^{b}(a + dn) now returns the textbook (b+1)(a + bd/2) instead of a four-term polynomial; \int\sqrt{1-x^2}dx returns \frac12(x\sqrt{1-x^2} + \arcsin x); \int e^x\sin x\,dx returns \frac12(\sin x - \cos x)e^x; solving x^2 - 2x\cos t + 1 = 0 for t returns \arccos\frac{x^2+1}{2x}, which now agrees with the validity condition reported beside it; and \frac{-b+\sqrt{b^2-4ac}}{2a} stays in closed form instead of splitting into -b/(2a) + \sqrt{b^2-4ac}/(2a).

  A rule that should apply regardless of cost is tagged purpose: 'transform', which bypasses the gate entirely — that, not a numeric tolerance, is the supported way to express "preferred even though larger".

  Making that work meant tagging the rewrites that had been surviving on the tolerance. Nine families now declare purpose: 'transform' explicitly: the power combinations x^n·x^m → x^{n+m}, x·x^n → x^{n+1} and x^n·x → x^{n+1} (whose equivalent for three or more factors already carried the tag, so the same rewrite had been obeying two different cost policies depending on which implementation caught it); collapsing nested radicals (√√12 → ⁴√12); removing a logarithm from under an exponential (e^{\ln x + y} → x·e^y and its \log_c sibling); \log_c(x^n) → n·\log_c(x); the geometric-series and shifted/falling-factorial closed forms for Sum and Product; the \sin/\cos(π ± x) argument reductions; rationalizing a radical denominator; and \sqrt{x^{2n+1}} → |x|^n\sqrt{x} (a branch-cut correctness rewrite, not a size optimization). Several were untagged only because the original string-matching exemption list never named them. Tagging them also makes them robust to a caller-supplied costFunction. Distributing a negation over a sum (-(x+1) → -1-x) is tagged for the same reason — it trades one negation for one per term, so it always scores worse, but it is the form the rest of the engine works in.

• ComputeEngine, expr.engine and ExpressionComputeEngine are now one interchangeable type — no more casts between them. The ComputeEngine exported from the package (and from the /core sub-path) is now a constructor value paired with the structural IComputeEngine interface, rather than the class itself, whose private fields made its type nominal. An engine obtained from expr.engine (typed ExpressionComputeEngine) can now be assigned or passed wherever a ComputeEngine is expected, and vice versa. new ComputeEngine(), instanceof ComputeEngine, InstanceType<typeof ComputeEngine> and the static ComputeEngine.getStandardLibrary() all work as before (the constructor's type is the new ComputeEngineConstructor interface). The interface also gained members that were previously only on the class: Two, toJSON(), suggestOperatorName() and functionProperties(). One typed-surface consequence: the legacy canonical/structural options of ce.expr() and ce.box() are not part of the interface — use the equivalent form option ({ canonical: false } → { form: 'raw' }, { structural: true } → { form: 'structural' }); the legacy options still work at runtime. The ExpressionComputeEngine type is now deprecated: it is interchangeable with ComputeEngine, which should be used instead.

Resolved Issues

• Serialization no longer writes to the current scope. toLatex() and toMathJson() internally re-canonicalize parts of the expression they lay out (for example to display a product with negative exponents as a fraction), and that re-canonicalization could declare an undeclared function head — or write an inferred type onto a declaration — in whatever scope was ambient at serialization time. In particular, serializing an expression that had been parsed against a per-call scope leaked its function heads into the surrounding scope, changing how later parses read (aQ_{z}(x,y) parsed before the leak as an implicit product, after it as a function application). Serialization now runs in a resolve-only region: names resolve against the scope chain but are never declared, and no inference is written. The same region now also covers the undeclared-head auto-declaration during partial-form boxing, which the resolve-only contract already promised but did not enforce.

• Cortex: a wrapped operator chain no longer ends in a dangling operator. When an infix chain was long enough to wrap, the formatter emitted the operator after every element, including the last. With no closing fence to absorb it the output ended on the operator, and the result did not re-parse: Add(accumulator, someLongVariableName, x, y) at a narrow margin produced accumulator + / someLongVariableName + x + y +, which reports unexpected-symbol "+". Only the separators between elements are kept now; those are fine across a line break, since an operator with whitespace on both sides stays infix. Fenced lists are unaffected — a trailing , before ] or ; before } is legal Cortex and still emitted.

• Cortex: a statement block that has to wrap is indented, not staircased. do { … } (and the new if block form) laid its body out with the generic fenced-list layout, which aligns continuation lines to the opening brace. For a statement block that pushed the body out to the brace's column, and at a realistic margin left so little width that the statements broke apart in turn. Both are now laid out anchored at the keyword: body one indent in, closing brace under the keyword, statements separated by the line break itself. An else if chain is flattened first, so every clause stays at the same column instead of nesting a level deeper per else if. Expressions and collections keep the existing brace-aligned layout.

• A subscript or bracket on a set constant is no longer read as an index. \mathbb{Z}_n parsed as ["At", "Integers", "n"] — indexing into a set — which is not a valid type, and serialized back to \Z[n], which parsed as an incompatible-type error, so the expression did not survive a round trip. Those spellings now produce the ring constructions above. Indexing a genuine indexed collection is unchanged.

• A bare N or D used as a variable now binds the same way everywhere in an expression. A single-uppercase-letter name that is also a standard library operator reads as a variable when it appears where a value is required (N + 1), and the engine declares that variable the moment it first sees such an occurrence. Occurrences boxed before that point — the first N of N, N+1 — kept the operator binding, so one expression carried two different bindings for one name (isSame() was false between two occurrences) and boxing the same input a second time produced a third. An expression such as N, N+1, N+2 or (DB+BC)^2 = AD^2+AC^2 therefore did not survive a serialize-and-reparse round trip. When the variable is declared partway through a boxing, the expression is now rebuilt against it, so every occurrence shares one binding. What the names mean is unchanged: N(2.3) and D(x^2, x) still apply the builtin operators, x^2 |> D still pipes into the derivative operator, and a bare mention on its own (ce.parse('N')) still leaves the operator definition intact.

• A symbol spelled by the generic (name-based) speller now reads back as the same symbol. Two cosmetic spellings changed the symbol's identity on a round trip. A plain trailing digit run became a subscript, so the symbol x2 serialized as x_2 and parsed back as the different symbol x_2 (and Arctan2 as \mathrm{Arctan_2} → Arctan_2). Such a name is now spelled verbatim and upright — \mathrm{x2}, \mathrm{Arctan2} — which reads back as the original symbol; a name that already uses the _ subscript convention is unaffected (x_2 still serializes as x_2). This changes the rendering of digit-suffixed names: they display upright as x2 rather than as x₂; write x_2 for the subscripted form. Separately, a name from the Greek-letter table whose command the LaTeX dictionary gives to a constant was spelled with that command: the symbol pi serialized as \pi and parsed back as Pi, zeta as \zeta → Zeta, phiLetter as \varphi → GoldenRatio. Those names are now spelled \mathrm{pi}, \mathrm{zeta} and \mathrm{phiLetter}. The set is derived from the dictionary, not hardcoded: a constant that yields its bare command to a declaration does not claim the spelling, so the symbol gamma still serializes as \gamma — that command reads back as EulerGamma only in an engine where gamma is undeclared, i.e. never in an engine holding the symbol.

• The imaginary unit now has a single canonical spelling: ["Complex", 0, 1]. A bare i parsed to the complex literal ["Complex", 0, 1], but \imaginaryI (and \mathrm{i}, \operatorname{i}, and the MathJSON symbol "ImaginaryUnit") canonicalized to the symbol ImaginaryUnit. Since the serializer emits \imaginaryI for both, ce.parse('i') did not round-trip, and two structurally identical expressions could compare unequal with isSame(). The ImaginaryUnit definition has always been declared holdUntil: 'never' — meaning its value is substituted at canonicalization — but the symbol was interned in the engine's common-symbol table, which short-circuited that substitution. It no longer is, so ce.parse('i'), ce.parse('\imaginaryI') and ce.box('ImaginaryUnit') now all canonicalize to the same complex literal. Migration: raw MathJSON "ImaginaryUnit" is still accepted and still round-trips non-canonically (ce.box('ImaginaryUnit', { canonical: false })), but code matching on the canonical form should test for the complex literal — expr.isSame(ce.I) or the isImaginaryUnit() helper — rather than for the symbol name.

• The interned imaginary unit is now an exact value, so exactness-gated folds fire on it. It was built from a float-lane numeric value while the identical ["Complex", 0, 1] literal boxed exact, which made canonicalization disagree with itself depending on how i reached it: ["Power", "ImaginaryUnit", 2] stayed i^2 (and ce.parse('i^2').simplify() did not reduce) while ["Power", ["Complex", 0, 1], 2] folded to -1 — so ce.box(expr.json) did not round-trip to expr. Small integer powers of i now fold on every route, and \sqrt{-1}, (-1)^{1/2} and \frac{a}{i} canonicalize to i, i and -ia respectively.

• A product of an infinity and the imaginary unit no longer collapses to NaN at canonicalization. ["Multiply", "PositiveInfinity", ["Complex", 0, 1]] canonicalized to NaN while the other operand order stayed symbolic: the n·i promotion (which folds 2·i to the exact 2i) accepted a non-finite left operand and built a value out of an infinite component. Infinities are excluded from that fold, so both operand orders now keep the symbolic product. evaluate() still returns NaN for the indeterminate form.

• Degrees() of a non-real argument no longer drops the imaginary part. \imaginaryI\degree canonicalized to 0 (and (2+3i)\degree to \frac{\pi}{90}) because the conversion read only the real part of its operand. The linear conversion is now applied to the whole value: Degrees(i) = i\pi/180.

• A canonical Multiply is now always flat. A product built by the invisible (juxtaposition) operator could keep a nested Multiply operand — for example ce.parse('2f(ab)') canonicalized to Multiply(2, f, Multiply(a, b)) instead of Multiply(2, a, b, f) — which broke the associativity contract and made two structurally identical products compare unequal with isSame(). Note the MathJSON serializer flattens on output, so expr.json printed the same ["Multiply", 2, "a", "b", "f"] either way; only expr.ops showed the difference. As a consequence exact numeric factors separated by parentheses now fold as they should: 2(3x) canonicalizes to 6x. One visible knock-on: two closed forms returned by Sum — (b+1)(a+bd/2) and its sibling — now come back from simplify() expanded (1/2·d·b² + a·b + 1/2·b·d + a) rather than factored. The values and the canonical forms are unchanged; flattening shaved the expanded rewrite's cost just under simplify()'s 1.3× acceptance gate (35 vs 35.1), so the rewrite is now accepted where it used to be rejected. That margin shows how finely the cost gate discriminates between a factored and an expanded form, and it is a candidate for tuning.

• An operator used as a value — unapplied, as in ["Tuple", "A", "Abs"] or as a callback in ["Map", xs, "Factorial"] — no longer serializes to a fragment of its own notation. The serializer reached for the operator's LaTeX notation, which is written in terms of operands, so with none it emitted \vert\vert for Abs, ! for Factorial or \sum for Sum; none of those re-parse. A LaTeX dictionary entry now declares whether its notation stands on its own with the new standaloneSymbol property — true of the function commands (\sin, \ln, \arctan) and of the constant and set notations (\Z, \emptyset, \varphi) — and only a flagged entry's notation is used for an unapplied symbol. Every other operator is spelled out as \mathrm{Abs}, \mathrm{Factorial}, \mathrm{Sum}, which re-parses to the same symbol. Leaving standaloneSymbol unset on a custom dictionary entry is always safe: it only costs the nicer spelling.

• The Fungrim identity artifact was regenerated against the canonical forms above (1445 rules: 1435 simplify + 10 solve). Five imaginary-unit identities were retired because canonicalization now performs them natively, which made each rule a no-op: \sqrt{-1} = i, i^2 = -1, 1/i = -i, i^3 = -i and i^4 = 1. Every rule matching on the imaginary unit was re-encoded to the ["Complex", 0, 1] canonical spelling; none was lost, and no rule's meaning changed. Separately, the offline rule compiler's self-test no longer rejects a rewrite whose result is structurally identical to the expectation but carries a different binder identity (the fallback compared with isSame and isEqual only, and isEqual no longer proves identities in free variables) — this recovers the Sinc derivative identity.

• A negated product now has a single canonical spelling. canonicalMultiply normalizes signs before folding exact numeric factors, so a fold that itself produced a negative real coefficient — only a product with complex factors can, e.g. i \cdot i = -1 — stranded a literal -1 operand: ["Multiply", "ImaginaryUnit", "ImaginaryUnit", "a", "b"] canonicalized to ["Multiply", -1, "a", "b"], which serializes as -(ab) and re-parses as the structurally different ["Negate", ["Multiply", "a", "b"]]. A fold-produced negative coefficient now re-enters the sign normalization, so both spellings converge on Negate(Multiply(a, b)) — the same form literal input has always produced. With this, every remaining serialize-and-reparse exception in the MathNet corpus ledger is a documented-lossy prettification: the ledger carries zero bug classes (385/391 round-trip).

• A function assigned at the top level is no longer mistaken for an un-applied builtin. The single-uppercase-letter fallback (see the N/D entry above) decided "standard library" by scope position, but ce.assign('F', ce.parse('x \mapsto x^2')) lands its definition in the same scope as the library — so F + 1 silently shadowed the function with an unknown variable, and from then on F(2) evaluated to the product 2F. The fallback now discriminates on the definition's origin: a user-defined function used as a numeric operand surfaces an incompatible-type error and the function stays intact. Devolution of the builtins themselves (N + 1, S/D) is unchanged.

• A raw (non-canonical) symbol now evaluates through its binding, matching the behavior raw and structural function nodes gained in 0.101.0. With x assigned 5, ce.box('x', { canonical: false }).evaluate() returned the symbol x; it now returns 5, while the receiver stays on its tier. A symbol with no assigned value still evaluates to itself.

• The ellipsis fold barrier holds at any depth, on every route. A product carrying a ContinuationPlaceholder (\dots) in a nested operand could still be spliced — and its factors folded or reordered across the ellipsis — when it arrived through raw MathJSON (ce.box), wrapped in a Sequence, or as an explicit \cdot/* chain hanging off a juxtaposed run: (px_1 + 1) \cdots (px_n + 1) \cdot p^m re-serialized with the \dots moved to the front. All flattening sites now share one depth-aware barrier, and an explicit multiplication folds a juxtaposed ellipsis run into a single flat notational product, which round-trips.

• An unknown function head whose name collides with a claimed constant spelling is spelled upright. A function head literally named pi serialized as \pi(x), which re-parses as Pi — the constant, a different symbol. The head-spelling path now consults the same claimed-spellings table as bare symbols: \mathrm{pi}(x), \mathrm{zeta}(x). A custom dictionary entry for such a head with a notation of its own is honored — only the colliding auto-generated spelling is bypassed.

• InterpolatingFunction used as a bare symbol no longer serializes with a trailing empty subscript (\operatorname{InterpolatingFunction}_{}); it is spelled \mathrm{InterpolatingFunction}, which re-parses to the symbol. The applied form keeps its domain-subscript notation.

0.101.0 2026-08-04

Breaking Changes

• A bare G is now a variable, not Catalan's constant. The LaTeX dictionary claimed the bare letter G for CatalanConstant ahead of any declaration, so a formula such as \int_{t_i}^{t_e}(G - F)\,dt silently read as Subtract(CatalanConstant, F), and declaring G could not reclaim it. As with e and i, the constant is now reachable only through explicit upright markup: \operatorname{G} — also its serialized form, so CatalanConstant still round-trips — and \mathrm{G}. Two consequences: G, G(2) and G_x are ordinary symbols, and \mathrm{G} no longer parses as the upright symbol G_upright.

• EulerGamma now serializes as \operatorname{EulerGamma}, not \gamma. Bare \gamma still parses as the constant, but it now yields to a declaration (see Improvements), which would have made the old serialized form ambiguous: in an engine where gamma is a variable, serializing an expression carrying the constant and parsing it back silently returned the variable. \gamma could not be disambiguated with upright markup the way G was — \operatorname{\gamma} already means the plain symbol gamma — so the constant serializes to its MathJSON name, which reaches EulerGamma through the generic symbol path regardless of what is declared. Rendered output that used to show γ now shows an upright EulerGamma.

• An ungrouped full-signature marker on a function literal is now always the literal's own contract. ["Function", ["Typed", body, "'(x: number) -> number'"], "x"] previously read the marker as a return type — the literal typed (unknown) -> (x: number) -> number — unless the signature carried an effect specifier. It now declares the literal's own signature (parameter types, arity — now checked — and return), whether or not effects are stated, matching the forall and effect-bearing readings. A plain arrow still states no effect contract. To ascribe a function-returning type, use the grouped spelling, which is unchanged: ["Typed", body, "'((x: number) -> number)'"].

• expr.isEqual() and = no longer prove symbolic identities. Equality is now arithmetic: the operands are evaluated, compared structurally, and — when their difference has no unknowns — compared numerically within ce.tolerance. No expansion, no simplification and no sampling is attempted. An identity between expressions with free variables, such as \sin^2 x + \cos^2 x = 1 or (x+1)^2 = x^2 + 2x + 1, therefore returns undefined instead of true, and the corresponding Equal expression stays inert (as it already did for any undetermined comparison, so an equation is still usable as an argument to Solve). Every = comparison is now cheap and predictable. Migration: wherever an identity was being proven, call expr.isIdenticallyEqual(other), or evaluate ["IdenticallyEqual", lhs, rhs] (LaTeX lhs \equiv rhs) instead of ["Equal", lhs, rhs]. Comparisons of numbers, of constant expressions, and of expressions that evaluate to the same structure are unaffected.

• expr.isSame() no longer follows the value of a symbol. It is now strictly syntactic everywhere: it compares canonical forms as written and never substitutes an assigned value. Previously a top-level symbol-versus-literal comparison dereferenced the binding — with one := 1, ce.symbol('one').isSame(1) was true — while the same comparison between two symbols, or nested inside a larger expression, did not, which made the method inconsistent with itself. With x := 5, ce.symbol('x').isSame(5) is now false. Migration: use expr.isEqual(value) — or expr.is(value), which adds a numeric fallback for constant expressions — to compare values; .isSame() answers "is this the same expression?" only. Internal checks against a literal operand, such as op.isSame(0), are unaffected. One canonicalization consequence: the x^0 → 1 fold is now a pure generic-symbol fold (like x/x → 1), so z^0 canonicalizes to 1 even while z := 0 — previously the assigned value was peeked and the fold was blocked. The literal 0^0 still canonicalizes to NaN.

• A bare \equiv now parses as IdenticallyEqual, not Equivalent. \equiv is the mathematical identity sign, so p \equiv q parses as ["IdenticallyEqual", "p", "q"] (see New Features). The biconditional keeps all of its other notations — \iff, \Leftrightarrow, \leftrightarrow, \Longleftrightarrow, \longleftrightarrow — and Equivalent still serializes as \iff, so logic round-trips are unaffected; only the reading of p \equiv q as a biconditional changes. Over boolean operands the two operators agree in any case, since two propositions are identically equal exactly when they are equivalent. Migration: write p \iff q where a biconditional is meant. A \equiv followed by \pmod{n} is still a Congruent, and \not\equiv still negates a congruence.

• ce.expr()'s scope option now RECEIVES the boxing's writes. It used to steer lookup only: auto-declared free symbols, undeclared call heads and type inference all still landed in the engine's current scope, so the option half-contained a boxing. The whole box now runs with the supplied scope as the current lexical scope, so every declaration and inference lands rooted there and discarding the scope discards the writes. Code that passed scope to redirect lookups while deliberately keeping the declarations in the engine's scope must now box without the option (or re-declare the harvested names). The same semantics are what the new scope option on ce.parse() provides — see New Features.

• evaluate() on a raw or structural expression now evaluates through its canonical form. Binder machinery — declaring a Sum index, normalizing Tuple → Limits — is a canonicalization step, so evaluating a structural binder ran its handler against an unbound index and returned a silently wrong value: ["Sum", "n", ["Tuple", "n", 1, 3]] boxed with structural: true evaluated to 9 instead of 6, and the same tree boxed raw evaluated to itself. Both routes (and .N() and async evaluation) now produce the canonical result: the receiver stays on its tier, but .evaluate() leaves the tier and every tier agrees on the value. Two consequences: a raw 2 + 3 now evaluates to 5 instead of echoing itself, and arity or type errors that only canonicalization checks can now surface from evaluating a raw tree (a raw 5 |> 3 evaluates to the same incompatible-type error the canonical route reports, where it used to stay an inert Pipe). A bare unbound symbol still evaluates to itself.

• Quantifiers now canonicalize their operands. ForAll, Exists, NotForAll, NotExists and ExistsUnique held their operands without a canonical handler, so a parsed quantifier kept raw parse sugar in its body: InvisibleOperator where Multiply was meant, Delimiter(Sequence(…)) where a Tuple was meant, stray HorizontalSpacing from \quad. The condition and body are now canonical (e.g. a condition x > 0 normalizes to 0 < x like every other comparison), which also makes the serialized form round-trip.

New Features

• Evaluate handlers now receive the expression being evaluated. The handler options carry an optional expression field — the canonical node, whose .ops are the raw (pre-numericization) operands, unlike the handler's first parameter which holds the evaluated operands (see the EvaluateHandlerOptions documentation for the caveats: positional correspondence does not survive associative flattening, ReleaseHold, or dropped operands, and for lazy operators the first parameter is also unevaluated). The first consumer is Power: under .N() a negative base's real-vs-complex branch is now decided from the exponent's exact rational — read from the raw operand, or through a symbol's binding — so .N(), the type handler, and the compiled constant fold agree for exact odd-denominator exponents of any term size ((-2)^{1000003/1000001} is now real on every leg; parity is decided on the exact bigint terms, so denominators beyond 2⁵³ do not corrupt the branch). Exponents with no exact provenance (floats, π, a lambda parameter, a Sum body) keep the rate-bounded float reconstruction.

• expr.hash is now a documented public property. A structural, bucketing-grade hash suitable as an in-memory cache or bucket key with a deep compare on hit. The documented contract: a.isSame(b) implies a.hash === b.hash (the hash is a pure function of the canonical tree — a symbol's assigned value never affects it); it is deterministic within a release but not stable across releases, so it must never be persisted; it is 32-bit-class, so a hash hit must be verified with isSame(); and it folds bound-variable names (binding-identity, not alpha-equivalence), matching isSame(). The property itself is unchanged — it was previously marked @internal.

• Parametric polymorphism: forall type variables in function signatures. The type language gains rank-1 (prenex) type variables with optional ground upper bounds: forall T. (list<T>) -> T, forall T: indexed_collection. (T) -> T, forall T, U. (list<T>, (T) any -> U) -> list<U>. Any identifier can be a variable — the clause declares it, and within its arm the quantified name shadows a nominal type of the same name (forall itself is now reserved in type strings). User functions can be declared generically (ce.declare('swap', 'forall T, U. (tuple<T, U>) -> tuple<U, T>')); each call solves the variables from the operands by local type inference (repeated variables join — forall T. (T, T) -> T at an integer and a real gives real), the result type is obtained by substitution, and a violated bound reports the instantiated expected type. Overload sets quantify per arm, with a ground arm beating an equally-specific generic one. On the query APIs, matches with a generic pattern answers existentially ((number) -> number matches 'forall T. (T) -> T') while couldMatch reads each variable as its declared bound. A generic declaration may be implemented by an evaluate handler or by a function body — see the two entries below. See the new "Generic Signatures" section of the types guide.

  Twenty-five library operators now state their contracts declaratively with generic signatures instead of imperative type handlers, preserving operand kinds and dimensions exactly: Identity, Prime, BaseForm, Chop, PlusMinus, Remainder, Conjugate, Inverse, Reverse, the tuple constructors (Single, Pair, Triple, KeyValuePair), and the collection family (Take, Drop, Slice, DeleteAt, Insert, ReplaceAt, Sort, Unique, RandomShuffle, Tally, Partition, ChunkBy) — e.g. Reverse of a matrix<integer^(2x3)> is now statically a matrix<integer^(2x3)>, and Take of it a list<vector<integer^3>>. As part of this, a dimensioned collection is now also a subtype of a collection of its rows (matrix<integer^(2x3)> <: indexed_collection<vector<integer^3>>), matching how single-index access has always evaluated.

• Generic function literals: a forall signature can now be implemented by an inline body. A whole-signature clause makes a ["Function"] literal generic — written as a signature string (["Function", body, "'forall T. (x: T) -> T'"]), as a Typed marker on the body, or as the declared type of the symbol the literal is assigned to — and it works on every route: ce.assign, the Assign operator, and an annotated Cortex const/let. Each call instantiates the clause, so on one engine f(5) types finite_integer and f("a") types string, bounds are enforced at the call, and a collection argument still broadcasts at the variable's bound. Declaring first and assigning after — ce.declare('nest', 'forall T. (x: T, n: integer) -> T') — makes generic recursion work for the first time. The body is canonicalized once with the quantified parameters erased: inside it, x: T is an ordinary unannotated parameter, a bound does not narrow it, two parameters sharing T are not known to have the same type, and the variable-correlated result is a trusted ascription rather than a run-time check. Four boundaries are rejected with dedicated diagnostics: partial application of a generic function, a generic clause in a multi-clause set (in either direction), a function-literal body for a generic overload set, and a forall clause on an individual parameter annotation.

• Cortex: generic function definitions, function f<T>(…). A definition takes a type-parameter clause between its name and its parameter list: function g<T: number, U>(x: T, k: (T) any -> U) -> list<U> { … }. Bounds must be ground types, the effect specifier and return type are unchanged, and the clause names scope over the definition's head only — its parameters, effect specifier and return type — so a body-local annotation such as let y: T is an ordinary unknown-type error. Unused variables, result-only variables and non-ground bounds are diagnosed at parse time by the type grammar itself; an empty clause, a duplicate name and a generic clause in a multi-clause set have their own codes (empty-type-parameter-clause, duplicate-type-parameter, generic-clause-unsupported). A generic definition serializes back to the sugared form losslessly. The math definition form does not take a clause: f<T>(x) = x remains an ordinary expression, since it is genuinely ambiguous with a relational one.

• Transparent generic type aliases: type alias Pair<T> = tuple<T, T>. A structural type alias can now take a type-parameter clause, in Cortex as above and from the host with ce.declareType('Pair', 'tuple<T, T>', { alias: true, typeParams: ['T'] }) — parameter names, { name, bound } records or one clause string ('T, U: number'). The applied spelling is usable anywhere a type is written: an annotation, a parameter, the element position of another type. It expands eagerly, at type resolution, into the substituted definition, so nothing downstream ever meets an applied reference: Pair<integer> is tuple<integer, integer>, and that expansion is what .type, toString(), matches() and error messages show — the source keeps the spelling it was written with, and a Cortex program round-trips let p: Pair<integer> = (1, 2) verbatim. Arguments nest (Pair<Pair<integer>>, list<Pair<integer>>) and aliases compose (type alias Wrap<T> = list<Pair<T>>). A parameter may carry a ground bound, enforced wherever the alias is applied; an argument that is itself a type variable — a forall clause's, or the enclosing alias's own parameter — is admitted by comparing bounds: the variable's declared bound must satisfy the parameter's (an unbounded variable is bounded by any, so a bare forall T. (Keyed<T>) -> T reports generic-alias-bound naming both bounds). Four limits, each with its own diagnostic: a generic alias may not refer to itself (recursive generic aliases are out of scope), every parameter must be used in the definition, a bare or wrongly-sized application is an arity error, and a parameterized nominal type is still unsupported. No constructor is minted for a generic alias and its name is not claimed in the value namespace at all, so a function of the same name stays legal, before or after. A dependent alias snapshots what it was built from: re-running a type statement replaces that alias, and re-running the cell re-declares the dependents in order. See the new "Generic Type Aliases" section of the types guide.

• IdenticallyEqual: a dedicated operator for mathematical identities. ["IdenticallyEqual", lhs, rhs], the method expr.isIdenticallyEqual(other) and the LaTeX notation \equiv (the ≡ character parses the same way, and the operator serializes back to \equiv) ask whether two expressions have the same value for every value of their free variables. This is the tier that proves an identity: it applies expansion and simplification and evaluates both sides at pseudo-random sample points, so a True verdict may rest on sampling — a very strong indication rather than a formal proof, and the only comparison in the engine that can answer this way. It is three-valued: an identity that can neither be established nor refuted stays unevaluated. The machinery itself is not new — it is the prover that Equal used to run — but it is now reached explicitly. On the compile targets, Equal keeps its tolerance comparison while IdenticallyEqual and Same decline to compile.

• Same (the Cortex === operator, also written ≣) is now specified as canonical-syntactic equality, matching expr.isSame(). It compares the canonical form of its operands as written and never dereferences the value of a symbol: with x := 5, ["Same", "x", 5] is False while ["Equal", "x", 5] is True. It remains total — always True or False, with no tolerance — and keeps no IEEE exemption for NaN: ["Same", "NaN", "NaN"] is True where ["Equal", "NaN", "NaN"] is False, and the same holds inside a collection (["Equal", ["List", "NaN"], ["List", "NaN"]] is False). Together with Equal and IdenticallyEqual, this gives three tiers of comparison — syntactic, same value, and same function of the free variables — described in the "Comparing Expressions" section of the Symbolic Computing guide.

• Per-call scope control: ce.parse(latex, { scope }) and ce.createScope(). Canonical parsing writes to the engine's lexical scope — free symbols are auto-declared, undeclared call heads become inferred functions, types are narrowed by usage — which consumers parsing untrusted or out-of-order input had to contain with pushScope/popScope discipline. scope makes the containment first-class: the whole parse runs with the supplied scope current, so name resolution walks scope → parents and every auto-declare and inference lands rooted there. ce.createScope(bindings?, parent?) builds one from a declarations table, which turns each parse into a function of (latex, dictionary, declarations):

  const scope = ce.createScope({ h: 'function', p: 'tuple<3>' });
  const expr = ce.parse('h(u) = u^2', { scope });

  One binding per definition head is enough to make a definition parse against a predeclared name of a different arity — or against a builtin (N(x, m, s) = …) — with no mutation and no ordering requirement, and a binding for a subscripted spelling ({ theta_z: 'number' }) is what \theta_z resolves to instead of being declared. Trigger-spelled names now participate fully: a subscripted Greek-letter base consults the same joined-name resolution as ASCII names, so a function-typed binding (or resolveSymbol answer) for alpha_1 makes \alpha_1(x) parse as a function application — previously only ASCII bases could commit a joined name. The scope is caller-owned and readable: declarations() returns its entries with their post-inference types (as BoxedType, so entry.type.toString() is the canonical, fingerprintable spelling) and an inferred flag, sorted by name; narrowings() reports definitions in enclosing scopes that a contained parse narrowed (the one write an ephemeral scope cannot contain); dispose() releases the scope's definitions from configuration-change tracking. A definition harvested from one scope can seed the next — ce.createScope({ f: def }) installs the same object, preserving binding identity — and the scope's definitions are never auto-disposed, so a harvested definition outlives the call.

Issues Resolved

• Real, Imaginary and Argument are real-by-definition for the compile targets (Tycho item 147). The complexness analysis judged these heads by their operands, so Mod(Im(z), 1) over a declared-complex z tripped the GLSL real-only helper gate and failed closed — even though the projections always lower to a real scalar ((z).y, atan(z.y, z.x), .im) on every target. The analysis now short-circuits these heads as real-shaped regardless of their operand's type, so real projections of complex interiors (Mod(Im(b + a·ln(x+iy)), Im(w)), domain-coloring rows) compile again. Provably complex operands (Mod(√-2, 1), Mod(i·x, 1), Mod(Conjugate(z), 1)) still fail closed.

• A function literal with an invalid explicit Block body no longer prints a bare internal TypeError while boxing (Tycho item 150). Canonicalizing ["Function", ["Block", ⟨body with an Error node⟩], …] dereferenced the invalid block's missing scope, and the caught Cannot read properties of undefined (reading 'bindings') was printed raw to console.error before recovering to a non-canonical literal. The block is now rebuilt so scope creation runs even over an invalid body: the literal boxes canonically (still isValid: false), with no console noise. Two siblings fixed alongside: the nullary form silently produced a canonical literal with an unscoped block, and an empty Block body threw the same TypeError (it now follows the annotated branch's convention: an empty body is Nothing). The two recovery catch sites in applyOperatorDefinition now attribute anything they print (ComputeEngine: error canonicalizing \op`: …`) instead of emitting a bare message.

• Machine-precision .N() of a negative base to a rational power took the wrong branch. At machine precision, (-2)^{100/3} numericized to -1.08e10 (wrong sign — p = 100 is even), (-2)^{7/3} took the complex branch where the real root exists, and (-2)^{7/6} — a genuine even-q case — wrongly produced a real value. The exponent is numericized at the engine's 15-digit precision before the real-root convention applies, which put it far outside the branch decision's reconstruction tolerance. The tolerance now scales with the engine precision, threaded identically through the type handler and the compiled constant fold. A 288-cell rational sweep against an independent reference went from 53 mismatches to 0.

  Fixing this exposed a deeper, longstanding defect in the same fallback: every irrational's continued-fraction convergents eventually fall within any fixed tolerance, so a negative base raised to an irrational exponent could silently take the real branch — (-2)^{\sqrt2} and (-2)^{1/\pi} were wrong-real on both precision lanes, (-2)^e on the bignum lane, (-2)^\pi on the machine lane. The reconstruction now also requires a coincidence bound (the candidate rational must be identifiable as the value the double was rounded from, not merely a nearby convergent): π, e, √2, √5, ln 2 and plain non-rational floats now take the complex principal branch identically at every precision, and the compiled constant fold follows ((-2)^{\sqrt2} folded to a wrong real constant; it now folds to the complex value, matching .N()). Exact Rational exponents are unaffected by the bound. Known limitation: once an exponent has been numericized, a rational whose terms exceed what a 15–17-digit double preserves (denominator ≳ 3·10⁵) is indistinguishable from an irrational and takes the complex branch.

• Symbols declared with a non-finite type now report their finiteness. A symbol declared non_finite_number answered isFinite/isInfinity with undefined; both predicates now decide from the declared type (isFinite is false, isInfinity is true), completing the type-consult work started for function expressions in 0.100.2. The workaround checks this blindness had required in the Add/Multiply/ Divide type handlers were retired after an instrumented full-suite run showed the getters now subsume them everywhere.

• A parse or box under a partial canonical form no longer declares free symbols. ce.parse(s, { canonical: ['Number'] }) — any partial form — auto-declared every free symbol into the current scope, so containing the writes of an untrusted or out-of-order parse required pushScope/popScope discipline even though the result is not fully canonical. A partial form now follows the same symbol contract as the structural route: names resolve against the scope chain — an existing declaration still binds, and a holdUntil: 'never' constant still substitutes its value — but a name that resolves to nothing stays unbound instead of being declared. canonical: true, canonical: false and structural: true are unchanged.

• A function declared with a scalar return type is now rejected when added to a tuple. scalar + tuple is an error when the scalar is provable, and a declared (non-inferred) numeric result counts as proof. That guard never fired for a head declared with a function type — ce.declare('f', '(number) -> number') — because such a declaration produces a value definition rather than an operator definition, and the guard consulted only the latter: f(x) + (1, 2) stayed a symbolic Add. It now falls back to the head's value definition. The inferred cases are unchanged and still stay symbolic, an inferred numeric type being retractable evidence rather than proof: a signature inferred from a := body, or a function type inferred from earlier use.

• A built-in operator name used as a callback no longer compiles to a broken artifact. Map(xs, Sin), CountIf(xs, IsPrime) and the like compiled "successfully" and then threw _f is not a function at run time: the callback symbol fell through to a free-variable lookup instead of resolving to a function. Such a name is now eta-expanded into a shared emitted wrapper (const _fn_Sin = (_tv1) => Math.sin(_tv1)) — the same machinery a user-defined function callback already used — so the artifact runs and agrees with the interpreter. The expansion happens at the operator's required arity, so an operator with an OPTIONAL tail works too (Sum(Map(xs, Ln)): a callback site applies Ln unary and the optional base defaults), as does a unary operator with a target operator mapping (Negate), which the bare-operator-symbol path used to refuse outright. Where a built-in cannot be expanded at all — a variadic tail (Less), no required parameter (Random), or a wrapper body with no lowering on the target (IsPrime on JavaScript, any function value on the shader targets) — compilation now fails closed at compile time and the caller falls back to the interpreter, instead of producing an artifact that throws. (A bare single-uppercase-letter operator name such as D or N is exempt: the engine reads those as variables when they appear un-applied, so they keep their free-symbol reading.)

• Six classes of LaTeX round-trip defects are fixed (found by the corpus round-trip lane, npm run check:roundtrip — 18 of its 37 recorded failures cleared):

  • A symbol naming an operator, used as a value, serialized to the empty string, silently deleting the operand: (A, +) — the tuple of a set and its operation — parsed to ["Tuple", "A", "Add"] but serialized as (A,). Such a symbol now falls back to \mathrm{Add}, which parses back to the same symbol.
  • A one-operand Tuple serialized as (x), which parses back as plain x. It now serializes with a trailing comma, (x,), a spelling the parser already accepted.
  • A product containing an ellipsis (ContinuationPlaceholder) serialized with mixed separators (ab\times\dots\times z), so the juxtaposed run regrouped into a nested Multiply when parsed back; and a product of rationals with an ellipsis was merged into a single \frac, moving factors across the ellipsis and folding them (3/2 · 6/5 · … · X came back with a spurious 9/5). An ellipsis product now joins every factor with an explicit multiplication sign and is never merged into one fraction.
  • Prime serialized its exponent unbraced (A^\prime), so a following letter was swallowed into the command name: \angle BA'C serialized back to \primeC, which does not parse. The exponent is now braced.
  • A quantifier body serialized without delimiters, so \forall k\ge0, a_0=9\land a_1=3 parsed back with the \land bound above the ForAll. The body is now parenthesized when its precedence requires it, and — with quantifier operands now canonical (see Breaking Changes) — the body round-trips structurally.

• Find now types as the element, not the collection. Its static type was the whole collection's (Find([1,2,3], p) claimed list<integer>) while evaluation returns a single element or Nothing; it is now element | nothing.

• BaseForm's declared result contradicted its behavior (-> string | nothing while echoing its numeric operand); it now echoes the operand type.

• If without an else branch keeps the nothing arm in its type. A false condition yields Nothing, but the static type silently dropped that arm.

Improvements

• Structural-tier freeVariables, unknowns and references now derive bound variables from the operator's binding sites. On a structural tree (ce.box(…, { structural: true })) a binder's bound variable leaked as a free variable whenever its operand carried a raw parse spelling the free-variable walk did not know: ["Sum", ["Power", "n", 2], ["Tuple", "n", 1, 10]] reported n as free, while the canonical route reported nothing. The bound names now come from the operator definition's binding-site selectors, which read every spelling the binder accepts (Tuple, Element, Limits, a bare symbol, held or not), so the structural and canonical routes agree. Consumers that pattern-match raw binder spellings to build capture-avoidance sets can retire those collectors. A node with no operator definition (canonical: false) has no binding sites to consult and keeps the previous spelling-based recognition, and a binder whose variable survives into its result (D, Series) still reports that variable as free.

• The bare-assign route now broadcasts like the value route. Assigning an annotated or generic function literal directly (ce.assign('f', …) with no prior declaration) installs an operator definition whose broadcastability is derived from its parameter types, so f([1,2,3]) with f: (x: number) -> number — or forall T: number. (T) -> T — maps over the list instead of rejecting, matching both the declare-then-assign route and what compiled code already did. Unbounded generic identities still return their operand whole, and an empty source answers [] on every route.

• Re-assigning a function literal no longer changes its representation. Assigning the same annotated literal twice (the notebook re-run pattern) used to silently convert the operator definition into a value definition, losing the derived broadcast behavior; re-assignment now rebuilds the same representation as the first assignment.

• Function-literal signature markers may reference user-declared types when serialized to Cortex, and anonymous literals carrying a signature marker round-trip losslessly instead of dropping the ascription.

• A broadcast argument at a generic parameter now binds the element type. When a collection is admitted against a scalar-bounded type variable — the lift that makes Conjugate([1, 2, 3]) legal — the variable used to bind the whole argument, and the result was un-wrapped again only where it was the bare variable. A result that merely mentioned the variable then came out one rank too high: forall T. (T) -> tuple<T, T> over [1, 2] typed list<tuple<vector<…^2>, vector<…^2>>> against the value [(1, 1), (2, 2)]. The bound is still checked at the scalar base — admission is unchanged — but the variable now binds the argument's element type and the call site's ordinary broadcast wrap re-adds the rank, which gives one rule for every result shape: an echo (Chop, Conjugate) types exactly as before, and a result that mentions the variable is the per-element result with the argument's shape around it. Two static types change: a mixed-rank or union-typed argument now takes the broadcast wrapper's (coarser) shape answer, the same one its ground counterpart gets, and Remainder(M, 7)-shaped calls no longer widen to a union at all — matrix<finite_integer^(2x2)>, where the whole-argument bind gave list<finite_integer | vector<finite_integer^2>^(2x2)>. Only the kinds a broadcast actually maps are peeled: a set argument is admitted but never mapped (Conjugate(Set(1, 2)) stays a set), and a tuple stays atomic. In the same pass, a rank ≥ 2 argument to a generic function literal now maps to the scalar leaves on the value route too, matching the operator route and compiled code: with f: forall T. (x: T) -> tuple<T, T> assigned x |-> (x, x), a 2×2 argument evaluates to a 2×2 of pairs of scalars instead of applying the literal to whole rows.

• A malformed type annotation no longer swallows the statement after it. Recovery from a bad annotation ran twice — once inside the type subparser and once in its caller — so a declaration such as let x: )bad( = 1 resynchronized one statement too far and discarded the line that followed. The subparser now only diagnoses, and each caller resynchronizes at the unit its own grammar uses: a statement boundary for a declaration, the next , or closing bracket for one element of a list. A malformed annotation in a function's parameter list, in a |-> parameter list or in a match tuple pattern therefore costs only its own annotation — the parameter survives untyped and the rest of the list still parses. The |-> case is the one that was silently wrong rather than merely noisy: the parameters after the malformed one were dropped, so the lambda went on to parse at a different arity. The resync honors <…> nesting, so an unclosed applied alias (f(x: Pair<integer, string)) no longer mints a bogus parameter out of the type's own argument list.

• Generic values describe themselves, and an anonymous generic application is typed. A function literal assigned to a symbol declared with a polytype now carries that polytype as its own type on all three routes (ce.assign, the Assign operator, an annotated Cortex const/let), so the stored value reports forall T. (x: T) -> T instead of the arrow its erased body would infer — as long as every parameter of the clause mentions a quantified variable (a literal with a ground parameter still self-describes as its inferred arrow). Applying a generic literal anonymously — ["Apply", literal, arg], and the bare [literal, arg] it canonicalizes from — instantiates the clause too, so the application types finite_integer at 5 and string at "a" rather than falling back to unknown. And re-assigning an untyped literal over a signature that was itself derived from an earlier assignment now fully replaces it, arity included; a signature the author declared (any ce.declare form) stays sticky, as before.

• Euler derivative notation and \gamma now yield to a declaration. Both spellings are claimed by a parselet that runs ahead of symbol resolution, so no declaration could reclaim them: D_x + 1 parsed as the derivative of the constant function x ↦ 1, and \gamma was always the Euler-Mascheroni constant. Each parselet now consults the symbol oracle first — the engine scope, supplemented by the resolveSymbol parse option. D_x reads as a symbol when the joined name D_x is declared (the same sibling-name rule subscripted spellings already use) or when D itself is shadowed by a non-function declaration; a function-typed D keeps the derivative reading. \gamma reads as the symbol gamma when gamma is declared. Left undeclared, both notations parse exactly as before. This gives a host embedding the engine a way to say "these are my variables" without having to replace the LaTeX dictionary.

• Degenerate big operators (Σ_{i=a}^{a}, Π_{i=a}^{a}) now reduce. A big operator whose lower and upper bounds are structurally equal has a one-point domain, so it has exactly one term. Two reductions follow. First, at evaluation: a symbolic bound made the domain non-enumerable, so \sum_{i=x}^{x} i^2 stayed inert; one point needs no enumeration, and it now evaluates to x^2 (Product likewise). Second, at canonicalization: when the index does not occur in the body, the indexing set carries no information at all, so the "identity wrapper" spelling \sum_{i=d}^{d} f(x) folds to f(x) — the same generic-symbol fold family as x/x → 1. With several indexing sets only the degenerate, unused ones are dropped — and never one whose index a sibling indexing set's bounds reference. The canonicalization fold compares the bounds strictly syntactically: it never reads a symbol's assigned value (\sum_{i=a}^{5} with a := 5 keeps its Sum structure — values belong to evaluation, where the reduction does follow them). Bounds that are not provably a fixed one-point domain are unaffected: ±∞ and NaN bounds keep their previous behavior, impure or invalid bounds (two syntactically identical RandomInteger(1,6) draws) stay symbolic, and literal equal bounds with the index used (\sum_{i=5}^{5} i^2 → 25) still enumerate as before. The evaluate-path reduction substitutes the bound for the index under a capture guard; a body whose inner binder could capture it stays symbolic rather than being silently corrupted (and the guard's refusal is final — it does not fall through to the closed-form rewrites, which carry no such guard).

• Compile-time CSE now merges repeated pure user-function calls everywhere, including named callbacks. The 0.100.0 admission of pure user-function applications applied only inside emitted definition bodies (where a repeated recursive self-call made compiled recursion exponential); a repeated call at the top level of the compiled expression — f(x+1) + f(x+1)^2 — still compiled to two calls. Both compiler harvest routes now admit them, behind the same transitive callee-body validation (each level's purity is re-derived against current bindings at compile time, so a callee that draws, writes, or splices caller-supplied source stays un-merged). In addition, a named callback that resolves to a validated pure function literal no longer blocks eligibility: two identical Map(xs, f) applications with a pure user-defined f now compile to one traversal, and the same applies to typed callbacks of eager operators such as CountIf (a drawing f still compiles to two — draw streams and call counts are preserved; a callback or callee name shadowed by an enclosing parameter is conservatively never merged). Callbacks naming built-in operators (Map(xs, Sin), CountIf(xs, IsPrime)) are now admitted as well: the compiler eta-expands them into a shared emitted wrapper, so what they do is the built-in's own deterministic, effect-free emission. They merge when the operator is pure, is the engine's own definition for that name, and has at least one required parameter and no variadic tail — a drawing built-in (Random), a variadic one (Add, Less), a name a user definition shadows, a name a caller vars entry maps, and a name a caller functions / operators mapping overrides all stay conservatively excluded. Opt out as before with compile(expr, { cse: false }).

• FindFit/FindRoot now report a setup-phase deadline in band (Tycho item 118 addendum). A time budget consumed entirely before the solver started — evaluating the data operand, differentiating the model, compiling it — used to escape as a bare CancellationError the caller could only duck-type, which an interpreted model over a few hundred rows hits routinely under a tight ambient budget. Such an expiry now answers the same record shape a mid-solve expiry does, with timedOut: True and a new phase entry naming how far the call got: "setup" (nothing was fitted — iterations is 0, residualNorm is NaN, and the reported parameters are the starting guesses) or "solve" (genuine best-so-far, as before). Only an expired time budget converts: every other failure during setup — a malformed model, bad data, an abort signal — behaves exactly as it did. The keys appear only on a timed-out record, so a successful fit is unchanged.

Performance

• Symbolic equality of free-variable expressions now samples before it simplifies. The eq() free-variable branch ran expand+simplify on both sides to try a structural proof, and only then fell back to stochastic sampling — on large trees the symbolic pass costs hundreds of milliseconds and, whenever the sides genuinely differ, contributes nothing the sampler doesn't decide alone. The order is now reversed: sample first (a compile plus ~50 deterministic point evaluations), and run the expand+simplify proof only when sampling is uninformative (no compilable/finite sample points). Verdicts are unchanged: a sampled agreement was already accepted as true, a sampled disagreement already degraded to undefined under the truth-under-constraints contract, and an identity provable by simplify cannot genuinely disagree at a shared sample point. On a consumer's Voronoi document whose piecewise rows compare each broadcast element against min(⟨list⟩) (18 identical expand+simplify passes of the same min-expression), the document build drops from 14.6 s to 6.2 s — faster than releases that predate the 0.100.2 \bmod serialization fix, which had made the comparison trees honest (and bigger).

• Compile-time complexness analysis is no longer quadratic on large expressions (Tycho item 148). The 0.100.2 operand-consulting fixes (items 144/143) made isComplexValued walk a node's whole subtree per query, and the GPU emitters query per node — a deeply nested expression (a textually inlined user-function chain) paid O(n²): a depth-6 nested chain spent 82% of its GLSL compile (1.5 million analysis calls) in the walk, roughly doubling large shader compiles relative to 0.100.1. The analysis is now memoized per compilation with a LAYERED memo that mirrors the context's lexical nesting: entering a block frame or binder mask pushes a fresh layer (an answer cached under a mask can never be reused outside it), and leaving restores the enclosing layer — so binder-dense bodies (nested Sum/Product) memoize too, instead of wiping the cache at every mask crossing. Compiled output verified byte-identical across targets. The depth-6 chain compiles 7× faster than unmemoized (and ~2× faster than 0.100.1, which did fewer, cheaper walks); scaling on the regressed class — deeply nested inlined expression chains — is near-linear in expression size again. (Deeply nested binder chains, Sum-in-Sum, keep a pre-existing superlinear analysis cost that 0.100.1 shares — measured, not part of this regression.)

0.100.2 2026-08-03

New Features

• Repeat compiles on the JavaScript target. Repeat(7, 3) previously failed closed (interpreted fallback); it now lowers to a native array construction with interpreter parity at the edges: the value is evaluated exactly once and replicated (Repeat(Random(), 3) yields three copies of a single draw — and consumes its draw even when the count is ≤ 0, matching the interpreter), a zero or negative count yields [], and the 1-argument infinite form and a statically non-finite count still decline — the interpreter leaves Repeat(7, ∞) unevaluated, so a compiled [] would be a valid-looking value with the wrong meaning.

• Binomial/Choose compile on the GPU targets. A literal k ∈ 0…8 unrolls to the falling-factorial form on GLSL and WGSL (Binomial(x+1, 2) → (((x + 1.0) * ((x + 1.0) - 1.0)) / 2.0)), matching the interpreter's generalized semantics for non-integer and negative first operands (Binomial(5.5, 2) = 12.375, Binomial(-1, 2) = 1). An impure (Random-family) operand is hoisted and drawn exactly once; Binomial(Random(), 0) declines rather than folding to 1.0 (the interpreter consumes that draw); a statically non-finite first operand declines (Binomial(∞, k) is NaN in the interpreter for every k, including k = 0); non-literal, negative, non-integer, or larger k fail closed.

Improvements

• Products and quotients with a provably non-finite real factor now type non_finite_number. New ratified rule: a provably non-finite real factor is implicitly nonzero — proven signs are required only of the finite factors. 2\ln(0) and \ln(0)/2 now type non_finite_number (previously the top type number); shapes admitting 0·∞, ∞/∞ or ∞·i keep the sound widen. Structurally, Ln(0) now reports isFinite === false and isInfinity === true from its static type (both were undefined), valueOf() projects a direction-proven infinity (Ln(0).valueOf() is -Infinity; ~oo is reserved for provably non-real values), \ln(0)/\pi canonicalizes to \ln(0) and 2/\ln(0) to 0, and an unfolded finite-real-over-±∞ quotient types finite_integer (the value is exactly 0). Compiled emissions are unchanged.

Resolved Issues

• .subs() on a structural expression now preserves the structural form. A structural receiver requested a CANONICAL rebuild, so substituting into one silently canonicalized it: the parse vocabulary structural form exists to preserve (Subtract, Divide, InvisibleOperator, Delimiter, operand order) was erased and exact literals were folded — substituting k+1 for x in a structural 2(x+1)-\frac{y}{3} returned Add(Multiply(2, Add(k, 2)), …) instead of the structural Subtract(InvisibleOperator(2, Delimiter(Add(Add(k, 1), 1))), Divide(y, 3)). The receiver's form is now preserved three ways (canonical → canonical, structural → structural, raw → raw); an explicit canonical option is unchanged. The binder-rebuild internals (rewriteWithBinders, used by the escaping-scope re-bind and by binding-keyed substitution) had the same conflation and are fixed the same way. .map() had the mirror-image defect — a structural receiver fell into the RAW rebuild arm, keeping the shape but silently losing the binding — and now preserves the receiver's form under the same three-way rule.

• GLSL compile of Mod over wide-typed real expressions. The shader targets' real-only helper gate refused operands whose type merely could be complex: the complex type a Sqrt/Ln of unknown sign carries since 0.100.0 propagated to enclosing arithmetic (10^5·√(⌈x⌉²+⌈y⌉²)), through boolean nodes, and into piecewise conditions, so Mod expressions over plot variables failed closed (D6) where 0.99.0 compiled them. The complexness analysis now short-circuits boolean- and string-typed nodes and, for arithmetic heads that only propagate complexness (Add, Subtract, Multiply, Divide, Negate), consults the operands — honoring the unknown-sign Sqrt/Ln/Log real-kernel contract — instead of the widened type. Provably complex operands still fail closed.

• Min/Max over a collection whose element type is unknown. With a base declared indexed_collection, Distance(S, p)'s result type degraded to scalar number (the broadcast arm was invisible through the elementless type), and Min then compiled to the variadic-scalar Math.min(...) — which returns NaN when handed the runtime array, silently, behind success: true. Distance now reports number | list<number> when an operand's collection element type is undecidable, and the Min/Max JavaScript lowering emits a runtime shape projection (reduce an array, pass a scalar through) for operands that could be collections, matching the interpreter both ways.

• invisibleMultiply serialization option vs \bmod. With invisibleMultiply: '\\cdot', Mod(k·f, 1) serialized as k\cdot f\bmod1, which re-parses as k·Mod(f, 1) — the Mod serializer decided parenthesization assuming juxtaposition, which binds tighter than \bmod while an explicit \cdot binds looser. A Multiply operand of an infix \bmod is now parenthesized whenever the option is set (products that serialize as \frac remain unwrapped).

• Ordering comparisons over a provably complex operand now fail closed at compile time. Less/LessEqual/Greater/GreaterEqual with a complex-valued operand (i·x < 0) compiled to a raw JavaScript comparison of a {re, im} object — a silent false behind success: true — while the interpreter correctly leaves the comparison symbolic (the complex numbers are not ordered). Such comparisons now decline with a clear diagnostic on every compile target. Equal/NotEqual keep their complex support, and real-kernel expressions of unknown sign (√x < 2) still compile.

• Non-canonical trees are no longer restructured by pretty serialization. Serializing a canonical: false expression (toMathJson/toLatex with prettify) could rebuild a Multiply containing a symbolic Divide factor through the canonical product machinery: explicit Delimiter fences were dropped, factors reordered, and the round-trip changed the expression (Mod((k·f), 1)/n + y re-parsed with the dividend split). Pretty rewrites on a non-canonical tree are now shape-preserving — order-preserving numerator/denominator split, no factor sorting, fences kept. Canonical serialization is unchanged.

• Impure operands spliced by multi-use compile templates drew more than once. A lowering that splices a compiled operand string into its emitted code more than once re-evaluates a Random-family operand at run time — a silent wrong value that also shifts every later draw. A full audit of the JavaScript, GLSL, WGSL, interval, and Python targets fixed twelve such sites: Equal/NotEqual over a complex operand and across n-ary chains, Range(start, stop, step) (which re-drew once per element), Round, odd-degree Root, Variance (12 draws where the interpreter makes 2), complex Argument/Conjugate, chained relations and element-wise selection masks, Match subjects, and a double-compile in the complex Add fallback that orphaned a hoisted draw. Impure operands are now bound to a temporary exactly once and in argument order — binding only the middle of Random() < Random() < 0.9 had executed the second draw first, inverting the comparison. Pure emissions are byte-identical throughout, and regression tests count draw sites in the emitted code.

• ContrastingColor emitted invalid WGSL. Both the 1-argument and 3-argument forms lowered to a GLSL-only ?: ternary on every GPU language; WGSL now emits select(…) (the GLSL emission is byte-identical), and an impure (Random-family) color operand fails closed instead of being spliced twice.

0.100.1 2026-08-02

Breaking Changes

• PointZ applied to a 2-D point is now a typed incompatible-dimensions error instead of the NaN absence marker — at type-check time when the operand's type statically proves 2-D (tuple<number, number>, or a list or set of such points, in either the tuple or coordinate-row spelling), and at evaluation time otherwise. The broadcast over a list of 2-D points errors identically, and the JavaScript compile of a statically 2-D operand declines. A statically-absent component is a type-level fact: the silent NaN masked upstream pipeline defects. PointX/PointY, all 3-D behavior, and the compiled runtime NaN marker for dynamically-shaped bases are unchanged.

New Features

• Multi-clause function definitions can now be declared before they are defined. declare("J", "(number, complex) -> complex") followed by clauses such as J(0, z) = z previously failed with incompatible-type (a literal-parameter clause is a narrowed arm of the declared signature, not a function subtype) — and failed silently, leaving the symbol undefined. Clauses are now checked arm-shaped against the declaration (parameters and result must be subtypes of the declared ones), a genuinely incompatible clause errors loudly without corrupting the definition, and the declared signature is preserved on the installed function — making declare-then-define usable for recursive clause sets (let fact: (number) -> number; fact(0) = 1; fact(n: integer) = n * fact(n-1)).

• Complex-valued multi-clause function definitions now compile. A recursive clause set such as J(0, z: complex) = z; J(n: integer, z: complex) = J(n-1, z)^2 + z_0 evaluated correctly but declined the JavaScript compile target while its real-valued twin compiled; the clause dispatcher now carries the complex {re, im} convention through parameter guards, call-site coercion, and mixed real/complex clause bodies.

• The LaTeX serialization style options (rootStyle, fractionStyle, indexStyle, powerStyle, logicStyle, numericSetStyle, groupStyle, applyFunctionStyle) can now be specified as a constant, in addition to a function of the expression and of its nesting level:

  ce.latexOptions = { rootStyle: 'solidus' };
  expr.toLatex({ fractionStyle: 'inline-solidus' });

  Previously, only the function form was supported, and a string value serialized to an empty string. A string that is not one of the values accepted for that option now throws when the option is set, instead of producing an empty serialization.

• Distance broadcasts over point lists. Distance(S, p) (either argument order) where S is a list of points — spelled as tuples [(0,0),(3,4)] or as coordinate rows [[0,0],[3,4]] — returns the list of per-point distances, on both the interpreted and compiled routes, so min(Distance(S, p)) computes the nearest-point distance directly. Distance(S, T) over two point lists is pairwise with strict length matching. Scalar and string arguments are still rejected.

• Point operators are aligned over point lists: PointX/Y/Z project a coordinate-row list (PointX([[10,11],[20,21]]) is now [10, 20], not the first row), and Norm over a list of point tuples returns per-point norms [0, 5, 10] on both routes, matching Abs. Norm over a plain matrix (list of lists) keeps its Frobenius meaning. Compiled Norm and Abs previously disagreed with the interpreter on these shapes (a flattened norm and componentwise values behind success: true).

Performance

• Registering many interdependent user functions (declare + assign chains, the shape a document importer produces) was quadratic in the number of functions: the type accessor's cache key evaluated the effect projection (isPure, made binding-aware in 0.100.0) before the cheap constant-operand short-circuit, forcing every stored function literal's full signature to re-derive down the call chain on each registration. Reordering the check and adding a per-generation fast path makes registration near-linear — 9–14× faster at 120 functions, 13.7× at 240 — with byte-identical isPure/effects/type answers. New benchmark: benchmarks/effects-registration.ts.

Resolved Issues

• isPure and .effects no longer claim a seed frame contains a lazy view that escapes it. WithRandomSeed(42, Map(xs, x => Random())) — and the [Random() for k = [1...6]] comprehension spelling of it — reported isPure: true with no effects, even though a lazy view draws at materialization, from whatever frame is active then, so the values were genuinely live. Such an expression now reports isPure: false and ["random"], including when the view leaves the frame as a cell of a returned List/Tuple/Pair or as a block's result. The runtime semantics are unchanged: materializing inside the frame (WithRandomSeed(42, ListFrom(Map(...))), an index, a reducer) still replays, and still reports pure — as do a view whose element body draws nothing and a seeded frame around an ordinary draw.

• Two-sample bracket ranges with decimal anchors now compute their step exactly. [1.008, 1.016...5] previously differenced its anchors in binary floating point, baking the step 0.008000000000000007 into the range — 499 elements ending at ~4.992 instead of 500 landing on the 5 anchor. The step is now derived from the anchors' decimal digits in exact integer arithmetic (0.008), matching the compound-anchor spelling [1+0.008, 1+0.016...5]. Anchors with no short decimal form keep their float step.

• Symbolic differentiation declines honestly instead of blowing up. The derivative of a deeply self-nested expression (e.g. a 12-deep √(x+√(x+…)) chain, whose second derivative is a 6.5-million-character expression taking 45+ seconds) doubles its tree per level; differentiation now tracks the size of what it builds and, past an internal budget (25,000 nodes, ~30× the largest derivative in the test corpus), aborts and leaves the D/Derivative inert — the established decline convention — in bounded time rather than hanging.

• A divergent definite integral is no longer reported as a confident measurement. \int_0^1 \frac{1}{x}\,dx numericized to 709.08956571281 ± 0.00000000074 — the value at which the adaptive quadrature's refinement toward the singularity ran out of floating-point range, dressed up as a measured quantity. Numeric integration now checks the series of dyadic shells it sheds while refining toward each endpoint: a convergent improper integral's shells shrink geometrically, a divergent one's do not. Divergent integrals return NaN on all three numeric routes (.N(), iterated .N(), and compiled code), and detection also stops the refinement early, so \int_0^\infty x\,dx resolves in ~1 ms instead of ~560 ms. Legitimate improper integrals — \int_0^1 \frac{1}{\sqrt{x}}\,dx, \int_0^1 \ln x\,dx — are unaffected, as are proper integrals, whose results are bit-for-bit unchanged.

• The order in which functions are defined no longer changes their meaning. A definition body that referenced a function before it was defined — for example g(t) \coloneq 2a(t) registered ahead of a(t) \coloneq [\cos t, \sin t] — froze the reference as a multiplication (2 \cdot a \cdot t), producing scalar results interpreted and null/NaN from compiled functions behind success: true. Such provisional readings are now re-derived when the referenced name later gains a function definition, so every registration order yields the same interpreted and compiled results. Genuinely scalar juxtaposition (2x(t+1) where x never becomes a function) is unchanged.

• Function parameters whose bodies index them are now treated as collections. A definition such as h(v) \coloneq v_1 + v_2 with h declared (list<real>) -> real parsed the subscripts as unrelated symbols, an At-indexing body inferred a scalar parameter (so list arguments broadcast elementwise instead of applying), and the JavaScript compile target declined At over a declared list parameter. All three are fixed: parameters are bound at their declared types while the body is parsed, an indexing body infers a collection parameter, and the compile target admits indexing over such parameters with a runtime shape guard.

• The derivative of trigonometric functions now honors angularUnit. In degree mode, D(Sin(x), x) evaluates to \frac{\pi}{180}\cos x (and inverse trigonometric derivatives are divided by the conversion factor), on both the interpreted and compiled routes, for both by-reference functions (f' where f := x \mapsto \sin x) and inline \frac{d}{dx} expressions. Previously the interpreted derivative and the compiled by-reference derivative returned radian-convention values, and compiled ND double-converted.

• Sqrt no longer claims a complex result type for a radicand whose non-negativity only constant folding can establish. Machine floats are not folded at canonicalization, so \sqrt{1-0.2^2} reached the type handler with an undecided sign and typed as finite_complex, while the folded \sqrt{0.96} typed as finite_real. A radicand that is pure and has no unknowns is now folded to decide the sign. A negative radicand (\sqrt{0.2^2-1}) still types as complex, and a radicand with unknowns (\sqrt{x}) is unchanged.

• Declared types are enforced for symbols named with a single uppercase letter. An argument-validation repair intended for bare standard-library operator symbols such as N and D used in value position also fired for any user-declared single-letter symbol that failed a parameter type check, silently skipping the declared-type check. The repair is now gated on the provenance of the shadow it rebinds to.

• Arguments with free variables are no longer exempt from declared-type checking when their type is provably incompatible. Applying a function whose parameter is, e.g., tuple<number, number, number> to an unassigned symbol declared string (or any provably disjoint type) now reports incompatible-type instead of silently accepting the call. Arguments whose type could still turn out compatible — unknown, inferred symbols, unions with a matching arm, same-category collections — continue to defer to runtime exactly as before.

Performance

• GPU compilation no longer re-merges its function table on every compile. The table reached V8's fast-property limit, so the per-call object spread allocated ~45 KB of transient dictionary-mode garbage per compilation; the merged table is now memoized per target instance.

0.100.0 2026-08-02

Breaking Changes

• Assignments now enforce declared types consistently. Declare-with-value, Assign, and ce.assign() all reject values incompatible with an explicit symbol type. Inferred types retain their widening behavior.

• Purity and effects reporting is more precise. expr.isPure and expr.effects now account for which operands an operator evaluates and for the current bindings of referenced functions. Consequently, held expressions and seeded-random blocks can be pure, function literals do not inherit the effects of calling them, and named effectful callbacks are no longer reported as pure. Unresolved forward references conservatively have unknown effects.

• Effects from partially applied functions occur only when all required arguments have been supplied. Partial application no longer evaluates an effectful body early or repeats its effects.

• Contradictory operator declarations are rejected. In particular, pure: true cannot be combined with drawsRandom: true; inconsistent legacy flags, effect annotations, and effects: declarations also fail registration.

• Errors now propagate through strict built-in operators and function application. An invalid operand generally produces the underlying Error value instead of leaving an inert expression. Collection constructors and error-observing or lazy operators can still contain or inspect errors. Assume returns string status values such as "ok" and "not-a-predicate".

• Literal Nothing arguments are removed consistently. f(Nothing), Apply(f, Nothing), and Nothing |> f now all behave like f(). An expression that evaluates to Nothing is still passed as an argument. Invalid Pipe callees now return the error directly.

• Positional collection operators require indexed collections. First, Second, Third, and Last now reject sets and other non-indexed collections. Empty indexed collections still return Missing.

• Several elementary functions now report complex result types when their real-valued domain cannot be proven. This includes Sqrt, Ln, Log, Arcsec, and Arccsc. Provably in-domain operands retain a real type; provably negative operands use complex helpers when compiled.

New Features

Functions, types, and effects

• Multi-clause function definitions can dispatch by arity, literal value, and parameter type. The most specific clause wins, declaration order breaks ties, and redefining the same parameter domain replaces that clause. Recursive clause sets compile to JavaScript. Partial application of a clause set is not supported.

• Cortex programs can declare nominal types and structural aliases with type name = … and type alias name = …. Declarations provide checked constructors where appropriate; nominal values are opaque, support structural equality by tag and payload, and can expose named fields with value.field. Record-shaped nominal types can use a same-name constructor function for validation or normalization. Type tags are erased by compiled output.

• Function signatures can declare effects. Types and Cortex definitions accept pure, any, or effect labels including random, scope, network, time, and file-system effects. Inferred effects follow the function body; explicit effects are checked contracts. Callback parameters can use effect annotations to restrict accepted functions.

• Operator definitions accept an effects: field. The existing pure and drawsRandom properties remain supported as shorthand.

• Expressions and function types expose effects directly through expr.effects and type.effects. Effect-discharging operators such as WithRandomSeed can absorb an effect, while Hold defers the effects of its contents until release.

Cortex language

• Function definitions accept literal parameters, including strings, booleans, finite numbers, NaN, and infinities. Unicode mathematical symbols such as π, ∞, ⅈ, and ℝ now resolve to their standard constants or sets. Non-finite literal names are reserved; verbatim identifiers remain available when those spellings are needed as names.

• match supports inclusive numeric range patterns, including negative and infinite bounds. Range patterns can be combined with alternatives and guards and compile on every target.

• Errors can be handled in Cortex. match can catch error values, and the new held predicate IsError(x) detects errors without propagating them. Propagated errors include a non-rendered trace available through MathJSON and the error-trace APIs.

• The structural equality operator === now evaluates. It performs total, exact structural comparison, including for symbolic expressions, Missing, and NaN.

• Count accepts either a value or a predicate, and Table accepts tuple iterator specifications. Tuple and brace iterator forms now behave consistently for Sum, Product, and Integrate.

• Static type errors are reported by cortex check and before program evaluation. Checking canonicalizes without executing effects.

Compilation

• PointList with collection-valued components compiles to JavaScript. GPU coordinate accessors can also project compatible point-list components.

• At compiles to GLSL and WGSL for statically sized numeric collections, including guarded dynamic scalar indexes and literal gathers or masks.

• D, Derivative, and ND compile on all targets when they can be reduced to a compilable form.

• Non-finite numeric literals compile to GLSL and WGSL.

• Python compilation supports collection equality and inequality. Norm(matrix, 2) now compiles with the interpreter's Frobenius-norm semantics.

• Added Cortex transition guides for Python and Mathematica users.

Improvements

• Unknown Wolfram Language and JavaScript collection-function names now suggest the corresponding Cortex operators.

• Static result types are tighter for trigonometric, hyperbolic, extrema, Pochhammer, and collection-rank operations when the result can be proven real or finite.

• User-declared type names now resolve consistently in annotations, signatures, type predicates, collection operators, and MathJSON declarations. Literal value types and bounded numeric refinements now accept their own values.

• Range and Linspace can enumerate exact symbolic bounds and steps that have a numeric value, such as multiples of π. Bracket range syntax also recognizes more exact arithmetic progressions.

• Sum and Product evaluate pure, closed bound expressions such as Length(P) while retaining symbolic behavior for genuinely free bounds.

• Distance accepts the same point-or-point-list union types as related point operators and reports a clear error if a point list reaches scalar distance evaluation, in both interpreted and compiled code.

Performance

• Numeric integration retains a sufficiently accurate deterministic quadrature estimate instead of replacing it with a much slower Monte Carlo estimate. This substantially improves nested and difficult integrals.

• Lazy function-applying collections, including Map, Filter, FlatMap, Scan, Tabulate, and Iterate, memoize evaluated elements per instance. Cache invalidation now follows actual symbol and configuration dependencies, so unrelated assignments no longer discard cached collection elements.

• Exact evaluation of sufficiently large, bounded integer Map broadcasts can use compiled float64 arithmetic when exactness is statically guaranteed.

• Compiled function bodies reuse repeated pure user-function calls, avoiding exponential work in recursive definitions. Common-subexpression elimination now also recognizes engine-provided compiled operators.

Issues Resolved

Parsing and serialization

• Chained postfix indexes parse uniformly as nested At expressions for symbols, subscripted expressions, and literal collections.

• Ranges with compound first anchors, such as n+1..n+10, [2n..3n], or x = m+n...m+n+4, bind the whole anchor expression instead of absorbing part of it into the surrounding addition, multiplication, or negation. This applies in brackets, bare expressions, and relations. A parenthesized range opts out: n+(1..10) remains a broadcast addition.

• Dictionary-valued function results preserve parameter bindings through MathJSON round trips, including in recursive functions.

• Roots serialized with solidus or quotient notation now delimit a Power base correctly.

• Invalid multi-argument Exp and Exp2 expressions remain well-formed inert expressions with an arity error.

Evaluation and functions

• At reports incompatible dimensions when more indexes are supplied than a collection can consume.

• match inside a function uses the current call's parameters rather than retaining values from the first call.

• Seeded-random frames no longer incorrectly mark enclosing function literals or Map callbacks as random.

• Purity and effect results for recursive functions are stable and no longer depend on which expression is queried first.

• Compiled lambdas with unbound symbols, including symbols reached through assigned values or function bodies, decline cleanly instead of throwing a JavaScript ReferenceError. Interpreter fallback numericizes symbolic results and is also available for failed interval compilation.

• Numeric use of a list-valued function no longer widens its inferred result to a scalar or causes compiled reductions to use scalar arithmetic on arrays.

• FindFit and FindRoot observe ambient time limits during expensive iterations and return their best result with timedOut: True when possible.

• Differentiating control-flow or binding operators such as Which, Sum, and Integrate stays symbolic instead of throwing or producing an invalid slot-wise derivative.

Collections

• Bounded Take expressions are recognized as finite even when the source length is unknown, allowing finite prefixes of infinite filtered collections to materialize.

• Bare _ works as the identity function in function slots, and wildcard predicate shorthand is accepted consistently by eager and lazy collection operators.

• Unary Iterate functions receive the accumulator, and indexed access now agrees with iteration about the first emitted value.

• Collection operators resolve evaluable numeric arguments such as N-1. They remain symbolic, rather than using a default, when a required numeric argument is unresolved.

• Partition distinguishes unresolved integer sizes from predicates and no longer throws on an unbound size.

• Seeded comprehensions materialize consistently with other lazy collections.

Numerical evaluation and types

• Numeric roundoff cleanup is independent of ce.tolerance; comparison and explicit Chop operations continue to honor the configured tolerance. Compiled Chop now uses that tolerance, and exact operands remain exact.

• The reported uncertainty of an iterated numeric integral includes inner-level quadrature error. Previously only the outermost level's own estimate was reported, which could present a result with inner error as exact.

• Bignum Sin and Cos preserve small representable values near zero crossings.

• Static types for non-finite, complex, and out-of-domain results were corrected across arithmetic, special functions, elliptic functions, inverse trigonometric functions, and complex component operators.

• Domain boundaries are classified with exact comparisons for exact literals.

• Non-radian angle conversion preserves the imaginary component of complex results.

• LCM(0, 0) returns 0, and SigmaMinus1 preserves exact results under evaluate().

Compilers

• Compiled Mod, Remainder, division, and negation now parenthesize compound operands correctly across JavaScript, GPU, and Python targets.

• Impure operands used by compiled remainder, modulus, and selected GPU functions are evaluated exactly once.

• Broadcasts over literal lists emit target-appropriate code for JavaScript, Python, GLSL, and WGSL instead of leaking JavaScript syntax into other targets.

• GPU compilation rejects unsupported alpha colour constructors, non-finite loop bounds, mismatched shapes, and scalar-only operations on vectors or matrices instead of emitting invalid or silently incorrect shaders.

• GPU Max and Min over one collection reduce to a scalar instead of returning the input vector.

• JavaScript compilation now broadcasts Sign, Arctan2, Hypot, and Sinc over lists.

• Complex constant folding is consistent with realOnly: non-real constants produce NaN in real-only mode and principal complex values when complex compilation is supported.

• Compiled InverseHaversine supports complex results on JavaScript and reports an appropriate complex static type for symbolic inputs.

Benchmarks

Numeric performance (200-digit precision)

Median time per call, in microseconds — lower is better. — means the tool returned no usable result at that precision.

Expression	CE 0.100.0	CE 0.99.0	SymPy	math.js	Mathematica
\pi^2	8.2	8.7	201	190	3.7
\sin 1	25	23	253	599	5.8
\cos 1	24	23	240	668	7.6
\ln 2	15	15	375	4,939	4.1
e^{\pi}	15	15	238	5,375	5.1
\zeta(3)	1,707	1,716	290	—	54
\Gamma(\tfrac13)	915	933	382	—	233
\psi(\tfrac13)	797	786	3,051	—	192

Symbolic capability & performance

Each cell is how many times faster than Mathematica that engine is on the case (Mathematica ÷ engine, so higher is better; Mathematica itself is 1×). — means the engine can't do the case; ✓ means it solves a case Mathematica can't. Compare the CE 0.100.0 and CE 0.99.0 columns to see what is new this release (a — under 0.99.0 next to a number under the current build). The CE + R/F column is the current build with the opt-in Rubi integrator + Fungrim identities loaded (loadIntegrationRules / loadIdentities), on the same minified bundle.

Operation	CE 0.100.0	CE + R/F	CE 0.99.0	SymPy	math.js	Mathematica
Antiderivatives
\int\frac{1}{\sqrt x}\,dx	3.5×	1.8×	2.8×	0.4×	—	1×
\int\frac{x}{\sqrt{1-x^2}}\,dx	6.4×	1.0×	5.7×	0.08×	—	1×
\int\frac{1}{x^3+1}\,dx	3.3×	0.4×	2.7×	0.3×	—	1×
\int\frac{\sqrt x}{1+x}\,dx	—	1.5×	—	0.1×	—	1×
\int\frac{x}{(1+x)^{1/3}}\,dx	—	0.9×	—	0.009×	—	1×
\int\frac{x^2}{(1+x)^{1/3}}\,dx	—	0.8×	—	0.006×	—	1×
Derivatives
\tfrac{d}{dx}\sqrt{1-x^2}	0.03×	0.03×	0.03×	0.001×	0.003×	1×
Simplification
\sqrt{3+2\sqrt2}	40×	29×	28×	—	—	1×
\sqrt6\,x+\sqrt2\,x	73×	38×	51×	3.1×	16×	1×
Evaluation
\lim_{x\to0}\tfrac{\sin x}{x}	36×	17×	34×	3.0×	—	1×
\lim_{x\to\infty}(1+\tfrac1x)^x	4.9×	3.6×	4.7×	2.2×	—	1×
\int_1^2\tfrac1x\,dx	18356×	20053×	17734×	310×	—	1×
\int_{-\infty}^{\infty} e^{-x^2}\,dx	258×	110×	251×	2.6×	—	1×
Solving
x^4+x^2-1=0	0.3×	0.2×	0.3×	0.06×	—	1×
x^3-x-1=0	1.4×	1.6×	1.3×	0.04×	—	1×

Across the cases both solve, Compute Engine is a median 3.5× faster than Mathematica (up to 18356×) — in the browser, not a proprietary kernel.

_{Measured 2026-08-02 · Compute Engine 0.100.0 (current build @ 14fc06c2) · published 0.99.0 · SymPy 1.14.0 · math.js 15.2.0 · Mathematica 14.3.0 for Mac OS X ARM · Node v22.13.1. Correctness is verified numerically against an independent mpmath reference, never another tool. Reproduce with npm run build production && ./venv/bin/python3 benchmarks/gen_cases.py && node benchmarks/report.mjs && node benchmarks/report_changelog.mjs.}

0.99.0 2026-07-30

New Features

• User-defined functions now compile to GLSL and WGSL. A function declared in the engine (f(u,v) := …) and called from a GPU-compiled expression is emitted once as a real shader function and called by name — matching what the JavaScript and interval targets already did — instead of failing as an unknown operator and forcing callers to inline the body at every call site. Definitions arrive on the compilation result's preamble alongside the _gpu_* helpers, so existing shader assembly keeps working; when f calls g, g is declared first. Signatures are synthesized statically (declared parameter types, or the body's inferred shape: float, bool, vec2–vec4, complex as vec2); anything a shader cannot express fails closed with a diagnostic naming the parameter — including recursion (GLSL/WGSL forbid it; it still compiles on the JavaScript target), collection-valued arguments beyond the static vec2–vec4 shapes, and argument shapes that disagree with the declared parameter type. Undeclared parameters default to float. Caller-declared types on compileFunction parameters and on shader inputs/uniforms are authoritative for these checks, with element-aware matching across both languages' spellings (bvecN, ivecN, vec2<i32>, …) — a vec2<bool> argument no longer passes for a vec2<f32> parameter. A WGSL body referencing a declared input now correctly emits input.<name> (previously a bare, undeclared identifier).

Performance

• Compiled code now shares repeated subexpressions (common-subexpression elimination) on the javascript, interval-js, and python targets: a pure subtree occurring several times inside one compiled expression is bound to a temporary once and reused, instead of being recomputed at every site. Corpus-extreme shapes (a 500-node subtree repeated 128×) compile ~50% faster because the emitted source collapses, and run up to ~1.9× faster when the repeats involve runtime helpers the JS engine cannot eliminate itself. Sharing is conservative by construction: random draws, user-defined function applications, named callbacks, caller-supplied custom lowerings, and anything inside a conditionally-evaluated position (unselected Which/If arms, short-circuited And/Or tails) are never merged, so values, draw streams, and selection laziness are unchanged. Opt out per call with compile(expr, { cse: false }). Design notes: docs/plans/2026-07-28-compile-cse-design.md.

• Compiled output is now byte-for-byte deterministic on every target: compiler-generated temporaries (chained-relation operand bindings, loop accumulators, complex power chains — and the new CSE temps) draw deterministic _tvN/_cseN names from a per-compilation counter instead of Math.random(), and the allocator avoids capture against every symbol in the expression. Two compilations of the same expression now emit identical source, on the GPU targets included.

• Random draws are up to 100x faster when the engine runs inside a secondary JavaScript realm — a vm context, a sandboxed worker, or an embedder-supplied global. V8 compiles Math.imul(...) down to a single machine instruction only when Math is the host realm's; reached through another realm's global it becomes a property lookup plus a call, and the PCG3D hash behind every draw performs six of them. Binding the function once at module scope removes the lookup. Measured on node 22: 10 million draws went from ~880 ms to ~40 ms in a vm context (and from ~36 s to ~0.5 s for a 10-million-sample Monte-Carlo integral under Jest). Draw values are unchanged — the stability vectors are untouched.

Resolved Issues

• A Leibniz derivative no longer swallows the comparison that follows it. \frac{d}{dx}x^2 > 0 parsed as D(x^2 > 0, x) — the derivative of a boolean — which then evaluated to an incompatible-type error. The operand of \frac{d}{dx} / \frac{\partial}{\partial x} is now parsed as a term: it still takes a trailing sum (\frac{d}{dx}x^2 + 1, matching the \int … dx integrand convention), but stops before a relational, assignment or arrow operator, so the expression parses as D(x^2, x) > 0.

• An undecidable comparison no longer discards its evaluated operands. x^2 + x^2 > 0 evaluated to 0 < x^2 + x^2 and \frac{d}{dx}x^2 > 0 to 0 < D(x^2, x): Equal, NotEqual, Less and LessEqual evaluate their own operands (they are lazy, so their canonical handlers can see raw operands for chain decomposition), then threw that work away when the comparison itself could not be decided. They now report the evaluated operands — 0 < 2x^2 and 0 < 2x — which is what the non-lazy relations (Approx, Tilde, Precedes…) already did. The comparison itself is unchanged: an undecidable one still stays inert, since x^2 = 4 is a condition rather than a falsity, and decidable ones still fold to True/False.

• Monte-Carlo integration and stochastic equality now replay under WithRandomSeed(). Both sampled Math.random() directly, so a seeded block did not reproduce: WithRandomSeed(42, \int_0^1 \sin(1/x) dx) returned a different estimate on every evaluation, and a seeded isEqual() verdict could not be replayed at all.

  They now draw from a derived sub-stream — a private stream seeded from the ambient frame that consumes none of its indices. This matters because an integral takes up to 1e7 samples and the sampling loop is deadline-truncated: charging those to the frame would make adding an integral shift every later Random() draw in the block, and would make replay depend on wall-clock time. Adding or removing an integral now leaves sibling draws untouched, and the same integral samples the same points wherever it appears in the frame.

  Outside a frame both remain live, as before. monteCarloEstimate() (exported from @cortex-js/compute-engine/numerics) gains an optional trailing draw parameter defaulting to Math.random, so existing callers are unaffected.

  This applies to evaluation, not to compiled code. An integral inside a compiled function still samples live, even within a seed frame, and that is deliberate: the generated code emits one independent quadrature call per limit, so a sub-stream would restart at the same sample points on every outer node of a nested integral — trading reproducibility for a biased estimate. In practice a smooth integrand never reaches the stochastic estimator anyway (it folds to a constant at compile time, or converges under deterministic Gauss–Kronrod); only a pathological one does. Use .N() when a seeded integral has to reproduce. See docs/plans/2026-07-28-derived-substreams.md §5.1.

  An integral that cannot finish inside a seed frame — a bound or parameter is still unbound — now keeps the frame, so completing it later reproduces the estimate the frame would have produced. Operator definitions gain a readsRandomFrame flag for this: it means "reads the seed frame, consumes none of its indices", and is inferred for a user function whose body reaches an estimator. Unlike drawsRandom, it does not make the operator impure.

• A collection operator now resolves an unevaluated numeric argument. Collection handlers are consulted on the canonical expression — .at(), .each() and .count are available on any canonical expression, and the broadcast that zips Add/Multiply operands runs before they are evaluated. An argument spelled N-1 was therefore still an unevaluated sum at that point, and each operator silently fell back to its default: with N assigned 6, RotateLeft(S, N-1) + RotateLeft(S, N-2) returned 2·RotateLeft(S, 1), and Take(S, N-2) + Take(S, N-2) returned nothing at all. Literal arguments were never affected. Fixed for Take, Drop, RotateLeft, RotateRight, Repeat, Fill, Partition, Tabulate, Insert, DeleteAt, ReplaceAt, Slice, Permutations and Combinations.

• Take, Drop, RotateLeft, RotateRight and Slice now stay symbolic when their numeric argument has no value. An argument that is absent and one that is present but unresolved (a free variable) were treated alike, so each operator answered a collection it does not denote: Take(S, n) returned [], Drop(S, n) returned all of S, and RotateLeft(S, n) and RotateRight(S, n) rotated by one. They now report an unknown count and evaluate to themselves until n has a value — matching Repeat, Insert, DeleteAt, ReplaceAt, Permutations, Combinations, Tabulate, Chunk and Range, which already did. An omitted argument still takes the operator's default, so RotateLeft(S) still rotates by one; a NaN argument counts as unresolved, not omitted. Fill joins them: Fill(f, (n, 3)) used to produce an empty matrix and Fill(f, (2, n)) two empty rows. IsEmpty of such a Take over an infinite collection no longer answers False either — a zero count would make it empty.

  Membership is deliberately unaffected: Contains(RotateLeft(xs, n), x) still answers, because a rotation is a permutation.

• Partition(xs, n) with an unbound n no longer throws. Partition accepts either a chunk size or a predicate, and chose the arm by whether the size read as an integer — so an argument that is typed integer but has no value yet was applied as a predicate, and the resulting exception escaped evaluate(). The arm is now chosen by type, and an unresolved predicate (a symbol declared function with no value) leaves the expression unevaluated as well. A predicate that does resolve, to something other than a boolean, still reports the error with its spelling hint.

• WithRandomSeed(seed, [… for …]) draws again. A comprehension is a lazy view, like Map: its body draws per element when the collection is materialized. The rule that keeps the seed frame around a body that could not finish its draws (0.98.0) read the comprehension's unevaluated body as unfinished work, so the expression evaluated to itself — the collection was inert and yielded no elements, and nothing would ever complete it. A comprehension now follows the same convention as Map: materialize it inside the frame (WithRandomSeed(1, ListFrom([Random() for k = [1...6]]))) for reproducible draws. A comprehension whose clause still owes draws keeps the frame, as before.

• A user-defined function now infers its pure and drawsRandom flags from its body. Previously every user function was born pure, so f() := Random() reported isConstant: true while drawing from the random stream on every call. Two things broke as a result: N(f() + \pi) consumed two draws where N(\mathrm{Random}() + \pi) consumed one, and a partially evaluated WithRandomSeed() body calling f lost its seed frame — silently resuming with live, unseeded draws.

  The flags are derived from the heads the body applies: a body reaching a known-impure head is impure, and one reaching a stream-drawing head also draws. A head with no definition at the point of definition — a higher-order parameter (f(g) := g()), or a callee defined after its caller — is still assumed pure; set the flag explicitly on the definition when that matters.

• RandomPrime() is now reproducible under WithRandomSeed(). It declared drawsRandom: true, but its draws bypassed the seed frame, so WithRandomSeed(42, RandomPrime(1000)) returned a different prime on every evaluation. It now draws from the frame like the rest of the random family.

• RandomExpression() is now declared impure. It was declared pure, which made isConstant true for a generator that returns a different expression on every call, and admitted it to common-subexpression elimination.

Improvements

• A loop-form Sum/Product can now be a sub-expression in GLSL and WGSL. A shader has no expression-level loop, so a Sum with a symbolic bound was emitted as statements — valid as a whole function body, but nothing more: \sum_{k=0}^{n}kx compiled while 1 + \sum_{k=0}^{n}kx and 0.03\sum_{k=0}^{n}kx failed closed, which demoted the whole class to the CPU path (corpus rows are almost never a bare sum). The loop is now hoisted ahead of the value it feeds and referenced through its accumulator, so these compose. Constant bounds still unroll to an expression as before, and a loop nested inside an unrolled term hoists alongside it. Nested sums hoist into their enclosing loop body, not out of it.

  A loop inside a conditionally-evaluated branch (an If/When/Which/ Match arm) still fails closed, with a diagnostic that says so. A shader conditional is an expression, not a statement, so hoisting the loop out of the branch would run it unconditionally — and because a compiled Random() advances a counter at run time, a loop stranded ahead of a branch it never feeds would change the value of every later draw.

  Loop and Block still fail closed as sub-expressions on these targets.

• A compile decline now names its actual cause. Unknown operator \X` was reported for three different situations, so a failing compile band could not be triaged by message. They are now distinct: an operator whose compile handler declined a particular _operand shape_ says which operand and why (PointList: cannot compile — component 1 is collection-valued …); a head the engine knows but the target cannot lower says so (Integrate: cannot compile — the operator is known to the engine but target 'glsl' has no lowering for it.); and Unknown operatoris now reserved for a head with no operator definition at all. The reason reaches callers asCompilationResult.erroron thesuccess: false` paths, and as the thrown message on the direct-target path.

Benchmarks

Numeric performance (200-digit precision)

Median time per call, in microseconds — lower is better. — means the tool returned no usable result at that precision.

Expression	CE (current)	CE 0.98.0	SymPy	math.js	Mathematica
\pi^2	6.8	7.4	173	106	3.9
\sin 1	20	21	218	442	5.3
\cos 1	20	21	219	573	7.1
\ln 2	14	14	332	4,344	3.8
e^{\pi}	12	13	216	4,751	4.7
\zeta(3)	1,511	1,547	265	—	49
\Gamma(\tfrac13)	831	812	346	—	213
\psi(\tfrac13)	712	712	2,762	—	171

Symbolic capability & performance

Each cell is how many times faster than Mathematica that engine is on the case (Mathematica ÷ engine, so higher is better; Mathematica itself is 1×). — means the engine can't do the case; ✓ means it solves a case Mathematica can't. Compare the CE (current) and CE 0.98.0 columns to see what is new this release (a — under 0.98.0 next to a number under the current build). The CE + R/F column is the current build with the opt-in Rubi integrator + Fungrim identities loaded (loadIntegrationRules / loadIdentities), on the same minified bundle.

Operation	CE (current)	CE + R/F	CE 0.98.0	SymPy	math.js	Mathematica
Antiderivatives
\int\frac{1}{\sqrt x}\,dx	4.6×	2.5×	3.5×	0.5×	—	1×
\int\frac{x}{\sqrt{1-x^2}}\,dx	8.3×	1.5×	6.6×	0.09×	—	1×
\int\frac{1}{x^3+1}\,dx	5.3×	0.9×	4.2×	0.3×	—	1×
\int\frac{\sqrt x}{1+x}\,dx	—	1.9×	—	0.1×	—	1×
\int\frac{x}{(1+x)^{1/3}}\,dx	—	1.1×	—	0.01×	—	1×
\int\frac{x^2}{(1+x)^{1/3}}\,dx	—	1.1×	—	0.007×	—	1×
Derivatives
\tfrac{d}{dx}\sqrt{1-x^2}	0.04×	0.04×	0.03×	0.001×	0.004×	1×
Simplification
\sqrt{3+2\sqrt2}	41×	28×	31×	—	—	1×
\sqrt6\,x+\sqrt2\,x	79×	45×	48×	3.1×	19×	1×
Evaluation
\lim_{x\to0}\tfrac{\sin x}{x}	39×	16×	34×	3.0×	—	1×
\lim_{x\to\infty}(1+\tfrac1x)^x	5.2×	4.0×	6.9×	2.1×	—	1×
\int_1^2\tfrac1x\,dx	4458×	4875×	4193×	77×	—	1×
\int_{-\infty}^{\infty} e^{-x^2}\,dx	316×	128×	274×	2.5×	—	1×
Solving
x^4+x^2-1=0	0.3×	0.3×	0.3×	0.07×	—	1×
x^3-x-1=0	1.6×	1.8×	1.4×	0.04×	—	1×

Across the cases both solve, Compute Engine is a median 5.2× faster than Mathematica (up to 4458×) — in the browser, not a proprietary kernel.

_{Measured 2026-07-30 · Compute Engine 0.98.0 @ ef394659 (current build) · published 0.98.0 · SymPy 1.14.0 · math.js 15.2.0 · Mathematica 14.3.0 for Mac OS X ARM · Node v22.13.1. Correctness is verified numerically against an independent mpmath reference, never another tool. Reproduce with npm run build production && ./venv/bin/python3 benchmarks/gen_cases.py && node benchmarks/report.mjs && node benchmarks/report_changelog.mjs.}

0.98.0 2026-07-28

New Features

• Which and If now broadcast over list-valued conditions. Selection is performed element by element, with the first matching clause winning. Scalar conditions and results are broadcast as needed, and list-valued inputs must have the same length. Results are evaluated at most once and only when needed by at least one element. A position with no matching clause returns NaN.

  Which([3, 2, 1, 3] == 3, 1, True, 0) → [1, 0, 0, 1]

  JavaScript compilation supports the same behavior. GLSL and WGSL compilation support fixed-size vectors of two to four elements.

• New operator-definition attribute drawsRandom marks operators that consume the engine's random stream (Random, RandomShuffle, …). It is narrower than pure: false, which also covers side-effecting operators such as Assign, and is used by WithRandomSeed to decide whether a partially evaluated body still owes draws to its seed frame.

Breaking Changes

• Multiplying lists of different lengths now returns an incompatible-dimensions error, consistent with other element-wise operations. Matrix multiplication is unchanged.

• Assigning a value to a builtin operator name now creates a symbol in the current scope instead of replacing the builtin globally. For example, ce.assign("Sin", 30) changes the value of the symbol Sin, but Sin(0) continues to call the builtin function.

• The unused BindingSite.shield field has been removed. Setting it had no effect.

Improvements

• JavaScript compilation now supports more collection-valued expressions: user-defined function calls, comparisons and logical operators, and indexed Sum/Product bodies all broadcast element-wise. Length mismatches compile to NaN.

• Unsupported collection-valued conditions and vector operations now fail compilation with a diagnostic instead of producing incorrect Python, interval-js, GLSL, or WGSL code.

• RandomChoice now infers more precise result types, including finite_real for finite intervals and finite_integer for integer ranges.

• Repeated compilation with the same external target is now deterministic and produces identical generated code.

• Evaluating chained broadcast operations over lazy collections is faster: approximately 3× faster with evaluate() and 4× faster with .N() at default precision in the benchmark used for this release.

Issues Resolved

• Partially evaluated WithRandomSeed expressions now retain their seed, so later substitutions produce the same result as substituting before evaluation.

• Nested N and Evaluate expressions now preserve numeric approximation and precision. For example, N(Evaluate(pi)) now returns a numeric value instead of the symbolic pi.

Benchmarks

Numeric performance (200-digit precision)

Median time per call, in microseconds — lower is better. — means the tool returned no usable result at that precision.

Expression	CE (current)	CE 0.97.0	SymPy	math.js	Mathematica
\pi^2	7.9	8.5	203	288	4.5
\sin 1	23	23	253	598	6.0
\cos 1	21	22	253	781	8.1
\ln 2	16	16	386	5,471	4.7
e^{\pi}	15	15	273	7,477	5.9
\zeta(3)	1,738	1,702	337	—	56
\Gamma(\tfrac13)	924	913	407	—	268
\psi(\tfrac13)	781	790	3,496	—	199

Symbolic capability & performance

Each cell is how many times faster than Mathematica that engine is on the case (Mathematica ÷ engine, so higher is better; Mathematica itself is 1×). — means the engine can't do the case; ✓ means it solves a case Mathematica can't. Compare the CE (current) and CE 0.97.0 columns to see what is new this release (a — under 0.97.0 next to a number under the current build). The CE + R/F column is the current build with the opt-in Rubi integrator + Fungrim identities loaded (loadIntegrationRules / loadIdentities), on the same minified bundle.

Operation	CE (current)	CE + R/F	CE 0.97.0	SymPy	math.js	Mathematica
Antiderivatives
\int\frac{1}{\sqrt x}\,dx	4.5×	2.2×	3.5×	0.5×	—	1×
\int\frac{x}{\sqrt{1-x^2}}\,dx	8.1×	1.6×	6.8×	0.08×	—	1×
\int\frac{1}{x^3+1}\,dx	4.9×	0.6×	3.7×	0.3×	—	1×
\int\frac{\sqrt x}{1+x}\,dx	—	1.7×	—	0.08×	—	1×
\int\frac{x}{(1+x)^{1/3}}\,dx	—	1.0×	—	0.008×	—	1×
\int\frac{x^2}{(1+x)^{1/3}}\,dx	—	1.0×	—	0.006×	—	1×
Derivatives
\tfrac{d}{dx}\sqrt{1-x^2}	0.04×	0.03×	0.03×	0.001×	0.004×	1×
Simplification
\sqrt{3+2\sqrt2}	37×	27×	31×	—	—	1×
\sqrt6\,x+\sqrt2\,x	74×	42×	44×	3.1×	17×	1×
Evaluation
\lim_{x\to0}\tfrac{\sin x}{x}	41×	17×	38×	2.9×	—	1×
\lim_{x\to\infty}(1+\tfrac1x)^x	7.2×	4.6×	6.8×	1.9×	—	1×
\int_1^2\tfrac1x\,dx	5778×	6836×	5520×	107×	—	1×
\int_{-\infty}^{\infty} e^{-x^2}\,dx	352×	146×	298×	2.7×	—	1×
Solving
x^4+x^2-1=0	0.3×	0.3×	0.3×	0.05×	—	1×
x^3-x-1=0	2.2×	2.4×	1.9×	0.05×	—	1×

Across the cases both solve, Compute Engine is a median 4.9× faster than Mathematica (up to 5778×) — in the browser, not a proprietary kernel.

_{Measured 2026-07-28 · Compute Engine 0.97.0 @ bff3c3b1 (current build) · published 0.97.0 · SymPy 1.14.0 · math.js 15.2.0 · Mathematica 14.3.0 for Mac OS X ARM · Node v22.13.1. Correctness is verified numerically against an independent mpmath reference, never another tool. Reproduce with npm run build production && ./venv/bin/python3 benchmarks/gen_cases.py && node benchmarks/report.mjs && node benchmarks/report_changelog.mjs.}

0.97.0 2026-07-27

Breaking Changes

• A broadcast operand is evaluated ONCE. Broadcasting is an operation on values (the NumPy/Julia/R model), so an operand that is lifted into every cell is evaluated a single time and the operation then maps over the cells:

  L < Random()        // ONE draw, compared against every element of L
  Map(L, l ↦ l < Random())   // a draw per element, written explicitly

  Comparisons and logical connectives used to re-evaluate a lifted operand per element, so [0.5, 0.5, 0.5] < Random() could answer [True, False, True]. Arithmetic (L + Random()) already drew once; the disagreement was an implementation artifact of the element-wise zip, not a semantic. Only impure scalar operands under a broadcast are affected — an impure operand that IS the traversed collection ([Random(), Random()] < 0.5) still draws per cell, because those draws are the cells.

• A length mismatch in a broadcast is an error, not a truncation. Zipping to the shortest operand silently discarded the tail of the longer one:

  [1,2,3] < [2,2]     // was: [True, False]   now: incompatible-dimensions

  Add already answered incompatible-dimensions for this shape, so the engine disagreed with itself depending on the head. One check now governs every broadcast path — the eager zip, the arithmetic broadcast, and the lazy Map form — so the ordering relations, the logical connectives, Divide/Power/Mod, Add/Multiply, and ElementMax/ElementMin/ Clamp all answer alike. In particular the SIZE of a collection no longer decides the semantics (a mismatch used to error below the eager threshold and truncate above it), and neither does the shape of the source: Add(Filter(…), L) used to truncate where Less on the same operands errored.

  An unbounded operand against a finite one is a mismatch too (count is Infinity, which agrees with no finite length). A scalar operand is a LIFT, not a participant, so it never mismatches; an operand whose length is not yet known is not compared, since there is nothing to compare until it resolves. An empty operand alongside a non-empty one is a mismatch, while a lone empty operand still broadcasts to Nothing (Not([])).

  PointList is deliberately unaffected: it ZIPS components rather than broadcasting an operator over them, and its shortest-zip (PointList([1,2,3],[10,20]) → two points) is an existing consumer contract. The general rule: an operator LIFTED over collections requires length agreement, while an explicit PAIRING constructor (Zip, the variadic Map, PointList) defines its length as the shortest input. See docs/BROADCAST-MODEL.md for the full policy.

New Features

• Compiled comparisons and logical connectives broadcast element-wise. The ordering relations (<, <=, >, >=) and the connectives (And, Or, Not) compile over collection-valued operands on the JavaScript target instead of failing closed, so the Desmos filter form compiles rather than falling back to the interpreter:

  compile(ce.parse('[10,20,30][|[1...3]-k|>0]')).run(); // [10, 30]

  These heads lower to raw JS infix operators, which are silently wrong on an array (0 < [1,0,1] stringifies it; an array is truthy, so m1 && m2 returns a whole operand). They now wrap the head's own scalar codegen in the _SYS.bcast runtime helper, which recurses per POSITION — an empty or mismatched position projects to NaN without poisoning its siblings (Not([[], [True]]) → [NaN, [false]], matching the interpreter's [Nothing, [False]]).

  The scalar path is untouched: x < 3 with an unknown-typed plot variable still emits _.x < 3, with no runtime guard. Three shapes deliberately keep failing closed, because the compiled answer would disagree with interpretation: a CHAINED ordering (0 < xs < 5, whose pairwise && is sound only over scalars); Equal/NotEqual over two collections (whole- collection equality, which keeps its _SYS.eq dispatch and stays element-wise for the list-vs-scalar case); and an operand that types as a list but does not COMPILE to one — notably a user-function application over a collection argument (q(L) < y), since a user function's body compiles as scalar code and returns NaN on an array, which a comparison would turn into a plausible false.

Issues Resolved

• Covariance/Correlation report a length mismatch as incompatible-dimensions. They were already strict — a ragged pair of data collections errored — but with their own unexpected-argument ("collections differ in length") rather than the incompatible-dimensions error every broadcast path answers. One error tag now covers every length-mismatch diagnosis (see docs/BROADCAST-MODEL.md). The other argument errors (at least 2 data points required, the shape error, zero variance) are unchanged.

• Multiply over operands of mixed collection kinds is element-wise again. A List literal packs as a tensor value while a Range/Filter/Take/ Reverse result does not, so the non-tensor operand was classified as a SCALAR factor — and applying a scalar multiplies every CELL by it, turning each cell into a list:

  [1...3] · [4,5,6]        // was: [[4,8,12],[5,10,15],[6,12,18]]   now: [4,10,18]
  Range(1,3) · [1,2,3]     // was: [[1,2,3],[2,4,6],[3,6,9]]        now: [1,4,9]
  Range(1,3) + [1,2,3]     // [2,4,6] — `Add` was always element-wise

  Not a length problem: it misfired at matched lengths, and same-kind pairs (List×List, Range×Range) were always element-wise, because neither operand reached the scalar bucket. Add avoids it by declining its tensor kernel when fewer than two operands pack; Multiply now declines when a non-tensor collection would land among the scalars, and falls through to the same element-wise broadcast — which also brings mixed-kind mismatches under the length ruling above. Scalar×list scaling, matrix products, and component-wise tuple scaling are unchanged.

• A symbol bound to a derived collection serializes as its name again. Regression in 0.96.0. .latex and toString() materialize a lazy collection before serializing, and a symbol whose assigned value is a derived collection (the result of evaluating a Join, a comprehension, …) delegates isLazyCollection to that value — so the symbol serialized as its materialized value while .json still answered the symbol's name:

  ce.assign('L', ce.box(['List', 1, 2, 3]));
  ce.assign('L', ce.box(['Join', 'L', ['List', 4]]).evaluate());
  ce.symbol('L').json;   // 'L'
  ce.symbol('L').latex;  // was: '\bigl\lbrack1, 2, 3, 4\bigr\rbrack'  now: 'L'

  A literal List value (eager, not lazy) never triggered it, which is what made the failure value-provenance-dependent and hard to spot: a consumer that serialized a symbol to persist a document wrote the value where the name belonged. A symbol now never takes the materialize-before-serialize path — a name's spelling does not depend on what the name currently holds. Lazy collection expressions (Range, Map, comprehensions) serialize as before. Reported by the Tycho team.

• Add and Multiply fold a Measurement operand on the first .N(). Every unary/binary arithmetic head already folded a quadrature result (Power, Divide, Sqrt, Sin, Negate, Abs of a Measurement yield another Measurement), but the two n-ary heads checked for Measurement operands against the plainly evaluated operands. A quadrature operand only becomes a Measurement under numeric approximation — \int_0^1 \sin x\,dx evaluates to the exact 1 - \cos 1 — so the check saw nothing, and the first .N() returned an inert Add/Multiply whose .re was NaN (the same dead numeric read fixed for Measurement itself in 0.96.0, one level up). A second .N() folded it:

  const once = ce.parse('1 + \\int_0^1 \\sin(x) dx').N();
  once.re;      // was: NaN     now: 1.4596976941318605
  once.operator // was: 'Add'   now: 'Measurement'

  The handlers now re-dispatch the Measurement fold on the numericized result, so the parse route agrees with the (already correct) ce.box(['Add', <measurement>, 1]).N() route exactly — value and error bar. The sibling Quantity check in the same handlers had the identical structural gap (an operand that only becomes a Quantity under numeric approximation was invisible to it); it is closed the same way. evaluate() is unaffected and still returns the exact symbolic form. Reported by the Tycho team.

• Compiling a definite integral that cannot close symbolically is now bounded. The JavaScript compilation target first attempts to resolve an Integrate to a closed form (so a plotted ∫₀ˣ f costs ~µs per sample instead of a quadrature); that attempt ran under whatever deadline the caller had armed — and under none by default. An integrand with a symbolic exponent (y^{k/2-1} with k a free document parameter) sent the integration-by-parts search into a cycle with no shrinking measure: bounded in depth but not in cost, it did not return in over 5 minutes, synchronously, with no way to interrupt it:

  // χ² tail with shape parameter k left free: >5 min → ~2 s, success: true
  ce.getCompilationTarget('javascript')
    .compile(ce.parse('\\int_x^\\infty \\frac{e^{-y/2} y^{k/2-1}}{(k/2-1)!\\, 2^{k/2}} dy',
             {strict: false}), {realOnly: true});

  The antiderivative-first attempt now arms its own 2-second span (an enclosing caller span still tightens it, per the timeout model's min() nesting — it can only shorten, never extend, a caller's bound) and degrades to the GK15 quadrature emitter on expiry, so the compiled integral samples numerically with the parameter bound at run time — which is the desired behavior for a non-elementary integrand. Note that ce.timeLimit was retired in 0.89.0: to bound evaluate() itself, use ce.withTimeLimit({ms, label}, () => …), which this shape honors to the millisecond. Reported by the Tycho team.

• Compiled Factorial of a non-integer computes Γ(x+1) instead of NaN. The interpreter has always extended the factorial to the reals ((\frac12-1)! → Γ(\frac12) = √π), but the compiled runtime used the integer-only helper, so the same expression compiled successfully and returned NaN — silently zeroing out, e.g., a χ² density's normalizing constant (k/2-1)! at odd k:

  const f = ce.getCompilationTarget('javascript')
              .compile(ce.parse('(\\frac{1}{2}-1)!'));
  f.run({}); // was: NaN    now: 1.7724538509055159 (= √π, identical to .N())

  Non-integer arguments route through the same gamma the interpreter uses, so compiled and interpreted values are bit-identical; the non-negative integer fast path is unchanged, and a negative integer stays at the Γ pole (NaN, the real projection of the interpreter's ComplexInfinity). The Python target likewise now emits scipy.special.gamma(x + 1) instead of scipy.special.factorial (which answers 0 for a negative non-integer). The GPU targets already extended via Γ; the interval target remains deliberately integer-only. Reported by the Tycho team.

Benchmarks

Numeric performance (200-digit precision)

Median time per call, in microseconds — lower is better. — means the tool returned no usable result at that precision.

Expression	CE (current)	CE 0.96.0	SymPy	math.js	Mathematica
\pi^2	7.3	7.5	180	110	3.9
\sin 1	20	21	218	439	5.2
\cos 1	20	20	221	456	7.1
\ln 2	14	14	347	4,406	3.7
e^{\pi}	13	13	214	4,655	4.5
\zeta(3)	1,534	1,562	272	—	49
\Gamma(\tfrac13)	840	834	350	—	213
\psi(\tfrac13)	724	720	2,776	—	174

Symbolic capability & performance

Each cell is how many times faster than Mathematica that engine is on the case (Mathematica ÷ engine, so higher is better; Mathematica itself is 1×). — means the engine can't do the case; ✓ means it solves a case Mathematica can't. Compare the CE (current) and CE 0.96.0 columns to see what is new this release (a — under 0.96.0 next to a number under the current build). The CE + R/F column is the current build with the opt-in Rubi integrator + Fungrim identities loaded (loadIntegrationRules / loadIdentities), on the same minified bundle.

Operation	CE (current)	CE + R/F	CE 0.96.0	SymPy	math.js	Mathematica
Antiderivatives
\int\frac{1}{\sqrt x}\,dx	4.6×	2.3×	3.4×	0.5×	—	1×
\int\frac{x}{\sqrt{1-x^2}}\,dx	7.9×	1.5×	6.1×	0.08×	—	1×
\int\frac{1}{x^3+1}\,dx	5.2×	0.8×	4.0×	0.3×	—	1×
\int\frac{\sqrt x}{1+x}\,dx	—	1.9×	—	0.1×	—	1×
\int\frac{x}{(1+x)^{1/3}}\,dx	—	1.1×	—	0.01×	—	1×
\int\frac{x^2}{(1+x)^{1/3}}\,dx	—	1.1×	—	0.007×	—	1×
Derivatives
\tfrac{d}{dx}\sqrt{1-x^2}	0.04×	0.03×	0.03×	0.001×	0.003×	1×
Simplification
\sqrt{3+2\sqrt2}	42×	29×	29×	—	—	1×
\sqrt6\,x+\sqrt2\,x	77×	45×	51×	3.0×	18×	1×
Evaluation
\lim_{x\to0}\tfrac{\sin x}{x}	15×	16×	33×	3.1×	—	1×
\lim_{x\to\infty}(1+\tfrac1x)^x	7.3×	4.9×	6.8×	2.2×	—	1×
\int_1^2\tfrac1x\,dx	4819×	4740×	4259×	80×	—	1×
\int_{-\infty}^{\infty} e^{-x^2}\,dx	293×	124×	250×	2.6×	—	1×
Solving
x^4+x^2-1=0	0.3×	0.3×	0.3×	0.06×	—	1×
x^3-x-1=0	1.7×	1.9×	1.5×	0.04×	—	1×

Across the cases both solve, Compute Engine is a median 5.2× faster than Mathematica (up to 4819×) — in the browser, not a proprietary kernel.

_{Measured 2026-07-27 · Compute Engine 0.96.0 @ 9eb6538a (current build) · published 0.96.0 · SymPy 1.14.0 · math.js 15.2.0 · Mathematica 14.3.0 for Mac OS X ARM · Node v22.13.1. Correctness is verified numerically against an independent mpmath reference, never another tool. Reproduce with npm run build production && ./venv/bin/python3 benchmarks/gen_cases.py && node benchmarks/report.mjs && node benchmarks/report_changelog.mjs.}

0.96.0 2026-07-26

Breaking Changes

• The 0.95.0 random-family tombstones are deleted, completing the one-release migration window. In 0.95.0, evaluating a removed head (RandomInteger, RandomList, RandomSeed, Sample, Shuffle) threw an operator-removed error naming its replacement, and ce.randomSeed was a throwing accessor. Those guards are now gone: the removed heads behave like any other unrecognized operator — a valid, inert expression — and randomSeed is no longer a property of the engine. Migrate on 0.95.0 (where every legacy call site fails loudly with its replacement named) before adopting this release; the migration table is in the 0.95.0 notes below.

• evaluate() on a symbol now resolves the free symbols of its stored value. A symbolic value was returned verbatim, so a symbol assigned after it was stored never reached it:

  let d = 3x^2 + 1
  let x = 2
  d          // was: 3x^2 + 1     now: 13
  N(d)       // 13 — unchanged

  N() and compile() already resolved it, so plain evaluate() was the outlier and disagreed with both; a second evaluate() used to resolve one more level ("one-evaluate-late"). Assignment remains eager, so declaration order still decides what a value snapshots: let x = 2; let d = 3x^2 + 1; x = 3; N(d) is 13, while let d = 3x^2 + 1; let x = 2; x = 3; N(d) is 28. The residual for a cyclic binding is unchanged (s = s + 1; s → s + 1).

• A stored value's free symbols are no longer captured by a same-named parameter. They now denote the binding they were canonicalized against, not whatever an inner scope calls that name:

  let a = x + 1
  g(x) = a
  g(5)       // was: 6            now: x + 1
  N(g(5))    // was: 6            now: x + 1

  With a global x = 100, g(5) is 101 — the lexically correct binding — where it used to be 6. This closes the same defect on three paths that disagreed with each other: the parameter substitution applied after a call, the constant dereference path, and the numeric (N) re-evaluation inside a call frame. A dictionary-valued symbol is covered too. Behavior that was already correct is unchanged: a block-local let does not leak into a stored value, and a renamed parameter never captured.

• Only a value SHIELD hides a stored value's own binding. Dereferencing a symbol's stored value deferred to the ambient lookup whenever ANY valueless shadow of a free symbol's name was in scope. That was a proxy for the shield idiom — Solve, D, Integrate, Limit and simplify() all hide a symbol's value by shadow-declaring it valueless — and it swept in ordinary declarations, which shield nothing:

  a = x + 1                          // x still valueless: a captures the global x
  x = 100
  a + 5                              // 106 — unchanged
  Block(Declare(x, "real"), a + 5)   // was: x + 6        now: 106

  a captured a binding; the inner Declare created a different variable, and a different variable has no business intercepting. The asymmetry that made the old rule indefensible: give that shadow a value — Block(Declare(x, "real"), Assign(x, 7), a + 5) — and it already did not intercept, evaluating to 106 before and after. The genuine shields are unaffected: with a global x = 100, Solve(x + 1 = 0, x) is still [-1], and simplify() remains value-blind.

• A union type is assignable only to a target that covers EVERY arm. isSubtype() (and therefore type.matches()) used the any-branch rule when the target was a composite type and the all-branch rule when it was a primitive, so the answer depended on the shape of the target: list<tuple<number,number,number>> | tuple<number,number,number> matched tuple<number, number, number>, while number | list<number> matched neither number nor collection. Assignability is now all-branch throughout — the dual of the intersection rule, where A & B is assignable as soon as one arm is. A union still matches any target covering all its arms (integer | real ⊑ real, number | list<number> ⊑ collection | number).

  Code that asked unionType.matches(T) to mean "could this be a T" was asking the wrong question and now gets false; ask it the other way round (ce.type(T).matches(unionType) — "would a T satisfy this"), which is how the matrix-inference repair gate on union-typed parameters such as LinearSolve's matrix | vector is now spelled.

Issues Resolved

• Adaptive quadrature no longer reports a sharply-peaked integral as a converged zero. The adaptive loop started from a single 15-node panel over the whole interval, and could never recover from a first panel that read as zero: with every node returning ~0, the Gauss/Kronrod difference also vanished, met the absolute tolerance, and the integral "converged" after 15 evaluations. The witness is a peak far narrower than the interval whose weight vanishes at the center node:

  ce.parse('\\int_{-50}^{50} x^2 \\frac{1}{\\sqrt{2\\pi}} e^{-x^2/2} dx').N();
  // was: 3.2e-21 ± 5.1e-21     (true value: 1)
  // now: 1.0000000000000002

  The whole Gaussian moment family failed this way (moments 1, 2, 4 and 6 were each 100% wrong while claiming an error bar around 1e-21); the bare ∫φ over the same interval was correct, because φ(0) is sampled. Quadrature now starts from 16 equal panels, which fixes the family exactly and brings the comb ∫₋₁₅¹⁵ φ(x)³⁵⁰ to 0.14% relative error. This is a mitigation, not a guarantee — a peak narrower than a starting panel can still be missed, and the error estimate still cannot report it — a peak-discovery strategy (shifted or low-discrepancy probes) would close the family rather than move the threshold, and is not attempted here. Iterated integrals reduce the per-level count so the floor applies to the whole integral rather than multiplying per dimension. The cost falls on integrals that used to converge on the first panel: ∫₀¹ sin x goes from 15 to 240 evaluations, so a compiled integral called per plotted sample goes from ~40 µs to ~130 µs. Reported by the Tycho team.

• A definite integral whose integrand is identically non-finite now fails fast instead of burning the full Monte-Carlo budget. Adaptive quadrature can never converge on a NaN integrand, so it fell through to the Monte-Carlo estimator, which spent 1e7 samples — 250–450 ms per call — to return the NaN that was decidable up front. The usual trigger is an unbound variable reaching a compiled artifact as undefined; a plotted curve then froze the thread and drew nothing. monteCarloEstimate now probes a few samples and returns NaN immediately when they are all non-finite. Measured: 100 calls on such an integrand went from 30.5 s to 22 ms. Integrands with a genuine endpoint singularity are unaffected — the bail requires every probe to be non-finite. Reported by the Tycho team.

• .N() of a definite integral with a free symbol in the integrand no longer throws a raw ReferenceError out of generated code. The integrand was handed to the implicit compiler regardless, and the generated body read the free symbol from a scope slot the numeric caller never supplies:

  ce.parse('\\int_0^1 (x+q)\\,dx').N();
  // was: throws ReferenceError: _ is not defined
  // now: stays symbolic

  A parameter with no value leaves nothing to integrate numerically, so the expression now stays symbolic; it evaluates as before once the symbol has a value. Both the single-limit and the iterated multi-limit paths are covered.

• A Measurement now answers the numeric accessors. Measurement(value, error) types as its nominal's scalar type, so Integrate(…).N() reports finite_real and isNumber === true — but every numeric read was dead, which silently poisoned any consumer reading .re, and propagated, since Measurement + 1 is another Measurement:

  const n = ce.parse('\\int_0^1 \\sin(x) dx').N();
  n.re;            // was NaN,       now 0.45969769413186023
  n.numericValue;  // was undefined, now the nominal
  n.valueOf();     // was the string '0.4596… ± 0.0000…', now the number
  n.sgn;           // was undefined, now 'positive'

  .re, .im, .bignumRe, .bignumIm, .sgn and valueOf() project the nominal — all of them on the public Expression type, so .re is the channel and no cast is needed. The uncertainty stays reachable through the MathJSON, or through op2 after narrowing with isFunction(), and toString() still shows ±.

  Three members are deliberately NOT projected, all for the same reason: a Measurement is a function expression, and the number-literal surface belongs behind the isNumber() guard, which narrows on expression kind. numericValue stays undefined — it is declared only on NumberLiteralInterface, and projecting it would advertise an exact numeric representation that a quadrature result does not have; isNumberLiteral and the isNumber() guard stay false; and .value stays undefined (it is the expression for a literal and undefined for a symbolic expression). .re/.im are sufficient for a Measurement precisely because its nominal is never an exact number. Reported by the Tycho team.

• A shorthand-lambda placeholder in a pipeline stage no longer picks up a same-named global. Pipe is lazy, so it canonicalizes its right operand in the caller's scope — before the operand is wrapped into the implicit lambda it denotes. A global _1 holding a value therefore captured the placeholder, and the stage silently failed to canonicalize:

  ce.box(['Assign', '_1', 7]).evaluate();
  ce.parse('[1,2,3] \\rhd \\mathrm{Map}(\\_1, k \\mapsto k^2)').evaluate();
  // was: Map([1,2,3], (k) |-> k^2)   now: [1,4,9]

  The placeholders (_, _1…_9) mentioned by the stage are now bound to fresh, valueless locals for the duration of that canonicalization. A genuine free variable in the stage still resolves — and auto-declares — in the caller's scope, unchanged.

• The empty list is now a member of every list type. [] typed list<nothing>, which made [] <: list<integer> false — an empty list satisfied no list type at all:

  ce.parse('[]').type.toString(); // was: "list<nothing>"   now: "list<never>"
  ce.parse('[]').type.matches('list<integer>'); // was: false        now: true

  The cause was in the type lattice rather than in lists: widen() (a join) returned nothing for an empty input, where the join of no types is the bottom type. nothing is the unit type of the value Nothing, not the bottom type — never is. With never, covariance does the rest, since never <: X gives list<never> <: list<X>. narrow() had the mirror bug and now returns the top type for an empty input.

  Visible effects: the rendered type of an empty collection changes (list<nothing> → list<never>, list<list<nothing>> → list<list<never>>), and an empty list now satisfies a list-typed parameter of any element type. Note that couldMatch() deliberately ignores the empty list as a witness — list<integer>.couldMatch('list<string>') stays false, since that question is about element shape.

• nothing and missing were reported as disjoint from any union containing them — nothing vs boolean | nothing claimed disjointness, refuted by the value Nothing, which inhabits both. The unit-type short-circuit ran before the union was examined and compared a type name against a composite type object. This surfaced as an unsound nothing <: !(boolean | nothing).

• A quantifier's variable is no longer captured by a same-named global value. ForAll, Exists, NotExists, ExistsUnique and NotForAll declared a local scope that was created and then stayed empty, so the quantified variable was bound wherever the caller had it. With x assigned, the bound occurrence resolved the assigned value and the proposition was discharged from it:

  x = 5
  \forall x, x > 4      // was: True     now: ForAll(x, x > 4)
  \exists x, x > 4      // was: True     now: Exists(x, x > 4)

  Quantification over a finite domain (\forall x \in \{1,2,3\}, x > 0) is unchanged, and the global x is untouched in both cases. The quantifiers now use the sanctioned binder mechanism (scoped: limitsIndexSites(0)).

• Integrate's integration variable is bound by the integral. The index of a Limits operand was bound nowhere: it was left raw on the parse route and carried the caller's binding on the ce.function route, so the same integral written two ways did not compare equal.

  const parsed = ce.parse('\\int_0^1 x^2 \\,dx');
  const built = ce.function('Integrate', [
    ce.parse('x^2'),
    ce.function('Limits', [ce.symbol('x'), ce.number(0), ce.number(1)]),
  ]);
  parsed.isSame(built); // was: false    now: true

  Integrate now uses the sanctioned binder mechanism (scoped: indexingSetSites(1)), so expr.localScope reports the integration variable(s), and an indefinite integral's open result is re-bound to the enclosing scope on the way out.

• D's differentiation variables are bound by the derivative. D declared a local scope that was minted and then never populated, so its variable operands were bound wherever the caller had them and the same derivative written two ways did not compare equal:

  const parsed = ce.parse('\\frac{d}{dx} x^2');
  const built = ce.function('D', [ce.parse('x^2'), ce.symbol('x')]);
  parsed.isSame(built); // was: false    now: true

  D now uses the sanctioned binder mechanism, with a new variadic selector (scoped: operandsFrom(1)) for an operator whose bound variables are a trailing list of arbitrary length. expr.localScope reports every differentiation variable, and a derivative — an open expression in that variable — is re-bound to the enclosing scope on the way out.

  Visible consequence: a body handed to D already boxed is re-bound to the derivative's own scope, so it stops comparing equal to the expression it was built from — ce.box(['D', body, 'x']).op1.isSame(body) is now false, as it has been for a Sum's index since that operator was migrated. Code that lifts a differentiand back OUT of a D node into the ambient scope has to re-bind it; expr.explain('D'), which presents the differentiand as a free-standing expression, now does.

• A Function literal's parameter operand denotes the literal's own parameter. The body Block's binding was already the authority for occurrences in the body, but the parameter operand itself was left raw on the parse and ce.box routes and carried the CALLER's binding on the ce.function route — the same route disagreement Series and Integrate were migrated to fix. A canonical literal now has exactly one binding per parameter, referenced from both the parameter operand and the body.

• A bound variable named after a library constant (Pi, e, i, ...) is now bound like any other. A binder's variable was resolved by NAME, and a name owned by a constant short-circuits to the interned constant before the scope chain is consulted — so the binder declared a binding that nothing referenced. With an already-canonical body, the parameter was silently lost:

  const f = ce.function('Function', [ce.parse('\\pi + 1'), ce.symbol('Pi')]);
  ce.box(['Apply', f, 10]).evaluate(); // was: 1 + pi    now: 11

  The parse and ce.box routes gave 11 throughout, so this was also a route disagreement. Bound variables are now built from the binder scope's own binding, which fixes the binding identity for every binder that accepts a bare symbol — including D(Pi^2, Pi), whose value was already right while the binding underneath was the constant's.

• Interval now survives a LaTeX round-trip. A closed interval serialized as \lbrack a, b\rbrack, which the parser reads as a two-element List, and a fully-open one as \lparen a, b\rparen, read back as a parenthesized Sequence. The substitution was silent and could be destructive: under RandomChoice — the documented migration target for the removed RandomList(n) — RandomChoice(Interval(0, 1), n) came back as RandomChoice(List(0, 1), n), demoting a uniform real draw to a Bernoulli pick of the two values 0 and 1, with nothing downstream able to detect it.

  Serialization now depends on whether the position disambiguates the interval:

  • In a set position of a set operator — the right side of \in/\notin, either side of \cup, \cap, \setminus, \subset, \subseteq, \supset, \supseteq — the conventional bracket notation is kept (x\in\lbrack0, 1\rbrack), because the operator forces the set reading when it is parsed back. This is unchanged, and stays the common display case.
  • Anywhere else the serialization has to stand on its own: half-open intervals use the unambiguous American spellings (\lbrack a, b\rparen, \lparen a, b\rbrack), an open interval uses ISO reversed brackets (\rbrack a, b\lbrack), and a closed interval uses the function form \mathrm{Interval}(a, b) — [a, b] is also how a two-element list is written, so no bracket spelling is available for it.

  Consumers storing a uniform draw as LaTeX can now use the closed RandomChoice(Interval(0, 1), n) from the migration table directly. The half-open Interval(0, Open(1)) remains the more precise spelling for what RandomList(n) meant (upper-exclusive) and also round-trips.

• ListFrom now compiles. It had no compile handler, so the one eager materializer that works over an arbitrary collection body could not be used in compiled code. That matters under WithRandomSeed: a frame around a lazy comprehension does not make it replayable, because the view materializes after the frame has exited and the draws escape it. ListFrom inside the frame makes it eager, and the compiled form now agrees with the interpreter draw-for-draw:

  WithRandomSeed(12345, ListFrom([Random() for k=[1...6]]))   // replays, compiles

  On the JavaScript and Python targets the splice is decided at runtime, per operand, so an operand typing as unknown — a free symbol in a compiled body — works. The GPU targets have no runtime splice (their lists are fixed-size vecN/array literals), so an all-scalar ListFrom compiles exactly like the equivalent List and anything with a provably collection operand fails closed. Note that Repeat(Random(), n) is eager and round-trips but draws once, yielding n copies of a single value — it is not a uniform batch.

• A cycle between symbol values no longer overflows the stack. A pair of bindings such as a := b with b := a is individually well-formed — each value mentions no symbol of its own name — so the self-reference guard never fired and any query that resolves a symbol's value and delegates to it recursed until the stack blew. This reached far more than the reported isFiniteCollection: count, at, each, N(), isEqual, isSame, and the scalar predicates (sgn, isFinite, isNaN, re/im) all crashed on a cyclic binding. Such a query now fails closed — undefined/false, never a throw. An enumeration that traverses a cycle yields the elements it gathered before closing it, matching what a direct self-reference (d := Append(d, 1) → [1]) has always produced.

• Comparison, ordering and rationalization no longer numericize an argument they are about to reject. isEqual/Equal, Sort, assume, ApproxEqual, Rationalize and the .N() trig path each called .N() on an operand and then discarded the result when it turned out not to be a number literal. An operand with unknowns can never become one, and over nested applications of a user function that discarded walk is exponential in the nesting depth: at depth 12, isEqual against such a chain took ~1.8 s and Sort of five of them far longer; both are now milliseconds. Arguments that can numericize — including partially numericizable symbolic ones such as sin(2) + x — are unaffected. (Same class as the elementwise-Sin fix below; that one was the exact path, these are the .N() path.)

• Round/Floor/Ceil/Truncate of a symbolic term keeps its integer type. Round(4Q) typed number while both the less informative Round(Q) and the fully known Round(4.7) typed finite_integer: an operand typed finite_number reports isReal === false, which means "not provably real", and the handler read it as "provably complex". Non-realness is now proven from a number literal or from a type that excludes the reals, so an operand of unknown realness keeps the generic-point convention. A wrap asserting integrality (RandomChoice(Interval(0,1), Round(4N)) with N unbound) now type-checks.

• sin/cos/tan… of a deeply nested symbolic argument no longer blows up. The constructible-value lookup called .N() on every argument, including one with free symbols that can never numericize; over nested applications of a user function that wasted traversal re-walked shared sub-chains and grew exponentially with the nesting depth. Evaluating sin(πR/8) over a 17-element list whose element k is a user function applied k times took 44 s and now takes ~0.1 s. Arguments that can numericize — including a symbol with an assigned value — reduce exactly as before.

• A function literal built around an already-canonical body now binds its named parameters. Canonicalizing an expression that is already canonical is a no-op, so a body constructed before the literal existed kept the bindings it was built with — and its parameter occurrences went on denoting the enclosing scope's variable of the same name instead of the literal's own parameter. The repair previously covered only the anonymous placeholders (_, _1, …) produced by the pipe/shorthand desugaring; it now covers named parameters as well, so anything keyed on a symbol's binding — the post-application substitution of a partially-symbolic result, symbol equality — sees the parameter for what it is:

  const body = ce.box(['Add', 'y', 1]); // canonical: `y` is the caller's
  const f = ce.function('Function', [body, ce.symbol('y', { canonical: false })]);
  // the body's `y` is now this literal's parameter, not the caller's `y`

  Values are unchanged; what changes is which variable an occurrence refers to.

• An antiderivative is expressed in the caller's symbols. Integrate binds its integration variable, and an integrand's free coefficients are declared alongside it — but the antiderivative machinery (and the Rubi rule driver installed by loadIntegrationRules) works on the bare integrand and creates its own occurrences of those names in the caller's scope. The two sets of occurrences denoted different variables, so the result could compare unequal to the same expression written by hand. The integrand is now re-bound as it is lifted, matching how a Jacobian's body is lifted from its literal.

New Features

• An operator definition can now declare its bound variables. The scoped flag accepts a binding-site selector in addition to true:

  import { operandSites } from '@cortex-js/compute-engine';
  
  ce.declare('MyBinder', {
    lazy: true,
    scoped: operandSites(1), // operand 1 is my bound variable
    signature: '(expression, symbol) -> number',
  });

  A selector implies a scope, so scoped remains the complete inventory of scope-creating operators. When one is given, the engine declares each site's symbol in the operator's own scope before the canonical handler runs, and binds every occurrence of those names to that scope afterwards — so the parse, ce.box() and ce.function() routes agree about which binding a bound variable denotes, whichever route built the expression. The prebuilt selectors (operandSites, indexingSetSites, limitsIndexSites, lambdaParamSites) and the BindingSite/BindingSiteSelector types are exported from @cortex-js/compute-engine. Sum, Product, Loop, Comprehension, Series and NDSolveFunction now use it in place of six hand-rolled conventions.

  An indexing-set selector marks its sites clauseLocal: later clauses see earlier bindings, but an earlier clause's collection resolves a name a later clause binds in the enclosing scope (Comprehension(…, Element(i, [j, j+1]), Element(j, […])) draws i from the ambient j).

• BoxedType.couldMatch() — "could a value of this type be a target?", the predicate for classifying a value by shape. matches() answers the other question — "is every value of this type a target" — so it reports false for a union whose members include exactly the shape being asked about, which is the steady state for a variable declared with more than one admissible shape:

  const t = ce.type('tuple<number, number> | list<tuple<number, number>>');
  t.matches('list<tuple<number, number>>'); // false
  t.couldMatch('list<tuple<number, number>>'); // true

  Unions are distributed at every depth, so a union nested inside a parameter is handled too: list<integer | tuple<number, number>> could be a list<tuple<number, number>> — witness [(1,2)].

  The relation is symmetric and decisive for the composite shapes it models: a tuple<number, number> could not be a list<tuple<number, number>>, list<integer> could not be a list<string>, and a set<T> could not be a list<T>. List dimensions and tuple arity and element names are compared. Shapes it does not model fall back to assignability in either direction, so the answer is never narrower than matches() — with one deliberate exception: never is uninhabited, so nothing could be a never. unknown could be anything; consumers that treat an inconclusive type as "no" should check isUnknown themselves.

• BoxedType.unionMembers — the members of a union type, each boxed, or [this] for any other type. Lets a consumer reason arm-by-arm without reading the raw Type AST. It does not reach a union nested inside a parameter; couldMatch() covers that case directly.

• BoxedType.isDisjointFrom() — a type-overlap predicate for consumers that classify an expression by comparing its type against a set of candidates. matches() answers subtyping (this <: other), so two types that share values without either containing the other look unrelated in both directions:

  const a = ce.type('integer | string');
  const b = ce.type('integer | boolean');
  a.matches(b); // false
  b.matches(a); // false — yet they share `integer`
  a.isDisjointFrom(b); // false: they may overlap

  The predicate is conservative in the safe direction: when disjointness cannot be established the answer is false ("may overlap"), never a false claim of disjointness. unknown — the type of an undeclared symbol — therefore overlaps everything. It accepts a Type, a type string, or a BoxedType, and throws on a string that is not a valid type.

Improvements

• Disjointness now distributes over unions, so integer | string is recognized as disjoint from boolean instead of falling through to "may overlap". This also makes a union a subtype of a negation when no member meets the negated type (integer | boolean <: !string).

• Disjointness is now decided by comparing the primitive categories of the two types, so composite types are separated from each other and from primitives instead of falling through to "may overlap": a list<integer> is not a string, a tuple<number, number> is not a list<tuple<number, number>>, a set is not a list, and a record is not a dictionary. Broad categories still contain the narrow ones, so integer and value, or list<integer> and collection, correctly report "may overlap". This generalizes and replaces the narrower numeric-vs-non-numeric rule.

  Two same-category composites whose parameters cannot coincide (list<integer> vs list<string>) are deliberately not claimed disjoint: list<never> is a subtype of both, so the claim would rest on how the empty list is typed rather than on the type lattice. couldMatch() answers that question decisively.

0.95.0 2026-07-25

Breaking Changes

• The random family is redesigned around block-scoped seeding. Seeding moves out of argument lists and engine state entirely: there is no seed argument anywhere in the family, and no ambient seed to set. A WithRandomSeed(seed, body) frame makes every draw inside it — including draws in user-function calls (dynamic scoping) and in compiled code — deterministic and replayable, while draws outside any frame are live.

  ce.box(['WithRandomSeed', 42, ['List', ['Random'], ['Random']]]).evaluate();
  // → two DIFFERENT values, and the same two values on every re-evaluation

  The n-th draw of a frame is hash(seed, n) — PCG3D, a pure function of the seed and the draw index, computed identically by the interpreter, the JavaScript target, and (as f32) the GPU targets. Draws are IEEE float64 regardless of the engine's precision mode, and the seed→stream mapping is a cross-version contract pinned by published test vectors. Frames nest (innermost wins) with independent per-frame counters, so one document cell's frame cannot perturb another's. See Random Numbers (and docs/RANDOMNESS-MODEL.md in the repository) for the full contract, including the draw-consumption table and the rule that only evaluation consumes draw indices (an untaken branch, an unmaterialized lazy view, or a canonicalized-away wrapper consumes none).

  The surface changes:

  • Random is domain-only. Random() draws a real in [0, 1); Random(Interval(a, b)) a real in [a, b); Random(Range(…)) an element of the (normalized, inclusive) range; Random(xs) an element of a finite collection. The old Random(seed)/Random(m, n) forms — where the first argument meant a seed or a bound depending on its numeric type — are rejected by the signature.
  • Sample → RandomSample, Shuffle → RandomShuffle — renamed, seedless. RandomSample's domain must be an indexed collection (a Set or an Interval is now invalid), and k < 0 or k > n is now an out-of-range error rather than undefined. Without-replacement remains over positions, not values: sampling a multiset can repeat a value.
  • RandomInteger, RandomList, and RandomSeed are removed, along with the ce.randomSeed property. For one release, evaluating a removed head throws an operator-removed error naming its replacement, and ce.randomSeed is a throwing accessor — nothing fails silently.

  Migration: Random(seed) → WithRandomSeed(seed, Random()); Random(n)/Random(m, n) → Random(Range(0, n-1))/Random(Range(m, n-1)) (for the non-degenerate ranges — the old bounds were upper-exclusive); RandomInteger(a, b) → Random(Range(a, b)); RandomList(n[, seed]) → RandomChoice(Interval(0, 1), n), framed if seeded; Shuffle(xs[, seed])/Sample(xs, k[, seed]) → RandomShuffle(xs)/RandomSample(xs, k), framed if seeded; RandomSeed(s)/ce.randomSeed = s → WithRandomSeed(s, …) around the work.

New Features

• RandomChoice(domain, k) — k independent draws with replacement, the twin of RandomSample (without replacement). The domain may be a bounded Interval, a Range, or any finite collection, and is never materialized: RandomChoice(Range(1, 10^9), 5) is O(k). The count is typed number (a computed count need not be pre-rounded; it is rounded on evaluation), and k may exceed the domain size — that is what replacement means.

• Random draws now compile — including inside auto-compiled Map bodies and in shaders. Every compiled draw goes through the same engine primitive as the interpreter, deciding framed-vs-unframed at call time, so a function compiled outside any frame is deterministic when later called inside one, bit-identical to the interpreter. On the GPU, WithRandomSeed frames compile lexically (per-invocation counters): per-pixel seeding is WithRandomSeed(perPixelSeed, Random()). Unsupported forms fail closed at compile time rather than drawing silently.

• Overload sets: an intersection of function signatures is now resolved at the call site. A function that can be called in several different ways is declared with &, and the arm whose parameters accept the arguments is selected. When several arms accept them the most specific one wins; incomparable arms are tried in declaration order.

  ce.declare('Draw', {
    signature: '((set<real>) -> real) & ((collection) -> any)',
    evaluate: (ops) => {
      /* dispatch on ops at run time */
    },
  });
  
  ce.box(['Draw', ['Interval', 0, 1]]).type; // → "real"  (set<real> is the more specific arm)
  ce.box(['Draw', ['List', 1, 2, 3]]).type; // → "any"
  ce.box(['Draw', 5]).isValid; // → false

  Previously such a signature parsed but was inert: applications were never arity- or type-checked and always typed unknown. Argument validation, result typing and type inference now all understand overload sets, on the operator definition and the symbol declaration routes alike.

  When an argument's type is not yet known, it is inferred as the union of the parameters the surviving arms accept at that position — the constraint the call actually carries. Above, an unknown x in Draw(x) is inferred collection, not set<real>: assuming the more specific arm would wrongly reject a later list.

  Note that -> binds looser than &, so each arm must be parenthesized: (number) -> real & string is a single signature returning real & string. See Overload Sets.

Issues Resolved

• .N() evaluated the operands of Add and Multiply twice. The numeric path evaluated every operand exactly, discarded the results, and re-evaluated them numerically — observably wrong for impure operands (a framed Random() consumed two draw indices under N() and one under evaluate()) and a 2× evaluation tax otherwise. Impure operands now evaluate exactly once; pure operands keep the substitute-once-guarded path.

• Sample materialized its whole source to draw a few elements. Sample(Range(1, 1000000), 3) allocated a million boxed numbers and ran a full Fisher-Yates (~300 ms) to return three; large sources were an uncatchable OOM. RandomSample now runs a sparse Fisher-Yates over the index space — O(k) time and memory — and RandomShuffle refuses sources past the element cap instead of exhausting the heap.

• Compiled random draws bypassed the engine's stream. A compiled Random() emitted a bare Math.random(), so a compiled Map silently stopped being reproducible under a seed. Compiled and interpreted draws now share one code path (and the auto-compile gate that excluded impure bodies from Map compilation is gone — per-sample-point draws stay in the hot path).

• compileShader emitted shaders that referenced undefined helpers. A shader whose body used any _gpu_* helper (Gamma, the fractal helpers, and now the random draw) compiled in CE but failed at GPU shader-compile time, because the helper preamble was never spliced into the emitted source. compileShader now derives the preamble from the compiled body and inserts it ahead of the entry point, on both GLSL and WGSL.

• A function signature nested in a union or an intersection lost its parentheses when serialized, and re-parsed as a structurally different type with an identical string. ((number) -> real) & ((string) -> boolean) came back as the single signature (number) -> (real & ((string) -> boolean)). Type serialization now parenthesizes a signature wherever it is a member of a union, intersection or negation.

• An intersection was not a subtype of its own members when the members were composite types. ((number) -> real) & ((string) -> boolean) did not match (number) -> real. A & B is now a subtype of R whenever any arm is, matching the behavior that already applied to primitive types.

• ce.assume() threw a TypeError when applied to an operator whose signature had no single result type.

• A function literal assigned to a symbol declared with an overload set was not arity-checked, so a two-parameter literal could be stored against one-argument arms and every declared call would silently partial-apply.

• Shuffle and Sample were treated as pure functions. Neither declared pure: false, so isPure — and therefore isConstant — was true for a random permutation or sample of a literal collection, and Sample was not gated out of Map auto-compilation. Both are now declared impure.

• A masked GPU branch could draw. The When/Which fall-through NaN compiled to 0.0 / 0.0, whose value is implementation-defined in GLSL: a driver may fold it to a finite, renderable value. The _gpu_nan() helper now returns intBitsToFloat(0x7FC00000), a guaranteed quiet-NaN bit pattern.

• Desktop GLSL 4.x shaders were emitted with GLSL ES 1.00 syntax. compileShader chose in/out over attribute/varying by testing whether the version string began with 3, so 450 core fell through to the ES 1.00 keywords. The version is now parsed rather than prefix-matched, and a version below 300 is rejected: the emitted code uses ES 3.00 constructs throughout, so a lower #version header could not have compiled.

Improvements

• The result type of the bare function type is now reported as unknown rather than any. function is shorthand for (any*) -> unknown and carries no information about its result, so an application of an undeclared function types unknown. This is visible in derived types — [h(x)] for an undeclared h now types list<unknown> instead of list<any>.

• The result type of a union or intersection of function signatures is now the union of the arms' result types, instead of being undetermined.

Benchmarks

Numeric performance (200-digit precision)

Median time per call, in microseconds — lower is better. — means the tool returned no usable result at that precision.

Expression	CE (current)	CE 0.92.1	SymPy	math.js	Mathematica
\pi^2	6.1	6.5	175	101	3.8
\sin 1	19	20	219	483	5.2
\cos 1	20	20	218	567	7.0
\ln 2	14	14	340	4,558	3.8
e^{\pi}	12	12	226	4,770	4.5
\zeta(3)	1,519	1,542	268	—	49
\Gamma(\tfrac13)	831	827	348	—	212
\psi(\tfrac13)	717	713	2,778	—	169

Symbolic capability & performance

Each cell is how many times faster than Mathematica that engine is on the case (Mathematica ÷ engine, so higher is better; Mathematica itself is 1×). — means the engine can't do the case; ✓ means it solves a case Mathematica can't. Compare the CE (current) and CE 0.92.1 columns to see what is new this release (a — under 0.92.1 next to a number under the current build). The CE + R/F column is the current build with the opt-in Rubi integrator + Fungrim identities loaded (loadIntegrationRules / loadIdentities), on the same minified bundle.

Operation	CE (current)	CE + R/F	CE 0.92.1	SymPy	math.js	Mathematica
Antiderivatives
\int\frac{1}{\sqrt x}\,dx	6.2×	2.9×	5.6×	0.5×	—	1×
\int\frac{x}{\sqrt{1-x^2}}\,dx	11×	1.6×	8.8×	0.1×	—	1×
\int\frac{1}{x^3+1}\,dx	6.1×	0.9×	4.7×	0.3×	—	1×
\int\frac{\sqrt x}{1+x}\,dx	—	2.0×	—	0.1×	—	1×
\int\frac{x}{(1+x)^{1/3}}\,dx	—	1.2×	—	0.01×	—	1×
\int\frac{x^2}{(1+x)^{1/3}}\,dx	—	1.2×	—	0.007×	—	1×
Derivatives
\tfrac{d}{dx}\sqrt{1-x^2}	0.04×	0.04×	0.04×	0.0009×	0.003×	1×
Simplification
\sqrt{3+2\sqrt2}	41×	29×	36×	—	—	1×
\sqrt6\,x+\sqrt2\,x	83×	45×	58×	3.2×	15×	1×
Evaluation
\lim_{x\to0}\tfrac{\sin x}{x}	44×	18×	49×	3.1×	—	1×
\lim_{x\to\infty}(1+\tfrac1x)^x	8.5×	5.2×	8.5×	2.1×	—	1×
\int_1^2\tfrac1x\,dx	6294×	6380×	6457×	92×	—	1×
\int_{-\infty}^{\infty} e^{-x^2}\,dx	388×	143×	389×	2.4×	—	1×
Solving
x^4+x^2-1=0	0.3×	0.3×	0.3×	0.06×	—	1×
x^3-x-1=0	1.7×	1.8×	1.5×	0.04×	—	1×

Across the cases both solve, Compute Engine is a median 6.2× faster than Mathematica (up to 6294×) — in the browser, not a proprietary kernel.

_{Measured 2026-07-26 · Compute Engine 0.95.0 @ 6c188402 (current build) · published 0.92.1 · SymPy 1.14.0 · math.js 15.2.0 · Mathematica 14.3.0 for Mac OS X ARM · Node v22.13.1. Correctness is verified numerically against an independent mpmath reference, never another tool. Reproduce with npm run build production && ./venv/bin/python3 benchmarks/gen_cases.py && node benchmarks/report.mjs && node benchmarks/report_changelog.mjs.}

0.94.0 2026-07-24

Breaking Changes

• Nothing now erases inside collections, as it already did inside operator argument lists. Nothing is the ERASURE marker — an empty-sequence splice — so a Nothing element is spliced out of a List, Set or Tuple literal instead of being retained. Length, arity, type and indexing all follow:

  ce.box(['List', 12, 'Nothing', 34]); // → [12, 34]   (length 2, was length 3)
  ce.box(['Set', 1, 'Nothing', 3]); // → Set(1, 3)
  ce.box(['Tuple', 1, 'Nothing', 3]); // → (1, 3)
  ce.parse('(a,,b)'); // → (a, b)   (an empty slot is `Nothing`)

  A key–value pair tuple is a NON-erasing position, so a dictionary/record entry whose value is Nothing is dropped as a whole entry, but a caller that needs a fixed-arity positional pair whose slot may hold an absent value must build it with ce._fn('Tuple', …) and use Missing (below) for the hole. This also applies to lazy iteration: an element that evaluates to Nothing is dropped (Map(xs, _ ↦ Nothing) is the empty collection — the mapMaybe idiom).

• New Missing marker and missing type for an absent-but-positioned value. Missing is the complement of Nothing: "a position exists, its value is absent" (Julia missing, R NA). It is never erased — [1, Missing, 3] is a 3-element list<integer | missing> — and missing is a primitive unit type (a subtype only of itself and any, mirroring nothing), reachable as ce.Missing.

  Absence is domain-normalized at value construction: absence flowing through an operator into a NUMERIC result cell becomes NaN (the numeric absent element), while a non-numeric result cell keeps Missing. So a numeric operator ABSORBS a Missing operand into NaN rather than carrying a missing arm:

  ce.box(['Add', 'Missing', 1]).evaluate(); // → NaN   (Add(Missing, 1) : number)
  ce.box(['Sin', 'Missing']).evaluate(); // → NaN
  ce.box(['Sin', ['List', 1, 'Missing', 3]]).evaluate(); // → [Sin(1), NaN, Sin(3)]

• Out-of-band access preserves position instead of yielding Nothing or dropping the entry. An out-of-range index, or a dictionary key that is not present, now yields a position-preserving marker chosen by the collection's element domain: NaN when the elements are numeric, Missing otherwise.

  ce.box(['At', ['List', 10, 20, 30], 9]).evaluate(); // → NaN   (numeric)
  ce.box(['At', ['List', 'a', 'b'], 9]).evaluate(); // → Missing (non-numeric)

  Gather is now length-preserving — At([a, b], [1, 9, 2]) is [a, hole, b] (length 3), where the baseline dropped the out-of-range entry — and a boolean mask whose length differs from the collection is now an error, where the baseline silently applied the prefix.

• The 15 data-consuming aggregates return NaN on an absent datum or empty input. Mean, Variance, PopulationVariance, StandardDeviation, PopulationStandardDeviation, Kurtosis, Skewness, Median, InterquartileRange, Quartiles, Max, Min, Supremum, Infimum, and Mode — over both call shapes (Max(1, Missing, 3) and Max([1, Missing, 3])) — now evaluate to NaN when any datum is absent (Missing or NaN) or the input is empty (Quartiles → (NaN, NaN, NaN)). Max([])/Min([]) are therefore NaN (was ∓∞), and these operators now type as number rather than finite_real, since their result may be NaN.

  ce.box(['Max', 1, 'Missing', 3]).evaluate(); // → NaN
  ce.box(['Mean', ['List']]).evaluate(); // → NaN

• Comparisons follow IEEE 754 for NaN and Kleene for the Missing symbol, across the whole relational family (Equal, NotEqual, Less, LessEqual, Greater, GreaterEqual). This is the Julia model:

  • The Missing symbol is Kleene — a comparison with a Missing operand is itself Missing: Equal(x, Missing) = Missing, NotEqual(Missing, x) = Missing, Less(Missing, 1) = Missing. (An ordering with a Missing operand previously stayed symbolically unevaluated.)
  • NaN follows IEEE — NaN is unequal to everything (including itself) and unordered: Equal(NaN, NaN) = False, NotEqual(NaN, x) = True, and every ordering with a NaN operand is False. The Equal/NotEqual results match native float ==/!=; the ordering comparisons with NaN previously stayed symbolic (NaN < 1 was inert), so they now resolve to False.

  ce.box(['Equal', 'NaN', 'NaN']).evaluate(); // → False    (IEEE)
  ce.box(['Less', 'NaN', 1]).evaluate(); // → False         (IEEE unordered)
  ce.box(['Equal', 2, 'Missing']).evaluate(); // → Missing  (Kleene)
  ce.box(['Less', 'Missing', 1]).evaluate(); // → Missing   (Kleene)
  ce.box(['Equal', 2, 2]).evaluate(); // → True             (unchanged)

  Absence for discharge (IsMissing, Coalesce) and aggregates (Max, Mean, …) is unaffected — a NaN is still absent there (IsMissing(NaN) = True, Coalesce(NaN, d) = d, Max(1, NaN, 3) = NaN). Broadcast comparisons apply the rule per cell. Because NaN follows IEEE, compiled and interpreted comparisons now agree by construction on numeric operands (plain == is the IEEE semantics — no guard is emitted, and NaN == NaN compiles to false). A numeric-domain missing arm (number | missing) does not widen a comparison's result type — that slot's absence value is NaN, so the result is a plain boolean, a Missing value read through such a slot compares as NaN (IEEE), and the comparison compiles on float-only targets (GLSL/WGSL). A missing-arm operand over an object domain (e.g. string | missing) still types boolean | missing and lowers via the guarded form (isAbsent(a) || isAbsent(b) ? null : a == b) so a Missing becomes the target null. A scalar If/Which condition that evaluates to Missing yields a catchable error expression (The condition is absent…) rather than crashing evaluate() — absence is a runtime data state, so it must be renderable and catchable; discharge with Coalesce/IsMissing to branch on possibly-absent data. (A condition that is not boolean at all, e.g. If(3, …), keeps the existing spell-check throw.) A NaN-comparison condition yields a plain boolean (IEEE) and branches normally.

• Compiled Max([]) / Min([]) now return NaN, matching the interpreter (previously -Infinity / +Infinity from the identity-seeded reduce). Non-empty folds are unchanged.

New Features

• cortex check — validate a program without evaluating it. Parses the source (a file, --eval, or stdin) and reports diagnostics; exit status is 0 when there are no errors. With --json it emits a machine-readable envelope ({ ok, diagnostics } with severities, codes, messages, 0-based source offsets, 1-based line/column, and fix-its). The same structured diagnostics are available during evaluation with --diagnostics json.

• cortex doc — library documentation from the terminal. cortex doc Sin shows a definition's kind, signature or type, description, keywords, and (for constants) value; a non-name argument searches the library by identifier, description, curated keywords, and LaTeX commands (cortex doc greatest common divisor → GCD, …). --limit <n> controls the number of matches and --json emits a structured { query, matches } envelope.

• Cortex for AI Agents language card. A condensed, machine-verified reference for LLMs and coding agents writing Cortex (/cortex/for-agents/): core semantics, an operator-precedence summary, a table of Python/JavaScript reflexes that don't transfer, and verified idioms. Every example on the page is executed by the documentation test suite, so the card cannot drift from the implementation.

• cortex mcp — a Model Context Protocol server for Cortex. Starts an MCP server on stdio (the default) or native Streamable HTTP, giving AI agents structured access to the same operations as the CLI: an evaluate tool (each call runs a complete, self-contained program in a fresh session and returns the value as display text, Cortex source and MathJSON, plus diagnostics), check, doc, parse and serialize tools, and the Cortex for AI Agents language card as the cortex://docs/for-agents resource. Register the stdio transport with, e.g., claude mcp add cortex -- npx -y @cortex-js/compute-engine mcp, or start the URL endpoint with cortex mcp --transport streamable-http. The protocol implementation is self-contained: the package gains no new dependencies.

• Spread arguments — f(...t) splices a tuple into a call's arguments. New Cortex prefix syntax ... (call argument lists only) and engine Spread marker: the elements of a tuple become ordinary positional arguments, so a point can be fed to a component function without manual indexing — F(p) = (a(...p), b(...p), c(...p)) instead of a(p[1], p[2], p[3]). Several spreads splice in order (g(...p, ...q)), variadic built-ins accept them (Max(...t)), and the syntax round-trips through the Cortex serializer. A literal tuple splices at canonicalization; a symbolic argument defers — argument validation and the operator's canonical handler wait — until evaluation resolves the tuple and re-validates the real arguments. Tuples only: spreading a List or a scalar is an incompatible-type error, and an unresolved argument leaves the call symbolic.

  Spread also compiles, on every target: a literal tuple splices directly, and a tuple-typed argument (p: tuple<number, number>) is rewritten statically to positional accesses (f(At(p,1), At(p,2))). An argument whose tuple arity is not statically known fails closed — a dynamic JS/Python spread would silently mis-bind on an arity mismatch instead of erroring like the interpreter.

  ce.box(['Add', ['Spread', ['Tuple', 1, 2, 3]]]).evaluate(); // → 6

• Destructuring declarations — let (q, r) = divmod(17, 5). A Cortex let/const may bind the components of a tuple in one statement. Patterns are irrefutable in form — bare symbols, _ to skip a position, nested tuple patterns; no literals or pins (use match for conditional destructuring) — and require an initializer. The value is evaluated once; a shape mismatch (wrong length, or not a tuple) yields an incompatible-type error value and binds nothing. const (x, y) = … makes each binding constant. Lowers to the Declare primitive with a Tuple pattern in the name position, which now accepts it on all routes.

  In compiled code, a destructuring declare with a literal tuple value desugars to per-leaf declares (each element bound once, in order); a non-literal value or a shape mismatch fails closed so the interpreter takes over. (This also fixes a silent divergence: the pattern previously compiled as a single let _ = …, and every pattern name read as NaN behind success: true.)

  ce.box(['Declare', ['Tuple', 'x', 'y'], d]).evaluate(); // d = {value: (3, 4)}

• IsMissing and Coalesce — absence testing and discharge. IsMissing(x) is True when x is absent (the Missing symbol OR a NaN, R’s is.na); IsNaN remains a NaN-specific test. Coalesce(a, b, …) returns the first non-absent operand, evaluated left-to-right with short-circuit; if every operand is absent it returns the last one verbatim. Both discharge primitives work identically in the interpreter and compiled (JS/Python) — a numeric hole (NaN) and an object hole (Missing/null) are handled uniformly. On a target that cannot observe its absent element (a GPU shader, where fast-math may not preserve isnan), IsMissing/Coalesce fail closed with a compile error; propagation still works natively.

  ce.box(['Coalesce', ['At', ['List', 10, 20], 9], 0]).evaluate(); // → 0
  ce.box(['IsMissing', 'NaN']).evaluate(); // → True

• HoldValues(body) — the value-blind evaluation route from the operator surface. A binder that shields the assigned free symbols of its body: for the duration of the evaluation each such symbol becomes a pure symbol (its declared type and in-scope assumptions apply, its assigned value does not), analogous to Mathematica's Block[{x}, …]. Now that the Simplify operator evaluates its argument first, HoldValues is the only value-blind route reachable from Cortex (the .simplify() method is not exposed there). Shield every assigned symbol, or a listed subset with a second List/Set/Tuple or single-symbol operand. Constants are never shielded, in-scope assumptions survive the shield, and the global values are intact afterwards.

  ce.assign('x', 5);
  ce.assign('a', 3);
  ce.box([
    'HoldValues',
    ['Together', ['Add', ['Divide', 1, 'x'], ['Divide', 'a', ['Power', 'x', 2]]]],
  ]).evaluate(); // → (a + x) / x²   (without the wrapper: 8/25)
  ce.box(['HoldValues', ['Add', ['Power', 'x', 2], 'a'], ['List', 'a']]).evaluate();
  // → 25 + a   (x resolves, a shielded)

Improvements

• The Cortex serializer reconstructs let/const syntax. A Declare node now serializes back to its statement form — let x = 5, const c = 6.28, let x: real, and destructuring patterns let (x, y) = p — instead of the generic Declare(x, {value -> 5}) function spelling, so declarations round-trip source → MathJSON → source. Shapes with no let spelling (a holdUntil attribute, a computed name) keep the generic form.

• Compiled comparisons and connectives look through provably-scalar user functions. A helper declared with an open signature ((unknown) -> unknown, the shape that keeps list-broadcasting working) no longer makes a scalar comparison uncompilable: q(x) < y with q(t) = n·t+1 compiles when every argument is scalar and the function's body provably maps scalar parameters to a scalar result (arithmetic/transcendental operators, scalar-typed captured symbols, and nested user helpers — analyzed recursively, with self-recursion declining). A call whose argument may be a collection (q(L) < y) still fails closed, so the sound half of the 0.93.0 rule is preserved — and unlike a -> number return annotation, the look-through never mis-compiles the broadcast call. Element-wise compiled comparisons over collections remain a separate roadmap item.

• declare() accepts inferredSignature: true, to vouch that a name is an operator without pinning its types. Declaring a signature normally makes it a contract, so a wide placeholder such as (unknown) -> unknown keeps every call typed unknown even after a function literal is assigned. That is the right default for a fixed API, but not for a name that must be declared before its body exists — most often so that f(x) parses as an application rather than a multiplication:

  ce.declare('q', { signature: '(unknown) -> unknown', inferredSignature: true });
  ce.assign('q', ce.parse('t \\mapsto 2t+1'));
  // signature is now `(unknown) -> finite_number`
  // `q(x) < y` types `boolean` and compiles;
  // `q(L) < y` over a list `L` types `list<boolean>` and still fails closed

  The flag was already honored at run time and is now part of the OperatorDefinition type, so it no longer needs a cast. A declaration that omits signature entirely behaves the same way.

• An unapplied Derivative(f) now evaluates to a named-parameter function literal. Derivative(Sin) evaluates to x ↦ cos(x) (["Function", ["Cos","x"],"x"]) instead of the hole-form cos(_), which was typed finite_number — so a stored derivative is now callable: let g = Derivative(f); g(2) works instead of erroring with incompatible-type. Results with no closed form stay symbolic, and the multivariate mixed-partial form no longer throws Function body must be a scoped Block expression when applied.

• The Cortex CLI's --json output materializes finite lazy collections. Range, Map/Filter results, and loop-built Join chains now serialize as their elements (["List", 1, 2, …], up to 10,000) instead of their unevaluated recipe; infinite collections keep the structural form.

• Cortex trap lints and better "did you mean" suggestions. Three common cross-language reflexes that previously failed silently now produce an advisory warning (the parse and value are unchanged): = inside a call argument (Solve(x^2 = 4, x) is assignment, not an equation — use ==), a literal index 0 (indexing is 1-based; xs[0] is NaN), and a // comment that reads as floor division (7 // 2 is 7 followed by a comment; use Floor(a / b)). Calling print (or println, printf, puts, echo) now explains that a program's output is the value of its last statement. The "did you mean" matcher gained a curated cross-language tier: split → StringSplit, push → Append, ceiling → Ceil. The agent language card gained a verified library quick-roster, output-rendering notes (quoted booleans, list preview elision), and binder-variable semantics for D/Integrate.

• Pipe/Apply reject or defer a non-function right operand more sensibly. x |> f (Pipe) now returns an incompatible-type error when f is a number, string, or boolean literal (which can never be applied), instead of staying silently inert; a symbol or unevaluated f still defers (definitions may arrive later). Apply and Pipe no longer throw an uncaught Invalid function literal when given a string function operand — they decline gracefully. Applying a function-valued expression such as InverseFunction(f) now stays symbolic (Apply(InverseFunction(f), 2)) instead of misinterpreting it as a lambda body and substituting the argument for f.

• Rubi integrator: Euler-substitution lever for √(quadratic)-nested radicals. The experimental Rubi integrator now closes nested radicals whose inner radical is a square root of a quadratic with a positive leading coefficient — e.g. ∫ 1/(√(x+√(x²+1))+1) dx (Bondarenko #9) — via an Euler I substitution t = √a·x + √Q that rationalizes √Q and reduces the integrand to a form the existing linear-radical machinery closes. This raises the Bondarenko benchmark to CE+R/F 21/35.

• simplify() is value-blind for a symbol's sign and parity. .simplify() no longer reads an assigned symbol's value when applying a sign- or parity-driven rewrite: with w := 5, |w|.simplify() now stays |w| and √(w²).simplify() is |w| (previously both collapsed to w, silently baking in w ≥ 0 and evaluating wrong after a later w := -3). Sign and parity are taken only from a symbol's declared type and in-scope assumptions — so assume(w > 0) still licenses |w| → w — never from its assigned value. .evaluate() and .N() are unchanged — they still substitute the value.

• The Simplify operator now evaluates its argument before simplifying. Simplify(expr) is now evaluate-then-simplify — the operator counterpart of the expr.evaluate().simplify() recipe — so it computes handler-driven results the value-blind .simplify() method never touches: Simplify(Max(3, 5)) → 5, Simplify(D(x²+ax, x)) → a + 2x, Simplify(∫x² dx) → x³/3. Because evaluation substitutes assigned symbol values, the change is visible from Cortex: let x = 5; Simplify(x^2 + x) now gives 30. The other transformers (Expand, Factor, Together, Distribute) are unchanged — they keep reduce-not-evaluate.

• The .simplify() method no longer evaluates structural operators. Determinant, Trace, Transpose and Length are no longer reduced by the .simplify() method (an unreleased whitelist added earlier in this cycle): the method is rule-driven and value-blind, and running an operator's evaluate handler is .evaluate()'s job. Use expr.evaluate() (or the Simplify operator, which now evaluates) to reduce them — e.g. Determinant([[a,b],[c,d]]).evaluate() → a·d − b·c.

0.93.0 2026-07-23

New Features

• Cortex CLI and interactive REPL. Installing @cortex-js/compute-engine now provides a cortex executable. It evaluates inline source (cortex -e '1 + 2'), .cx/.cortex files, or a program read from standard input; with no input argument in a terminal it starts a stateful REPL.

  The REPL retains declarations across inputs, supports multiline programs and persistent history, and adds .load, .clear, .ast, and .time commands alongside Node's standard REPL commands. Non-interactive output can be emitted as text, MathJSON (--json), or Cortex source (--cortex). Diagnostics include source locations and are written to standard error. Evaluations have a 10-second deadline by default; --time-limit changes it and --time-limit 0 disables it.

• JacobianMatrix(fs, vars) — the matrix of partial derivatives ∂fᵢ/∂xⱼ, one row per function and one column per variable.

  ce.box(['JacobianMatrix', ['List', fs…], ['List', 'x', 'y', 'z']]).evaluate();
  // → [[∂f₁/∂x, ∂f₁/∂y, ∂f₁/∂z], …]

  • vars may be omitted: the free variables of fs are used, in lexicographic order, so the column order is predictable.
  • A single (non-list) fs is the gradient case and yields a flat vector [∂f/∂x₁, …, ∂f/∂xₙ], directly usable as one.
  • JacobianMatrix(F) accepts a bare function reference: its body is the system and its parameters are the differentiation variables, in declared order (JacobianMatrix((z,y,x) ↦ …) has columns z, y, x — an order free-variable inference could not preserve). Explicit variables then rename the parameters.
  • A square system composes with Determinant, which now also reduces under simplify() — so JacobianDeterminant is one composition away and is not provided separately.

  Beyond brevity this removes a real trap: the obvious hand-rolled form Map([x,y,z], v |-> D(f, v)) silently returns zeros, because the lambda parameter shadows and D differentiates with respect to v.

  The operands are held — evaluating the variable list would replace a symbol carrying a value (x := 5) by that value, leaving nothing to differentiate against. Whether fs is a system or a single function is decided on what the operand denotes, not its syntax, so a user-defined function returning a list and a symbol bound to a list are both treated as systems.

Improvements

• The LaTeX/ASCII-math pipeline operators now preserve Pipe. A bare pipeline stage such as x |> f previously lowered immediately to ["Apply", "f", "x"]; it now produces ["Pipe", "x", "f"], matching the Cortex parser and retaining Pipe's held-operand semantics. Chained stages remain left-associated, prefix stages use Function(Pipe(_, f), _), and a programmatic Pipe serializes with \rhd. Pipeline topic markers remain direct substitutions, so x |> f(\square) still becomes f(x).

• simplify() reaches a distributed form when it is cheaper. Rules tagged purpose: 'expand' are excluded from its scan, because expansion usually grows an expression — but that left simplify() unable to reach a strictly cheaper result. It was not cost-rejecting the better form; it never generated the candidate.

  const f = ce.parse('(1 + x y)^3 z + y^2 (1 + x y) (4 + 3 x y)');
  f.subs({ z: ce.parse('-\\frac{3y}{x} + \\frac{2}{x^2}') }).simplify();
  // Before → unchanged (cost 76)      Now → y^2 + 3y/x + 2/x^2 (cost 27)

  simplify() now tries expansion once, at the fixpoint, and keeps the result only when the cost function says it is strictly cheaper. That cannot cycle (the inner call has the trial disabled) and cannot blow up (the cost gate is the acceptance test), so a factored form survives: (x+1)^5, (a+b)(c+d) and x(y+z) are all left alone. A structural pre-check keeps the trial off expressions with no product or power for expansion to act on.

• simplify() now evaluates structural operators. It is rule-driven and ran no operator evaluate handler at all, so an operator whose result comes from a handler rather than a rule was handed straight back.

  ce.parse('\\det\\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix}').simplify();
  // Before → Determinant(Matrix([[a,b],[c,d]]))    Now → a * d - b * c

  The members are a closed list — Determinant, Trace, Transpose, Length — chosen by a membership rule: the handler reduces its operands to a closed form determined by their structure (a matrix to a scalar, a collection to a measure) rather than rewriting the expression, and the head carries no simplification rule of its own. Max/Min deliberately fail that rule and are not members: they reduce their operands' values, which is evaluation's job.

  simplify() remains value-blind. With a := 5, (a + 2).simplify() is still a + 2. A structural head whose operands mention a symbol carrying a value is left alone rather than substituting it.

  Operators outside the list are unchanged, so the ordering rule still stands: evaluate() then simplify(); simplify() alone is not a superset.

• The Simplify operator resolves symbols bound to a value. An operator normally evaluates its arguments; the transformers are lazy only to keep the operand's structure from being rewritten early, not to keep its values symbolic. This applies to Expand, ExpandAll, Factor, Together and Distribute as well.

  ce.assign('v', ce.parse('\\frac{x^2-1}{x-1}'));
  ce.box(['Simplify', 'v']).evaluate();
  // Before → v      Now → x + 1

  This is the operator, not the method: ce.symbol('v').simplify() is still v, because .simplify() is value-blind.

• Together reduces its result to lowest terms. It folds the terms over the product of the denominators, which is correct but not reduced; the result is now divided through by the numerator/denominator GCD. This also yields the least common denominator, since product / gcd is the LCD.

  ce.box(['Together', ce.parse('-\\frac{3y}{x}+\\frac{2}{x^2}')]).evaluate();
  // Before → (-3y * x^2 + 2x) / (x * x^2)    Now → (-3x * y + 2) / x^2

  The multivariate case needs Brown's algorithm: the univariate Euclidean GCD treats y as an opaque coefficient and reports 1 for that pair. cancelCommonFactors now falls back to the multivariate GCD when the univariate one comes back trivial and more than one unknown is present. The order matters — the cheap path runs first, so the common case is unaffected.

  The same-denominator simplification rule still uses the unreduced fold: it runs inside the simplify() fixpoint, where output stability and cost matter more than presentation.

• CircularIntegrate (\oint) gained an operator definition. It previously had only a parser entry, so it typed as any and its limits stayed a raw Tuple. It now types as number and canonicalizes its limits into Limits expressions, matching Integrate so a limits-consuming caller sees a uniform shape. CircularIntegrate remains inert — there is still no contour- integration evaluation.

  ce.parse('\\oint_C f').json;
  // Before → ["CircularIntegrate", "f", ["Tuple", "Nothing", "C", "Nothing"]]  (type: any)
  // Now    → ["CircularIntegrate", "f", ["Limits", "Nothing", "C", "Nothing"]] (type: number)

• RandomList now compiles on the JavaScript target. RandomList(n) draws fresh values on every call of the compiled function, matching evaluate(). For draws that stay the same from call to call, use the explicit-seed form RandomList(n, seed). A count that is negative, non-finite, or above the 1,000,000 cap makes the compiled function throw, rather than silently clamping or returning NaN. (evaluate() reports an out-of-range error expression for the same input.)

• declare() accepts a spread of an existing operator definition, so you can override one handler and keep the rest:

  ce.declare('At', { ...ce.lookupDefinition('At').operator, evaluate });

  The built-in's other handlers (type, signature, canonical, …) are preserved. Overriding with a bare { signature, evaluate } instead drops them — prefer the spread.

• Compiled At supports a collection-valued index. An index that is a list of indices or a boolean mask now works when compiled, matching evaluate():

  const p = ['List', 10, 20, 30];
  
  ce.box(['At', p, ['List', 3, 1]]);                    // → [30, 10]
  ce.box(['At', p, ['List', 'False', 'True', 'True']]); // → [20, 30]
  
  ce.assign('p', ce.box(p));
  ce.assign('X', ce.box(['List', 1, 2, 3]));
  ce.parse('p_{X}'); // → [10, 20, 30]

  Negative indices count from the end and out-of-range entries are dropped, so a gather may be shorter than its index list. Previously these returned undefined, a wrongly-shaped scalar (p[[2]] gave 20 rather than [20]), or threw.

  An At with a collection-valued index now has type list<T> rather than the element type T. If you dispatch on .type, expect the new value. This is what lets surrounding operations compose: At(p, I) + 1 broadcasts element-wise, and Length(At(p, I)) compiles.

Resolved Issues

• Exact integer tensors stay exact. A shadowed branch in the tensor dtype classifier routed every integer literal into a float64 buffer, which silently disabled the exact-arithmetic path for integer matrices. With the repair, Inverse([[1,2],[3,4]]) evaluates to exact rationals ([[-2,1],[3/2,-1/2]]) and .N() still produces the float form; an integer outside the float-safe range stays expression-backed instead of rounding.

• assume() after assign() now records the assumption. The predicate was evaluated through the symbol's assigned value before the assumption system saw it, so with w := 5, assume(w > 0) folded to True, returned 'tautology', and recorded nothing — the assumption was silently discarded on arrival (only the assume-before-assign order worked). Predicates mentioning assigned symbols are now recorded value-blind, as facts about the symbol: assume(w > 0) after w := 5 returns 'ok' and |w| then simplifies to w. A predicate that contradicts the current value (e.g. assume(w > 0) with w := -2) still returns 'contradiction' and is rejected. 'tautology' now means tautological relative to types and existing assumptions, never relative to an assigned value — re-asserting an equality a symbol's value already satisfies returns 'ok', not 'tautology'.

• .N() on a multi-limit Integrate no longer drops all but the first limit. The numeric-approximation branch read only the first Limits operand, so Integrate(f, Limits(x,0,3), Limits(y,0,2)) numericized as a single-variable integral — ∫∫ 1 over [0,3]×[0,2] gave 3 (a wrong value, not a decline) and a multivariate integrand gave NaN. Multiple limits now perform iterated adaptive Gauss–Kronrod quadrature (Monte-Carlo fallback per level, as for a single limit), following the Mathematica iterator convention the symbolic path already used: the FIRST limit is the OUTERMOST integral, so an inner bound may reference the outer variables — Integrate(1, Limits(x,0,1), Limits(y,0,x)) (the triangle) numericizes to ½. A bound that references an inner integration variable or a foreign free symbol declines (the integral stays inert) rather than integrating wrongly. The same fix applies to compile(): a multi-limit integral that does not close symbolically now compiles to nested quadrature calls — dependent bounds included — where it previously truncated to the first limit.

• Transpose/ConjugateTranspose report the transposed static type. They had no type handler, so an unevaluated Transpose(m) typed as the generic value and was rejected by matrix arithmetic: Multiply(Transpose(J), J) for J = JacobianMatrix(…) — the Gram-matrix idiom JᵀJ — errored with incompatible-type unless J was evaluated first. Both now preserve the element type and swap the shape's axes (matrix<T^(2x3)> → matrix<T^(3x2)>).

• simplify() reaches trig identities inside a quotient. Recursion into Divide operands is deliberately withheld to preserve factored structure for common-factor cancellation, but that also blocked operand-local trig reductions: (r·cosθ)/(r·sin²θ + r·cos²θ) didn't simplify even though the denominator alone reduces to r. Trig-bearing operands of a Divide now get the full recursion (the same carve-out Add/Multiply already made), so it simplifies to cos(θ); factored-polynomial cancellation is unaffected.

• Value-resolution overreach in the lazy-operand fixes. The machinery that lets Solve/Simplify/JacobianMatrix see through a held operand resolved bound symbols too aggressively. Fixed:

  • resolveBoundSymbols is now binder-aware: it no longer resolves a variable bound by a Function, Block, Sum, … to a same-named global value (Simplify(x ↦ x+1) no longer corrupts the body's bound x).
  • A transformer nested in a Solve equation could substitute an unknown that also carries a value — Solve(Simplify(x-2)=0, x) with x:=5 returned []. The unknown is now shielded across transformer reduction.
  • JacobianMatrix differentiated a system in which a diff variable had already been replaced by its global value (JacobianMatrix(g,[x,y]) with x:=5, g:=[x²y,x+y] gave a wrong matrix). It now resolves the operand's shape without substituting values, and differentiates against a fresh symbol when a diff variable carries a value.
  • simplify() evaluated a structural head's whole operand tree, running an impure descendant — simplify(Transpose([[Random()]])) drew a random number. It now declines when the expression is impure.

• D no longer evaluates its result at the differentiation variable's assigned value. With x := 5, D(x², x) is now 2x (was 10). Under the ratified binding convention (ARCHITECTURE.md, "Bound variables, free symbols, and assigned values"), the variable a binder owns is a pure symbol — its declared type and assumptions apply, its assigned value never does, INCLUDING in the result. A caller who wants the value evaluates the result again or substitutes explicitly. Free symbols are unaffected: with a := 3 too, D(a·x², x) is 6x.

• Integrate, Limit, and bundled Solve shield a value-bound bound variable. The same convention: a same-named global assignment no longer leaks into these binders or their results. With x := 5, Integrate(x², x) is x³/3 (was 125/3), ∫₀¹ x² dx is 1/3 (was 0), and Limit(Simplify(x²), x, 0) is 0 (was 25, on both the box and parse routes). A bundled Solve(\{Simplify(9 − w²) = 8, w ∈ -3..3\}) with w := 9 now returns [1, -1] (was []): the value-bound bundled unknown is now discovered before the shield is computed. A shared withValueShield helper implements the shield and replaces JacobianMatrix's per-variable rename.

• Distribute fold, numeratorDenominator, Together, transformers, and toString() grouping — as previously listed.

• Beta-reduction completeness. A finite self-composition (g(g(x)) for a non-recursive g) only inlined one level, so Solve(g(g(x))=0, x) returned []; the recursion guard is now a total-count budget that still terminates genuine recursion. Typed function parameters ((x: real) ↦ …) are unwrapped, so a typed function inlines like a bare one.

• Pipe (|>) declines a non-function right-hand side (5 |> 3 stays inert) and its handler now delegates to apply instead of duplicating its named-operator path.

• Pipe (|>) holds its operands, so x |> f behaves exactly like f(x). It evaluated the topic eagerly, regardless of f — which broke a chain whose right-hand side is a lazy operator that needs its argument unevaluated. A bare function reference was the sharp case: F |> JacobianMatrix passed an evaluated F, stripped of its definition, so the Jacobian could not see the map's body.

  // F |> JacobianMatrix |> Determinant |> Simplify
  ce.assign('F', /* the counterexample map */);
  // Before → Pipe(Pipe(F, JacobianMatrix), Determinant)   Now → -2

  f now decides whether the topic is evaluated (a lazy f receives it unevaluated). A chained topic — the inner pipe of a |> g |> f — is plumbing whose value flows on, so it is evaluated before reaching f. Eager stages, lambda right-hand sides, N(), and numeric chains are unaffected.

• Indexing into a computed list hid the unknown from Solve. A held equation containing At(List(…), k) was opaque to the solver — it saw no unknown and answered [], which by contract means "proven no solutions". This is the last of the lazy-operand family: At is now projected structurally in a held operand.

  ce.box(['Solve', ['Equal', ['At', ['List', 'Y', 2], 1], 5], 'Y']).evaluate();
  // Before → []      Now → [5]

  Projection, never evaluation: with Y := 99, At([Y, 2], 1).evaluate() is 99, which inside a held equation would replace the unknown being solved for. Only a literal List with a literal in-range index is projected (negative indices count from the end); a symbolic list or index is left alone. The transformers (Simplify, Expand, …) get the same treatment.

• Solve returned [] for an equation it could not see into. A lazy operator holds its equation and takes only .canonical, which binds structure without resolving values, so two kinds of operand stayed opaque: a call to a user-defined function, and a symbol whose value contains the unknown. Because [] means "proven no solutions", these were silent wrong answers rather than visible inertness.

  ce.assign('g', ce.parse('t \\mapsto t^2 - 4'));
  ce.box(['Solve', ['Equal', ['g', 'x'], 0], 'x']).evaluate();
  // Before → []      Now → [2, -2]
  
  ce.assign('s', ce.parse('\\frac{9-w^2}{4}'));   // `s = 2` has no `w` in it
  ce.box(['Solve', ['Equal', 's', 2], 'w']).evaluate();
  // Before → []      Now → [1, -1]

  A transformer nested inside the equation (rather than at its root) is now reduced too, so Solve(Simplify(u) = 2, w) works.

  The unknown is never substituted, even when it has a value: the reduction is structural — .subs on the lambda body, .value on a binding — never .evaluate(). With x assigned 5, Solve(g(x) = 0, x) still returns [2, -2]. A recursive definition expands one level and stops.

• Expression transformers ignored a user-defined function in their operand. Same root cause: Simplify(g(a)), Expand, Factor, Together and Distribute returned g(a) unchanged, and Integrate(g(t), t) stayed inert.

  ce.assign('g', ce.parse('t \\mapsto t^2 - 4'));
  ce.box(['Factor', ['g', 'a']]).evaluate();
  // Before → g(a)    Now → (a - 2) * (a + 2)

  Beta-reduction substitutes the function body, so an assigned value for a symbol elsewhere in the operand is still left alone.

• Distribute returned a product where it should return a sum. The helper recombined the branches of a distributed sum with Multiply instead of Add, so (a + b)·c became (a·c)·(b·c). Every input the operator acted on came back with a different value. The operator had no test coverage, which is why this survived; it is now covered by a numeric oracle.

  ce.box(['Distribute', ce.parse('(a+b)c')]).evaluate();
  // Before → a * b * c * c      Now → a * c + b * c

• numeratorDenominator reported a denominator of 1 for a bare negative power. Canonical form writes 1/x^2 as Power(x, -2), and the Power branch never moved a negative exponent into the denominator. The same factor inside a Multiply (y/x^2) routes through Product.asNumeratorDenominator, which splits on exponent sign and was already correct — so the two disagreed. This also affected the NumeratorDenominator operator and .denominator. A symbolic exponent is still left alone: its sign is not decidable there.

  ce.parse('\\frac{1}{x^2}').numeratorDenominator;
  // Before → [x^(-2), 1]        Now → [1, x^2]

• Together dropped denominators written as negative powers. It treated only a Divide node as carrying a denominator, so such terms were folded into the numerator and the combined fraction kept negative powers.

  ce.box(['Together', ce.parse('\\frac{1}{x}+\\frac{1}{x^2}')]).evaluate();
  // Before → (x * x^(-2) + 1) / x    Now → (x + 1) / x^2

• Expression transformers ignored a ReplaceAll in their operand. Expand, ExpandAll, Factor, Together, Distribute and Simplify are lazy and took only .canonical of their held operand, so a ReplaceAll reached them as an unevaluated call with no polynomial structure and was silently returned unchanged. The reduction is recursive, so a producer head nested inside the operand is handled too.

  ce.parse('\\mathrm{Expand}(\\mathrm{ReplaceAll}(x^2+x, x \\to a+1))').evaluate();
  // Before → ReplaceAll(x^2 + x, To(x, a + 1))    Now → a^2 + 3a + 2

  Note this is deliberately a different set from the transformer heads reduced by Solve/Integrate/Limit: ReplaceAll ends in .evaluate() and so substitutes assigned symbol values, which those algorithms must avoid.

• toString() dropped the parentheses around a product-of-sums denominator, producing text that reads back as a different expression. The MathJSON and LaTeX serializations were correct throughout; only the ASCII form was affected. The grouping check treated any string starting with ( and ending with ) as already parenthesized, which (x + 1) * (x^2 - 1) satisfies without being a single group.

  ce.box(['Divide', 1, ['Multiply', ['Add', 'x', 1], ['Add', ['Power','x',2], -1]]]).toString();
  // Before → 1 / (x + 1) * (x^2 - 1)    Now → 1 / ((x + 1) * (x^2 - 1))

• Map now evaluates over a source that only becomes a collection when evaluated. Map(X - 1, f) stayed in its unevaluated lazy form while Map(X, f) and Map([0, 1, 2], f) both evaluated. The trigger was any source whose collection-ness is not visible before evaluation — a broadcast arithmetic result over a list, or an eager collection operator such as UnicodeScalars.

  ce.assign('X', ce.box(['List', 0, 1, 2]));
  ce.box(['Map', ['Subtract', 'X', 1], sq]).evaluate();
  // Before → Map(X - 1, (x) |-> x^2)     Now → [1, 0, 1]
  
  ce.box(['Map', ['UnicodeScalars', { str: 'ab' }], sq]).evaluate();
  // Before → Map(UnicodeScalars("ab"), …) Now → [9409, 9604]

  Applies to the zipWith (multi-source) form as well, and to .at() — which previously returned undefined for such a source, so a result longer than the materialization head was silently rendered head-only instead of head-and-tail. Every other collection operator (Filter, Take, Sort, Reverse, First, …) already accepted these sources. Expressions that are genuinely not collections are unchanged: Map(5, f) still stays symbolic.

• Compiled comparisons and logical connectives no longer return a wrong answer for a list operand. <, <=, >, >=, And, Or and Not could produce a scalar false (or one of their operands) where evaluate() returns an element-wise list of booleans — a wrong result from a successful compile. Such expressions now decline to compile and fall back to interpretation, which gives the correct answer.

  Mostly affects a filter whose condition is computed, such as L[|[1...n]-k|>0]: it now evaluates correctly but is no longer compiled, so expect interpreter performance for that shape. Scalar comparisons and connectives, and list literals such as Not([True, False]), still compile.

• Complex values are handled correctly when compiled. At(p, 1+2i) and RandomList(n, 7+3i) now agree with evaluate(), which uses the real part of a complex index or seed. Previously the compiled forms returned NaN and a different random sequence respectively.

  A compiled scalar comparison against a complex value is still wrong (it returns false rather than declining); this is unchanged and tracked in ROADMAP.md.

• At with several indices reports the correct type. For a 2×3 matrix M, At(M, 1, [1,2]) is the 2-element list [1,2] and At(M, 1, 2) is a single element — both previously reported the type of a whole matrix row.

0.92.1 2026-07-22

Breaking Changes

• Join appends a tuple operand as a single element instead of splicing its components. A tuple is an indexed_collection, so Join used to iterate it and concatenate its components. But a tuple is a value — a point or vector — and the engine treats it as one everywhere else (Abs(point) → Norm, component-wise point arithmetic). Splicing silently degraded the point-list accumulation idiom L → Join(L, P): the result grew by 2 instead of 1 and stopped matching list<point>.

  // Before
  ce.box(['Join', pointList, ['Tuple', 2, 5]]).evaluate();
  // → ["List", …, 2, 5]        length +2, heterogeneous `number` tail
  // After
  // → ["List", …, ["Pair", 2, 5]]   length +1, still a list of points

  Join of collections is unchanged (Join([1,2],[3,4]) → [1,2,3,4]; Join([(0,3)],[(2,5)]) → [(0,3),(2,5)]). The one reversal is a tuple in operand position where it was previously flattened: Join((1,2),(3,4)) was [1,2,3,4] and is now [(1,2),(3,4)]. Join now agrees with Append on a tuple element.

• Join now reports the joined ELEMENT type instead of a bare list. Its type handler returned list whatever it was given, so a joined point list did not match list<tuple<…>> and type-directed dispatch downstream stopped recognizing it. Each operand now contributes either its own type (an atomic tuple, which becomes one element) or its element type (a collection, which is spliced), widened into the result:

  Join(pointList, point); // was: list      now: list<tuple<number, number>>
  Join([1, 2], [3, 4]); //   was: list      now: list<finite_integer>

  An operand whose element type is unknown still yields the bare list. This is a type-surface change: code pinning the exact string 'list' for a Join result sees the narrowed type instead — but matches()-based queries only gain precision.

Issues Resolved

• evaluateAsync() no longer tears down a scoped operator's local scope while the operator is still running. An evaluateAsync handler returns at its first suspension point, not at completion, so the dispatcher popped the operator's local evaluation context too early; everything the resumed handler did then ran against the enclosing scope. A big operator whose reduction outlived one time slice (roughly, more than ~16ms of work) assigned its loop index globally — leaking it, overwriting an outer binding of the same name, and throwing Cannot assign a value to the constant "i" for the commonest index spelling of all, since global i is ImaginaryUnit:

  await ce.parse('\\sum_{i=1}^{200000} i').evaluateAsync();
  // Before: throws `Cannot assign a value to the constant "i"`
  // After:  20000100000  (matches the synchronous lane)
  
  ce.assign('n', 7);
  await ce.parse('\\sum_{n=1}^{200000} n').evaluateAsync();
  ce.box('n'); // Before: 200000 (clobbered). After: 7

  The synchronous lane was fixed in 0.87.1; this brings the asynchronous lane — the cancellation path withTimeLimit documents — into line with it. Cancellation behavior is unchanged.

  Because the context is now held across the await, a scoped operator's frame is no longer necessarily on top of the evaluation-context stack when it unwinds — a second evaluation started while the first is suspended pushes above it. The frame is therefore removed by identity rather than popped, so an unwinding evaluation cannot discard (and dispose the bindings of) one that is still running.

  One known limitation remains, unchanged by this release: while an async evaluation is suspended, its scope is the engine's current one, so code that enters the same engine in that window can see the operator's local bindings (e.g. a loop index). Enclosing bindings still resolve correctly through the scope's parent chain, and the local scope is gone once the evaluation settles, but a name that collides with an in-flight index resolves to that index. Removing this needs per-evaluation (task-local) context propagation; until then, use one engine per concurrent evaluation.

0.92.0 2026-07-21

Breaking Changes

• The parser's three symbol hooks are replaced by one oracle: resolveSymbol. The ParseLatexOptions handlers getSymbolType and hasSubscriptEvaluate (and the internal isSymbolDeclared) are replaced by a single optional handler:

  resolveSymbol?: (symbol) => { type, subscriptEvaluate? } | undefined

  Return undefined for a symbol the handler does not know; return a record (with a BoxedType or type-string type) for one it does. Declaration is the presence of the record — { type: 'unknown' } is a declared symbol of unknown type — so the previously inexpressible distinction between "undeclared" and "declared, type unknown" is now first-class, and inconsistent answers (a typed-but-undeclared symbol) are unrepresentable.

  Semantics also changed from replace to supplement: through ce.parse() the handler is consulted first and any symbol it does not resolve falls back to the engine scope's definitions. Handlers no longer need to re-implement scope delegation (previously required boilerplate with getSymbolType):

  // Before: vouch + hand-written scope delegation
  getSymbolType: (id) => {
    if (vouched(id)) return 'function';
    const def = ce.lookupDefinition(id); // boilerplate, easy to forget
    /* ... map def to a type ... */
  };
  // After: vouch only; unresolved symbols fall back to the scope
  resolveSymbol: (id) => (vouched(id) ? { type: 'function' } : undefined);

  Similarly, the Parser interface (custom LaTeX dictionary entries) replaces getSymbolType()/hasSubscriptEvaluate() with parser.resolveSymbol(), e.g. parser.getSymbolType(id).matches('function') becomes parser.resolveSymbol(id)?.type.matches('function').

Improvements

• A declared name now outranks subscript-index capture. Once a symbol B is bound to an indexed-collection value (a point, list, tuple…), the parser reads B_{2} as indexing (At(B, 2)), which made every subscripted sibling name (B_2, B_3, … alongside the point B) unspellable — and, since B_{2} and B[2] produce identical trees, unrecoverable after the parse. A subscripted spelling whose joined name is declared or assigned in scope now parses as that symbol; index capture applies only to undeclared joins, and bracket indexing (B[2]) is unaffected. Note that with the non-default indexStyle: 'subscript' serialization, At(B, 2) serializes as B_{2}, which re-parses as the symbol B_2 when such a declaration exists.

• Rubi integration (experimental) — nested-radical substitution fallback (R31). loadIntegrationRules now closes nested-radical and sum-of-two-radical integrands the bundled algebraic rules leave inert. A nested radical (√(x+√(x+1))/x², √(1/x+√(1/x+1)), the double-radical √(x+1)/(x+√(√(x+1)+1))) is rationalized by iteratively substituting u = (a+b·x)^{1/k} (or the Laurent (a+b/x)^{1/k}) at the innermost radical linear in x, keeping the resulting rational's denominator factored so the bundled partial-fraction rules close it; the conjugate shape (√(x+1)+√(1−x))⁻² is rationalized by its conjugate. Each result is accepted only after a domain-aware numeric derivative check against the integrand, so out-of-scope shapes stay cleanly unsolved. On the Bondarenko benchmark this lifts CE+Rubi from 12/35 to 20/35 (closing 8 previously unsolved nested-radical integrals; a ninth, #16, closes only under the production bundle's compiled rule set). Structurally inert off its family, and disableable with RUBI_NO_R31.

0.91.0 2026-07-21

New Features

• FindFit — general nonlinear least-squares fitting. FindFit(data, model, params, vars) fits an arbitrary model expression to data by Levenberg–Marquardt, returning a record {parameters, converged, residualNorm, iterations}. Unlike the closed-form LinearRegression/PolynomialFit, the model may be any composition (a·e^{b·x} + c, Gaussians, power laws, cosines, …). data is a list of (x…, y) tuples or a plain list of y values (x = 1, 2, …). Each parameter spec is a bare symbol (start 1, unbounded), (a, a0) (explicit start), or (a, a0, lo, hi) (start plus a box constraint, with ±∞ allowed for one-sided bounds). Convergence is first-class: non-convergence within the iteration budget is reported as converged: False with the best-so-far values, never a silent wrong answer. Jacobians are analytic (via D), with a per-column forward finite-difference fallback for components that cannot be differentiated symbolically. A joint form fits several models to several datasets sharing parameters (a list of models paired with a list of datasets, residuals stacked).

• FindRoot — numerical equation solving. FindRoot(equations, params) finds parameter values that zero one or more residuals, sharing the same parameter-spec grammar, box constraints, and result record as FindFit (root-finding is the zero-residual case of the same Levenberg–Marquardt core). equations is an equation (lhs == rhs), a bare residual expression (read as = 0), or a list of either.

Resolved Issues

• Applying a declared-then-assigned function to a large collection is now lazy, matching the hybrid-laziness contract. Since 0.84.0, element-wise operations over collections of more than 100 elements (or of unknown length) evaluate to a lazy Map — but a function registered via ce.declare('g', '(number) -> number') + ce.assign('g', x ↦ …) resolved through a value definition whose application-site broadcast still zipped eagerly, walking the whole collection at evaluate() time. The value-definition application path now goes through the same laziness gate as parse-assigned functions (g(x) := …) and built-in broadcasts: past the eager threshold g(X) returns a lazy Map, results of ≤100 known-finite elements are byte-identical to before, collection-typed parameters still bind their argument whole, and tuples stay atomic. (Not a recent regression: this path had been eager on every release since laziness shipped in 0.84.0.)

Benchmarks

Numeric performance (200-digit precision)

Median time per call, in microseconds — lower is better. — means the tool returned no usable result at that precision.

Expression	CE (current)	CE 0.86.1	SymPy	math.js	Mathematica
\pi^2	6.2	7.0	175	173	3.9
\sin 1	20	20	220	506	5.2
\cos 1	20	21	219	612	6.9
\ln 2	14	14	348	4,616	4.0
e^{\pi}	12	12	221	5,086	4.6
\zeta(3)	1,580	1,619	261	—	49
\Gamma(\tfrac13)	842	839	345	—	211
\psi(\tfrac13)	728	720	2,806	—	168

Symbolic capability & performance

Each cell is how many times faster than Mathematica that engine is on the case (Mathematica ÷ engine, so higher is better; Mathematica itself is 1×). — means the engine can't do the case; ✓ means it solves a case Mathematica can't. Compare the CE (current) and CE 0.86.1 columns to see what is new this release (a — under 0.86.1 next to a number under the current build). The CE + R/F column is the current build with the opt-in Rubi integrator + Fungrim identities loaded (loadIntegrationRules / loadIdentities), on the same minified bundle.

Operation	CE (current)	CE + R/F	CE 0.86.1	SymPy	math.js	Mathematica
Antiderivatives
\int\frac{1}{\sqrt x}\,dx	6.7×	3.1×	5.2×	0.5×	—	1×
\int\frac{x}{\sqrt{1-x^2}}\,dx	10×	1.6×	8.4×	0.08×	—	1×
\int\frac{1}{x^3+1}\,dx	6.3×	0.9×	4.4×	0.4×	—	1×
\int\frac{\sqrt x}{1+x}\,dx	—	2.0×	—	0.1×	—	1×
\int\frac{x}{(1+x)^{1/3}}\,dx	—	1.4×	—	0.01×	—	1×
\int\frac{x^2}{(1+x)^{1/3}}\,dx	—	1.3×	—	0.007×	—	1×
Derivatives
\tfrac{d}{dx}\sqrt{1-x^2}	0.1×	0.1×	0.06×	0.003×	0.01×	1×
Simplification
\sqrt{3+2\sqrt2}	41×	31×	32×	—	—	1×
\sqrt6\,x+\sqrt2\,x	104×	54×	59×	3.3×	16×	1×
Evaluation
\lim_{x\to0}\tfrac{\sin x}{x}	58×	29×	47×	3.0×	—	1×
\lim_{x\to\infty}(1+\tfrac1x)^x	9.3×	5.6×	8.0×	2.1×	—	1×
\int_1^2\tfrac1x\,dx	7405×	6951×	6437×	77×	—	1×
\int_{-\infty}^{\infty} e^{-x^2}\,dx	423×	161×	363×	2.6×	—	1×
Solving
x^4+x^2-1=0	0.3×	0.3×	0.2×	0.06×	—	1×
x^3-x-1=0	1.7×	2.0×	1.4×	0.04×	—	1×

Across the cases both solve, Compute Engine is a median 6.7× faster than Mathematica (up to 7405×) — in the browser, not a proprietary kernel.

_{Measured 2026-07-21 · Compute Engine 0.90.0 @ 8740998f (current build) · published 0.86.1 · SymPy 1.14.0 · math.js 15.2.0 · Mathematica 14.3.0 for Mac OS X ARM · Node v22.13.1. Correctness is verified numerically against an independent mpmath reference, never another tool. Reproduce with npm run build production && ./venv/bin/python3 benchmarks/gen_cases.py && node benchmarks/report.mjs && node benchmarks/report_changelog.mjs.}

0.90.0 2026-07-21

New Features

• RandomList(n) — an eagerly-materialized list of n independent uniform reals in [0, 1). Draws come from the engine's random stream (honoring ce.randomSeed for reproducibility); the two-argument form RandomList(n, seed) produces a deterministic list from an explicit seed, independent of the engine stream. Eagerness is deliberate: the result is a concrete List, so every reference to it sees the same draws (a lazy collection of Random() calls re-draws on each traversal). With a literal count the length is part of the type (RandomList(5) types vector<finite_real^5>). The count is capped at 10⁶ elements; a larger count — or a negative one — returns an out-of-range error rather than silently misbehaving.

Improvements and Resolved Issues

• Abs of a fixed-arity point is now the Euclidean norm. |(3, 4)| evaluates to 5 — the single-bar spelling of the vector magnitude, consistent with the \lVert…\rVert (Norm) parse and with tuple arithmetic treating points as vectors in ℝⁿ. Previously the expression stayed inert under evaluate(), and compiled to garbage: the JavaScript target returned the bare component array (which concatenated into a string in downstream arithmetic), and the shader targets emitted invalid code. All targets now compile it as the norm (_SYS.norm on the js target, length() on GLSL/WGSL for 2–4 components; other arities and points with a broadcasting component fail closed to interpretation). Detection is type-based, so a tuple-typed symbol or parameter routes as a point too. Abs over a List is unchanged and still broadcasts elementwise. Hypot with a point argument now squares through the norm as well: Hypot((3,4), 1) = √26 (previously an inert Power of a tuple). And Norm/Abs of a point now honor the exactness contract: evaluate() keeps the exact √2, .N() numericizes.

• Norm compiles on the interval-js target for fixed-arity points (default L2 norm; hypot-based in 2-D for a tighter enclosure). Implicit curves and line series expressed with \lVert…\rVert or |…| now get interval-arithmetic break detection instead of silently degrading to point sampling.

• Norm/Abs of a point with a broadcasting component reports an honest list<number> type. ‖(x+[0.5,1], y, z+2)‖ evaluates to a List (one norm per zipped element); its static type now says so instead of number, matching the equivalent \sqrt{(x+[0.5,1])^2+…} spelling.

• A Comprehension serializes with bracket delimiters so it survives its own round trip in every position: H=[k^2 \operatorname{for} k=[1...4]] now serializes as H=\left[k^2 \operatorname{for} k = 1..4\right] (previously the unfenced form re-parsed with the assignment swallowed into the comprehension body, and under an Add the trailing term was absorbed into the iteration range). The fence is unconditional — [body for …] parses back to the same Comprehension, so it is lossless. (Non-canonical serialization-shape callout, same class as the 0.83 big-operator body fence.)

• Expression.isValid is now O(1) after the first query. Validity is a structural property of an immutable expression, so it is computed once and cached; previously every query re-walked the whole subtree (and a parent's query re-entered each child's), which dominated large-document workloads — one profiled 75-row import spent 77% of its time in repeated isValid walks.

0.89.0 2026-07-21

Breaking Changes

• ComputeEngine.timeLimit is removed, completing the deprecation begun in 0.88.0. There is no implicit ambient deadline anymore: an evaluate() or simplify() outside a span runs unbounded, and ce.withTimeLimit(ms, fn) / ce.withTimeLimit({ ms, label }, fn) spans are the only way to arm a deadline. If you relied on the old 2000 ms ambient default, wrap your evaluation entry point in a span:

  const r = ce.withTimeLimit({ ms: 2000, label: 'my-app:eval' }, () =>
    expr.evaluate()
  );

  The engine.timeLimit:<operator> attribution synthesized for ambient timeouts is gone with it — every CancellationError.attribution now names a span you created (or is undefined for an unlabelled span).

• The BoxedTensor class is removed. Tensor values (vectors, matrices) are now ordinary canonical List expressions; "tensor-ness" is a property (shape-regularity), not a distinct representation. The BoxedTensor type export and the TensorInterface.tensor accessor (which exposed the internal packed Tensor object) are gone. The isTensor() guard remains and now answers the representation-independent question "is this a shape-regular list?"; .shape and .rank remain and now report honestly on every tensor-shaped list — including broadcast results, which previously reported []/0. Code that used expr.tensor should operate on expr.ops (the elements) or use the public linear-algebra operators.

• Lists carry honest, shape-aware types. A shape-regular list's type reports its actual element type and dimensions: [1, 2, 3] types vector<finite_integer^3> (previously vector<3>, i.e. number elements), [[1, 2], [3.5, 4.5]] types matrix<finite_real^(2x2)>, and a list of non-numeric values is no longer mistyped as a numeric vector — [rgb(1,0,0), rgb(0,1,0)] types list<color^2>, not vector<2>. The new types are strict subtypes of the old ones, so type.matches('vector<3>') and similar queries continue to answer true; only code comparing exact type strings is affected. An evaluated broadcast result now carries its shape too (Sin([0,1]).evaluate() types vector<2>), and the declared type of a broadcast expression is always a sound upper bound of its evaluated value's type.

• Signature validation of collection arguments is two-stage. An operand whose static type could still conform to a collection parameter (a symbol declared plain list, list<unknown>, a broadcastable<…> intermediate) is accepted at canonicalization and checked against its actual value when the operator evaluates. Provably-wrong operands (Determinant("abc"), Determinant(v) with v: list<number> — a flat vector can never be a matrix) still error immediately. Consequently some incompatible-type errors that used to appear at parse/canonicalization time now surface at evaluation time instead (as the operator's specific error, e.g. expected-square-matrix, or as an inert expression).

Improvements

• Matrix operations work on computed matrices. The static type of a broadcast application now mirrors its operand's shape (Sqrt(M) with M a 2×2 matrix types as a 2×2 matrix), so expressions like Determinant(Sqrt(M)), MatrixMultiply(Sqrt(M), Sqrt(M)), or Inverse(A + B) — which used to fail with incompatible-type at canonicalization — now evaluate. Symbols declared list participate the same way once a conforming value is assigned.

• Structural matrix operations work on any cell type. Transpose, ConjugateTranspose, Reshape, Flatten, and the shape predicates (IsSquareMatrix, IsSymmetric, IsDiagonal) operate on any shape-regular list — a matrix of colors, tuples, or unevaluated function applications — not just numeric ones. Numeric kernels (Determinant, Inverse, …) still require numeric cells and decline others gracefully.

• Exact integer matrices stay exact. Linear-algebra kernels over exact integer matrices now use exact arithmetic under evaluate(): Determinant([[9007199254740991, 0], [0, 3]]) returns the exact 27021597764222973 (previously rounded through float arithmetic). Under .N() results are floated, as before (Inverse([[2,1],[1,3]]).N() → [[0.6, -0.2], [-0.2, 0.4]]).

• List equality is tolerant and NaN-aware in all cases. [1,2,3].isEqual([1,2,3+1e-11]) is true (engine tolerance), [x,2].isEqual([y,2]) is undefined (symbolic), and a list containing NaN is never isEqual to itself (mirroring scalar NaN ≠ NaN) — uniformly for every list, whatever produced it. Ordering comparisons (isLessEqual, …) between equal tensors of any cell type now answer true instead of undefined.

• Faster broadcast arithmetic. Removing the eager tensor construction from the boxing path makes broadcast-heavy evaluation measurably faster (~30% on elementwise matrix expressions), with no per-expression packing cost until a numeric kernel actually runs.

Issues Resolved

• A function call over a collection-valued expression keeps its broadcast type. With h a function over numbers (declared or inferred) and L a list, h(L) already typed as a list — but h(L + 1) or h(2L) typed as a scalar while still evaluating to a list. Both now type as the shaped vector (h(L+1) → vector<3> for a 3-element L), and a declared scalar signature no longer rejects a collection argument with incompatible-type — scalar-parameter functions are threadable, so the argument broadcasts, matching what evaluation always did. Scalar applications and genuinely invalid arguments (h("abc")) are unchanged.

Benchmarks

Numeric performance (200-digit precision)

Median time per call, in microseconds — lower is better. — means the tool returned no usable result at that precision.

Expression	CE (current)	CE 0.86.1	SymPy	math.js	Mathematica
\pi^2	6.7	7.5	176	137	4.1
\sin 1	21	21	220	444	5.1
\cos 1	20	20	224	536	6.9
\ln 2	16	16	352	5,754	3.8
e^{\pi}	14	15	215	4,765	4.5
\zeta(3)	1,734	1,790	282	—	51
\Gamma(\tfrac13)	957	950	356	—	209
\psi(\tfrac13)	806	795	2,818	—	170

Symbolic capability & performance

Each cell is how many times faster than Mathematica that engine is on the case (Mathematica ÷ engine, so higher is better; Mathematica itself is 1×). — means the engine can't do the case; ✓ means it solves a case Mathematica can't. Compare the CE (current) and CE 0.86.1 columns to see what is new this release (a — under 0.86.1 next to a number under the current build). The CE + R/F column is the current build with the opt-in Rubi integrator + Fungrim identities loaded (loadIntegrationRules / loadIdentities), on the same minified bundle.

Operation	CE (current)	CE + R/F	CE 0.86.1	SymPy	math.js	Mathematica
Antiderivatives
\int\frac{1}{\sqrt x}\,dx	6.6×	3.1×	5.4×	0.5×	—	1×
\int\frac{x}{\sqrt{1-x^2}}\,dx	8.0×	1.3×	6.8×	0.09×	—	1×
\int\frac{1}{x^3+1}\,dx	5.6×	0.8×	4.3×	0.3×	—	1×
\int\frac{\sqrt x}{1+x}\,dx	—	1.7×	—	0.1×	—	1×
\int\frac{x}{(1+x)^{1/3}}\,dx	—	1.2×	—	0.01×	—	1×
\int\frac{x^2}{(1+x)^{1/3}}\,dx	—	1.2×	—	0.007×	—	1×
Derivatives
\tfrac{d}{dx}\sqrt{1-x^2}	0.04×	0.04×	0.02×	0.001×	0.003×	1×
Simplification
\sqrt{3+2\sqrt2}	44×	34×	34×	—	—	1×
\sqrt6\,x+\sqrt2\,x	131×	67×	72×	5.0×	23×	1×
Evaluation
\lim_{x\to0}\tfrac{\sin x}{x}	53×	25×	43×	3.0×	—	1×
\lim_{x\to\infty}(1+\tfrac1x)^x	8.7×	5.1×	7.2×	2.1×	—	1×
\int_1^2\tfrac1x\,dx	6954×	6552×	5758×	93×	—	1×
\int_{-\infty}^{\infty} e^{-x^2}\,dx	382×	133×	341×	2.6×	—	1×
Solving
x^4+x^2-1=0	0.2×	0.3×	0.2×	0.06×	—	1×
x^3-x-1=0	1.5×	1.7×	1.4×	0.04×	—	1×

Across the cases both solve, Compute Engine is a median 6.6× faster than Mathematica (up to 6954×) — in the browser, not a proprietary kernel.

_{Measured 2026-07-21 · Compute Engine 0.88.1 @ afde4f88 (current build) · published 0.86.1 · SymPy 1.14.0 · math.js 15.2.0 · Mathematica 14.3.0 for Mac OS X ARM · Node v22.13.1. Correctness is verified numerically against an independent mpmath reference, never another tool. Reproduce with npm run build production && ./venv/bin/python3 benchmarks/gen_cases.py && node benchmarks/report.mjs && node benchmarks/report_changelog.mjs.}

0.88.1 2026-07-20

Issues Resolved

• Color converters (AsRgb, AsHsv, AsHsl, AsOklab, AsOklch) broadcast over lists, like the color constructors already did: AsRgb([Hsv(120,1,1), Hsv(0,1,1)]) → [Rgb(0,1,0), Rgb(1,0,0)] instead of an incompatible-type error. A non-color element produces a per-element error rather than rejecting the whole call. Out-of-range channels continue to pass through unchanged — they can represent valid out-of-sRGB-gamut colors, so constructors and converters neither clamp nor error.

• ce.box() no longer throws on malformed MathJSON input. A non-MathJSON plain object (ce.box({foo: 1})) or an array whose head is not a symbol used to throw a JavaScript Error; both now return a boxed ["Error", "'unexpected-mathjson'", …] expression (with the offending input as context), consistent with how every other input problem is reported. The engine remains fully usable afterward.

0.88.0 2026-07-20

Deprecations

• ComputeEngine.timeLimit is deprecated in favor of ComputeEngine.withTimeLimit(). timeLimit arms a hard-to-scope implicit deadline around each evaluate()/simplify(); wrap the work you want bounded in a span instead:

  // Before
  ce.timeLimit = 500;
  const r = expr.evaluate();
  
  // After
  const r = ce.withTimeLimit({ ms: 500, label: 'my-app:eval' }, () =>
    expr.evaluate()
  );

  timeLimit still functions exactly as before in this release; it will be removed in a future minor version.

Improvements

• ComputeEngine.withTimeLimit() accepts an attribution label. In addition to the numeric form withTimeLimit(ms, fn), an object form withTimeLimit({ ms, label }, fn) (preferred for new code) records a label on the span. Nesting still composes as min() — a labelled inner span can only shorten the effective deadline, never extend it past an enclosing one.

• CancellationError now carries attribution and spans. When a timeout fires, attribution is the label of the span that owns the deadline that fired (so a caller can compare it against the label it passed to distinguish "my sub-budget expired" from "my caller's budget expired"), and spans lists all active span labels, outermost first. Timeouts armed by the deprecated ambient ce.timeLimit are attributed as engine.timeLimit:<operator> (e.g. engine.timeLimit:Integrate).

• Divisor functions are now O(√n) instead of O(n). Sigma0, Sigma1, SigmaMinus1, IsPerfect, and Totient compute from the prime factorization rather than trial iteration: Sigma1(1000000007) went from ~14s to ~1ms, and 11+-digit inputs that previously ran essentially forever return instantly.

• Integer factorization is interruptible. The Pollard-rho factorizer now honors the active deadline (a withTimeLimit span interrupts a hard semiprime factorization on time, with attribution) and is backstopped by an iteration budget (cause: 'iteration-limit-exceeded') so it terminates even with no time limit set.

• Symbolic integration no longer swallows a caller's timeout. If a withTimeLimit span enclosing an integration expires mid-attempt, the CancellationError now propagates to the caller (identified via attribution) instead of being silently converted into "no antiderivative found". Rubi's own internal sub-budgets still degrade gracefully.

Fixes

• Unary D applications no longer serialize to a bare D. An arity-1 (or otherwise unrecognized) D application — e.g. a document-defined function named D — used to serialize with its argument silently dropped (["D","w"] → "D"). It now serializes as \operatorname{D}(w), which round-trips exactly. Recognized derivative shapes (D(f,x) → Leibniz notation) are unchanged.

• A known-value uppercase symbol before a parenthesized group now parses as multiplication. The predicate-notation heuristic (a single uppercase letter before (...) reads as a function application, e.g. P(x)) applied even when the scope knew the symbol was a value: with K assigned -32.3, K(2-0.1) parsed as a call of K and evaluated to an incompatible-type error. The heuristic now consults the scope — a symbol with a known non-function type falls through to multiplication (K(2-0.1) → -61.37). Unknown and function-typed symbols are unchanged.

• Juxtaposed-multiply serialization is round-trip safe for single-uppercase factors. Multiply(K, group) serialized as K(group), which re-parses as a function call of K — corrupting the expression when K is a numeric symbol. A single-uppercase-letter factor directly against a parenthesized group now emits an explicit \times (K\times(…)). Other factor shapes (x(y+z), 2(x+1), \mathrm{abc}(x+1)) already round-tripped and are untouched.

• ce.box() now accepts native bigint values. bigint was declared in ExpressionInput but unhandled by the boxing dispatch, so ce.box(123n) — bare or as an operand in ce.box(['Add', 10n, 5]) — silently became the Undefined symbol. Bigints now box as exact integer literals, preserving exactness at any magnitude (never routed through a float), identical to the ce.number(bigint) path.

• Unbounded collection walks over infinite lazy sources. First/At on a Filter or TakeWhile whose predicate never (or eventually never) matches, and any walk over a Dedup of a source with infinitely repeating duplicates (e.g. Dedup(Cycle([1,1]))), could previously spin until the ambient time limit fired — or forever with timeLimit = 0. These walks are now bounded by iterationLimit and degrade gracefully (Nothing/undefined), consistent with the documented lazy-collection contract. Note: as a consequence, Count(Dedup(...)) over a finite source larger than iterationLimit now returns undefined (unknown) instead of walking the whole source, matching Filter's existing count behavior.

0.87.2 2026-07-20

Breaking Changes

• Add no longer widens an unreachable scalar arm into a broadcast collection type. A sum mixing a scalar with a list-shaped operand typed as a union — matrix + 1 was finite_integer | matrix, 2·[1,2,3] + a was number | vector<3> — even though the value ALWAYS broadcasts elementwise and can never be a scalar ([[1,2],[3,4]] + 1 → [[2,3],[4,5]]). These now type as the collection: matrix, vector<3>. The behavior was inconsistent as well as imprecise — a dimensionless list<number> + 1 was already repaired to list<number> downstream, so only dimensioned shapes carried the artifact.

  This is a type-surface change: code pinning the union spelling will see the narrowed type instead. It is a strict improvement for consumers that dispatch on the type, and that is the motivation — union matching is all-members, so type.matches('collection') returned a confident false on a value that is always a collection, silently routing list-valued rows down scalar paths (reported by Tycho as item 67). It also unblocks expressions CE itself rejected: MatrixMultiply([[x, y]], aM₁ + M₂) failed signature validation on the union operand and now evaluates.

  Generic collection/set-typed operands are deliberately unchanged and keep the honest union — a non-indexed collection is never broadcast by the value path, so a scalar outcome stays reachable there.

Resolved Issues

• isValid now honors its documented contract through a list/tensor. isValid is specified as false if the expression "or any of its subexpressions" is an ["Error"], but BoxedTensor.isValid returned true unconditionally — so a List whose every element was an Error reported isValid: true. (1,2)+[3,4] broadcasts to a list of incompatible-type errors and passed the gate. An embedded Error element now poisons the enclosing expression, matching what BoxedFunction.isValid already did.

  Behavior change worth noting even though the contract is unchanged: consumers using isValid as an admission gate before compiling or plotting will now correctly reject expressions they previously admitted.

  Only expression-dtype tensors are scanned; float64/complex128/bool fields cannot hold an Error by construction and keep the O(1) answer, so this does not add a per-element walk to large numeric tensors.

  The isValid documentation has been expanded to spell out the contract: the check is deep (including list elements and held operands), and it tests well-formedness rather than meaningfulness — free symbols, undeclared functions, NaN and ±∞ are all valid.

0.87.1 2026-07-20

Breaking Changes

• compile(..., { realOnly: true }) now rejects a complex-valued tuple/list component. realOnly coerced only the top-level result, so a { re, im } object sitting in a component slot passed through untouched and reached the caller in a number slot — (t, i t) compiled successfully and returned [0.5, { re: 0, im: 0.5 }]. A complex component now fails the compile with a diagnostic naming the component, matching what the GPU targets already do and the existing Sqrt(-1) "no real value" error. The component's type cannot decide this — with t undeclared, (t, i t) and (t, t²) both infer finite_number — so the check uses the same isComplexValued analysis the GPU targets fail closed on, and every target now rejects the same shapes. Real-valued tuples are unaffected, and the runtime realOnly coercion now also recurses into array results for values that only become complex when called ((t, √t) at t = -4 → [-4, NaN]). The check follows only positions that can produce the compiled result, so a complex value consumed by an operation with a real result still compiles (At([i, 2], 2) → 2).

Resolved Issues

• A When (restriction) whose value was a COLLECTION did not expose the collection interface. isCollection was false, count undefined and each() yielded nothing, even though .type already reported vector<N> / list<tuple<…>> — the type system and the collection interface disagreed about the same value. Only a LIST-valued condition broadcast; the common case of one scalar restriction over a whole list left an opaque wrapper in place. A collection-valued When now behaves as [When(L₁,c), …, When(Lₙ,c)]: at or below MAX_SIZE_EAGER_COLLECTION it distributes into a List, and above the threshold it stays a held When that is nonetheless fully enumerable (count/each()/at()), following the same hybrid-lazy convention as PointList. A scalar When still reports as a scalar, and a Tuple-valued one is not split into its components so a restricted point stays a point. Operators whose collection-ness depends on their operands can now declare it with the new optional BaseCollectionHandlers.isCollection predicate. Every handler-backed collection API honors that opt-out — count, each(), at(), get(), indexWhere(), subsetOf(), contains(), isLazyCollection and isIndexedCollection — so none of them can report a collection answer for a value that says it is not one. The opt-out is deliberately narrower than isCollection === false: an eager collection operator such as UnicodeScalars has no collection handlers at all until it is evaluated, and still goes through the materialize-then-iterate path.

• A Sum/Product over three or more indexing sets silently dropped every index after the second. The cartesian product of the indexing sets was built by a fold that returned tuples of the wrong length — for a 2×2×2 product, eight tuples of length 2, with pairs duplicated — and since the reducer reads the tuple positionally, the third and later loop indexes were assigned undefined. Any triple sum or product was therefore wrong, without a diagnostic. The full n-dimensional product is now iterated, with the last index varying fastest. One and two-index big-ops are unaffected.

• A Sum/Product with a large finite bound exhausted the heap before it could be interrupted. The whole index product was materialized up front, so Σ_{i=1}^{10⁸} allocated 10⁸ one-element arrays before the reducer ran a single step — the process died before run()/runAsync() or any deadline could cancel it. Index tuples are now streamed one at a time, keeping allocation proportional to the number of indexes, and the engine deadline is checked between terms. Bounds whose magnitude exceeds Number.MAX_SAFE_INTEGER over a non-degenerate range can no longer be enumerated faithfully at number precision — adding one to such a value does not change it — and now evaluate to an ["Error", "out-of-range"] rather than silently truncating to a single term. A degenerate range (lower === upper) still yields exactly one term.

• A List element whose container type was revealed only by canonicalization was flattened into a numeric tensor. Tensor eligibility is decided on raw operands so nested Lists stay visible, but a wrapper such as a parsed Delimiter, If, Which, When or Hold reports its tuple/set/ dictionary/record type only after it is canonicalized. Such elements were taken for scalar tensor components, so a list of tuples collapsed into a vector<N> and lost its structure. Container-valued elements are now recognized both by operator name and by type — primitive and structured alike, including as a member of a union — and keep the expression a List.

• A Sum/Product index named i was read as ImaginaryUnit by the compiler's complex-valuedness analysis, silently corrupting the enclosing arithmetic on every target. The binder's bound name reached the engine-value fallback as if it were a free symbol, so the analysis complex-tainted the sibling operand of any enclosing arithmetic — the Sum itself emitted correctly. \sum_{i=0}^{2}\cos(it)+2.5 compiled (success: true) to NaN, and \sin(\sum_{i=0}^{2}\cos(it))+2.5 to a silently wrong 2.5; the interpreter was correct throughout. The index need not appear in the body, and \prod was affected identically. A binder's bound names are no longer analyzed as free symbols; loop/summation indices are treated as the integer counters they are, while function parameters keep their declared types (a complex parameter stays complex).

• A per-evaluate ce.timeLimit was silently inert inside a ce.withTimeLimit(ms, fn) span. The span deadline replaced, rather than min-ed with, every inner clamp, so a pipeline wrapped in a span lost its inner bounds — a 500 ms clamp ran the full 60 s span. The effective deadline is now min(ambient, now + timeLimit). Plain nested evaluations are unaffected. Note this applies to synchronous evaluate(): a span's deadline is still restored when its callback returns, so evaluateAsync() under a span remains unbounded.

• GPU targets emitted invalid shader source for vector-valued block locals and over-wide vecN constructors. A tuple/point-valued block local was declared float while being assigned a vecN/array, vector width did not propagate through an aliased local (q := p), and vecN constructor arity was chosen from the argument count when it is a component count — so a complex or nested-tuple element overflowed the constructor. Locals now declare the matching type, width propagates through aliases, and an aggregate-valued component fails closed with a diagnostic instead of emitting source no driver accepts. Failing closed now also covers a matrix-valued component, an empty tuple/list (neither language has a zero-length array type), and a block local bound to values of disagreeing shapes within one block (a shader local has a single declared type, and there is no declaration a scalar and a vecN assignment both satisfy).

• A delimited \mapsto body was read as a statement Block rather than a Tuple, so a point-valued lambda silently dropped all but its last component. t \mapsto (\cos t, \sin t) applied at t = 0.5 returned 0.479… — just sin 0.5 — on every target, js included; the equivalent g(t) := (\cos t, \sin t) was already correct. A delimited lambda body is now data (a Tuple) whatever its separator; a genuine statement block is built by the ; infix parser when the sequence contains an Assign and reaches the lambda parser already formed, so (x := 1; x+1) is unchanged.

0.87.0 2026-07-19

Resolved Issues

• Exponential blowup evaluating float-carrying symbolic bodies inside function applications. The "inexact operand numericizes a closed-constant sum/product" rule (0.5 + π → 3.64…) decided "closed constant" by resolving symbols through the dynamic scope chain, so inside a function application a bound-but-symbolic parameter counted as known: a body term like z² + 0.3 fired a full-subtree N() walk that could make no progress, at every nested level, mutually recursive with evaluate — ~×7.5 work per nesting level. The canonical victim was interpreted evaluation of a recursive function over a symbolic argument (Q(n, z) = Q(n-1, z)² + 0.3, the iterated-map shape): depth 7 took ~8 s and depth 8+ hit the time limit, where the same recursion with an exact constant (3/10) unwound in milliseconds. The gate is now the lexical isConstant (every symbol a constant binding) — depth 7 drops ~500× to ~15 ms, float and exact now cost the same, and 0.5 + π, 0.5 + √2, and 0.5 + x all behave exactly as before. Two neighboring sites sharing the wrong predicate returned flat-wrong values inside applications and are fixed the same way: KroneckerDelta(w) over a bound symbolic parameter returned 0 (now stays symbolic), and Degree(w²) returned 0 (now 2).

  Relatedly, many non-lazy evaluate handlers re-evaluated operands the evaluation driver had already evaluated. Each such call re-descends the whole operand subtree, so under nesting the waste compounded — a residual ×2-per-level re-walk on top of the bug above. All library handlers now follow the handler contract (a lazy operator's handler owns its operands' single evaluation; a non-lazy handler receives them already evaluated and must not re-evaluate): Power, Sqrt, Root, Divide, Ln, Log, Negate in arithmetic; the linear-algebra operators (Transpose, Determinant, Inverse, MatrixMultiply, Norm, the eigen/decomposition family, matrix constructors and predicates, ~30 sites); the statistics reducers (Mean/Median/Variance/… — 11 sites); and Text. Symbolic recursive unwinding is now linear: depth 80 unwinds in ~95 ms where depth ~20 previously hit the time limit.

• Timing now measures the actual evaluation. Timing was a non-lazy operator, so the engine evaluated its argument before the handler ran and the handler then timed a redundant second walk of the already-evaluated result — reported times measured cache-warm re-walks, not the computation. Timing is now lazy: the handler receives the raw argument, canonicalizes it outside the timed region, and times the real evaluation.

• One-time cache builds are no longer charged against the time limit. The engine builds some internal tables lazily on first use (constructible trig values on the first sin(π/6)-style evaluation, the standard simplification rule set, etc.). Previously this warm-up ran inside the caller's timeLimit/withTimeLimit() budget, so a tight deadline on a fresh engine could lose a large fraction of its budget — or fire mid-build — on the very first call. The deadline is now suspended while a cache builds and then pushed back by the build's duration, so a time budget measures only the caller's own evaluation. Relatedly, a timeout that did fire during a cache build was swallowed and resurfaced as an unrelated TypeError; an interruption now propagates as the CancellationError it is, leaving the cache unbuilt so a later call retries.

• Compiled real/complex convention mismatch in branch arms (js target). A provably-real branch arm (If/Which/When) alongside a complex-valued arm compiled to a plain number while consumers of the branch read { re, im } slots — so a constant base-case arm in a complex-ascribed recursive function (M(0, z) = 0, the canonical base-case shape) returned NaN at every point, including points that never left the base clause. Real arms are now coerced to the complex convention when any arm is complex (the no-match default likewise emits { re: NaN, im: NaN }); wide-typed pass-through arms (a z slot declared number carrying a complex value at run time) stay bare. The same coercion now applies at the two sibling seams: a Typed complex ascription over a provably-real operand (previously silently inert in compiled code — an all-real body under a declared -> complex return), and a provably-real call-site argument bound to a complex-typed parameter of a user-defined function (M(10, 0) — Complex(0, 0) canonicalizes to the real literal 0).

New Features

• New engine flag ce.jit: 'auto' | 'off' governing every implicit compilation path — the new lazy-Map auto-compilation (below) and the pre-existing compiled numeric kernels (NIntegrate/ND/NLimit, the Integrate/Limit numeric fallbacks, NDSolve right-hand sides, the solve-domain enumeration sieve, the stochastic-equality probes, the compiled Reduce fast path). Default 'auto': attempts run, and on the first environment-level EvalError (a strict-CSP host refusing dynamic code) the engine latches to 'off' engine-wide, capping CSP violation reports at one. Set 'off' up front on strict-CSP pages, MV3 extensions, or hardened runtimes — or as a diagnostic kill switch. Explicit compile() is exempt and keeps failing loudly. Implicit compile failures now fall back to the interpreter silently (previously some of these paths logged a Compilation fallback warning).

Performance

• Lazy-Map element lambdas auto-compile on numeric drains. Draining a lazy broadcast (f(Range(1, 10^5)).N(), a PointList sweep, an addN/mulN broadcast) whose element lambda applies interpreted user-defined functions previously paid the full symbolic pipeline per element (~ms/element). At machine precision (the gate: the default engine precision is bignum and never triggers this), such drains now compile the element lambda once per logical Map — eligibility-gated (pure bodies, literal-bounded loops, no unbound free symbols, ambient-scope captures only) — and serve elements from the compiled function, ~30–2500× faster with digit parity against the machine-precision interpreter. The compiled function is validated before every invocation against the same two-axis mutation keys as the comprehension memo, so reassigning a captured symbol (even mid-drain) recompiles, while unrelated assignments don't thrash the cache. Per-element fallback to the interpreter is silent and exact: non-numeric rows, NaN results (re-checked through the interpreter so √x over a sign-crossing source still yields complex values), and ineligible bodies (e.g. containing Random) behave exactly as before.

Benchmarks

Numeric performance (200-digit precision)

Median time per call, in microseconds — lower is better. — means the tool returned no usable result at that precision.

Expression	CE (current)	CE 0.86.1	SymPy	math.js	Mathematica
\pi^2	7.2	8.1	179	102	3.9
\sin 1	23	24	224	442	5.2
\cos 1	23	23	224	571	7.0
\ln 2	15	15	344	4,392	3.7
e^{\pi}	13	13	212	4,930	4.5
\zeta(3)	1,732	1,733	265	—	48
\Gamma(\tfrac13)	928	942	347	—	219
\psi(\tfrac13)	777	814	2,967	—	196

Symbolic capability & performance

Each cell is how many times faster than Mathematica that engine is on the case (Mathematica ÷ engine, so higher is better; Mathematica itself is 1×). — means the engine can't do the case; ✓ means it solves a case Mathematica can't. Compare the CE (current) and CE 0.86.1 columns to see what is new this release (a — under 0.86.1 next to a number under the current build). The CE + R/F column is the current build with the opt-in Rubi integrator + Fungrim identities loaded (loadIntegrationRules / loadIdentities), on the same minified bundle.

Operation	CE (current)	CE + R/F	CE 0.86.1	SymPy	math.js	Mathematica
Antiderivatives
\int\frac{1}{\sqrt x}\,dx	5.4×	2.4×	4.3×	0.5×	—	1×
\int\frac{x}{\sqrt{1-x^2}}\,dx	8.2×	1.1×	7.0×	0.09×	—	1×
\int\frac{1}{x^3+1}\,dx	4.3×	0.5×	3.3×	0.3×	—	1×
\int\frac{\sqrt x}{1+x}\,dx	—	1.4×	—	0.1×	—	1×
\int\frac{x}{(1+x)^{1/3}}\,dx	—	1.1×	—	0.01×	—	1×
\int\frac{x^2}{(1+x)^{1/3}}\,dx	—	1.1×	—	0.007×	—	1×
Derivatives
\tfrac{d}{dx}\sqrt{1-x^2}	0.03×	0.03×	0.02×	0.001×	0.003×	1×
Simplification
\sqrt{3+2\sqrt2}	33×	19×	25×	—	—	1×
\sqrt6\,x+\sqrt2\,x	88×	48×	63×	3.4×	16×	1×
Evaluation
\lim_{x\to0}\tfrac{\sin x}{x}	64×	31×	60×	3.3×	—	1×
\lim_{x\to\infty}(1+\tfrac1x)^x	8.5×	5.1×	8.0×	1.9×	—	1×
\int_1^2\tfrac1x\,dx	6388×	6536×	6482×	75×	—	1×
\int_{-\infty}^{\infty} e^{-x^2}\,dx	375×	144×	356×	2.2×	—	1×
Solving
x^4+x^2-1=0	0.4×	0.3×	0.3×	0.08×	—	1×
x^3-x-1=0	1.6×	1.7×	1.6×	0.04×	—	1×

Across the cases both solve, Compute Engine is a median 5.4× faster than Mathematica (up to 6388×) — in the browser, not a proprietary kernel.

_{Measured 2026-07-20 · Compute Engine 0.87.0 @ 0158397a (current build) · published 0.86.1 · SymPy 1.14.0 · math.js 15.2.0 · Mathematica 14.3.0 for Mac OS X ARM · Node v22.13.1. Correctness is verified numerically against an independent mpmath reference, never another tool. Reproduce with npm run build production && ./venv/bin/python3 benchmarks/gen_cases.py && node benchmarks/report.mjs && node benchmarks/report_changelog.mjs.}

0.86.3 2026-07-19

New Features

• Recursive user-defined functions now compile. Self- and mutually recursive functions (fact(n) := n ≤ 1 ? 1 : n · fact(n-1)) compile on the javascript and interval-js targets to true recursion, instead of failing closed. Termination is the caller's contract, matching compiled unbounded Loop: on the javascript target runaway recursion throws a catchable RangeError; on interval-js the runner converts runtime errors to the entire interval ("cannot bound"), per that target's error philosophy. (The interpreter throws CancellationError on its time limit instead.) A complex-valued recursive function needs a Typed complex return ascription on the function literal so the self-call types as a scalar — without it the application types broadcastable<number> and complex arithmetic over it does not compile. GPU targets (GLSL/WGSL) are unchanged: shaders cannot recurse, so recursion stays fail-closed there. Measured on a depth-10 iterated Julia map, the recursive form runs ~0.18 µs/pt — about an order of magnitude faster than the equivalent hand-unrolled closed form compiled before this release.

• Function literals accept a signature-string shorthand. ["Function", body, "'(n: integer, z: number) -> complex'"] desugars at canonicalization into the structural Typed form (typed parameters plus a return-type ascription) — one compact string instead of nested Typed wrappers, reusing the full type grammar. Signatures must name every parameter; optional/variadic markers are not yet supported and fall through to the standard parameter validation error.

Performance

• Small literal integer powers of complex values compile to inline multiply chains. z^k for literal k = 2…8 with a complex-valued z emitted the general polar-form power helper (hypot/atan2/exp/…) per evaluation; it is now an inline square-and-multiply chain, with the base bound to a const exactly once. Iterated-map workloads speed up ~9× (depth-10 Julia closed form: 1.25 → 0.14 µs/pt). The square is digit-compatible with the interpreter; for k ≥ 3 both routes go through different roundings and agree to ~1 ulp. Exponents ≥ 9, negative, and non-integer still use the general helper.

Resolved Issues

• Degree-mode compilation now reaches user-defined function bodies. The angular-unit rewrite (scaling trig arguments/results so radian-based compiled math reproduces angularUnit semantics) was applied only to the top-level expression: compiling t ↦ f(t) where f(x) := sin(x) under angularUnit: 'deg' emitted radian-based trig inside f's definition, while the inlined t ↦ sin(t) correctly scaled. The rewrite is now applied to each emitted user-function body. (Compiled-vs-interpreted agreement for unit-scaled trig is ~1 ulp, not digit-exact — the two routes round the unit conversion in different orders.)

• ce.assign(name, fn) ties the recursion knot for pre-boxed function literals. A Function literal canonicalized before ce.assign(name, …) (the programmatic box-then-assign route) left its self-reference bound to a stale auto-declaration, so the body's types were wrong — for most names the self-call typed any and compiling the function fail-closed on the collection guard, while a lucky subset of names (pre-declared shells such as K or J) masked the bug. ce.assign now pre-declares the target as function-typed and re-canonicalizes such a literal, matching the behavior of the Assign operator and of f(n) := … parsing. (Naming a function after a built-in operator, e.g. N, still collides — unchanged.)

0.86.2 2026-07-19

Resolved Issues

• An assigned complex symbol now compiles as complex without an explicit declaration. ce.assign("z_0", <complex value>) then compiling an expression using z_0 emitted the binding as a complex object literal while the operand analysis read only the DECLARED type (wide number / unknown ⇒ real) — number + {re, im} arithmetic, silently NaN at every point. The analysis now derives complex-ness from the assigned value, mirroring the fold. Compile-bound variables (loop indices, lambda parameters) shadow the engine, so an index named i does not pick up the imaginary unit.
• Block/\coloneq locals infer complex-ness from their assigned right-hand side. In w_1 ⩴ (x+iy)² + z_0; w_2 ⩴ w_1² + z_0 the local w_1 was emitted as a complex object but consumed as REAL by later statements (its type defaulted to real; outer declares don't reach block locals) — silent all-NaN. Locals' complex-ness is now inferred in statement order — a later local reading an earlier complex local is itself recognized — shared by every target (this also extends the GPU vec2 local hints to chained locals).
• Complex Add binds compound operands once — nested complex arithmetic now compiles in O(tree size). Each {re: …, im: …} slot spliced the full operand subexpression twice, doubling code size and runtime per nesting level: the depth-10 Julia closed form compiled to ~360 KB (~713 µs/pt). Compound complex operands are now bound to consts emitted exactly once: the same form compiles to ~1.9 KB and runs ~1.3 µs/pt, with digit-for-digit interpreter parity. Symbols and number literals stay inline, so simple shapes emit byte-identically.
• Max/Min (and Supremum/Infimum) now type as number. Their declared result type was the vestigial union number | list, so even Max(1, 2) typed as list | number, a comparison over one typed list<boolean>, and the compilation targets' scalar-condition assert fail-closed every When restriction containing a reduction (y = x \{\max(a,x) < 2\} masked its whole curve). These operators always REDUCE — including a collection argument's elements — to a single scalar extremum (ElementMax/ElementMin are the broadcasting variants), so the result type is number unconditionally. Evaluation is unchanged.

0.86.1 2026-07-19

Resolved Issues

• Materializing a list no longer restructures its eager elements. evaluate({materialization: true}) on a literal list spliced the contents of ANY collection-valued element into the parent — a list of Tuple pairs came back flattened ([("a",1), ("b",2)] → ["a",1,"b",2], so the result no longer fed DictionaryFrom), a nested list literal lost its nesting, and an infinite lazy element (Cycle) was spread until the evaluation deadline. Only finite lazy sub-collections are now flattened-and-materialized (the documented intent: [Range(1,3)] still materializes to [1,2,3]); eager literals are preserved, and an infinite lazy element stays put as a bounded preview.
• Take of an infinite collection with a finite bound is now finite. Take(Range(1,+∞), 3) reported count 3 but isFiniteCollection false (it propagated the source's finiteness), which left ListFrom(Take(<infinite>, n)) symbolic. Take now reports finite whenever its own element count is known-finite; ListFrom(Take(Range(1, +∞), 3)) → [1,2,3]. (When the source's count is genuinely unknown — Take(ChunkBy(<infinite>, f), 3) — finiteness stays unknown, keeping materialization previews honest.)
• Sort and Shuffle type as list<…>. Both always rebuild a List, but their static type claimed the source's type (Sort(Range(1,5)) typed as an indexed_collection-shaped Range). Both now report list<element-type>, matching Take.
• Slice facets are now coherent over infinite and unknown-length sources. Slice claimed isFiniteCollection unconditionally, and a negative start over an infinite source produced a NaN count while at(1) fabricated the element +oo (from source.at(Infinity)). The facets now share one bounds resolver: a negative end over an infinite source means "through the end" — an honest infinite tail (Slice(Range(1,+∞), 5, -1): count ∞, not finite, at(1) = 5, Take(…, 3) → [5,6,7]); a negative start over an infinite source ("the last k elements") is unresolvable and stays inert; an unknown-length source now reports finiteness as unknown rather than true. Bounded positive windows are unchanged (ListFrom(Slice(Range(1,+∞), 1, 5)) → [1,2,3,4,5]).
• Sum/Product bodies that bind looser than multiplication are now fenced when serialized. The big-op body is parsed back at multiplication precedence, so an additive body's trailing terms escaped the operator on re-parse: Sum(i + 1, i=1..3) serialized as \sum_{i=1}^3i+1, which re-parses as (\sum_{i=1}^{3}i)+1 — 9 became 7 — and a body-bound index in the escaped terms degenerated to a free symbol (i → the imaginary unit, turning real product expansions complex-valued). Additive (and other looser-than-multiplication) bodies now serialize parenthesized (\sum_{i=1}^3(i+1)); tighter-binding bodies (2i, \frac{1}{i}, a bare symbol) are unchanged.
• Fused stepped ranges with a fraction or compound second anchor now parse to the intended Range. [0,\frac{1}{6}...1] parsed to List(0, Range(1/6, 1)) — silently wrong values — because the sample reader did not recognize fraction literals; it now yields Range(0, 1, 1/6) with an EXACT rational step (a float 0.1666… step would drift and miss the end anchor; the range lands exactly on 1). And [m+n,m+n+15...m+n+60] parsed to a nested-Range List — the ... infix binds its left operand tight, so the continuation range was embedded in the additive tail (Add(m, n, Range(15, m+n+60))); the normalization now recovers the true second sample and end anchor, yielding Range(m+n, m+n+60, 15). Both rewrites keep the provenance guard: only ellipsis/..-written ranges participate — an explicit \operatorname{Range}(…) element (bare or embedded in a sum) stays a literal List entry.

0.86.0 2026-07-19

New Features

• ce.withTimeLimit(ms, fn) — run a block of work under a single evaluation deadline. Ordinarily each top-level evaluate() arms its own timeLimit budget, so a long sequence of short evaluations — e.g. draining a lazy collection element by element via each()/at() — can run unboundedly without ever tripping the limit. Wrapping the loop in withTimeLimit() arms one shared deadline for its full duration: any evaluation inside throws CancellationError (cause: 'timeout') once the deadline is exceeded. Re-entrant (an inner call can only shorten the effective deadline, never extend it).

Resolved Issues

• .N() of Tuple ± scalar·Tuple no longer throws at machine precision. When every term of a component sum was an integer-valued machine float (20 − 0.1·20), the exact summation path read bignumRe — which is undefined on a machine numeric value — and threw TypeError: Cannot read properties of undefined (reading 'toFixed'). At scale this killed composed lazy streams mid-drain (a 4001-point PointList − scalar·PointList died at the first integer-valued element) and made at(k) return undefined at the crashing indices. The integer fold now converts the integral machine value directly.
• A solidus-rendered fraction juxtaposed with following material keeps explicit grouping. Serializing a non-canonical InvisibleOperator(Divide(1, 2), Delimiter(…)) at nesting depth > 3 (where the default fraction style switches to an inline solidus) emitted 1/2(sq) — which re-parses as 1/(2·s·q), silently changing the value. The solidus form is now parenthesized ((1/2)(sq)); \frac-rendered fractions and trailing-position solidus fractions are unchanged.
• Multiply(symbol, Tuple) serializes with an explicit multiplication sign. s(1,2,3) re-parses as a function CALL ["s", 1, 2, 3] for any symbol the parser cannot prove non-applicable, silently turning a product into an application. A bare-symbol factor followed by a parenthesized comma-group now serializes as s\times(1,2,3). Single-expression groups (s(x+1)) and number-led products (2(1,2)), which re-parse as products, keep juxtaposition.
• CountIf, Position, Ordering, DictionaryFrom, and RecordFrom stay inert on an infinite or unknown-length collection. These operators require walking every element, so on an infinite input (Range(1, +∞), Cycle, Iterate) they previously consumed the entire evaluation time limit and then threw CancellationError instead of returning a result. They now detect the non-finite input structurally and stay symbolic immediately. (Ordering previously returned a spurious empty list, claiming a complete ordering it never computed.) Huge-but-finite inputs still walk under the deadline as before, and Find is unchanged: it streams and short-circuits, so Find(Range(1, +∞), x ↦ x > 5) still returns 6.

Collections

• Insert, DeleteAt, ReplaceAt, Partition (chunk and window forms), SlidingWindow, and ChunkBy are now hybrid-lazy. Inputs at or below the 100-element eager threshold evaluate to an eager List exactly as before (byte-identical shapes); larger, lazy, or infinite inputs stay symbolic and serve their elements on demand through count/at/ iteration, following the same convention as the hybrid-lazy broadcast forms. Insert into a million-element Range no longer materializes the whole list to answer Count or an index probe, and streaming prefixes of infinite results now work: Take(Partition(Range(1, +∞), 3), 2) → [[1,2,3],[4,5,6]], Take(ChunkBy(Cycle([1,1,2]), x ↦ x), 3) → [[1,1],[2],[1,1]]. Partition's predicate form (Partition(xs, pred) → [trueGroup, falseGroup]) requires a finite input and is unchanged. Materializing consumers (ListFrom, …) behave as before.

0.85.1 2026-07-18

Performance

• Large PointList transposes and point-coordinate projections are now hybrid-lazy, and scalar arithmetic over lazy broadcasts composes lazily instead of grinding (or staying inert). Numeric evaluation of scalar × (PointX(P), PointY(P), …) over a 4001-point PointList of broadcast components took ~600–1000 ms per product — each coordinate projection eagerly transposed the whole point list into n Tuples (at ~150 µs/element of per-element boxed re-canonicalization) only to project one slot back out — and could exceed the 2 s timeLimit on a full plot row. Three coordinated changes, all hybrid (collections at or below the 100-element eager threshold are byte-identical to the previous shapes):
  • PointList past the threshold transposes to the lazy Map form (consumable via at/each/count) instead of materializing every point-Tuple. BREAKING (shape): a >100-point PointList now evaluates to a lazy Map, not an eager List — element values are unchanged.
  • PointX/PointY/PointZ project lazily past the threshold, and project straight to the source collection when the operand is the lazy transpose form (PointX(PointList(a, b, c)) ≡ a for equal-length components; ragged or scalar slots keep transpose semantics).
  • addN/mulN re-dispatch their broadcast branches after numeric operand evaluation, so an operand that only becomes a collection through evaluation (Mod(L, 11) over a list L) now composes into the lazy Map form — previously the product/sum was silently left inert (0.2 · ⟨collection⟩ unreduced). The eager broadcast zip also streams its operands with hoisted iterators instead of per-index at() calls (which re-instantiated a lazy Map's mapping lambda on every access). The filed repro (a 4001-member 3D vector row) went from a 2 s timeout to ~10 ms of lazy composition, with materialization deferred to the consumer sweep.

Resolved Issues

• String vars values now splice into compiled JavaScript and Python as source, not as string literals. The vars compile option is the live-path contract: a mapped symbol always stays a runtime input instead of having its assigned value folded into the emitted code — so one engine state can serve both a compile-once path (sliders as runtime arguments) and a fold-early evaluate path. The GLSL and interval targets honored it, but the JavaScript and Python targets JSON-stringified the mapping, so compile(expr, { vars: { s: '_.s' } }) emitted Math.sin("_.s" * _.x) — a string literal yielding NaN at run time. A string value is now spliced verbatim (Math.sin(_.s * _.x)); a non-string value still bakes as a constant (vars: { a: 7 } → 7), unchanged.

0.85.0 2026-07-18

New Features

• DSolve frontier round — parity with SymPy on the ODE audit (50/51, 0 wrong). Four new solvable classes:

  • Nonhomogeneous Cauchy–Euler (x²y″ + bxy′ + cy = g(x)): an x-power indicial ansatz for power forcing (x²y″ + xy′ = x → c₁ + c₂·ln x + x), with a variation-of-parameters fallback for resonant or non-power forcing.
  • The Airy family y″ = (px + q)·y: solutions as c₁·AiryAi(t) + c₂·AiryBi(t) with t = ∛p·x + q/∛p² (real cube root, either sign of p).
  • Airy-type Riccati y′ = q₀(x) + q₂·y² (constant q₂, linear q₀): the y = −u′/(q₂u) linearization yields the one-parameter (Ai′ + C·Bi′)/(Ai + C·Bi) family — y′ = x + y² now solves (SymPy errors on it).
  • Repeated-eigenvalue first-order linear systems: diagonal systems of any size and defective 2×2 systems via a generalized eigenvector, gated on an exact (A−λI)² = 0 check so near-repeated numeric eigenvalues stay inert rather than producing an approximately-wrong solution.

• AiryAiPrime / AiryBiPrime operators (derivatives of the Airy functions), with machine-precision numerics across all three DLMF regimes and full derivative closure (Ai′ → AiryAiPrime, AiryAiPrime′ → x·Ai(x) — so repeated differentiation of Airy expressions evaluates and numericizes).

• NDSolveFunction — ODE solutions as applicable functions. Where NDSolve returns a sample List, the new NDSolveFunction (same arguments, without the sample count) returns the solution as a callable function — a Function literal wrapping the new InterpolatingFunction operator, which holds the adaptive solver's piecewise-quartic dense-output table. Assign it and evaluate anywhere in the integration interval (f := NDSolveFunction(y′ = y, y, (x, 0, 1), 1); f(0.5) → 1.6487…), at the integration accuracy; outside the interval the value clamps to the nearest endpoint, and a symbolic argument stays symbolic. The solution compiles to plain JavaScript — compile(f) yields a positional lambda (run(0.5)), and compile(f(t)) an expression over t (~1 µs per evaluation) — and LaTeX display elides the data table (\operatorname{InterpolatingFunction}_{[0, 1]}(x)); the full table round-trips through MathJSON. Scalar equations (first-order and higher-order) are supported; the multi-dependent system form stays inert.

Improvements

• NDSolve now uses adaptive stepping (Dormand–Prince 5(4)) with dense output. The output is unchanged in shape — a List of steps + 1 uniform [x, y] samples — but the values are now tolerance-controlled: integration adapts its internal step size (embedded 4th/5th-order error control) and the uniform grid is emitted from the quartic dense-output interpolant. Fixed-step RK4 silently lost accuracy near rapid transients (y′ = −50(y − cos x) over [0, 3] with 100 steps erred at ~4·10⁻⁵; now ~2·10⁻¹²). Non-integrable problems (finite-time blow-up, tolerance failure) leave NDSolve inert rather than returning inaccurate samples.

• Truncation dots after a repeating decimal tail now parse exactly. 0.999\ldots → 1, 0.333\ldots → 1/3, 0.1212\ldots → 4/33: a truncation marker after decimal digits ending in an evident repetend (a block repeated at least 3 times for single digits, at least twice for longer blocks) is read as the exact repeating decimal. Non-repeating tails (3.1415\ldots) keep the previous behavior (the marker is display-only).

• nPr(n, k) parses in lenient mode as the k-permutation count P(n, k) = C(n, k)·k!, joining the existing nCr(n, k) → Binomial.

• The infinite Sum/Product closed-form table grew substantially. New exactly-evaluated families (each numerically verified): alternating p-series (Σ (−1)^{k+1}/k → ln 2, Σ (−1)^{k+1}/k² → π²/12), odd p-series (Σ 1/(2k−1)² → π²/8), Dirichlet beta (Leibniz Σ (−1)^k/(2k+1) → π/4; β(2) → Catalan's constant; β(3) → π³/32; β(5) → 5π⁵/1536), the exponential series (Σ 1/k! → e, Σ xᵏ/k! → eˣ, shifted starts adjusted exactly), the first-moment geometric series (Σ k/2ᵏ → 2; symbolic ratio → x/(1−x)² guarded on |x| < 1), and the logarithmic series (Σ 1/(k·2ᵏ) → ln 2; symbolic ratio → −ln(1−x), same guard). New infinite-product entries: Π_{k≥a} (1 − 1/k²) → (a−1)/a, Π (1 − 1/(2k+1)²) → π/4, and Π (1 + 1/k²) → sinh(π)/π (previously numeric-only). Divergent or out-of-table shapes stay symbolic, as before.

0.84.2 2026-07-18

Performance

• Removed a per-call inference-snapshot tax that had slowed the whole engine by ~1.4× since 0.74.0. Every top-level boxing or parsing operation eagerly snapshotted the set of inferred symbols by walking every binding in every scope — including the entire standard library — to provide provenance for the fresh-matrix-inference repair (Determinant(A + B) inferring A, B as matrices), a consumer that runs only when a matrix-typed parameter mismatches. The provenance is now computed forward: BoxedSymbol.infer() records a definition when its type first transitions unknown → concrete during a boxing operation, and the repair's eligibility reads that log. Matrix inference behavior is unchanged (pinned by a 13-case matrix in matrix-operator-typing.test.ts); eligibility is now keyed on definition identity rather than name, so a name whose fresh inner-scope definition was popped no longer masks an outer definition. Measured recovery: π.N() at 200 digits 2.5 µs → 0.12 µs (21×, faster than 0.73.0); ∫ 1/(x³+1) 5.8 ms → 1.6 ms (3.7×); the drift vs 0.73.0 across the benchmark suite is eliminated.

Improvements

• The sign of integer powers of pure-imaginary bases is now determined. For z of type imaginary, z² reports negative, z⁴ positive (the cycle (βi)^p = (-1)^{p/2}·β^p for even p, including negative exponents: (2i)^{-2} = -1/4), and odd powers report unsigned (pure imaginary). Powers of a general finite non-real base report not-zero — a non-real value is necessarily nonzero. Previously all of these were indeterminate: the handler branch that addressed non-real bases was unreachable, and wrong as written (it claimed every even power of a non-real base was negative — but i⁴ = 1 and (1+i)² = 2i).

Improvements

• Abs typing now follows the operand's finiteness. |x| of a provably finite operand (real or complex) types finite_real instead of the signature's generic real; a provably infinite operand types non_finite_number, and a literal NaN types number. Finiteness also propagates structurally (|x| is finite iff x is), which makes signs of products of absolute values determinate: |x|·|y| for finite x, y now reports non-negative (this path previously hit a latent inverted parity claim in Multiply — see the sgn audit below — and before that was masked entirely).

Resolved Issues

• Lazy broadcast over a declared-unknown symbol no longer throws Not canonical (Tycho item 42). Evaluating mod(L, N)/N with L a declared-unknown symbol holding a >100-element list built the lazy Map(L, …) over the SYMBOL, whose static type is unknown; every lazy collection operator's canonical handler hard-rejected such a source and boxFunction fell back to a silently NON-canonical expression, which the first arithmetic composition rejected with a thrown assert. Lazy collection canonical handlers now admit operands whose type is merely indeterminate (unknown/any/value/broadcastable) — provably-scalar operands still reject — and Map over such a source keeps value-aware indexed-ness (type, at, count, and the display preview, which no longer renders with a misleading Set head). The composed lazy result is consumable and honors x.N() ≡ x.evaluate().N().

• A user symbol shadowing a builtin no longer breaks function application of that builtin. With N := 85 declared (ubiquitous in Desmos-style documents), any ["N", …] application — including the engine's own internal N(…) wrapper that makes lazy .N() elements float on access — resolved to the user's number and produced an incompatible-type error (surfacing as Nothing elements in lazy maps). Operator-position binding now defers a value definition that provably cannot be applied (a plain number, string, collection…) to an outer applicable definition of the same name; value-position references (N + 1) still resolve to the user's value. (Consequence: after prose-style devolution of an un-applied builtin — N + 1 — a later N(3.14159, 2) now numericizes instead of staying symbolic.)

• The JavaScript compile target's floored-Mod emission is parenthesized (Tycho item 43). The fragment ((a % b) + b) % b was emitted without outer parentheses; composed as a Multiply/Divide factor, JS's left-associative same-precedence % reduced the whole product mod b (c * ((x % 1) + 1) % 1 ≡ (c·(x%1+1)) % 1), silently value-wrong whenever the product's magnitude reached the divisor. The standalone form was correct, which is why it survived. Compiled and interpreted now agree on the Neyret-hash idiom Σ cos(i)·mod(10⁴sin(10⁴i), 1).

• Sum/Product over a collection-valued body type as the collection, and At extracts element types (Tycho item 44). A big-op whose body types vector<2> (e.g. summing scaled calls of a(t) := [cos t, sin t]) typed number, so indexing the sum baked an incompatible-type error at parse time; it now types vector<2>. At on a tuple-typed operand with a literal index types the selected slot, and an inference widen-guard stops a loose parameter type from coarsening an already-precise inferred function result (this made A(t)[1] type any; it now types number).

• At over a typed-collection application compiles; a collection-valued big-op body fails closed instead of emitting wrong code (Tycho item 45). a(x)[1] with a returning vector<2> now compiles (the collection gate is type-aware, so _SYS.at is emitted). A compiled Sum/Product whose body is collection-typed previously emitted scalar accumulation over arrays — NaN or string concatenation, silently wrong; it now fails closed (D6) with a hint to distribute the element access through the big op.

• Applying a function to a symbolic argument that mentions the parameter's own name no longer overflows the stack under .N() (Tycho item 46). a(t+1) for a(t) := [cos t, sin t] with t unbound: symbol values resolve by name through the evaluation context, so BoxedSymbol.N() recursed through the call-frame binding forever (t → t+1 → t → …). .N() now substitutes a self-referential context value once without numericizing through it — mirroring plain evaluate() — so nested helper-call expressions (the Tycho item-46 PointList(A(t)[1], A(t)[2]) repro) evaluate symbolically, verified against direct numeric evaluation.

• Desmos-style range ellipsis with an elided comma parses again (Tycho item 47, regression of the 0.76.0 "request 6" class). [0,...300] → Range(0,300) (was an inert List(0, ContinuationPlaceholder·300)), [1,...N] and [0,...3N^{2}-1] likewise, and the stepped [0,15...210] → Range(0, 210, 15) (was the silently WRONG List(0, Range(15,210))). Fully-comma'd, bare-fused ([1...5], [-3N...3N]), and nested-group ([f(a,b)...5]) forms are unchanged. Compound-symbolic stepped anchors sharing an identical additive base with numeric offsets now infer too: [m+n, m+n+15, ..., m+n+60] → Range(m+n, m+n+60, 15); differing bases or non-numeric offsets stay a literal List. Stepped-range inference only applies to ranges the ellipsis syntax itself produced — a list literal ending in an explicit \operatorname{Range}(a,b) element stays a List.

• A Range operand of a tighter-binding parent now serializes parenthesized (Tycho item 48). .. parses its end operand at a precedence below Add, so Add(Range(0, L-1), 3) serialized as 0..(L-1)+3, which re-parses as Range(0, L+2) — wrong values on any serialize→re-parse round-trip (with L = 5, an 8-element list instead of the shifted 5-element one). A Range under Add/Subtract/Multiply/ Power/solidus-Divide parents now wraps in parentheses ((0..(L-1))+3); bare and stepped ranges serialize unchanged. Same round-trip precedence class as the 0.83.2 Mod fix.

• GPU targets emit a shape-matched NaN for masked conditional branches (Tycho item 49). A When/Which whose value is a tuple body — a restricted parametric (x(t), y(t)) with \{0 \le t \le 1\} — compiles the value to a vec2, but the masked branch emitted a scalar NaN: GLSL has no implicit float→vecN conversion in a ternary, so the driver rejected the shader and every restricted parametric member lost its GPU sampling path. The NaN branch is now vectorized to the value's component count (vec2(_gpu_nan()) on GLSL, vec2f(bitcast<f32>(…)) on WGSL — WGSL's select requires matching operand types); scalar bodies are unchanged.

• Sign (sgn) handler audit. A mathematical-correctness pass over all ~69 sgn handlers fixed a dozen wrong claims (each could mislead simplifications or comparisons built on isPositive/isNegative): Gamma(0) and Gamma(-n) reported zero/indeterminate instead of recognizing poles; Log with a negative base claimed a real sign (the sign only flips for a base in (0,1)); Truncate(1/2) claimed positive (truncation of |x| < 1 is 0); Round(-1/2) claimed zero while evaluate rounds halves away from zero (−1); GCD(0,0) and LCM(0,n) claimed positive (both are 0); Floor/Ceil of a complex number used the sign of the raw real part instead of the rounded one (⌊0.5+0.5i⌋ = 0); Factorial(-1/2) claimed non-real (it is Γ(1/2) = √π; only negative integers are poles, same fix for Factorial2); Abs(NaN) claimed positive; Random(-5, 5) claimed non-negative; tensor Rank of a scalar claimed positive (it is 0); and a latent parity inversion in Multiply swapped non-negative/non-positive for products of sign-indefinite factors. Arctan now reports the sign of its argument (it previously never produced one).

• A single-letter builtin operator used as a variable now stays connected to later assignments. Prose-style input like N \equiv 1 \pmod 5 devolves the un-applied builtin N to an unknown symbol, but when every other operand validated cleanly the devolved symbol was discarded and the expression kept the original symbol, still bound to the builtin operator: a later N \coloneq 11 was invisible and the expression stayed stuck symbolic. The substituted operand is now retained (same fix for operands re-typed by matrix-context inference repair), so assigning the variable evaluates as expected.

Benchmarks

Numeric performance (200-digit precision)

Median time per call, in microseconds — lower is better. — means the tool returned no usable result at that precision.

Expression	CE (current)	CE 0.84.1	SymPy	math.js	Mathematica
\pi^2	7.0	12	184	110	4.0
\sin 1	21	26	226	479	5.3
\cos 1	21	25	230	667	7.1
\ln 2	14	18	356	5,093	3.1
e^{\pi}	12	17	215	4,862	4.5
\zeta(3)	1,573	1,620	268	—	49
\Gamma(\tfrac13)	914	921	366	—	225
\psi(\tfrac13)	749	743	6,854	—	188

Symbolic capability & performance

Each cell is how many times faster than Mathematica that engine is on the case (Mathematica ÷ engine, so higher is better; Mathematica itself is 1×). — means the engine can't do the case; ✓ means it solves a case Mathematica can't. Compare the CE (current) and CE 0.84.1 columns to see what is new this release (a — under 0.84.1 next to a number under the current build). The CE + R/F column is the current build with the opt-in Rubi integrator + Fungrim identities loaded (loadIntegrationRules / loadIdentities), on the same minified bundle.

Operation	CE (current)	CE + R/F	CE 0.84.1	SymPy	math.js	Mathematica
Antiderivatives
\int\frac{1}{\sqrt x}\,dx	6.4×	2.7×	3.4×	0.4×	—	1×
\int\frac{x}{\sqrt{1-x^2}}\,dx	9.6×	1.7×	6.0×	0.09×	—	1×
\int\frac{1}{x^3+1}\,dx	7.5×	0.9×	1.1×	0.4×	—	1×
\int\frac{\sqrt x}{1+x}\,dx	—	1.9×	—	0.09×	—	1×
\int\frac{x}{(1+x)^{1/3}}\,dx	—	1.3×	—	0.01×	—	1×
\int\frac{x^2}{(1+x)^{1/3}}\,dx	—	1.1×	—	0.007×	—	1×
Derivatives
\tfrac{d}{dx}\sqrt{1-x^2}	0.03×	0.02×	0.01×	0.001×	0.004×	1×
Simplification
\sqrt{3+2\sqrt2}	39×	29×	23×	—	—	1×
\sqrt6\,x+\sqrt2\,x	79×	45×	43×	2.8×	15×	1×
Evaluation
\lim_{x\to0}\tfrac{\sin x}{x}	59×	29×	21×	1.2×	—	1×
\lim_{x\to\infty}(1+\tfrac1x)^x	9.6×	5.1×	4.5×	2.3×	—	1×
\int_1^2\tfrac1x\,dx	6653×	6680×	2654×	63×	—	1×
\int_{-\infty}^{\infty} e^{-x^2}\,dx	402×	104×	159×	2.5×	—	1×
Solving
x^4+x^2-1=0	0.2×	0.2×	0.1×	0.07×	—	1×
x^3-x-1=0	1.6×	1.8×	1.0×	0.03×	—	1×

Across the cases both solve, Compute Engine is a median 7.5× faster than Mathematica (up to 6653×) — in the browser, not a proprietary kernel.

_{Measured 2026-07-18 · Compute Engine 0.84.2 @ 40cc077a (current build) · published 0.84.1 · SymPy 1.14.0 · math.js 15.2.0 · Mathematica 14.3.0 for Mac OS X ARM · Node v22.13.1. Correctness is verified numerically against an independent mpmath reference, never another tool. Reproduce with npm run build production && ./venv/bin/python3 benchmarks/gen_cases.py && node benchmarks/report.mjs && node benchmarks/report_changelog.mjs.}

0.84.1 2026-07-17

Resolved Issues

• Equal/NotEqual over a possibly-collection operand now compile on the javascript target with an interpreter-faithful runtime dispatch. A comparison like q(2) = 9 where q is declared (number) -> unknown — so the call may return a collection at run time — previously failed closed (success: false, interpreter fallback). The binary form now lowers to a runtime helper mirroring the interpreter shape by shape: scalar operands compare tolerantly (within engine.tolerance, complex via the modulus), an array-vs-scalar pair is element-wise ([1,4,4] = 4 → [false, true, true]), and an array-vs-array pair is whole-collection equality — a single boolean, false on a length or shape mismatch, recursive over nested arrays. The chained (n-ary) form over a possibly-collection operand still fails closed: its pairwise && conjunction is only sound over scalar booleans.

• Equality with an operand that only becomes a collection at evaluation no longer produces a cartesian nest. L(1) = [1,2] where L is declared (number) -> unknown and returns a list fanned the literal out before evaluation (the opaque call not yet being a collection), then broadcast again element-wise once it was — yielding a 2×2 list of lists of booleans. It now follows the documented, representation-independent rule that literal, symbol-bound and lazy collections already follow: two collections compare as a single boolean (L(1) = [1,2] → True), and a runtime scalar against a literal list still broadcasts element-wise.

• .N() of an already-evaluated lazy Map now yields numeric elements. The 0.84.0 fix wrapped a lazy broadcast's elements in N only when the broadcast was constructed under .N(); calling .N() on an already-evaluated lazy Map was an identity, so Sin(Range(1, 200)).evaluate().N() streamed exact elements (sin(1), sin(2), …) from both each() and At. Requesting a numeric approximation of a lazy Map now rewraps its mapping function so every element floats on access — restoring x.evaluate().N() ≡ x.N() — while evaluate() alone still keeps elements exact and the result stays lazy (O(1) on Sin(Range(1, 10^8))).

0.84.0 2026-07-17

Resolved Issues

• Sort and Shuffle of a non-List collection returned corrupt results. Both rebuilt their result with the source collection's operator, so the sorted/shuffled elements were reinterpreted as constructor arguments: Sort(Range(1, 10)) produced Range(1, 2, 3) — the one-element list [1] — and Shuffle(Range(1, 5)) could produce an empty collection. Both now return a List, matching every other eager collection operation. Sort of an infinite or unknown-length collection now stays inert instead of returning an empty collection.

• Negative indices now work uniformly across all indexed collections. Negative-index normalization (-1 = last element) was implemented per-collection and most lazy collections lacked it, so Last, At(xs, -1) and anything built on end-relative access silently returned Nothing — or worse: Reverse walks its source from the end, so ListFrom(Reverse(Range(1, 5))) returned [] instead of [5,4,3,2,1]. Normalization is now centralized in the index dispatcher and works for Range, Linspace, Zip, Scan, Differences and every other indexed collection with a known finite length; infinite or unknown-length collections correctly return Nothing without enumerating.

• A Filter over more than ~1000 elements crashed numeric canonicalization. Determining whether a Filter result was finite ran its count handler, which walks the source applying the predicate — and throws iteration-limit-exceeded past the iteration limit. Boxing an expression as simple as Filter(Range(1, 100000), p) + 1 threw. Finiteness and emptiness of a Filter are now answered structurally (a filter of a finite collection is finite) without running the predicate, and count of a filter of an infinite or unknown-length collection now correctly reports unknown instead of claiming Infinity (a filter of an infinite collection can be finite: Filter(Range(1, ∞), x < 5) has 4 elements).

• IsEmpty and Contains no longer answer False when the answer is unknown. Both coerced an undetermined result to a definite False: IsEmpty(Filter(Range(1, 10^5), x ↦ False)) — a collection that is empty, but whose emptiness can't be established within the iteration limit — returned False. Both predicates are now three-valued and stay inert (unevaluated) when the answer cannot be determined.

Collections

• A comprehension's element memo now survives unrelated evaluations. Elements of a comprehension ([x^2 for x in xs]) are cached per instance, but the cache was keyed on an engine-wide counter that every scoped evaluation — any \sum, Block or big operator — bumps on exit, so in a live document the cache never survived between two reads and every re-read re-evaluated the body per element (the Tycho/Graph Paper team measured a document that bound comprehensions lazily analyzing ~3× slower than one that eagerly materialized them). The cache is now invalidated only by semantic mutations: reassigning a free variable the body reads (directly or through a helper function it calls), redeclaring or inferring an operator, assume() and forget() — including the implicit revert when a scope that assumed exits — all refresh it, and a comprehension nested under a Sum/Product refills for each value of the enclosing binder. Unrelated evaluations between reads leave the cache intact.

• IdentityMatrix, ZeroMatrix, OnesMatrix and Diagonal of a vector are now safe at huge dimensions. These constructors eagerly built the full m×n matrix with no size limit — IdentityMatrix(10^6) attempted 10¹² elements. Above 10,000 total elements they now produce a lazy indexed collection with O(1) construction and element access; at or below, the result is the same materialized matrix as before, compatible with Determinant, Inverse and the rest of the dense linear-algebra operations.

• .N() of a lazily-broadcast operation yields numeric elements. When a broadcast returns a lazy Map (see below), requesting a numeric approximation now wraps the mapped function so each element is computed as a float on access: Sin(Range(1, 10^8)).N() elements are numbers, while evaluate() keeps them exact (sin(1), sin(2), …).

• Element-wise operations over infinite and unknown-length collections are now lazy instead of inert — or truncated. Broadcasting an element-wise operator over an infinite collection (Cycle), a finite collection of unknown size (Filter), or a symbolic-length Range now returns a lazy Map supporting First, At, Take and Length, rather than staying an inert expression: Add(Cycle([1,2]), 1) is the lazy [2,3,2,3,…], and Add(Range(1, n), 1) with n a declared, unassigned integer becomes Map(Range(1, n), _ ↦ _ + 1), which picks up n's value reactively when later evaluated. This also fixes a wrong-answer bug: the eager broadcast read a Filter's unknown length as 1, so Add(Filter(Range(1, 100000), x ↦ x > 2), 1) truncated to the single-element list [4]; it now yields the full lazy sequence [4,5,6,…]. A mixed broadcast folds with shortest-input semantics: Add([10,20,30], Cycle([1,2])) → [11,22,31].

• Element-wise operations over large collections are now lazy. Applying an element-wise operator (Add, Multiply, Sin, a user function literal, …) to finite indexed collections of more than 100 elements returns a lazy Map instead of materializing every element: Add(Range(1, 10^8), 1) and Sin(Range(1, 10^8)) evaluate instantly to a lazy collection supporting At, Take, First, Last and Length without enumeration. Collections of 100 elements or fewer are unchanged and still evaluate eagerly to a List.

• Length, Count, IsEmpty and Contains see through wrappers that cannot change their answer. Sort, Shuffle and Reverse preserve element count, and (together with Unique, for Contains) preserve membership, so these consumers now strip such wrappers at canonicalization: Count(Sort(xs)) becomes Count(xs) and no longer sorts — Count(Sort(Range(1, 10^5))) went from ~15 s to ~1 ms.

• Numeric argument validation no longer enumerates large lazy collections. A lazy collection passed to a numeric operator was fully enumerated at canonicalization to check or infer its elements — even a Map over millions of elements. Validation now decides on the static element type: provably numeric or provably non-numeric collections are accepted or rejected without enumeration, and a collection whose element type is genuinely indeterminate is accepted structurally and fails at evaluation time if an element turns out non-numeric. Only eager, literal collections still have their elements inspected individually.

0.83.2 2026-07-17

New Features

• Operator definitions can now supply a custom compilation handler. A compile handler on an operator definition emits target source for that operator when the expression is compiled; returning undefined falls back to the target's default lowering:

  ce.declare('MyGcd', {
    signature: '(number, number) -> number',
    compile: (args, compile, { language }) =>
      language === 'javascript'
        ? `_gcd(${compile(args[0])}, ${compile(args[1])})`
        : undefined,
  });

  The handler receives the canonical operands, a callback to lower sub-expressions, and the compilation context (branch on context.language — javascript or python). It takes precedence over the target's built-in operator mapping, so it can also re-map how a built-in operator compiles; structural and control-flow heads (Sum, If, Block, …) keep their bespoke lowering and ignore the handler.

Resolved Issues

• A Mod in a product or a power base now serializes parenthesized, fixing a round-trip corruption. Juxtaposition (invisible multiply) and superscripts bind tighter than infix \bmod on re-parse, so an unparenthesized Mod factor absorbed the adjacent notation into its trailing operand: ["Multiply", ["Mod", "A", 2], ["Mod", "B", 2]] serialized to A\bmod2B\bmod2, which re-parses as A mod (2B mod 2) = A mod 0 = NaN. It now serializes as (A\bmod2)(B\bmod2); similarly ["Power", ["Mod", "A", 2], "x"] is now (A\bmod2)^{x} instead of A\bmod2^{x} (which re-parsed as A mod 2^x). Same class as the 0.79.2 compound-operand Mod parenthesization fix, on the other side of the operator. (Reported by the Tycho/Graph Paper team — a hex-grid Desmos state rendered blank because a product of two Mod(Floor(…), 2) factors round-tripped to NaN.)

0.83.1 2026-07-17

Resolved Issues

• Fixed a 0.82.0 regression: an operand typed broadcastable<T> was rejected by operators with a plain scalar parameter, baking an unrecoverable incompatible-type error at canonicalization. An application of an undeclared function symbol (["f", "k"]) flowing through Add, Multiply, or Power lifts the surrounding expression to broadcastable<number>; a non-threadable operator with a number parameter (Binomial, Totient, and the rest of the number-theory family, among others) then rejected it — e.g. ce.box(["Binomial", ["Add", "n", ["f", "k"]], 2]) was invalid. A broadcastable<T> operand could be a plain scalar T at runtime, so validation now admits it whenever T matches the parameter, exactly restoring the pre-0.82.0 admission. LaTeX input was largely unaffected (undeclared f(k) parses as the product f \cdot k, and a declared function types precisely); expressions built directly from MathJSON were the exposed surface.

0.83.0 2026-07-17

Breaking Changes

• Collection indexing (At) now serializes with brackets by default: ["At", v, 1] → v[1] instead of v_1. The bracket form is the round-trip-safe notation: v[1] always parses back to At, while the subscript form v_1 only does when v is declared as an indexed collection — otherwise it re-parses as the unrelated subscripted symbol v_1, silently changing the meaning on a serialize→parse cycle. The previous behavior remains available engine-wide via ce.latexOptions.indexStyle = () => 'subscript' or per call via expr.toLatex({ indexStyle: () => 'subscript' }). (Requested by the Tycho/Graph Paper team, whose per-call indexStyle opt-ins were a recurring source of forgotten-call-site round-trip bugs.)

New Features

• Five linear-algebra operators now compile to the JavaScript and Python targets: ConjugateTranspose, Diagonal (rank-dispatched — a matrix gives its main-diagonal vector, a vector gives the diagonal matrix), MatrixPower (integer powers, with a negative power inverting first), RowReduce (reduced row echelon form), and Rank. Previously these threw at compile time and fell back to the interpreter. Note that Rank is the tensor rank — the number of axes (scalar 0, vector 1, matrix 2) — not the linear-algebra (row) rank.

Resolved Issues

• The javascript compile target now lowers a reduce (Sum/Product) and rank-dispatched multiplication over an unknown- or broadcastable-typed collection operand, where it previously failed closed and fell back to the interpreter. A collection Sum/Product over such an operand reduces under a runtime guard (a scalar at run time still matches Sum(scalar) = scalar), with an element-wise-aware combiner so a nested (matrix-valued) element reduces correctly rather than string-concatenating. A Multiply of two possibly-collection operands compiles to a runtime helper that dispatches on rank — element-wise for equal-length vectors, matrix product for matrices — matching the interpreter. This lets a compiled function whose evidence-derived signature is (…) -> unknown render through the compile path instead of only the interpreter.

• A function declared with a fixed-length list return type (e.g. (number) -> vector<11>) now compiles. The value assignment wraps the body in a Typed ascription, which the compile targets did not handle, so every compiled call threw Unknown operator `Typed` . Typed is a transparent, no-op-at-runtime ascription and now compiles to its value operand on every target.

• Multiply of two matrices is now the matrix product regardless of how the operands are presented. A matrix literal or a matrix-returning function application already contracted, but a symbol whose value is a matrix incorrectly broadcast element-wise (Hadamard) — the dispatch keyed on the node kind and missed a matrix-valued symbol. All three now contract consistently, so 0.82.0's per-step matrix-contraction rule holds for symbol operands too. Vectors remain element-wise.

• The Python compile target no longer unwraps a one-element ElementMax / ElementMin / Clamp broadcast to a scalar. ElementMax([1, 2], [3]) now compiles to a value that runs to [3] (zipping to the shortest operand), matching the interpreter and the JavaScript target, instead of the bare scalar 3.

• Broadcasting an element-wise operator over an N×1 column matrix now preserves its rank-2 shape. For example, \bold{v} = \begin{pmatrix} 5 \\ -3 \end{pmatrix} evaluates to the nested [[v === 5], [v === -3]] (a 2×1 result) instead of the flattened rank-1 [v === 5, v === -3]. A column vector is a matrix<Nx1>, and the broadcast result now mirrors the operand's shape, matching how row matrices (matrix<1xN>) and plain rank-1 vectors already broadcast. Consumers reading the broadcast result should expect one nested list per row.

• A Comprehension body containing a scoped subexpression now sees the iteration index correctly. When the body contained a Block (e.g. a with-style local), a big operator (Sum, Product), a nested comprehension, or a user-function application whose evaluation was deferred by any of these, the subexpression evaluated blind to the index value: the index was bound in a runtime scope that scoped subexpressions' lexical chains never reached. Results could be silently wrong — an applied function literal whose piecewise guard could not be decided without the index escaped with its parameters permanently unbound (e.g. [total(f(n, 4)) for n in 1..3] with f(a,b) := [{a>b: b, a}, a-b] returned expressions still containing a and b) — and, because the wrongly-symbolic elements never reduced, evaluation of such comprehensions cascaded into orders-of-magnitude excess work. Index values are now installed in the comprehension's own scope for the duration of each element's evaluation (isolated per walk, so interleaved iterations and .count reads during a paused iteration are unaffected), and evaluating a canonical Comprehension no longer re-creates it with a detached scope.

• Multiplying or adding a scalar to a piecewise (Which) no longer evaluates the selected branch twice. 2 \cdot \{A=1: X, Y\} evaluated the taken branch once during conditional-threading detection and again in the arithmetic handler, doubling the cost of every piecewise operand of Add/ Multiply (untaken branches were, and are, never evaluated).

• Fixed a stack overflow when evaluating Negate of an indexed collection that cannot be materialized, such as a Range with symbolic bounds reached inside a comprehension body (-Range(0, m + 5) with m unbound): the element-wise distribution retried the same non-distributable negation without progress.

0.82.0 2026-07-17

Breaking Changes

• Multiply of two vectors (rank-1 lists) is now element-wise, not a dot product. [1, 2, 3] \cdot [4, 5, 6] now evaluates to [4, 10, 18] instead of the scalar 32. This makes Multiply over lists consistent: element-wise is what Add, Power (k^2), scalar scaling (2k), and symbol-bound list operands (k \cdot k with k := [1,3,10]) already did — previously the same product could zip or contract depending on whether an operand was a literal list, a bound symbol, or a computed expression (\sqrt{k} \cdot k silently collapsed a 3-element family to one scalar). For the dot product, use the explicit Dot or MatrixMultiply operators, which are unchanged. A product is folded left-to-right, one pair at a time: a step involving a matrix (matrix·matrix, matrix·vector, vector·matrix) still contracts (matrix product, unchanged), while a step between two vectors is element-wise. Note this applies per step, so in a longer chain a contraction that produces a vector then combines element-wise with a following vector: M·u·v is (M·u) ⊙ v, no longer the scalar (M·u)·v. Vectors of differing lengths stay inert (no implicit zip-to-shortest). The compiled targets follow the same semantics: equal-length vector·vector compiles to the element-wise broadcast; statically mismatched lengths and matrix contractions fall back to the interpreter.

• A bounds-less big operator over LaTeX (\sum ⟨body⟩, \prod ⟨body⟩) now parses to its own head (["Sum", body] / ["Product", body]) instead of ["Reduce", body, "Add"/"Multiply"]. This is a (non-canonical and canonical) parse-shape change, called out per the pipeline-contract rules: consumers matching on the Reduce shape should match the big-op head instead. It makes the serialization round-trip lossless — ["Sum", body] serializes to a bounds-less \sum ⟨body⟩, which previously re-parsed to a different expression. Evaluation semantics are unchanged (a collection body still reduces).

• Broadcasting over a one-element collection now returns a one-element List instead of unwrapping to the scalar. \sin(2 \cdot [5]) now evaluates to [\sin(10)], previously the bare scalar \sin(10). Broadcasting a scalar function over an n-element indexed collection now produces an n-element List for every n ≥ 1, matching the expression's static list<…> type (a vector<1> operand no longer types list<number> while evaluating to a scalar) and the user-function broadcast path, which already returned a List for single-element collections. An empty broadcast still evaluates to Nothing.

Resolved Issues

• Sum/Product over a computed list-valued body now reduces instead of broadcasting. Sum(L) of a literal collection reduced correctly, but a body that only evaluates to a list — e.g. a broadcast chain over a list literal, \operatorname{Sum}(\operatorname{mod}(\operatorname{floor}(7/2^{[0...10]}),2)) — returned the broadcast list unchanged instead of its sum. The arity-1 reducer form now reduces the evaluated value when it is a collection.

• A symbol operand naming an operator no longer leaks into .unknowns. A function reference held as a symbol operand (e.g. the Add of ["Reduce", L, "Add"]) was reported as a free variable by .unknowns / .freeVariables, so consumers walking unknowns saw a phantom unbound name. Operator names now resolve as function references, not free variables.

• subs() no longer corrupts a bound index named i (or any name that collides with a constant). Substituting into a canonical big operator — ce.parse("\\sum_{i=1}^{n}2^{-i}").subs({n: 9}) — re-canonicalized the held Limits index outside its binding scope, re-typing i as the imaginary unit: the index slot became an incompatible-type error and serialization dropped the index (\sum_1^9…). Held (non-canonical) operands now stay raw through subs() and are re-bound by the parent's canonical handler, exactly as when the expression was first built.

• A bare numeric bounds pair on a big operator (\sum_1^9 ⟨body⟩) is no longer silently dropped at parse. It now parses to an index-less ["Limits", "Nothing", 1, 9]: a constant body iterates (\sum_1^9 2 → 18), and a body with free variables stays symbolic rather than losing its bounds.

• A divergent integral over (-∞, ∞) no longer numericizes to a clean 0. .N() of \int_{-\infty}^{\infty} x\,dx (and x^3, \sin x, any odd divergent integrand) returned an exact scalar 0: the Gauss–Kronrod quadrature mapped the doubly-infinite domain through a symmetric transform, so an odd integrand cancelled to exactly 0 on the first panel with a 0 error estimate — indistinguishable from a genuine result downstream. The doubly-infinite case is now split at 0 into two half-line integrals that must each converge (the definition of improper-integral convergence); a divergent half fails to converge and the result falls back to a Measurement with an honest (large) error bar that consumers can reject. Convergent integrands are unaffected (\int_{-\infty}^{\infty} x e^{-x^2}\,dx → 0, \int_{-\infty}^{\infty} e^{-x^2}\,dx → √π). Note this also means no Cauchy principal value is implied: a symmetric divergent integral reports non-convergence rather than its principal value.

• Compiled JavaScript arithmetic over a value that may be a list is now correct for both outcomes. Compiling 2h(x) - 1 where h may return a list produced scalar code that yielded NaN on a list value at run time (behind success: true). Such operands — typed broadcastable<…>, see New Features — now compile through the runtime broadcast helper: the same compiled artifact returns the scalar result for a scalar value and the element-wise list for a list value, matching the interpreter. Cases the helper cannot lower soundly now fail closed instead of emitting silently-wrong scalar code: a product of two or more possibly-list operands (a run-time matrix would need the matrix product, not an element-wise one), Equal/NotEqual over a possibly-list operand, and — on the Python target, where */+ repeat or concatenate a plain list — all arithmetic over possibly-list operands. compile() reports these as compilation failures (with the interpreter fallback available), rather than producing code that computes the wrong value.

New Features

• New broadcastable<T> type: honest static typing for values that may broadcast. The engine broadcasts element-wise at run time (2·[1,2,3] → [2,4,6]), and statically-visible collections have carried honest types (vector<3>, list<number>) for a while — but the same arithmetic over a value whose collection-ness is not statically visible (2h(x,y)-1 with h returning unknown) used to collapse to scalar number, even though evaluation broadcasts if h returns a list. Such expressions now type broadcastable<T> — "a T, or an indexed collection of T, applied element-wise". The type is produced by Add/Multiply and every broadcastable operator (Sin, Sqrt, Power, Abs, …) over an operand whose type is a top type (an unknown-return call) or already broadcastable; it propagates through nested arithmetic, juxtaposition (2(2h(x)-1) is a product, not a tuple), function application, and indexing ((2h(x,y)-1)[1] is valid, with element type number). Relatedly, a scalar function over a fixed-shape-typed intermediate no longer collapses either: \sin(10^4 \cdot [1,2,3]) — whose inner product types vector<3> — now types list<number> through every scalar-function hop (mod, scaling, …), so indexing the end state is valid. Operators that compute their own collection result (-M, M+N, matrix + scalar) are unaffected. Subtyping: number <: broadcastable<number> and list<number> <: broadcastable<number>, but broadcastable<number> is not a subtype of number (it may be a list). The type can be used in declarations (ce.declare('b', 'broadcastable<number>')) and signatures. Bare symbols are unaffected: an undeclared x in 2x still types scalar (inference pending), and tuples/points still bind atomically.

• Applying a scalar function to a collection-valued expression now broadcasts — for every function body. Broadcasting a user function over a literal collection (f([1,2,3]) → [f(1), f(2), f(3)]) is long-standing; it now also applies when the argument only evaluates to a collection (f(g(3)) where g returns a list), and for every body — previously a non-arithmetic body such as x \mapsto \operatorname{If}(x > 0, 1, -1) applied to a computed list stayed inert. The static type of such an application is honest as well: list<R> for a visible collection argument, broadcastable<R> for a possibly-collection argument, where R is the function's return type (a list-returning function maps to a list of lists — no flattening). Declaring a collection parameter type ((list<number>) -> …) still binds the argument whole, and tuple arguments still bind atomically.

0.81.0 2026-07-16

New Features

• New PointList operator (the point-list surface form). PointList is the explicit, importer-emitted operator that zips a point-with-collection into a list of points: PointList(-6, n) with n a 21-element list evaluates to the 21-element list of points, while PointList(1, 2) is just a plain point. Zip-to-shortest for multiple list components; scalars broadcast; an empty component yields an empty list; an infinite or unknown-length component fails closed (stays inert, no hang). It round-trips through LaTeX as \operatorname{PointList}(…). A plain Tuple now stays inert data — it never transposes — so tuples used as data are genuinely unaffected; tuple-with-collection arithmetic still scales component-wise, with no baked incompatible-type error, so a definition such as m(P) \coloneq P + s(P)\cdot(1, 0.3n) stays valid. (An earlier, evaluate-time Tuple-transpose of this idiom was replaced by the explicit PointList operator before it ever shipped in a release.)
• PointList compiles on the javascript, glsl, and wgsl targets. With all-scalar components (including free plot variables), it emits byte-identically to the equivalent Tuple ([x, y] / vec2(x, y) / vec2f(x, y)), so point literals rewritten to PointList stay on the compiled path (GPU grids, per-pixel bodies). Provably non-scalar components (collection- or tuple-typed, or a union with a collection member) fail closed to the interpreter, as does the interval-js target (where Tuple itself has no lowering).
• At defers a possibly-collection base to runtime instead of baking a type error. Indexing an expression whose type was over-narrowed to a scalar by arithmetic over an unresolved operand ((2h(x,y)-1)[1] with h returning unknown), or whose type is a union with an indexable member (number | list<number>), now stays symbolic and indexes once the base resolves to a collection — enabling structural substitution over vector-valued helper functions. Provably scalar bases ((5)[1], \pi[1], \sin(3)[1]) still report incompatible-type at canonicalization, and a base that turns out scalar at runtime errors at evaluation.
• Pipeline contract test suite. test/compute-engine/pipeline-contracts.test.ts pins the guarantees for MathJSON-carrying pipelines: non-canonical box(json).json structural fidelity (with its documented normalizations), non-canonical .latex round-trip (with its three documented exception classes), transform-then-canonicalize-once equivalence, cached-boxed re-binding rules, compile-from-boxed parity, and the non-canonical shape vocabulary. Breaking a test in this suite requires a CHANGELOG callout.

Breaking Changes

• Partition(xs, n) now returns chunks of size n, not n groups. Partition([1, 2, 3, 4, 5], 2) now evaluates to [[1, 2], [3, 4], [5]] (chunks of 2, trailing chunk short) instead of splitting the collection into 2 nearly equal groups. To split into a given number of groups, use Chunk instead (Chunk([1, 2, 3, 4, 5], 2) → [[1, 2, 3], [4, 5]]). A new sliding-window form Partition(xs, size, step) returns the complete windows of size elements whose starting positions are step apart (Partition([1, 2, 3, 4, 5], 2, 1) → [[1, 2], [2, 3], [3, 4], [4, 5]]). The predicate form Partition(xs, predicate) (split into matching / non-matching groups) is unchanged.

Resolved Issues

• The two declare forms are equivalent under declare-then-assign. Declaring a function head with the object form (ce.declare('f', {signature: …})) and then assigning a function literal (f(x) \coloneq … or ce.assign) silently discarded the declared signature — a scalar call to a tuple-typed parameter stopped type-erroring — while the string form (ce.declare('f', '(…) -> …')) preserved it. The object form now runs the same reconciliation: the declared signature is authoritative, arity mismatches error clearly, and the stored definition is identical to the string form's.
• Juxtaposition with a value-typed symbol is multiplication again. A symbol inferred or declared with the wide value type (e.g. any bare symbol that had passed through Max/Min-style (value*) signatures on the same engine) made subsequent parses of 2x silently produce Tuple(2, x) instead of Multiply(2, x) — an order-dependent wrong-parse present in released versions, affecting any warm engine. A wide type is not evidence of point-ness; the juxtaposition gate now treats value like unknown and multiplies. Locked by warm-engine order-independence tests in the pipeline-contract suite.
• \operatorname{sin} (and every lowercase spelled-out native function name) now binds as a function call. \operatorname{sin}(x)^{2} parsed as the unknown symbol sin times x^2 with isValid: true — silent wrong math; it now parses to \sin(x)^2 with call-binding identical to the native command (prefix minus after the call, postfix power on the result, ^{-1} inverse, base subscripts). Covers the trig/hyperbolic/inverse families, ln/log/lg/lb, and arg; bare identifiers (sin without \operatorname) are unchanged.
• A {…} group after a function is now its argument list. For a dictionary-registered function that takes parenthesized arguments, a brace group is accepted exactly as if it were (...): \gcd{a} → GCD(a), \gcd{2,4} → GCD(2, 4), \operatorname{floor}{2.5} → Floor(2.5), and consecutive groups are successive arguments (\mod{x}{2} → Mod(x, 2), the TeX multi-argument-macro habit). Previously the function parsed as a bare symbol and the group multiplied against it — silently wrong (\gcd{a} was GCD · a). Commands with implicit (unparenthesized) arguments keep the transparent-grouping convention — braces render invisibly, so the argument reads the way the rendered formula does: \sin{x}y is Sin(x·y) and \sin{x}^2 is Sin(x²), matching \sin x y and \sin x^2. A brace group after a generic declared or unknown name (f{x}) keeps its juxtaposition (multiply) reading.
• Function-style \operatorname{…} aliases now bind their call like natively-spelled functions. The parse-only aliases (\operatorname{mod}, var, cov, corr, count, length, nCr, random, shuffle, repeat, join, range, histogram, pdf, cdf) parsed as a bare symbol, so a prefix minus captured the function symbol itself (-\operatorname{mod}(x,1) → loud incompatible-type error) and a postfix power stole the argument group (\operatorname{mod}(-x,1)^{2} parsed silently as Mod · ((−x,1))², evaluating to NaN). They are now function-kind dictionary entries: the call binds before prefix minus, and a postfix power applies to the call result (\operatorname{mod}(-x,1)^{2} → Power(Mod(-x,1), 2)).
• target.compile() can return a failure instead of throwing. A fail-closed compile error (e.g. At on a non-indexed base) always threw out of a compilation target's compile(); passing the new { fallback: true } option returns the documented { success: false, error, run } shape instead, with run falling back to the interpreter — matching the engine-level compile() contract. The default remains throwing, so existing callers are unaffected.
• An over-arity function literal is rejected at registration. Assigning f(x, y) \coloneq x + y to a name declared (number) -> number was silently accepted, and f(3) then silently partial-applied; it now reports a clear error naming the literal's arity and the declared maximum. (The declared signature was already authoritative for types; this closes the arity gap.)
• Applying a function whose body stays partially symbolic no longer loses the argument. When a lambda body could not fully evaluate (e.g. a Which/If guard over an undetermined symbol), the application returned the body inert with the parameter unsubstituted — so Map([1,2,3], k \mapsto \operatorname{Which}(k = m, 10^9, k)) yielded three identical copies of the raw body with k leaked free and the elements gone. The parameter's value is now substituted into the held result, fixing Map, Filter, Tabulate, Zip-with-function, and direct Apply in one place.
• Lazy collections serialize faithfully. .latex of a canonical Map, Filter, Zip, Tabulate, Range, Linspace, or Comprehension no longer materializes an elided or value-baked preview (which could re-parse to a corrupt expression — a Map over a bound symbol serialized as N copies of its raw lambda body); each now emits its operator form, which re-parses to the identical expression. toString() still shows the materialized preview for display.
• Negative indexing on a lazy Take was off by one. Take(xs, n).at(-1) returned the second-to-last element of the taken prefix (at(-1) on Take([10, 20, 30], 2) was 10 instead of 20).
• The display preview of a lazy Take sampled the wrong tail. Displaying Take(xs, 50) over a lazily-enumerated source showed the source's last elements ([1, 2, …, 98, 99]) instead of the taken prefix's ([1, 2, …, 49, 50]): the operands were materialized to their own display preview — continuation placeholder included — before Take consumed them.
• A boolean Sort comparator now orders instead of silently doing nothing. A comparator returning True/False (e.g. (a, b) -> a > b) never reordered — only signed-number comparators worked. Boolean comparators are now interpreted Elixir-style: True means the first argument sorts first, so (a, b) -> a > b sorts descending.
• A mistyped GroupBy key function is now reported. GroupBy(xs, Even) (an unknown symbol auto-declared by its own use) silently placed every element in its own garbage group keyed "Even(1)", "Even(2)", …; it now throws with a spell-check suggestion, like Filter and Partition do for broken predicates. Grouping by explicitly declared symbolic functions is unaffected.
• The optimization form of ArgMax/ArgMin canonicalizes its function operand again. ArgMin(f, RealNumbers) (the "locations of the minimum over a domain" form, used by the identities library) short-circuited canonicalization, leaving the function literal in a non-canonical shape that no longer matched the identities library's stored rewrite patterns. The form remains inert under evaluation; the collection form (ArgMin([3, 1, 2]) → 2) is unchanged.
• A canonical Comprehension now serializes to LaTeX that round-trips. Its .latex was an elided display preview (\lbrack 1, 4, 9, \dots, 62\,500\rbrack) that silently re-parsed to a corrupt 11-element List containing a literal \dots; it now serializes through the faithful body \operatorname{for} var = domain form, which re-parses to the identical comprehension (including tuple bodies, dependent domains, and infinite domains).
• Lazy Comprehension elements are memoized. .at(n) re-walked the domain on every call and each .each() recomputed from scratch, making repeated indexed access quadratic (at(100)×100 on a 200-element comprehension: ~5 s → ~23 ms; a repeat .each() walk: ~110 ms → ~0.2 ms). The prefix cache is generation-stamped, so reassigning a free variable the body depends on invalidates it, and it is capped (100k elements) beyond which access streams as before.
• A broadcast condition no longer crashes Which/If. A condition that evaluates to a collection of booleans (e.g. a piecewise guard over a broadcast function application inside a comprehension) threw Condition must evaluate to "True" or "False"; it now stays symbolic (held), letting the surrounding expression evaluate.

Fixes from a review of the 0.78.0–0.80.0 changes:

• Compiled n-ary and collection GCD/LCM returned wrong numbers. The compiled form passed a third operand into the internal tolerance slot (GCD(2.25, 2.1, 0.6) compiled to 2.1, silently consuming the 0.6 as ε; GCD(12, 18, 8) → 6 instead of 2; LCM(4, 6, 10) → 12 instead of 60), and a collection argument (GCD([12, 18])) compiled to NaN. All forms now fold pairwise and match the interpreter; collection operands whose elements can't be enumerated at run time fail closed to the interpreter.
• A user-defined function sharing its name with a loop index hijacked compiled Sum/Product. With f(x) := x^2 declared, compiling \sum_{f=1}^{3} f emitted references to the function instead of the loop index (returning garbage with success: true; a null interval on interval-js). Bound names are now tracked explicitly through every binding form instead of being inferred from resolved code.
• Adaptive quadrature no longer poisoned by a NaN sample. A single integrand NaN at a quadrature node (e.g. the removable singularity of \sin(x)/x at the midpoint of a symmetric interval) permanently corrupted the convergence accumulators, silently falling back to slow, nondeterministic Monte Carlo. Non-finite panels are now excluded until subdivided away: \int_{-1}^{1} \sin(x)/x \, dx converges to 2\,\mathrm{Si}(1). Also, .N() on an integral without a closed form now uses adaptive Gauss–Kronrod before falling back to Monte Carlo, matching the compiled path's accuracy.
• Comprehension iteration state was shared across traversals. Two interleaved iterators over the same comprehension (or reading .count mid-iteration on a dependent comprehension) corrupted each other's index variables, yielding wrong elements. Each traversal now gets its own scope, and a function literal produced by a comprehension body now captures the per-iteration value of the loop variable ([x \mapsto x + i \text{ for } i \in 1..3] applied to 10 gives 11, 12, 13, not 13, 13, 13).
• GPU gcd regressed on large integers. The tolerant float loop shipped in 0.80.0 dropped the exact-integer path on glsl/wgsl: gcd(4000000, 2) returned 4000000. Exact Euclid is restored for integer inputs within f32 range.
• Tolerant GCD/LCM invariants. Scale-mismatched inputs violated gcd ≤ min / lcm ≥ max (\gcd(2.5, 10^{21}) → 10^{21}); zero-argument GCD()/LCM() crashed (now the identities 0/1); nested collections now fold in a single evaluation.
• Max/Min absorb NaN found inside collections: Max([1, NaN, 3]) now returns NaN, consistent with Max(NaN, 5) and with compiled code.
• IndexOf on infinite lazy collections hung indefinitely, ignoring ce.timeLimit. The search now streams (linear instead of quadratic on lazy collections) with deadline checkpoints. Compiled IndexOf also now uses the interpreter's tolerance-aware comparison instead of strict ===.
• Sequence interpretation (Interpret) regressions. The 0.79.0 anchor-search optimization rejected legitimate non-monotonic polynomial sums (e.g. 100 + 164 + 198 + 208 + \dots + 308, which is \sum_{k=1}^{14} k^3-21k^2+120k) and then ground for hours in an exact-rational recurrence search that ignored ce.timeLimit. The break heuristic now requires a sustained divergence streak, and the recurrence search honors the deadline.
• Parse-diagnostics false positives. f(x) \coloneq x^2 no longer emits juxtaposition-as-multiply/undeclared-symbol for the definition's own head, and symbols declared through the getSymbolType handler are no longer reported as undeclared.
• Custom compile handler contract. The handler now genuinely takes precedence over built-in operator mappings (e.g. Add) as documented (control-flow heads remain non-overridable, now stated explicitly), and analyzeReferences no longer reports custom-compiled operators as unsupported.
• Assorted: applying a non-numeric symbol to a collection (t(\{1,2\}) with t a string) reports an application type error again instead of a confusing Multiply error; PointX/PointY work on Sets of points (previously returned []); a provider timeout inside Integrate is re-thrown instead of swallowed; \operatorname{erf} at a directionless complex infinity stays symbolic instead of saturating to 1; compiled Map/Filter no longer leak JavaScript's 0-based callback index into two-parameter lambdas.
• Long flat operator chains no longer overflow the parser stack. A flat chain of a same-precedence associative operator (1+1+\dots, a\times b\times\dots, chained <) recursed one parselet frame per term and overflowed at ~1,300 terms; same-precedence continuations now iterate in the parser's infix loop (a 20,000-term sum parses; same-operator chains produce the same flattened n-ary trees as before). One deliberate tree change: mixed same-precedence operators now group left-to-right — the conventional reading — where they previously nested rightward as an artifact of the recursion (a\times b\otimes c now parses as CircleTimes(Multiply(a,b), c), not Multiply(a, CircleTimes(b,c))). Right-associative chains (a=b=c) still nest by construction.
• Only binary arithmetic operator symbols lower to first-class combiners. Passing a unary or relational operator symbol where a function is expected (Reduce([1,2,3], Negate, 0), Map(xs, Negate), Filter(xs, Less)) compiled to a wrong binary infix lambda (Negate folded like Subtract, returning -6 behind success: true). Non-arithmetic operator symbols and combiners of the wrong arity now fail closed to the interpreter. Also: compiled Reduce over an empty collection without an initial value returns NaN instead of throwing, and Tabulate with a statically non-positive dimension fails closed instead of returning [].
• Flatten with no depth now fully flattens ragged lists. Flatten([[1,x],[2]]) was a no-op (only uniform tensors flattened), which became visibly inconsistent once Flatten(expr, depth) shipped; the default now flattens completely, per the documented (Wolfram) semantics.
• Scan no longer silently drops an invalid initial value — the error is surfaced instead of computing the unseeded scan.
• ElementMax/ElementMin/Clamp on the python target now match the interpreter's broadcasting. Length-mismatched arrays previously raised a NumPy ValueError (or size-1-broadcast) where the interpreter zips to the shortest operand; an injected helper now aligns semantics while keeping the vectorized NumPy fast path.
• lambda.body is canonical for every declaration route. The public accessor returned a raw, non-canonical body for functions declared with a MathJSON evaluate handler (parse/assign routes were fine), tripping canonical-only asserts in consumers; it now returns the canonical scoped Block shape everywhere. And analyzeReferences now probes a custom compile handler per target language, so an operator whose handler only supports some targets is correctly reported in unsupported on the others.

New Features

• New collection operators. A batch of higher-order and structural collection operators (see the Collections reference):
  • Quantifiers — Any(xs, predicate?) and All(xs, predicate?) test whether some or every element satisfies a predicate (the elements themselves, treated as booleans, when no predicate is given). Both short-circuit, so they return a definite answer even on infinite collections, and stay symbolic when the result depends on undetermined elements. Any([]) is False, All([]) is True.
  • Cumulative — Scan(xs, f, initial?) is the running fold (same length as the input: Scan([1, 2, 3, 4], Add) → [1, 3, 6, 10]), and Differences(xs) gives the successive differences (length n − 1, computed exactly).
  • Prefix/suffix — TakeWhile(xs, predicate) and DropWhile(xs, predicate) take or drop leading elements while the predicate holds; both are lazy and compose with infinite collections.
  • Mapping — FlatMap(xs, f) maps f over xs and splices collection-valued results into a single list (a scalar result is kept as a single element).
  • Extrema — MaxBy(xs, f) / MinBy(xs, f) return the element with the largest/smallest key f(x); ArgMax(xs, f?) / ArgMin(xs, f?) return its 1-based index. The first occurrence wins ties, and all stay symbolic on empty or infinite collections.
  • Grouping — ChunkBy(xs, f) splits into maximal runs of consecutive elements sharing the same key f(x); Dedup(xs) collapses consecutive duplicates only (contrast Unique, which removes all duplicates).
  • Functional element updates — Insert(xs, index, value), DeleteAt(xs, index), and ReplaceAt(xs, index, value) return a new list with an element inserted, removed, or replaced at a 1-based index. Negative indexes count from the end (Elixir-style); Insert at -1 or n + 1 appends.
• Map is now variadic. Map(xs, ys, …, f) applies f element-wise across several collections (a zipWith), truncating to the shortest input; the function is always the last argument (Map([1, 2, 3], [10, 20, 30], (x, y) ↦ x + y) → [11, 22, 33]).
• Sort accepts a one-argument key function. In addition to a two-argument comparator, Sort(xs, f) with a unary f sorts ascending by the key f(x) (stable on ties), discriminated from a comparator by arity.
• Flatten accepts an optional depth. Flatten(xs, depth) flattens only depth levels of nesting; without it, the collection is fully flattened (Flatten([[1, [2]], [3]], 1) → [1, [2], 3]).
• More collection operators compile on the javascript target.
  • Reduce accepts a custom combiner: a Function literal (Reduce([1, 2, 3], (a, b) \mapsto a + 2b, 0) → 12), a user-defined function symbol, or an operator symbol such as Subtract — previously only the Add/Multiply/Min/Max folds compiled. A custom combiner requires an explicit initial value (without one the interpreter folds from Nothing, which has no numeric equivalent — that form still fails closed). Fold, which canonicalizes to Reduce, now compiles too.
  • Tabulate (1-D and 2-D) and Fill compile to native array construction with 1-based indexes — and therefore Table, in both its alias and Mathematica-style iterator forms. Compiling is the natural fast path for materializing these now-lazy collections.
  • CountIf, Find, IndexWhere, Position compile their predicate lambda to native array operations, with the interpreter's conventions preserved: 1-based indexes, IndexWhere → 0 and Find → NaN (Nothing projected onto a real target) when no element matches.
  • Append, Most, Slice, IsEmpty, Count, Contains, Unique, RotateLeft/RotateRight, Zip, Linspace, Chunk, Partition (integer and predicate forms), Ordering, and Shuffle compile to native array operations, each verified element-for-element against the interpreter (1-based inclusive Slice with negative-from-end indexes, rotation shift normalized modulo the length, Chunk/Partition producing k chunks of ⌈len/k⌉, stable Ordering ties, Zip truncating to the shortest input, Linspace including both endpoints). Non-finite runtime counts/indexes fall back to the interpreter's defaults instead of crashing or silently diverging, and Shuffle honors the engine's randomSeed (a deterministic, reproducible permutation, like Random). Forms with no numeric equivalent on the target fail closed: a custom Ordering function, the explicit-seed Shuffle form, statically non-positive Chunk/Partition counts, and Contains/Unique over compound (nested-list, tuple, complex) elements, whose JS equality is referential rather than structural.
  • Any, All, TakeWhile, DropWhile, FlatMap, and Scan compile (predicate/mapping lambdas → native some/every/flatMap and slices; Scan is the running fold, with the initial value not emitted and the seedless form emitting the first element as-is, matching the interpreter).
• Core scalar operators compile on the javascript target: Boole (Iverson bracket, with the same runtime boolean guard as Which/When), KroneckerDelta (n-ary, tolerance-aware like compiled Equal), Element(x, list) membership, Identity, and Apply of a function literal.
• Linear algebra compiles on the javascript target (closing the parity gap with the python target): Dot and MatrixMultiply (with the interpreter's dimensionality dispatch — vector·vector → scalar, matrix·vector → vector, matrix·matrix → matrix), Cross, Norm (scalar, 2-/Frobenius, and p-norms), Transpose, Determinant, Inverse (singular → NaN), Trace, Flatten (with optional depth), Shape, and Reshape (with the interpreter's cyclic padding).
• The python compilation target now covers collections and function literals. Function literals compile to Python lambdas, and the collection operators above (list access/slicing, Map/Filter and the other higher-order operators, Reduce/Scan, Tabulate/Fill, Linspace, Zip, Ordering, plus Flatten/Shape/Reshape/Trace via NumPy) emit Python with the same interpreter-verified semantics, validated by executing the emitted code (venv-gated parity suite). Also fixed: compiled Range was off by one on the Python target — np.arange excludes the stop value and is 0-based in the one-argument form, so Range(2, 6) compiled to [2..5] and Range(5) to [0..4]; CE Range is inclusive and 1-based.
• Compiled Range with no explicit step now auto-descends on both targets, like the interpreter: Range(5, 1) → [5,4,3,2,1] and Range(-2) → [1,0,-1,-2] previously compiled to [] on the JavaScript target (the implicit step was a fixed +1).

0.80.0 2026-07-16

New Features

• Custom per-operator compilation handler. An OperatorDefinition can now supply a compile handler — (args, compile, { language }) => string | undefined — that emits target-language source for a call to that operator. It mirrors a built-in compiled-function handler (recursively compile operands, branch on language for javascript/glsl/wgsl/python) and takes precedence over the target's built-in mapping, so a consumer can override how even a built-in operator compiles (e.g. a custom-tolerance GCD), or add compilation for an operator the target doesn't know. Returning undefined falls back to the default compilation. This replaces the never-wired, interpreter-only xcompile stub. See OperatorCompileHandler.
• ElementMax, ElementMin — element-wise (broadcasting) maximum and minimum (the NumPy maximum/minimum primitive). Unlike Max/Min, which reduce all operands — including a collection's elements — to a single scalar, these broadcast: a scalar over a collection returns a collection of the per-element extremum (ElementMax(0, [1, -2, 3]) → [1, 0, 3]), collections zip, and all-scalar arguments give a scalar. They are variadic (two or more arguments): ElementMax(0, [1, -2, 3], 2) → [2, 2, 3]. Exactness is preserved (ElementMax(√2, 1) → √2). They compile on every target: javascript (all-scalar → a direct call, a collection operand → a _SYS.bcast), interval-js (interval max/min — restoring break detection), glsl/wgsl (native max/min), and python (np.maximum/np.minimum).
• Clamp(x, lo, hi) — clamp a value to a range (min(max(x, lo), hi)), also broadcasting over collection arguments (Clamp([-1, 0.5, 2], 0, 1) → [0, 0.5, 1]). Compiles on all targets, including the native clamp on glsl/wgsl and np.clip on python.

Resolved Issues

• GCD/LCM now evaluate on non-integer real arguments. \gcd(2.25, 2.1) and \operatorname{lcm}(2.5, 1.5) previously stayed symbolic (and compiled to NaN, blanking any plot that used them); they now fold via a tolerant floating Euclidean algorithm — the standard "float GCD" that terminates when a remainder falls below ε · max(|a|, |b|) (ε = 1e-6). This honors the exactness contract: an inexact (float) argument numericizes, like \cos(5.1), while integer and exact-rational operands keep their exact (\gcd(4, 6) → 2) and symbolic (\gcd(9/4, 21/10) under evaluate(), 0.15 under .N()) behavior. The compiled javascript, glsl, and wgsl targets fold reals the same way (the GPU _gpu_gcd cutoff was integer-tuned and is now scale-relative). The tolerance is deliberately its own constant, not the engine's numeric tolerance — float-commensurability is a much looser notion, and the value that reproduces a given renderer's output is a consumer choice. (Requested by the Tycho/Graph Paper team for expressions such as r ≤ gcd(θ², θ + a).)
• GCD/LCM reduce a finite collection argument. \gcd([12, 18, 24]) → 6 and \operatorname{lcm}([4, 6]) → 12 (a list argument is folded over its elements); previously the whole call stayed symbolic even for integers. Mixed list-and-scalar arguments (GCD([12, 18], 8) → 2) and lists of reals are handled; an infinite or enumeration-declined collection stays symbolic rather than grinding to the evaluation deadline.
• Max/Min of a scalar and a collection no longer mis-compiles to NaN. Max(0, [1, -2, 3]) evaluates to 3 (the reduction folds the collection's elements), but on the JavaScript target it compiled to Math.max(0, [1,-2,3]) — passing an array as an argument — and returned NaN at run time. It now folds the scalar and every collection's elements into a single reduction, matching evaluate(). (Surfaced by the Tycho/Graph Paper team's max(0, …) plot expressions.)
• Max/Min compile correctly on the python target. They were mapped to the element-wise, strictly-binary np.maximum/np.minimum, so a collection operand was mis-reduced (Max(0, [1,2,3]) → [1,2,3] instead of 3) and a single-list or n-ary call errored at run time. Max/Min now reduce a collection operand with np.max/np.min and combine the per-operand results element-wise (keeping a scalar/array operand — e.g. the plot variable — vectorized), matching evaluate() and the JavaScript target.
• At(base, index) serialization now parenthesizes a compound base. At(x+1, 2) serialized to x+1_2 (subscript) / x+1[2] (bracket index style) — the index bound only to the trailing operand, dropping the base grouping so the LaTeX re-parsed to something other than element access. A base whose precedence falls below the postfix index operator is now wrapped: (x+1)_2 / (x+1)[2]. Symbol and function-application bases (v_1, H(x, y)[1]) are unchanged. (Serialization round-trip reported by the Tycho/Graph Paper team. The default subscript index style is unchanged; programming-style bracket output — which re-parses to At independent of the base's declared type — remains available via indexStyle: () => 'bracket'.)

Collections

• A list comprehension is now a lazy collection: evaluate() no longer walks its whole domain. A Comprehension ([body \operatorname{for} i=…]) behaves like Range/Map — evaluate() returns the comprehension itself, and its collection type, .count, and emptiness are reported from the iterator-clause counts without enumerating a single element. Elements are materialized only when actually consumed (a positive at(n) walks just the first n); indexing, iteration, and aggregation (Sum, Length, At, Take, Map, …) are unchanged. Binding an unread comprehension to a name is therefore ~O(1) instead of materializing its whole domain up front — e.g. 25 dead 225-element weight tables dropped from ~6 s to negligible. (Reported by the Tycho/Graph Paper team.)
• A bracket comprehension parses to the comprehension itself, not a one-element List wrapping it. [body \operatorname{for} i=…] previously produced ["List", ["Comprehension", …]], which reported count: 1 and mis-indexed (W[3] → NaN); it now returns the Comprehension directly, mirroring the Range/Linspace bracket passthrough. (Reported by the Tycho/Graph Paper team.)
• Tabulate (and its Table alias) is now a lazy indexed collection. evaluate() returns the Tabulate itself rather than building the whole array; .count is the outer dimension and an element is computed by applying the function only when indexed or iterated. A Tabulate(f, 1_000_000) that is bound but unread is now O(1) instead of hanging while it builds a million-element list. The Mathematica-style Table(i^2, {i, 1, n}) inherits this (it canonicalizes to Tabulate).
• Permutations and Combinations are now lazy collections with closed-form counts. Permutations(xs, k?) and Combinations(xs, k) no longer materialize their factorially-many elements to be bound, counted, or indexed: .count is P(n, k) / C(n, k) computed directly (previously Permutations of a 9-element list took ~33 s just to answer .count), elements stream from the iterator, and at(n) walks only as far as needed.

Compilation

• A user-defined function passed as a higher-order operand now compiles by reference. Compiling Map(list, f) / Filter(list, f) — where f is a function declared on the engine (f(x) := …, x ↦ …) rather than an inline lambda — previously emitted a dangling _.f and threw at run time. The function operand now resolves to the same shared local (_fn_f) the call-site path already emitted, so a user function used as a first-class value works wherever it is referenced, not only when it is called. Its freeSymbols are computed from the operand function's body (so a free symbol used only inside f is reported), and inline-lambda operands are unaffected. (Reported by the Tycho/Graph Paper team.)

API

• BoxedOperatorDefinition.lambda exposes a user-defined function's body and parameters. ce.lookupDefinition(name).operator.lambda returns { parameters, body } — the parameter names/types and the body as a boxed expression — for a definition created from a function literal (f(x) := …, x ↦ …, ce.assign('f', lambda)), or undefined for a built-in operator. This is a supported, stable accessor over the internal _lambdaLiteral, letting a consumer traverse or resolve a function reference structurally without re-parsing or textually inlining its source. (Requested by the Tycho/Graph Paper team.)

0.79.3 2026-07-15

Parsing

• A number-valued symbol juxtaposed with a parenthesized collection now parses as multiplication, not a function application. When a symbol known to be a non-function value — declared with a numeric type or assigned a value — was juxtaposed against (…) whose body referenced a collection (e.g. k(\cos(S)) with k a number and S a bound list), it parsed as k applied to the body — an illegal application of a number, yielding NaN — instead of k\cdot\cos(S). The single-argument invisible-operator rule only treated a scalar-numeric argument as multiplication; a collection-typed argument fell through to the function-call heuristic even when the leading symbol could not be a function. Such a symbol now scales over the argument, matching the scalar-argument and multi-operand cases. An undeclared or unknown-typed symbol stays ambiguous and keeps the f(x) function-application default. (Reported by the Tycho/Graph Paper team.)

• Parsing a call no longer un-assigns a bare-symbol argument. Parsing an application like f(S) runs argument-type inference on each operand. When the callee's parameter type was unknown or any and the argument symbol had been declared unknown and assigned a value, inference computed unknown (a no-op narrowing) and wrote it back to the symbol's type — and the value-definition type setter discards the held value whenever the type is set to unknown. So merely parsing f(S) (no evaluate/N) silently cleared S's assigned value, leaving it unbound and every dependent expression NaN. Inferring unknown adds no information, so it is now skipped entirely and never overwrites an existing binding; inference of a concrete parameter type still narrows an open argument as before. (Reported by the Tycho/Graph Paper team.)

Collections

• The .x/.y/.z point-coordinate accessors now broadcast over a list of points. They previously parsed to First/Second/Third — the collection element-indexing operators. On a single point that is correct ((3,4).x = 3, since the first element of a 2-tuple is its x-coordinate), but on a list of points the two diverge: [(1,2),(3,4),(5,6)].x returned the first point (1,2) instead of the list of x-coordinates [1,3,5]. The accessors now parse to dedicated PointX/PointY/PointZ operators that extract a coordinate and map element-wise over a list of points (matching the threadable .real/.imag accessors), while a single point still returns the scalar coordinate. First/Second/Third are unchanged and continue to index a collection. The new operators broadcast on the javascript compile target (L.x → (L).map((p) => p[0])) and, for a single point, swizzle on the GPU target as before. (Reported by the Tycho/Graph Paper team.)

Compilation

• Element-wise (scalar↔list) arithmetic now broadcasts on the javascript compile target. An arithmetic or element-wise math operator applied to a list-valued operand — x - L, 2L, L^2, -L, \sin(L), \sqrt{L}, or two lists L + M — previously compiled to scalar JavaScript that returned garbage (-_.L + _.x → NaN) behind a success: true, unless an operand was a concrete collection at compile time (which failed closed). A symbolic list-valued parameter (bound at run time — the normal compile case) slipped through entirely. These now compile to a _SYS.bcast runtime helper that maps the operator element-wise, matching the interpreter's broadcasting: scalars are reused for every element, two lists zip to the shorter length, and nested lists (matrices) recurse. The pure-scalar fast path is unchanged. A complex-valued list still has no coverage and now fails closed correctly (success: false → interpreter fallback) instead of silently returning garbage. Combined with the .x/.y point-broadcast change above, a compiled expression such as \min((x - V.x)^2 + (y - V.y)^2) over a list of points V now evaluates correctly. (Reported by the Tycho/Graph Paper team.)

0.79.2 2026-07-15

Compilation

• The collection form of Sum, Product, Max, and Min now compiles on the javascript target. Applied to a collection with no indexing set — e.g. [3,4,5].\operatorname{total} (which canonicalizes to Sum([3,4,5])), a list product, or \max(v) for a list v — these previously threw Sum: no indexing set (dropping the whole expression to interpretation) or, for Max/Min, compiled to Math.max([…]) and returned NaN. They now lower to a native .reduce, with empty-collection identities matching the interpreter (Sum([]) = 0, Product([]) = 1, Max([]) = -\infty, Min([]) = +\infty). This lets a compiled list comprehension whose body uses .total/.count/\min/\max compile end-to-end into a loop instead of falling back. The indexing-set forms (\sum_{n=1}^{5}) and the scalar variadic \max(a,b,c) are unchanged; a non-collection operand still fails closed. (Reported by the Tycho/Graph Paper team.)
• List-shaped collection operators now compile on the javascript target. Last, Rest, Take, Drop, Join, Reverse, Sort, IndexOf, Map, and Filter previously fell back to interpretation (Unknown operator); they now lower to native array operations (.slice, .reverse, .sort, .indexOf, .map, .filter, …). Take/Drop clamp a negative count to match the interpreter (Take(xs, -2) = [], Drop(xs, -2) = xs); Reverse and Sort copy first so the source is not mutated; Sort compiles the default ascending numeric order (a custom comparator fails closed); IndexOf is 1-based (0 when absent); Map/Filter compile their lambda operand. A non-indexed-collection operand fails closed. (Requested by the Tycho/Graph Paper team.)

Evaluation

• IndexOf / IndexWhere now work on a tensor-backed list. A rectangular numeric list is represented as a BoxedTensor, which inherited the abstract no-op indexWhere and so made IndexOf/IndexWhere always return 0 (not found) even for a present element. The base indexWhere now scans any finite indexed collection for the 1-based index of the first match.

Serialization

• Mod (\bmod) parenthesizes compound operands so its LaTeX round-trips. Infix \bmod binds tighter than +/- on re-parse, so Mod(x+5, 2\pi) serialized to x+5\bmod2\pi and re-parsed as x + (5 \bmod 2\pi). An operand at addition precedence is now wrapped — (x+5)\bmod2\pi — while juxtaposition products (3k, 2\pi), fractions, powers, and negation stay unwrapped since they already re-parse as tight units. A left-nested Mod is also now parenthesized (\bmod is right-associative, so Mod(Mod(a,b),c) → (a\bmod b)\bmod c). (Reported by the Tycho/Graph Paper team.)

Parsing

• Juxtaposition of a collection-typed symbol with a function call parses as multiplication. A symbol declared with an abstract indexed_collection or collection type but not yet assigned a value — e.g. y_r in y_r\sin(a) — grouped into a Tuple instead of a Multiply, even though its concrete subtypes (list, vector, matrix, numeric tuple) and the assigned-value case already multiplied. The abstract collection type now scales like its subtypes; non-indexed set and heterogeneous tuple operands still group as a Tuple. (Reported by the Tycho/Graph Paper team.)

0.79.1 2026-07-15

Evaluation

• ce.timeLimit is now enforced during symbolic integration and rule matching. Extends the 0.79.0 expression-tree-growth checkpoint to two paths that previously ran unbounded: the multinomial expansion of a power of a sum (expandPower) and the rule-set scan (matchAnyRules). The worst case is compiling a definite integral of a high-power integrand — e.g. \int_{-15}^{15} (2 + \sin(3y) + \cos(\pi^2 y))^p\,dy: the compiler's antiderivative-first attempt expands (trinomial)^p into a multinomial with C(p+2, 2) terms (≈ 6·10⁴ at p=350) and matches the integration rule set against it (~100 ms per rule), neither of which hit the per-node checkpoint, so compilation stalled for many seconds (or hung outright) instead of honoring the deadline. Both paths now cooperatively check the deadline, so a hard integrand degrades to Gauss–Kronrod quadrature at ce.timeLimit (default 2 s) as intended, and a bare evaluate() of such an integral throws a catchable CancellationError (cause: 'timeout') rather than running unbounded. (Reported by the Tycho/Graph Paper team.)

0.79.0 2026-07-14

Evaluation

• ce.timeLimit is now enforced during expression-tree growth. The evaluation deadline was only checked inside specific loops (collection enumeration, polynomial GCD, …), so an evaluation whose cost is dominated by expression construction never hit a checkpoint. The worst case is a nested user-function chain whose body references a parameter several times — e.g. a symbolic Newton iteration s(y, x_p) := \frac{y - f(x_p)}{f'(x_p)} + x_p applied as s(y, s(y, … s(y, x_0))) — which grows the result ×4 per nesting level: at depth 15 it previously exhausted an 8 GB heap without ever honoring timeLimit. A cooperative checkpoint on the per-node evaluation path now cancels such evaluations at the deadline with a catchable CancellationError (cause: 'timeout'), e.g. at the default 2 s limit the depth-15 chain aborts using < 60 MB. Numeric chains (each level folds to a number) are unaffected and remain fast. Note: long-running evaluations that previously completed after exceeding timeLimit now throw — raise ce.timeLimit (or set it to 0 for no limit) if you rely on multi-second symbolic evaluations.

Library and Definitions

• ce.searchDefinitions() now treats the query as OR-ed keywords. Previously a multi-word query only matched definitions containing every word, so keyword-bag queries like "floor quotient integer division" returned nothing. Any matching word now suffices, and results are ranked by how many words they match and how exactly (identifier match, then trigger or curated keyword, then description). The query may also be an array of strings — ce.searchDefinitions(['gcd', 'least common multiple']) — with each element treated as an OR-ed alternative.

Solving

• Solve accepts Mathematica-style constraint systems. The first argument may now bundle domain constraints together with the equation — \mathrm{Solve}(\{100a+10b+c=11(a^2+b^2+c^2), a\in\{1,\dots,9\}, b\in\{0,\dots,9\}, c\in\{0,\dots,9\}\}, \{a,b,c\}) → [(5, 5, 0), (8, 0, 3)]. Element items inside a Set/List/And first operand are lifted into per-variable domain specs (equivalent to passing a \in D as separate spec arguments), the remaining items form the equation/system, and a variable list written as a set (\{a,b,c\}) is accepted alongside the existing [a,b,c] list form. The variable list may be omitted entirely when the bundled constraints name the unknowns: Solve(\{eq, a\in\{1,\dots,9\}, …\}) solves for the constrained symbols in constraint order. A constraint for a variable that already carries a spec-position domain is merged conjunctively (both must hold), and a constraint naming a symbol absent from an explicit variable list leaves the expression unevaluated rather than guessing. A system of equations given as a Set (or And) now also solves like the equivalent List.
• Trailing domain argument: Solve(eq, x, \mathbb{Z}). A trailing set constant (Integers, RealNumbers, …) after the unknowns applies as the domain of every unknown, Mathematica-style: \mathrm{Solve}(x^2=4, x, \mathbb{Z}) → [2, -2]. Unknowns that already carry an explicit Element domain keep it. Over an unbounded integer domain a polynomial equation with no integer roots now decides [] (\mathrm{Solve}(2x=3, x, \mathbb{Z}), \mathrm{Solve}(x^2=2, x, \mathbb{Z})) instead of staying unevaluated; non-polynomial equations stay inert rather than risk over-claiming "no solutions" from a partial root set.
• Inequality side conditions in constraint sets. A relational or boolean predicate bundled in the first argument restricts the solution set instead of being mistaken for an equation: \mathrm{Solve}(\{x^2=4, x>0\}, x) → [2] (previously returned the incorrect []), and a multi-variable condition filters candidate tuples — \mathrm{Solve}(\{a+b=5, a\in\{0,\dots,5\}, b\in\{0,\dots,5\}, a<b\}, \{a,b\}) → [(0, 5), (1, 4), (2, 3)]. Filtering is conservative (a candidate is dropped only when a condition is definitely False) and applies across the symbolic, diophantine-fallback, and enumeration paths. A constraint set containing only predicates solves them directly by enumeration over the domain (\mathrm{Solve}(\{x \equiv 2 \pmod 5, x\in\{1,\dots,20\}\}, x) → [2, 7, 12, 17]).

Mathematica-Style Operator Forms

• Iterator triples: \{i, lo, hi\} and \{i, lo, hi, step\}. The Mathematica iterator spec is now recognized in the iterator/bounds slot of Sum, Product, Integrate and D: \mathrm{Sum}(i^2, \{i, 1, 10\}) → 385, \mathrm{Sum}(i, \{i, 0, 10, 2\}) → 30, \mathrm{Integrate}(x^2, \{x, 0, 1\}) → 1/3 (the bounds were previously silently dropped, yielding an indefinite integral), and \mathrm{D}(f, \{x, n\}) is the n-th derivative. Symbolic bounds work (\mathrm{Sum}(k, \{k, 1, n\}) ≡ \sum_{k=1}^n k). The interpretation is strictly positional — a brace set anywhere else keeps its literal set meaning — and operates on held (raw) operands, so the index symbol is scoped like a binder (an i index does not collapse to the imaginary unit).
• New Table operator, an alias for Tabulate. \mathrm{Table}(i^2, \{i, 1, 5\}) → [1, 4, 9, 16, 25], with general iterator bounds and step (\mathrm{Table}(i, \{i, 0, 10, 2\}) → [0, 2, 4, 6, 8, 10]) and multiple iterator specs for nested dimensions (\mathrm{Table}(i j, \{i, 1, 2\}, \{j, 1, 3\}) → [[1,2,3],[2,4,6]], first spec outermost). \{v, 1, n\} specs canonicalize directly to Tabulate; general bounds map to Map over Range. Tabulate also gained the table search keyword so ce.searchDefinitions('table') finds it.
• \mathrm{D}(f, x) differentiation. Applied to an argument list, \mathrm{D} / \operatorname{D} is the derivative operator: \mathrm{D}(x^3, x) → 3x^2, \mathrm{D}(x^2 y, x, y) takes sequential partials. The bare forms keep their previous meanings (\mathrm{D} is the upright-D glyph symbol; \operatorname{D} remains usable as a pipeline stage, x^2 \rhd \operatorname{D} → 2x).
• Limit(f, x \to x_0) rule-arrow form. \mathrm{Limit}(\frac{\sin x}{x}, x\to 0) → 1, equivalent to \lim_{x\to 0}. One-sided arrows carry the direction: \mathrm{Limit}(\frac{1}{x}, x\to 0^+) → +∞.
• Simplify(expr, assumptions). An optional second argument supplies one or more boolean assumptions (a bare predicate, or a List/And of them) that hold only for the duration of the simplification: \mathrm{Simplify}(\sqrt{x^2}, x>0) → x, \mathrm{Simplify}(|x|, x<0) → -x.
• New ReplaceAll operator. \mathrm{ReplaceAll}(x^2+x, x\to 2) → 6 (Mathematica expr /. rules). Rules are lhs \to rhs (or Rule(lhs, rhs)), given as extra arguments or bundled in a set/list: \mathrm{ReplaceAll}(x+y, \{x\to 1, y\to 2\}) → 3. Symbol rules are applied simultaneously in a single pass; non-symbol left-hand sides use the pattern-rule machinery. The result is evaluated after substitution.
• Tuple membership distributes: (a,b) \in \mathbb{Z}. Membership of a tuple of symbols in a scalar (number-element) collection now distributes to a conjunction — Element(a, Integers) ∧ Element(b, Integers) — instead of evaluating to False. Value tuples against product sets are unaffected.

LaTeX Parsing

• One-sided limits: \lim_{x\to 0^+} and \lim_{x\to 0^-}. The ^+/^- direction marker on a limit point was previously captured by the generic superscript entries as PseudoInverse(0) / Superminus(0), making every one-sided limit unevaluatable. The marker now maps to Limit's direction operand (["Limit", f, 0, 1] / …, -1]), which the limit evaluator already supported: \lim_{x\to 0^+} \frac{1}{x} → +∞, \lim_{x\to 0^-} \frac{1}{x} → -∞, \lim_{x\to 0^+} \ln x → -∞. Directions serialize back as ^{+}/^{-} (round-trip), symbolic points (\lim_{x\to a^+}) keep a correct representation, and superscript +/- everywhere else (A^+ pseudoinverse, 3^- signed value) is unaffected.
• \mapsto lambda bodies extend through comparisons. n \mapsto n > 102 now parses as n \mapsto (n > 102) — previously the body closed at the comparison, mis-parsing as (n \mapsto n) > 102, which made unparenthesized predicates like \mathrm{Filter}(\mathrm{Range}(100,105), n \mapsto n > 102) fail. The body now extends through comparisons and logical connectives (n \mapsto n > 2 \wedge n < 5), stopping at the comma/sequence level, so a lambda in an argument list still does not swallow the following argument.
• Ellipsis ranges in set braces. \{1,\dots,9\} now parses to ["Range", 1, 9], matching the existing bracket form \lbrack1,\dots,9\rbrack; the stepped form \{0, 2, \dots, 10\} yields ["Range", 0, 10, 2]. Previously the ellipsis was kept as a literal placeholder element (["Set", 1, "ContinuationPlaceholder", 9]), which made a \in \{1,\dots,9\} unusable as a domain. Enumerated sets of non-numeric or non-progression elements (\{a, b, c\}, \{1, 2, 3\}) are unaffected.

Performance

• Interpret no longer spends ~13 s rejecting a non-polynomial sequence. Interpreting a continuation such as 1 + 1 + 2 + 3 + 5 + 8 + \dots + 55 (Fibonacci) first tries the polynomial recognizer, which fits a degree-5 interpolant through the samples and searches for the index where it reaches the anchor. That interpolant has a negative leading coefficient, so it eventually decreases — but the search's overshoot test used the sample trend ("increasing"), which never fired, so it ground through all 100 000 candidate indices before falling through to the recurrence recognizer. The search now stops on the interpolant's local trend (a polynomial is eventually monotonic, so once it is past the anchor and still diverging it cannot return), cutting this interpretation from ~13 s to a few milliseconds. Legitimate polynomial sums (triangular numbers, squares) are unaffected.

Calculus

• Improper integrals of polynomial × exp-decay no longer return NaN. A definite integral to ±∞ whose antiderivative carries a term like y^2 e^{-y} was evaluated at the infinite bound by naive substitution, producing an ∞·0 indeterminate that collapsed to NaN — so \int_0^\infty y^2 e^{-y}\,dy (which is \Gamma(3) = 2) returned NaN. Such an endpoint is now resolved as the limit \lim_{y\to\infty} F(y) (exponential decay dominates polynomial growth), giving the exact closed form (2); with the Rubi integration rules loaded the same fix closes the χ²-tail \int_x^\infty y^{3/2} e^{-y/2}\,dy to 3\sqrt{2\pi} - F(x). An endpoint that still cannot be resolved keeps the integral inert (so N() quadrature applies) rather than leaking NaN.
• Limit at infinity resolves Erf/Erfc and \sqrt{}/\sqrt[n]{}. \lim_{x\to\infty}\operatorname{erf}(x) = 1, \lim_{x\to-\infty}\operatorname{erf}(x) = -1, \operatorname{erfc} saturating to 0/2, and \lim_{x\to\infty}\sqrt{x} = +\infty (likewise \sqrt[n]{x}) were previously left unevaluated. Besides being correct in their own right, these fill the gaps behind the improper-integral endpoints above — an antiderivative's \operatorname{erf}(\sqrt{y}) term needs both.
• The upper incomplete gamma reduces at infinity: \Gamma(s, +\infty) = 0. The tail \int_{+\infty}^\infty t^{s-1} e^{-t}\,dt vanishes for any finite s (the e^{-t} factor dominates), so \Gamma(s, \infty) — including symbolic s such as \Gamma(\frac{k}{2}, \infty) — now evaluates to 0 instead of staying inert (a provably infinite first argument stays symbolic). This closes the free-parameter χ²-tail antiderivative \int_x^\infty y^{\frac{k}{2}-1} e^{-\frac{y}{2}}\,dy to a form free of the leftover \Gamma(\cdot, \infty) term.
• A rational integrand with fully symbolic coefficients no longer hangs. \int_0^x \frac{u-a}{b_2 u^2 + b_1 u + b_0}\,du spun for ~109 s (ignoring a 3 s timeLimit) inside the polynomial-GCD used to cancel common factors: its Euclidean loop divided by a symbolic constant, which produced a spurious nonzero constant remainder that never tested as zero, so the loop iterated forever building ever-larger coefficient expressions. A nonzero constant remainder now correctly resolves the GCD to 1 (coprime over the coefficient field), and the loop carries a deadline checkpoint as a backstop. The integral closes in ~200 ms (to an ArcTanh/Ln form with the Rubi rules loaded); polynomialGCD of coprime symbolic polynomials returns 1 instead of spinning.

Compilation

• Compiled definite integrals now resolve symbolically before falling back to quadrature. compile() of an expression containing a definite Integrate first attempts a closed form (the same antiderivative machinery evaluate() uses); if one is found, the generated code is straight-line arithmetic rather than a per-call numerical integration. A plotted \int_0^x 0.1\sqrt{1+t^2}\,dt compiles to its closed form 0.05\,(x\sqrt{1+x^2} + \operatorname{arsinh} x), evaluated in microseconds per sample (exact and deterministic) instead of ~150 ms/sample of quadrature. The symbolic attempt is bounded by ce.timeLimit (default 2 s), so a hard integrand degrades to quadrature rather than stalling compilation, and it is skipped when the integral references a symbol supplied through the vars option (which must stay a live runtime input, not be folded to a constant).
• The compiled quadrature fallback is now deterministic adaptive Gauss–Kronrod (GK15), not Monte-Carlo. An integral that does not resolve symbolically is estimated with adaptive Gauss–Kronrod quadrature: near machine precision on smooth integrands, microseconds-to-milliseconds per call, and — unlike the previous 10⁷-sample Monte-Carlo estimator (~1e-4 error, a different value on every call, ~150 ms/call) — the same value on every call. Infinite bounds are handled by a smooth variable transform. Monte-Carlo remains an automatic fallback when the adaptive rule does not converge, and can be forced with the new compile(expr, { quadrature: 'monte-carlo' }) option ('adaptive' is the default).

0.78.1 2026-07-14

Parse Diagnostics

• juxtaposition-as-multiply now covers unit-lexed and letter-run application shapes. Two application-shaped sources that produced no diagnostic on 0.78.0 are now reported: a symbol lexed as a unit applied to a group (\mathrm{N}(2), where N reads as the newton unit) fires with detail.name set to the source symbol and a new additive detail: { lexedAs: "unit" } hint; and a letter-run applied to a group (divisors(60), which segments into d·i·v·i·s·o·r·s) fires a single diagnostic whose name is the joined run ("divisors") and whose span covers the full divisors(60) source shape. Run reconstruction stops at numbers and multi-character commands (2x(3) reports x; \pi r(2) reports r).

0.78.0 2026-07-14

Parse Diagnostics

• New opt-in diagnostics parse option. ce.parse(latex, { diagnostics: true }) attaches a parseDiagnostics array to the top-level result, flagging charitable parse decisions that are usually errors in machine-generated LaTeX (LLM output, OCR). Four codes are reported: undeclared-symbol (a symbol reference with no declaration — detail: { name, type }), juxtaposition-as-multiply (a symbol immediately followed by a delimited group (…) or a matrix environment read as multiplication — detail: { name, declaredAs }), comment-discarded (an unescaped % dropped input — detail: { discardedLength }), and recovered (trailing noise silently skipped by non-strict error recovery). Each carries a start/end source span: undeclared-symbol and juxtaposition-as-multiply are offsets into CE's normalized LaTeX, while comment-discarded (and best-effort recovered) use original-input coordinates. The feature is purely additive — enabling it never changes the parse output — and works under { canonical: false }. parseDiagnostics is present (a possibly-empty array) only on the result of a diagnostics: true parse, and undefined otherwise.

0.77.1 2026-07-13

Breaking Changes

• f'(x) now parses to ["Apply", ["Derivative", "f", 1], x] instead of ["D", ["f", x], x]. Prime notation on an applied function denotes the derivative function evaluated at the argument — Lagrange semantics, as in Mathematica and Desmos — not the derivative of the applied expression with respect to an inferred variable. The previous representation produced silently wrong results whenever the argument was not a bare variable: f'(2) evaluated to 0 instead of f' evaluated at 2 (e.g. 6 for f(x) := x^2+2x+1), and f'(2x) picked up a spurious chain-rule factor (8x+4 instead of 4x+2). Higher orders (f''(2) → 2) and the f^{(n)}(x) superscript form follow the same rule. Serialization is unchanged (f^{\prime}(x)), so LaTeX round-trips are unaffected; only consumers pattern-matching the MathJSON parse shape need updating.

Resolved Issues

• Symbolic derivatives of declared-then-assigned functions (0.77.0 regression). Declaring a function symbol before assigning its body — ce.declare("f", "function") (or with an explicit signature such as "(number) -> number") followed by f(x) := x^2 + 2x + 1 — left the function's derivative inert: f'(x) evaluated to itself instead of 2x + 2, and ["D", "f", "x"] evaluated to 0, even though direct calls like f(2) worked. The declared-signature reconciliation introduced in 0.77.0 keeps the assigned function literal in the symbol's value definition (preserving the declared signature) instead of converting it to an operator definition, and the symbolic differentiation path only expanded function bodies from operator definitions. Differentiation now expands user-defined function bodies from both definition shapes.

0.77.0 2026-07-13

Pattern Matching

• New Match operator for structural pattern matching. ["Match", subject, ["MatchCase", pattern, body], …] selects the first case whose pattern matches the structure of the subject and applies its body to the captured values: ["Match", ["List", 3, 4], ["MatchCase", ["List", "_a", "_b"], ["Add", "a", "b"]]] → 7. Cases may carry a guard (["MatchCase", pattern, guard, body]); ["Pin", expr] matches the value of an expression (a constant like Pi, or the current value of a variable); ["Alternatives", p1, p2, …] shares one body among several binding-free patterns. Unlike Which, which stays unevaluated while a condition is undecidable, Match always decides — a symbolic subject falls through to a wildcard case. No matching case yields an ["Error", "'match-no-case'"] value.
• Cortex: match expression. The reserved match keyword is now a full pattern-matching expression: match x { 0 => "zero"; 1 | 2 | == Pi => "small"; [first, ...rest] => first; n if n > 3 => n; _ => "other" }. Bare identifiers in a pattern always bind (a non-final catch-all like Pi => … is a parse error suggesting == Pi to match the constant); == expr pins a value; | gives or-alternatives; […], (…) and {key -> pat} destructure lists, tuples and dictionaries (open matching); n: integer adds a type guard; ...rest captures the tail.
• Constant-time dispatch and compilation. Matches over constant cases dispatch through a cached table instead of the general pattern matcher, and fixed-shape destructuring compiles to direct positional checks. compile() emits comparison chains or a JavaScript switch for constant cases and destructuring closures for fixed shapes; symbolic patterns (e.g. a + b) fail closed with a clear error rather than producing incorrect code.

Typed Function Literals

• Function literals can declare parameter and return types. A Function parameter may be annotated — ["Typed", "x", "'integer'"] — and the body may carry a return-type ascription, so ["Function", ["Add", "x", 1], ["Typed", "x", "'integer'"]] now has type (x: integer) -> integer instead of (unknown) -> number. Annotations feed body type inference, and in strict mode arguments are checked at application: applying 2.5 to an integer parameter yields an incompatible-type error instead of silently computing. Partial application preserves the remaining annotations and the return type. Assigning an annotated literal to a symbol gives it the full typed signature — including the declared return type, which is an ascription (authoritative, like a TypeScript annotation) rather than a check against inference. Untyped literals are unchanged.
• New Typed operator for type ascription. ["Typed", expr, type] asserts the type of an expression for the type system and is transparent at evaluation. It accepts a type string ("'integer'") or a type-name symbol (integer). LaTeX serialization drops annotations (no typed-parameter notation in v1); MathJSON round-trips them.
• Cortex: typed function definitions are enforced end to end. f(x: integer) -> real = x + 1, function g(n: integer) -> integer {…}, and the anonymous form (x: integer) |-> x + 1 (new grammar) all parse to native annotated literals; mistyped calls error, declared return types are carried, and serializeCortex reconstructs the typed syntax faithfully. Recursive typed definitions work (fact(n: integer) -> integer = if n <= 1 {1} else {n * fact(n - 1)}).

Programming and Collections

• Closures capture per-call state. A zero-parameter closure returned from a factory function now captures its own instance of the factory's local variables, so separate invocations no longer share mutable state. For example, two counters built from the same makeCounter() factory advance independently. Parameterized factories already behaved this way; the fix extends the same per-call scope instantiation to nullary functions.
• Lazy collection operations iterate eager sources. A lazy operation such as Map or Filter applied to a collection that only materializes on evaluation (e.g. UnicodeScalars(s), Characters(s)) now iterates its elements instead of behaving as empty. For example, StringFrom(Map(UnicodeScalars(s), c -> c + 1), "unicode-scalars") now produces the shifted string rather than "".

Symbolic Computation

• Arithmetic and function application thread through conditional values. A restricted value When(v, cond) and a piecewise Which(c_1, v_1, …) now flow through scalar operations instead of staying inert: sin(When(x, x > 0)) → When(sin(x), x > 0), guards combining by conjunction (When(x, x > 0) · When(y, y < 1) → When(x·y, x > 0 ∧ y < 1)), and Which(x > 0, 1, x < 0, -1) + 2 → Which(x > 0, 3, x < 0, 1). Logic operators are excluded (so And(When(A, g), False) still short-circuits to False), and piecewise products above 16 combined branches stay unevaluated.
• Restriction guards survive arithmetic cancellation. Evaluating When(x, c) − When(x, c) previously folded to plain 0, silently discarding the restriction on the (fat) region where c fails; it now yields When(0, c). Similarly 0 · When(x, c) → When(0, c) and When(x, c) / When(x, c) → When(1, c).
• When respects numeric approximation. When(π, cond).N() with a true condition now numericizes (previously the option was dropped and the value stayed symbolic).
• DMS angles stay exact. (contributed by yelliver) Degrees-minutes-seconds notation now parses to an exact rational number of degrees instead of a float (9°30' is 19/2°), and Degrees converts any rational — not just integers — to an exact multiple of π, so 5°37'30" simplifies to π/32. Decimal components are recovered exactly when possible (9°30'15.5" → 68431/7200°) and otherwise fall back to floats. Degrees of values beyond 2⁵³ no longer loses precision. In raw (non-canonical) parsing, DMS angles that previously produced a float now produce a Rational. (#321)

Calculus

• Exponentials with any linear exponent integrate. ∫e^{−ax}dx with a symbolic a previously stayed unevaluated (only e^{a·x}-shaped exponents were recognized); it now returns −e^{−ax}/a. Any linear exponent works: ∫e^{3−2x}dx, ∫5e^{−x/2}dx.
• Improper integrals with symbolic parameters return convergence-guarded results. ∫₀^∞ e^(−ax)dx with a free a previously stayed unevaluated; it now returns 1/a {0 < a}. Results that formerly leaked indeterminate endpoint forms are fixed: ∫₀^1 xⁿdx returned an expression containing 0^(n+1), and ∫₁^∞ x^(−s)dx one containing ∞^(1−s); they now return 1/(n+1) {0 < n+1} and 1/(s−1) {1 < s}. Integrals whose endpoint behavior cannot be classified stay unevaluated rather than leaking indeterminates. Numeric-parameter integrals are unchanged.
• Series expands at algebraic branch points (Puiseux series). A series expansion may now carry fractional powers: Series(√(sin x), x) → √x − x^{5/2}/12 + x^{9/2}/1440 + O(x^{13/2}). This covers √x, 1/√x, x^{3/2}·e^x, Root(g, r) and rational powers g^{p/r} where the base vanishes (or has a pole), as well as compositions: cos(√x), csc(√x) (→ 1/√x + √x/6 + …), and Γ(√x) (→ 1/√x − γ + …).
• Series expands through logarithmic singularities. Expanding about a zero or pole of a logarithm's argument now yields a log-carrying series: Series(ln(sin x), x) → ln x − x²/6 − x⁴/180 + O(x⁶), and Series(x^x, x) → 1 + x·ln x + x²·ln²x/2 + …. Base-b logarithms (log₂ x, log₁₀ x) expand through the same path. Nested or reciprocal logarithms (ln(ln x), 1/ln x) and essential singularities (e^{1/x}) still stay unevaluated rather than returning a partial expansion.
• Series of an irrational power no longer returns an invalid expansion. Series(x^π, x) previously produced coefficients containing unresolved 0^{π−1}-style indeterminates; it now stays unevaluated at 0 (the expansion about a regular point, e.g. x^π about 2, is unchanged).
• Logarithmic asymptotic expansions at ±∞. A log-carrying expansion at +∞ now resolves back to x instead of deferring: Series(ln x, x, +∞) → ln x, and Series(ln(x²+x), x, +∞) → 2 ln x + 1/x − 1/(2x²) + …. At −∞ (a logarithm of a negative quantity) such expansions still stay unevaluated.
• Stirling asymptotics for the log-gamma. Series(GammaLn(x), x, +∞) (and the parsed ln Γ(x)) now returns Stirling's series x·ln x − x − ½ln x + ½ln(2π) + 1/(12x) − 1/(360x³) + O(1/x⁵). The series is asymptotic (divergent), so the BigO is placed at the true remainder order. Series(Γ(x), x, +∞) — the exponential of a trans-series — still defers.
• GammaLn evaluates to +∞ at the poles of Γ (the non-positive integers, where ln|Γ| → +∞ — as in Mathematica's LogGamma and SymPy's loggamma); it previously stayed inert there. Consequently Series(GammaLn(x), x) at such a pole now returns the log-aware expansion −ln x − γ·x + (π²/12)·x² + … (matching the parsed ln Γ(x)) instead of an invalid expansion with inert GammaLn(0), Digamma(0), … coefficients.
• Provably-exact expansions drop the BigO remainder. When the truncated sum is symbolically equal to the whole function, Series now returns it without a remainder term: Series(√x, x) → √x, Series(ln x, x) → ln x, Series(x/(x−2)², x, 2) → 2(x−2)⁻² + (x−2)⁻¹. Genuinely-truncated series (1/sin x, Γ(x) at 0, ζ at 1) keep their BigO.

Sums and Products

• Geometric series closed form. Σ_{n=0}^∞ rⁿ now evaluates: exactly for a numeric ratio (Σ(1/2)ⁿ → 2, Σ(1/√2)ⁿ → 2 + √2), and with its convergence condition for a symbolic ratio (Σxⁿ → 1/(1−x) {|x| < 1}). Constant multiples and integer start indices are handled (Σ_{n=2}^∞ xⁿ → x²/(1−x) {|x| < 1}); divergent numeric ratios stay unevaluated.

Solving Equations

• Radical equations with a symbolic right-hand side return guarded roots. Solve(√(x+3) = a, x) previously returned []; it now returns a² − 3 {0 <= a} (a square root is non-negative, so a real solution exists only for a ≥ 0). Substituting a concrete value resolves the guard: a = 2 gives 1, a = −2 gives Undefined. Numeric right-hand sides are unchanged.
• Trigonometric and hyperbolic equations with symbolic coefficients record their validity condition. Solve(sin(x) = a, x) previously returned arcsin(a) and π − arcsin(a) unconditionally — wrong whenever |a| > 1. Roots now carry their domain-of-validity guard — a When restriction, displayed arcsin(a) {|a| <= 1} (LaTeX \arcsin(a)\left\{|a|\le 1\right\}). The guard resolves as soon as it is decidable: substituting a = 1/2 collapses the root to π/6, while a = 3 yields Undefined, and a guard known false at solve time prunes the root (down to []). Numeric-coefficient equations are unchanged. The same applies to cos (|ratio| ≤ 1), cosh (ratio ≥ 1), and tanh (|ratio| < 1) equations.
• Solve of an inequality stays inert instead of returning an empty list. Solve(x^2 < 4, x) previously returned [], which reads as "no solutions"; univariate inequality solving is unsupported, so the expression now stays unevaluated. Solving equations is unchanged.

Compilation

• The deprecated interval-glsl compilation target has been removed. GPU interval evaluation only pays off when the entire pipeline stays on the GPU, and the target could not compile relational operators, so it could not host restriction conditions. Use interval-js (CPU interval arithmetic) or the scalar glsl/wgsl targets instead. The IntervalGLSLTarget export from @cortex-js/compute-engine/compile is gone, and compile(expr, { to: 'interval-glsl' }) now throws an unregistered-target error.

Parsing

• ce.parse() no longer blows up on repeated \command[opt]{} groups. Adjacent bracketed groups — a garbage \begin{tikzpicture}katu[scale=0.6]{}… run, or even plain index notation a[6]a[6]… — triggered exponential-time backtracking, so a few-hundred-character string could hang the parser for tens of seconds (and timeLimit, which bounds evaluation, did not stop it). The reversed-bracket ISO interval notation ]a, b[ opens on ], the same token that closes an index bracket, so every stray ] speculatively re-parsed the rest of the input as an interval body, nesting exponentially. Parsing is now polynomial; the interval (]0, 1[) and indexing (a[6]) notations are unchanged.
• \operatorname{nPr}(n, r) now has a definition. Matching \operatorname{nCr} (the binomial coefficient), the Desmos permutation-count notation lowers to Choose(n, r)·r! (= n!/(n−r)!), so nPr(5, 2) evaluates to 20. Previously it parsed to an inert symbol and stayed symbolic under N(), silently producing NaN in a compiled function.

Arithmetic

• Round accepts an optional precision argument. Round(x, n) rounds x to n decimal places — Round(2.567, 2) → 2.57, Round(1234.5, −2) → 1200 — matching the Desmos/spreadsheet round(x, n) convention. Previously the second argument produced an unexpected-argument error. The single-argument round-to-integer form is unchanged, and the two-argument form compiles on the javascript and interval-js targets.
• New Rationalize operator for rational approximation. Rationalize(x) approximates a real number by a rational at full working precision (like single-argument Rational); with a tolerance, Rationalize(x, tolerance) returns the rational with the smallest denominator within the bound — Rationalize(√3, 1/500) → 26/15, Rationalize(π, 1/100) → 22/7 — a continued-fraction convergent cut.

Number Theory

• New StirlingS1 operator: signed Stirling numbers of the first kind. StirlingS1(n, m) is the coefficient of xᵐ in the falling factorial x(x−1)…(x−n+1); its absolute value counts the permutations of n elements with m disjoint cycles — StirlingS1(5, 2) → −50. Complements the existing Stirling (second kind).

Relational Operators

• Equal and NotEqual broadcast over a named list operand. With R = [1, 2, 3], x² + y² = R² now broadcasts to a list of three element-wise equations — matching the literal-list form (x² + y² = [1, 2, 3]) and the inequality operators (<, ≤, …), which already broadcast. Previously the named form collapsed to a single False. Whole-list equality, where two or more operands are collections ([1, 2] = [1, 2]), still returns a single boolean.

0.76.0 2026-07-11

Programming and Collections

• Dictionary lookups have the value's type. At(dict, key) — and thus d["a"] in Cortex — was statically typed as the key-value pair (tuple<string, T>), so using a lookup directly in arithmetic (d["a"] + 10) failed with an incompatible-type error. It is now typed as the value; a record indexed by a literal string gets that field's precise type.

• Reduce/Fold honor the exactness contract. The compiled floating-point fast path no longer runs under plain evaluate(): exact operands fold exactly (Fold((a, k) ↦ a + 1/k, 0, Range(1, 5)) → 137/60 instead of 2.2833…). The fast path is reserved for .N() and already-inexact inputs. Complex-valued folds no longer silently drop imaginary parts (Product(Map(Range(1, 3), k ↦ k + i)) → 10i, previously 6).

• Map infers its element type from the mapped function. The result of Map(Range(1, 3), k ↦ k + i) was typed with the source element type (integer); it now reflects the lambda's result type, so downstream operations dispatch correctly.

• Elementwise broadcasting is uniform across lazy and eager collections. A finite lazy Range now broadcasts like an eager List in tuple products: with R = Range(-2, 2), R·(2,3) yields a list of five scaled points instead of distributing the range inside the tuple components. A scalar also folds into a collection produced by an inner broadcast step: L^2 - 2 evaluates to [-1, 2, 7] instead of the unevaluated Add(-2, [1, 4, 9]), and evaluation is idempotent again on these shapes. Infinite or unknown-length ranges stay symbolic rather than transposing.

• Scalar operations accept lazy collection operands during validation. Mod([0,\ldots,kN], N) with a symbolic bound produced an incompatible-type error at canonicalization even though the eager-list form broadcast fine; the argument validator now recognizes parametrized indexed_collection<T> wherever broadcasting applies. Declared types follow the values: broadcast results type list<…> (previously a scalar-or-list union, or a scalar type for symbolic-length ranges).

• Big integers survive numeric list literals. A list literal promoted to a tensor stored oversized integers in float64 and truncated them ([100!] lost digits, breaking exact iterative algorithms such as a Fibonacci pair accumulator). Integers beyond the float-safe range now keep their exact representation.

• StringFrom joins collections. With a list argument and the "unicode-scalars", "utf-8", or "utf-16" format, StringFrom([100, 101, 102], "unicode-scalars") returns "def"; it previously broadcast element-wise and returned a list of one-character strings.

• One-step function definitions bind inside function bodies (Cortex). function outer(n) { sq(m) = m * m; sq(n) } left sq(n) unevaluated because the call site resolved a value placeholder while the runtime created an operator definition; function application now falls back to the operator definition. The example-program suite grew by 18 programs covering control flow, number theory, complex numbers, linear algebra, and exact sums (mirrored in the Cortex documentation).

• Cortex: do { … } block expressions and zero-parameter lambdas. A statement block can now appear in any expression position with the explicit do prefix — its value is its final statement — so multi-statement closure bodies are expressible: x |-> do { let t = x * x; t + 1 }. Set literals are unchanged (x |-> {1, 2} still returns a set). Zero-parameter lambdas (() |-> …) now parse and apply, enabling the stateful makeCounter-style closure documented in the examples.

• Cortex: a named inner function escapes its scope as a value. function make() { helper(x) = x + 1; helper } now returns a callable first-class function — previously the returned symbol went inert once the call frame popped. Captured locals and parameters of the enclosing call are preserved; returned |-> lambdas are unchanged.

• Cortex: lowercase true/false, ASCII .. ranges, and StringJoin over a list. true/false are now input aliases for True/False (and reserved as binding names); 1..n is a range (for k in 1..5), equivalent to the existing ‥, without disturbing decimal literals like 1.5; and StringJoin accepts a single collection of strings — StringJoin(Reverse(Characters("hello"))) → "olleh".

• Cortex: "did you mean" warnings for near-miss function names. Calling an undeclared function whose name is close to a library operator no longer fails silently symbolic: executeCortex emits a warning diagnostic with the suggestion (Quartile(xs) → did you mean Quartiles?). The match is conservative (case-insensitive, singular/plural, small edit distance, unique prefix) and the diagnostic only fires when a suggestion exists, so intentionally symbolic calls are never flagged; the returned value is unchanged. The matcher is also available as ce.suggestOperatorName(name). Arg is now an alias for Argument.

Compilation

• Calls to user-defined functions compile. After f(x) := e^{-x^2/2}, compiling f(2) — or any expression referencing f — emits the definition as a named local function instead of throwing Unknown operator `f` . Nested user functions are emitted in dependency order; recursive definitions fail closed with an explanatory error. This also removes a silent interpreted fallback in numeric integration: definite integrals of user-defined functions now run compiled quadrature (10⁷ samples instead of 10⁴ — comparable wall time, ~30× tighter error estimate). Applies to the javascript and interval-js targets.

• Collection-valued conditions fail closed instead of compiling wrong code. Equal/NotEqual over a collection-typed operand, and If/Which/When with a collection-typed condition, previously compiled with success: true and returned null or the wrong branch at run time; they now throw an explanatory compile-time error. Interpreted evaluation is unchanged: comparisons broadcast elementwise, conditionals require a scalar boolean.

• Reduce, Length, and At compile on the javascript target. Reduce(xs, Add|Multiply|Min|Max, init?) compiles to a loop; Length returns the element count; At follows the interpreter's 1-based, negative-from-end indexing (out-of-range yields NaN).

• GLSL: Length no longer collides with the length() builtin. CE's Length (element count) compiled to GLSL length() — the Euclidean norm — reporting success while computing the wrong value, or emitting invalid GLSL for lists longer than four. length() is now emitted for Norm; collection Length fails closed on the GPU targets.

• GLSL/WGSL: literal integer powers are sign-correct on the GPU. x^3 compiled to pow(x, 3.0), which the GLSL specification leaves undefined for negative bases — real GPUs returned pow(-2, 3) = +8, silently flipping the sign of odd-power terms. Small integer exponents now emit repeated multiplication; larger and compound-base cases use a sign-preserving _gpu_powi preamble helper; negative integer exponents wrap the reciprocal. Fractional exponents still emit pow.

• The interval-glsl target is deprecated. GPU interval evaluation only pays off when the entire pipeline stays on the GPU, and the target cannot compile relational operators, so it cannot host restriction conditions. A once-per-process warning now points to interval-js and the scalar glsl/wgsl targets. It will be removed in a future release.

Numeric Evaluation

• Numeric infinite products use tail acceleration. .N() now Richardson-extrapolates the logarithms of positive real factors instead of returning a plain finite truncation. This gives accurate values for products such as Product(1 + 1/k², k, 1, +∞) = sinh(π)/π. Products with non-real, non-positive, or non-convergent factors decline the accelerator and retain the existing bounded-truncation behavior.

• Machine Gamma keeps full relative accuracy through the overflow edge. Positive real arguments now use a balanced recurrence from the Lanczos core instead of reconstructing large values from exp(gammaln(z)), which preserves about 15-16 digits up to the IEEE-754 limit near Gamma(171.624).

Linear Algebra

• Matrix operators infer fresh symbolic operands from context. An expression such as \det(A+2B) no longer fails because bottom-up arithmetic canonicalization provisionally typed A and B as numbers before Determinant required a matrix. Validation now repairs only inferences made while constructing the current expression and canonicalizes the argument once more with matrix context. Explicit declarations and inferences from earlier expressions are never overwritten, and ambiguous products remain unchanged rather than guessing which factor is the matrix.

Symbolic Computation

• Exact cube-root arithmetic handles more algebraic forms. Positive perfect-power bases with rational exponents normalize to a common base and extract their integer part (4^(2/3) → 2·2^(1/3)), allowing compatible cube-root powers to combine exactly. Real nested cube roots of the form ∛(a+b√c) are denested when exact integer conjugate identities prove a result; in particular, ∛(90+34√7) → 3+√7 (and the conjugate form with minus signs). The Wester-28 cube-root identity now simplifies directly to exact zero (no explicit Expand required) and numericizes without NaN.

• Infinite p-series support positive-integer lower bounds beyond 1. Sum(k^(-s), k, a, +∞) now returns Zeta(s) − Sum(k^(-s), k, 1, a−1) for exact real s > 1; for example, Sum(1/k², k, 3, +∞) evaluates to π²/6 − 5/4. The existing lower-bound-1 behavior and divergence guards are unchanged.

• e^{iθ} stays exact for constructible angles. e^{i\pi/3} now evaluates to 1/2 + (√3/2)i instead of a machine float (the exact cosine/sine values were being recombined through float-folding arithmetic). .N() numericizes as before, and the degenerate angles (e^{i\pi} → -1, e^{i\pi/2} → i) are unchanged.

Serialization

• AsciiMath prints series in textbook order. Taylor-series terms are serialized in ascending degree and asymptotic series in descending degree, with the BigO remainder last, matching LaTeX output. Canonical expression order and ordinary sums without a BigO term are unchanged.

Parsing

• Stepped ellipsis ranges accept negative and symbolic samples. [-9,-6,\ldots,9] now parses to Range(-9, 9, 3) — previously any negative leading sample fell back to a literal list containing a ContinuationPlaceholder that enumerated as NaN. Symbolic stepped forms infer a symbolic step when the samples are numeric multiples of one common symbol ([-3N,-2N,\ldots,3N] → Range(-3N, 3N, N), progression-validated on the coefficients); generic sequence notation ([x_1,x_2,\ldots,x_n]) intentionally still parses as a plain list.

Engine Lifecycle

• Popping a scope releases configuration listeners owned by its constants. Constant definitions now retain and invoke the unsubscribe closure returned when they register for precision and angular-unit changes. Local constants from discarded scopes therefore no longer remain reachable for the lifetime of the compute engine.

• Cancellation errors carry a structured cause. A cap breach reports 'timeout', 'iteration-limit-exceeded', or 'recursion-depth-exceeded' via the exported CancellationCause type. In Cortex, a final-statement breach carries the cause as a second operand on the Error value, and non-final statements emit a dedicated evaluation-canceled diagnostic. Error messages are unchanged, so existing string matching keeps working.

0.75.0 2026-07-11

Numeric Evaluation

• Inverse trigonometric and hyperbolic functions now evaluate outside their real domain. .N() returns the complex principal value when no real value exists, for example \arcsin(2) → 1.571 − 1.317i and \operatorname{arcosh}(0.5) → 1.047i. Complex arguments such as \operatorname{arsinh}(1+i) are also supported. Exact arguments remain symbolic with evaluate(). Incorrect complex values from Arcoth on part of its branch cut and from Arsech have also been fixed.
• Products and quotients of square roots no longer throw on large radicands. Evaluating an expression such as \sqrt{1234}\cdot\sqrt{1235}, whose combined radicand (1234·1235) exceeds the exact-radical limit, no longer raises an internal "Unexpected value for radical part" error. Any perfect-square factor is extracted (√(k²·r) = k·√r), keeping the result exact when the square-free part is small enough and otherwise returning the numeric value.

Solving Equations

• Solve handles systems of equations. Solve([eq1, eq2, …], [x, y, …]) returns each solution as a tuple of values in the order of the variable list: Solve([x + y = 3, x - y = 1], [x, y]) → [(2, 1)], and a nonlinear system such as [x^2 + y^2 = 25, x + y = 7] returns both solutions [(3, 4), (4, 3)]. Linear systems solve exactly (rational values), an underdetermined system returns a parametric tuple ([(5 - y, y)] for x + y = 5), and a system the solver cannot decide stays unevaluated. This matches the tuple shape already used when solving over explicit domains.
• solve() correctly rejects trigonometric and hyperbolic equations with no real solutions. Equations such as \sin x = 2, \tanh x = 2, and \cosh x = 1/2 now return no solutions. Symbolic equations and equations with complex polynomial roots are unaffected.

Integration (opt-in Rubi rules)

• More integrals containing binomial radicals are supported. This includes mixed even- and odd-power numerators such as \int\frac{c+dx}{\sqrt{-a-bx^4}}dx, \int\frac{x^2(c+dx+ex^2+fx^3)}{(a+bx^4)^{3/2}}dx, and Laurent variants with denominators of the form (a+b·x^n)^{3/2}.
• More integrals that are algebraic in a hyperbolic function are supported. Half-integer powers of hyperbolic expressions such as \int\coth(x)(a+b\sinh^2 x)^{3/2}\,dx, \int\coth^2 x\sqrt{a+b\tanh^2 x}\,dx, \int\frac{\operatorname{csch} x}{(a+b\sinh^2 x)^{3/2}}\,dx, and \int\sqrt{a+b\operatorname{csch}^2 x}\,dx now close in elementary form via a hyperbolic substitution. This also fixes a wrong-answer case, \int\frac{\sqrt{\coth(a+b\ln(cx^n))}}{x}\,dx.
• More integrals that are rational in a hyperbolic function are supported. Ratios of hyperbolic functions with a squared-or-higher power, such as \int\frac{\tanh^2 x}{a+b\tanh x}\,dx, \int\frac{\tanh x}{a+b\sinh x}\,dx, and \int(a+b\tanh^2 x)^3\tanh^4 x\,dx, now close in elementary form.

Library and Definitions

• Definitions can be searched by concept. ce.searchDefinitions(query) returns a ranked list of matching { id, kind } entries. It searches names, descriptions, synonyms (for example, average finds Mean), and LaTeX commands (for example, \gcd finds GCD). Use the optional limit argument to control the number of results (default: 10). Custom definitions declared with ce.declare() can provide an optional keywords list, and returned IDs can be passed to ce.lookupDefinition().
• Definition descriptions filled in. About 80 operators and constants that lacked a description now have one (the trigonometric family, logic and relational operators, collection primitives such as List, Range, and Fold, and constants like Pi and ExponentialE), and Sec's description was corrected (secant is the reciprocal, not the inverse, of cosine). These surface in ce.searchDefinitions() and ce.lookupDefinition().
• New builtin operators are available: Pipe(x, f), Append(collection, element), Fold(f, init, collection), StringJoin(s1, s2, …), and RandomInteger(n) or RandomInteger(a, b). Pipe enables evaluation of x |> f; StringJoin enables the existing <> notation; and RandomInteger uses inclusive bounds and honors the seeded random-number generator.

Calculus

• Limit accepts the explicit-variable form. Limit(expr, var, point) — e.g. ["Limit", ["Divide", ["Sin", "x"], "x"], "x", 0] → 1 — now canonicalizes to the same internal form as Limit(expr, point), matching the convention Series already uses. The (function, point, direction) reading is preserved when the middle operand is not a free variable of the expression.

Linear Algebra

• Inverse of an exact matrix is exact. An integer or rational matrix now inverts over the rationals — Inverse([[2,1],[1,3]]) → [[3/5,-1/5],[-1/5,2/5]] instead of floats — with .N() and inexact matrices using the numeric path as before.
• New LinearSolve(A, b) operator solves the linear system A·x = b, exactly for exact input. Composed forms like Dot(Inverse(A), b) also work now: Inverse's result is typed as a matrix, so matrix operators accept it.

Units and Quantities

• Quantity accepts a string unit. Quantity(30, "km/h") parses the string through the same unit grammar as the LaTeX path and canonicalizes identically to the symbolic form; a malformed unit string produces a clear error instead of a partially-built expression.

Programming and Collections

• Recursive functions can be defined without a separate declaration. A function assignment that refers to itself, such as ce.parse('f(n) := n \\cdot f(n-1)'), now works directly.
• N() numericizes through user-defined functions. For f(x) := x/3, N(f(2)) now returns 0.666… instead of the exact 2/3; plain evaluate() still returns the exact form, and the approximation is applied within the function's own scope, preserving lexical scoping.
• Keys(dict) and Values(dict) evaluate, returning the keys (as strings) and values in the dictionary's iteration order — the same order for kv in dict yields.
• Intersection accepts lists (any finite collection), deduplicating into a Set; Union already did.
• A 2-element MathJSON List in a set operation is a collection, not an interval. ["Intersection", ["List",1,2], ["List",2,3]] (Cortex: Intersection([1,2], [2,3])) now intersects the two-element collections — Set(2) — instead of reading the lists as closed intervals. The interval reading of ambiguous bracket pairs is now applied where it belongs, at the LaTeX boundary: x \in \lbrack 1, 5 \rbrack, (-\infty, 0) \cup (0, \infty), and the subset relations parse to Interval exactly as before, and \setminus now gets the same interval reading (previously \R \setminus (0, 1) kept a raw pair). Unambiguous interval notations ([a, b), ]a, b[, …) are unchanged.
• Collection equality no longer depends on representation. A computed collection — an Intersection or Union result, a lazy Map, Filter, or Join pipeline, or a symbol assigned a collection — now compares equal to a literal with the same elements: Intersection({1,2,3,4}, {2,3,5}) = {2,3} is True (it was False unless the operand was evaluated first). Sequences compare element-wise in order, sets by membership, and a set is never equal to a sequence. In addition, Equal between two collections now always returns a scalar boolean instead of sometimes broadcasting element-wise ({1,2} = [1,2] returned ["True","True"]); broadcasting still applies to list-vs-scalar comparisons such as L = 4.
• Intersection of two Filter collections no longer overflows the stack. Membership tests on a lazy Filter recursed without bound; Intersection(Filter(…), Filter(…)) now evaluates normally.
• Indexing a matrix once returns a correctly typed row. Expressions such as At(At(m, 2), 1) now validate and evaluate correctly for matrices and other rank-2-or-higher collections.
• List elements and dictionary values are evaluated by evaluate() and .N(). For example, ["List", "y", ["Add", "y", 1]] with y = 7 evaluates to [7, 8], and [1/3].N() is numericized. Lazy collections such as Range, Map, and Filter remain lazily enumerated.
• Ellipsis lists with symbolic bounds parse to Range. \left[-N,\ldots,N\right] and \left[-3N,\ldots,3N\right] now parse to Range(-N, N) and Range(-3N, 3N), matching the numeric-start forms ([1,\ldots,N]). Previously a symbolic start fell through to a raw List containing a literal ContinuationPlaceholder, which enumerated as NaN. In addition, a Range whose bounds bind looser than the .. operator now serializes with parentheses ((-N)..N) so it round-trips through LaTeX (an unwrapped -N..N reads back as -(N..N)).
• Using a symbol bound to a symbolic list no longer corrupts builtin definitions. After ce.assign('L_1', ce.parse('\\left[N,2N\\right]')), constructing 2 L_1 — via subs(), ce.box(), or ce.function() — permanently broke the builtin N operator for the lifetime of the engine: every subsequent parse of the token N returned an unexpected-symbol error. Type inference on the list elements no longer overwrites an operator definition with an unsatisfiable (never) type.

Cortex

• % and postfix ! operators: a % b is Mod(a, b) (multiplicative precedence) and n! is Factorial(n) (the ! must directly follow its operand; prefix !x is still Not and x != y is still NotEqual).
• Chained indexing: m[2][1] now works alongside m[2, 1].
• String escape sequences are processed correctly. "a\tb\nc" now contains a real tab and newline (escapes were previously double-processed in plain and multiline strings; interpolated strings were already correct).
• The examples suite roughly doubled (src/cortex/docs/examples.md), adding units and uncertainty, calculus, linear systems, dictionaries, sets, closures, seeded randomness, errors-as-values, and string formatting — every example verified by an executable test.

0.74.0 2026-07-10

This release significantly expands CE's calculus capabilities. Limits, residues, and series now handle many poles of special functions exactly; infinite sums and products gain more closed forms and substantially better numeric convergence; and the optional Rubi integration rules support more integrands, any integration variable, and reliable time limits. Step-by-step explanations now cover integration and systems of equations or inequalities, with clearer traces for simplification.

It also improves exact and symbolic computation throughout the engine. Linear algebra gains exact ranks, null spaces, eigenvectors, matrix square roots, and singular values; assumptions and simplification prove more identities; and several correctness issues involving canonicalization, fractions, symbolic collections, compilation, and LaTeX parsing are fixed. New special-function support, reproducible seeded randomness, and more useful Cortex diagnostics round out the release.

Calculus

• Limits and residues at special-function poles evaluate exactly. Expressions at poles of Gamma, Digamma, Trigamma, PolyGamma, and Zeta that previously stayed symbolic now resolve in closed form: \lim_{x\to-1}(x+1)\psi(x) = -1, \lim_{x\to0}(\Gamma(x)-1/x) = -\gamma, \lim_{s\to1}(s-1)\zeta(s) = 1, \operatorname{Res}_{s=1}\Gamma(s)\zeta(s) = 1, \operatorname{Res}_{x=0}\Gamma(x)^2 = -2\gamma, and higher-order poles that previously deferred (\operatorname{Res}_{x=-2} \Gamma(x)/(x+2) = 3/4 - \gamma/2). Deferral behavior is unchanged where no exact expansion exists (branch points, essential singularities, two-sided pole limits).
• The polygamma family expands, differentiates, and integrates through the ladder. Series now produces correct Laurent expansions of Trigamma and integer-order PolyGamma(m, x) at their poles (previously a spurious regular expansion could be produced), and D knows \psi_1' = \psi^{(2)} and the general d/du\,\psi^{(m)}(u) = \psi^{(m+1)}(u). Series at a Digamma pole is also about 20× faster.
• Residues at infinity evaluate. Residue(f, x, \infty) — any infinite point names the Riemann-sphere point at infinity — computes -\operatorname{Res}_{s=0} f(1/s)/s^2 through the exact Laurent kernel: \operatorname{Res}_\infty 1/x = -1, \operatorname{Res}_\infty \frac{3x^2+2}{x^3+x} = -3 (the negated sum of the finite residues).
• Limits at poles resolve to signed infinities. A directional limit at a pole now evaluates to \pm\infty from the exact Laurent data: \lim_{x\to0^+} 1/x = +\infty, \lim_{x\to0^-}\Gamma(x) = -\infty, \lim_{s\to1^\pm}\zeta(s) = \pm\infty, \lim_{x\to0^+}\ln x = -\infty. A two-sided limit resolves only when both sides agree (even pole order): \lim_{x\to0} 1/x^2 = +\infty, \Gamma(x)^2 \to +\infty, \ln(x^2) \to -\infty. Disagreeing two-sided limits (\lim_{x\to0} 1/x, \Gamma, \ln x at their poles) deliberately stay inert — the engine does not produce ComplexInfinity limits.
• Beta joins the meromorphic pole family. The Laurent kernel expands \operatorname{B}(a,b) through the \Gamma-quotient identity, so residues, limits and Series at Beta poles evaluate: \operatorname{Res}_{x=0} \operatorname{B}(x,3) = 1, \lim_{x\to0} x\cdot\operatorname{B}(x,3) = 1.
• Numeric limits containing sums now converge instead of hanging. N() respects evaluation limits when probing a Limit at \infty whose body contains a variable-length Sum or Product. Examples that now converge quickly include \lim_{n\to\infty}(\sum_{k=1}^{n} 1/k - \ln n), which evaluates to the Euler–Mascheroni constant, and \lim_{n\to\infty}\frac{4}{n^2}\sum_{k=1}^{n}\sqrt{n^2-k^2}, which evaluates to \pi.
• Numeric limits with odd-power error terms now converge correctly. This fixes cases such as H_n - \ln n - \gamma \sim 1/2n, which previously returned NaN. Decaying oscillations such as \operatorname{sinc} at -\infty now resolve to 0, while divergent oscillations such as \sin x at \infty still return NaN.

Step-by-Step Explanations

• explain('Integrate') traces symbolic integration through the Rubi rule chain. With the opt-in integration rules loaded (loadIntegrationRules(ce) from @cortex-js/compute-engine/integration-rules), ce.parse('\\int x\\sqrt{1+x}\\,dx').explain('Integrate') replays the driver's derivation as whole-expression states — term-by-term splits (integrate.sum), constant factors moved out (integrate.constant-factor), each corpus rule application (a stable rubi:… id with a compact description such as "Apply integration rule 1.1.1.2#19 (Rubi)"), reductions to special functions (integrate.si-ci, integrate.partial-fractions, …), and a closing simplification. A definite integral is presented via the Fundamental Theorem of Calculus: the antiderivative derivation, then the bracket F |_a^b (integrate.fundamental-theorem), the bounds substituted unevaluated (integrate.evaluate-bounds — skipped for improper integrals, where the bracket is a limit), and the value. Symbolic bounds are supported. The result is identical to evaluate(). Without the rules loaded, or when the rules cannot close the integral, a precise error is thrown. (Also fixed: the LaTeX serialization of the two-bound \left. F \right|_a^b EvaluateAt form dropped the upper bound.)

• explain('solve') traces systems of inequalities and mixed systems. A List/And of linear inequalities in two variables is traced through constraint normalization (solve.system.normalize-inequality), boundary intersection (solve.system.intersect-boundaries), and the feasible vertices (solve.system.vertices); mixed equality/inequality systems show the elimination steps, then each candidate checked against the constraints (solve.system.check-constraints, solve.system.reject). Both previously threw "not supported" errors.

• explain('simplify') surfaces the work done inside operands. Simplifications applied while descending into the operands of a sum, product or function argument — previously summarized by an opaque bookkeeping step — now appear as labeled steps with their own rule ids (\tan x\cot x + \frac{x^3+x^2}{x^2} shows the \tan x\cot x \to 1 rewrite before the expansion). At default verbosity, consecutive applications of the same rule are coalesced into a single step; pass verbosity: 'all' for the raw chain.

Integration (opt-in Rubi rules)

• Integration consistently respects timeLimitMs. Nested integration attempts now share the original time limit and recursion safeguards, avoiding runaway evaluation on cyclic or difficult subproblems.
• Any integration variable works—not just x. Integrals using another variable could previously return an expression in x; for example, \int t^2\,dt now correctly returns t^3/3.
• Symbolic-coefficient quartic-denominator rationals close. \int \frac{d+e\,x^2}{a+b\,x^4}\,dx — and shapes that reduce to it, such as \int \frac{x^6}{(a+c\,x^4)^3}\,dx — now reach the trinomial terminal rules instead of ping-ponging between integrand expansion and binomial splitting.
• Symbolic-coefficient reciprocal hyperbolics close. \int \frac{1}{a+b\sinh x}\,dx and the cosh/tanh/coth/sech/csch variants resolve via a rational-normal-form retry in the exponential-substitution fallback.
• Complex special-function closures. Rational integrands with irreducible quadratic denominators split over complex-conjugate roots in the Si/Ci fallback, reciprocal-argument integrands like \int x^m \sin(a + b/x)\,dx close, and inverse-trig antiderivatives producing complex-argument Erfi evaluate (riding the new complex error-function kernels).
• \int F(\ln(a\,x^n))/x\,dx closes via a function-of-logarithm recognizer (substitution u = \ln(a\,x^n)).
• Products of sines and cosines reduce via product-to-sum before integration, closing mixed-angle products the term-by-term rules could not reach.

Arithmetic

• Canonical expressions no longer depend on a variable's current value. A mutable symbol holding 0, 1, or -1 could be folded into an expression while it was boxed, producing stale and sometimes incorrect results after the symbol changed. Canonicalization now folds only literal numbers; symbol values are substituted during evaluation. Numeric const symbols follow the same evaluation behavior as Pi.
• Huge scientific exponents no longer crash. Parsing or serializing a number literal whose exponent exceeds what the bignum layer can represent (1e999999999) threw; it now overflows cleanly to +\infty (and -\infty for negative mantissas), matching float semantics.
• Complex values with an infinite component type as complex. A Complex whose real or imaginary part is infinite was typed finite_complex, so type-gated paths mishandled it; it now reports the non-finite complex type.

Sums and Products

• Telescoping sums and products evaluate in closed form. A sum whose body is a k \to k+1 shift pair collapses exactly, for arbitrary symbolic bounds and either orientation: \sum_{k=0}^{n} \bigl(g(k+1) - g(k)\bigr) evaluates to g(n+1) - g(0). The product counterpart recognizes a shift-quotient body after combining it over a common denominator: \prod_{k=1}^{n-1}\left(1 + \frac{1}{k}\right) evaluates to n.
• \prod_{k=1}^{n} k evaluates to n!. The bare-index product with a symbolic upper bound returns Factorial(n) instead of staying inert.
• Classic infinite series and products evaluate to their exact closed forms. p-series reduce to the zeta function — \sum_{k=1}^{\infty} \frac{1}{k^2} evaluates to \frac{\pi^2}{6}, \sum \frac{1}{k^2} + \frac{1}{k^3} to \frac{\pi^2}{6} + \zeta(3) (term-wise splitting applies only when every summand has a closed form) — and the Wallis product \prod_{k=1}^{\infty}\left(1 - \frac{1}{(2k)^2}\right) evaluates to \frac{2}{\pi}. Series with no known closed form stay symbolic under exact evaluate(), per the infinite-domain contract.
• .N() of convergent infinite sums reaches near machine precision. The numeric path Richardson-extrapolates the partial sums instead of returning a plain 10⁴-term truncation: \sum 1/k^2 now numericizes to ~2·10⁻¹⁶ of π²/6 (previously ~10⁻⁴ off), and series without closed forms benefit equally (\sum 1/(k^2+1) to ~2·10⁻¹⁴). Divergent or non-smooth series are detected and fall back to the capped truncation.

Equation Solving

• Trigonometric equations with symbolic coefficients solve correctly. For example, x^2 - 2x\cos t + 1 = 0 solved for t now returns \pm\arccos\left(\frac{x^2+1}{2x}\right).

Assumptions

• Transitive closure over assumed inequality chains. Assumptions now chain: a \ge b, b \ge c, c \ge d entails a \ge d, strictness propagates (p > q > r entails p > r and p \ne r), and an antisymmetric cycle collapses to equality — Wester 21's x \ge y, y \ge z, z \ge x now proves x = z is True. A chain without a back-edge deliberately does not prove equality.
• Even-power monotonicity on ordered positives. Wester 22's x > y, y > 0 \vdash 2x^2 > 2y^2 now evaluates to True (a difference of equally-scaled squares factors as k(x-y)(x+y) with both factor signs settled from the assumptions). x > y alone deliberately does not conclude x^2 > y^2, and solve()'s conservative root-filtering behavior is unchanged.

Simplification and Exact Arithmetic

• The Fu strategy reduces same-power sin/cos differences. simplify({ strategy: 'fu' }) now rewrites \sin^4 x - \cos^4 x to -\cos 2x (and the mirrored/2nd-power forms): a difference of squares whose Pythagorean sum factor is 1, which the exponent-2-only TR5/TR6/TR7 transforms could not reach. Verified numerically; the default simplify() path is deliberately unchanged (pinned by test).
• Exact modulus of complex expressions with radical parts. Abs of a constant a + b\,i with radical/rational parts computes the exact \sqrt{a^2 + b^2} when it genuinely folds: Kahan's \left|3-\sqrt{7}+i\sqrt{6\sqrt{7}-15}\right| simplifies to exactly 1 (its .N() alone carries a 1.0000000000000000315 float residue), |5-12i| = 13, |2+\sqrt{5}\,i| = 3, |1+2i| = \sqrt{5}. A split whose "imaginary part" is itself imaginary (a negative radicand) is rejected by a numeric cross-check, and symbolic |x+iy| never folds.
• Matrices differentiate elementwise. D over a vector/matrix List literal maps over the elements (recursively for nested lists) instead of producing a nonsensical scalar chain-rule expansion: the second derivative of the rotation matrix [[\cos t, \sin t], [-\sin t, \cos t]] is -M, as it should be. Derivative shares the fix.
• Together combines fractions correctly. It now uses a common denominator instead of adding numerators and denominators independently: \frac{a}{b} + \frac{c}{d} \to \frac{ad + bc}{bd}, 1 + \frac{1}{k} \to \frac{k+1}{k}, reusing the denominator when terms already share it.

Linear Algebra

• Exact null spaces, ranks, and eigenvectors. The exact bigint-fraction elimination introduced for RowReduce in 0.73.0 now backs Kernel (null-space basis vectors come out as exact rationals: [[2,3],[0,0]] → basis [-3/2, 1]), MatrixRank (rank = exact pivot count, with no float-tolerance ambiguity), and eigenvector computation (when the matrix and the eigenvalue are exact rationals, A - \lambda I is solved exactly — the eigenvectors of [[4,1],[2,3]] are the exact [1, 1] and [-1/2, 1]). Inexact or symbolic entries fall back to the numeric path unchanged.
• M · M^{-1} simplifies to the identity for symbolic matrices. Two fixes combine: simplify() now recurses into List elements (matrix entries were previously unreachable by any simplify rule), and a new rule combines a sum of fractions sharing an identical denominator into a single fraction so the diagonal entries \frac{a^2 b}{a^2 b - b} + \frac{-b}{a^2 b - b} cancel to 1.
• Symbolic matrix rank via the determinant. MatrixRank of a small symbolic matrix now concludes when the simplified determinant settles the question: the trigonometric matrix [[\sin 2t, \cos 2t], [2\sin t\cos t, \cos^2 t - \sin^2 t]] has rank 1 (its determinant vanishes under TrigReduce). Indeterminate cases stay symbolic, as before.
• Vandermonde determinants return the difference product. The determinant of a symbolic Vandermonde matrix (either orientation) is produced directly in its factored closed form \prod_{i<j}(x_j - x_i) instead of an unfactored expansion.
• The numeric eigensolver converges on hard spectra. The QR iteration was rebuilt as Householder reduction to Hessenberg form followed by the Francis double-shift algorithm with deflation. The classic 8×8 Rosser stress matrix — double eigenvalue 1000, a ±10\sqrt{10405} pair, and a tiny eigenvalue ≈0.098 — now yields the true spectrum (the unshifted iteration returned wrong values), and non-symmetric matrices get proper complex-conjugate eigenvalue pairs ([[0,-1],[1,0]] → \{i, -i\}).
• MatrixPower(M, 1/2) — principal matrix square root. Half-integer powers of an exact 2×2 positive-semidefinite matrix evaluate exactly via the closed form \sqrt{M} = (M + \sqrt{\det M}\,I)/\sqrt{\operatorname{tr} M + 2\sqrt{\det M}}: MatrixPower([[10,7],[7,17]], 1/2) → [[3,1],[1,4]], and 3/2, -1/2 etc. compose with the integer path.
• New operator: SingularValues — the singular values of a matrix, descending, zeros included; exact when the Gram matrix is at most 2×2 with rational entries (SingularValues([[1,1],[2,2],[3,3]]) → \{2\sqrt{7}, 0\}), numeric via the SVD machinery otherwise. (Across this release's Wester rounds the wester.test.ts skip ledger drops from 21 to 3 — the remaining three are the radical-denesting tail.)

Core

• String(…) joins values, not serialized forms. A string operand's quotes leaked into the result: String("x = ", 3) evaluated to a string whose content was "x = "3. It now evaluates to x = 3. This also fixes Cortex string interpolation, which lowers to String — the documentation's headline example "\(x) has type \(Type(x))" now produces "2047 has type integer".
• Type reports the type of symbols and expressions. The Type operator holds its operand unevaluated, but an unevaluated operand is not canonical and a non-canonical expression has no type — so Type(y) returned "unknown" even for a symbol bound to an integer, and Type(1 + x) returned "unknown" instead of "number". The operand is now canonicalized (still not evaluated) before its type is read.

Cortex Language (Experimental)

• Runtime problems in non-final statements are no longer silent. Only the last statement's value is returned from executeCortex, so an error value produced by an earlier statement used to vanish — an unsupported indexed assignment (xs[2] = 9) or a mid-program const reassignment went completely unreported. Each non-final statement that evaluates to an error value now emits a runtime-error diagnostic carrying the statement's source range; the final statement's errors stay in value, per the errors-are-values contract.
• Verbatim symbols are truly literal. The content of a backtick-quoted symbol (`while`) receives no escape processing and must be a valid MathJSON symbol name — the verbatim form exists to name reserved words. Previously, string escape sequences were applied inside the backticks (`\sin` silently cooked \s into a space) even though no valid symbol name contains an escapable character, so every such escape could only produce an invalid name.
• New “Examples” documentation page. Eighteen complete Cortex programs — iteration and accumulation, recursion, numeric methods, exact and symbolic computation, collections — from FizzBuzz-as-a-Map to Newton's method on exact rationals, the Basel problem against \pi^2/6, and a golden-ratio continued fraction checked against a $…$ LaTeX island. Every program on the page is verified by an executable test suite.

Collections

• Symbolic-bound Range and Linspace stay inert instead of collapsing. A symbolic bound was silently coerced to 1, so Range(1, n) behaved as the one-element range [1] everywhere: Count(Range(1, n)) evaluated to 1, Sum(Range(1, n)) to 1, Range(1, n) = Range(1, m) to True, and materialization produced the literal [1]. All of these now stay symbolic/indeterminate, across the scalar accessors (Count, At, equality, SubsetOf, element sign), iteration, materialization, and the extrema (Supremum/Infimum/Min/Max). Likewise for Linspace: a symbolic point count is indeterminate (only a missing count selects the default of 50), and symbolic endpoints no longer materialize as NaN literals or fold Sum(Linspace(a, 1, 3)) to 0 — a collection that reports a size but cannot compute its elements now keeps its lazy form rather than fold to the reduction's initial value. Concrete bounds are unaffected.
• Min/Max/Supremum/Infimum keep unenumerable collections symbolic. The extrema used to iterate any collection operand: an infinite one (a Map over a continuous Interval) ground through the interval's dense sampler until the evaluation deadline, and one that reports elements it cannot compute (a Map over a Linspace with a symbolic endpoint) silently vanished from the result — Min(Map(...), 5) returned 5 even though the mapped values could be smaller. Both now stay in the symbolic result. A genuinely empty lazy collection (a Filter with no matches) still folds away, and finite collections fold as before.

Compilation

• New iterationBudget compile option. expr.compile({ iterationBudget: 1e6 }) caps the trip count of emitted Sum/Product loops: a loop whose iteration count would exceed the budget — including an infinite bound, which previously compiled to a loop that never terminated — evaluates to NaN instead of running. Compilation without the option is unchanged (unbounded loops, zero overhead); the engine's numeric limit probes use it internally to stay interruptible.
• The interval-js target compiles every operand of n-ary nodes. Chained relations (1<x<4) compiled to only their first binary comparison, and n-ary And/Or dropped every operand past the first pair — for Or this was unsound in the exclusion direction (an interval admitted only by a dropped branch reported a definitive "false", so a mask-driven consumer would wrongly cull it). Chains now emit the tri-state conjunction of all pairwise comparisons, and And/Or fold all operands; the javascript/glsl targets were always correct.
• The javascript target fails closed on scalar arithmetic over a list-valued operand. L + x with a list-valued L previously compiled with success: true to JS array coercion (returning a string). It now reports success: false with an explanatory error, and the interpretation fallback returns the correct broadcast list. Supported list compilation — broadcast (\sin([x, 2x])), literals, ranges, GPU vectors, custom vector operators — is unchanged.
• Seeded, reproducible randomness: ce.randomSeed. Assigning a number or string seed makes Random()/Random(n) (and Shuffle, Sample) draw from a per-engine deterministic PRNG stream; re-assigning the same seed resets the stream so identical evaluation sequences reproduce, and null (the default) restores non-deterministic behavior. With a seed set at compile time, each Random node in a javascript-target compilation bakes to a constant derived from the seed and the node's position — a compiled plot function returns the same value at the same call site on every invocation (one draw per compilation), instead of flickering per sample. The explicit per-call Random(seed) overload is unchanged.
• GLSL masked branches emit an overridable _gpu_nan() helper. The else-branch of a compiled When/Which/If was a bare 0.0 / 0.0, whose NaN semantics GLSL ES 1.00 leaves implementation-defined. The literal now lives in a single selective-preamble helper that ES 3.00 hosts can replace with intBitsToFloat(0x7FC00000) for a guaranteed bit pattern.

Parsing

• Bare-command function names \abs, \floor, \mod, \sign parse as function calls. \abs\left(x\right) → Abs(x), \floor(x) → Floor(x), \mod(a, b) → Mod(a, b), \sign(x) → Sign(x) — common informal shorthand (and Desmos output) that previously errored with unexpected-command. The infix a \mod b (synonym of \bmod) is unchanged. Also, \operatorname{sign} now aliases to Sign like sgn (it previously parsed silently as a free symbol sign multiplied by the argument).
• A dot-number after a closing group multiplies. \left(1-t\right).9\left(2\right) and t^{i}.4 parse the .9/.4 as a decimal literal juxtaposed with the preceding operand (implicit multiplication), instead of erroring with unexpected-operator. Degenerate dot sequences after a number (1.2.3) still error, and member access (v.x), ranges (1..2), and trailing-dot numbers ((1., 2)) are unaffected.
• \frac{d}{X} is a division unless the denominator is a differential. Leibniz-derivative parsing now requires an actual d-marker in the denominator (\frac{d}{dx}, \frac{dy}{dx}, \frac{d^2}{dx^2}…). A bare-d numerator over a plain denominator — \frac{d}{L} where d is an ordinary variable, common in pedagogy graphs — previously parsed to a malformed derivative D(missing, L); it is now Divide(d, L).
• A matrix environment parses as a function argument. \operatorname{Trace}\left(\begin{pmatrix}1&2\\3&4\end{pmatrix}\right) — and any library or user-declared function called on a pmatrix-family environment, with or without \left/\right — parsed the argument as a missing-argument error, so Trace, Eigenvalues, Eigenvectors, etc. appeared broken from LaTeX while working from MathJSON. The matrix (alone or among other arguments) now parses, evaluates, and round-trips.

API

• ce.operatorInfo() reports computability. The returned record now carries canEvaluate: boolean — true when the operator's definition has an evaluation rule, false for a registered-but-inert head that only parses/serializes (e.g. To, Tilde). Together with an undefined return (no operator definition), integrators can gate free-form input on "can this actually compute" instead of hand-maintaining allowlists. Note: heads that reduce via canonicalization to another operator (Exp → Power, Greater → Less) report false; query the canonical form.

Special Functions

• New operators: SinhIntegral and CoshIntegral — the hyperbolic sine and cosine integrals Shi and Chi, with numeric evaluation for real and complex arguments (Shi(2) ≈ 2.50157, Chi(2) ≈ 2.45267; validated against mpmath) and derivatives (\frac{d}{dx}\operatorname{Shi}(x) = \frac{\sinh x}{x}, \frac{d}{dx}\operatorname{Chi}(x) = \frac{\cosh x}{x}). Exact arguments stay symbolic under evaluate(); .N() owns the numeric path.
• Erf and Erfi evaluate for complex arguments. Both error functions now have full complex-plane numeric kernels (\operatorname{erf}(1+i) ≈ 1.31615 + 0.19045i, validated against mpmath), instead of evaluating only on the real line.
• Subscripted special-function notation parses. \operatorname{W}_{-1}(x) now parses to the two-argument ["LambertW", x, -1] (branch last), and \operatorname{J}_{n}(x) / \operatorname{Y} / \operatorname{I} / \operatorname{K} parse to BesselJ(n, x) et al. (order first) — these forms previously serialized but did not parse back, so LaTeX round-trips of non-principal Lambert branches and indexed Bessel functions now close.
• The two-argument LambertW(z, k) differentiates. Every fixed branch satisfies the same functional equation, so d/dz W(z,k) = W(z,k)/(z·(1+W(z,k))) now carries the branch through (chain rule included); the derivative with respect to the discrete branch index stays inert. Verified against central differences on both real branches.
• Fungrim identities: W₋₁(x·ln x) → ln x fires. The upstream entry a172c7 published an empty assumption interval (OpenClosedInterval(0, −1/e)); the corrected band x ∈ (0, 1/e] was fixed in the corpus fork (submitted upstream), and the recompiled identities artifact now carries the rule: with loadIdentities(ce) and assume(0 < x ≤ 1/4), simplify(W(x·ln x, −1)) returns ln x.
• Fungrim identities: the polygamma family is live (+28 rules, artifact 1,442). The corpus' 2-argument DigammaFunction(z, m) (the order-m polygamma) now translates to CE's native PolyGamma(m, z) instead of a compat-shadowed 2-arg Digamma, so 28 previously skipped identities and special values compile and fire: simplify(PolyGamma(1, 1)) → π²/6, PolyGamma(1, 1/4) → π² + 8·Catalan, PolyGamma(1, 1/2) → π²/2, the digamma/polygamma recurrence and reflection identities, and more.
• Fungrim identities: set-builder comprehensions get a real encoding (+8 rules, artifact 1,450). Corpus formulas of the shape {f(x) : x \in S, P(x)} used to translate to a literal Set that CE read as a two-element enumeration — producing wrong scalars where one was consulted (Count of a set-builder returned its operand count). They now translate to the faithful Map(Filter(S, P), f) form, which both fixed the miscounts and recovered nine identities whose match side had been untranslatable — notably the prime-counting definition, so with loadIdentities(ce), simplify rewrites Count(\{p \in \mathrm{Primes} : p \le x\}) to \operatorname{PrimePi}(x). Extrema over comprehensions (\min\{f(x) : x \in S\}) get the same encoding. The full 2,551-entry corpus now validates with zero numerically false entries.

0.73.0 2026-07-09

New Operator: Interpret

• Interpret(expr) gives formal meaning to elliptical notation. Evaluating ["Interpret", expr] turns a continuation-bearing sum or product (the inert notational objects produced by the ellipsis fold barrier, see below) into a formal Sum/Product: Interpret(1 + 2 + \dots + n) → \sum_{k=1}^{n} k, Interpret(2 \cdot 4 \cdot \dots \cdot 2n) → \prod_{k=1}^{n} 2k, and Interpret(1 + 2 + \dots + 100) → a Sum that evaluates to 5050. Interpretation is an explicit opt-in — a plain evaluate() never guesses — and the gate is strict by design: at least two exact numeric sample terms in arithmetic progression and a single anchor whose implied upper bound is integral (so 1 + 3 + \dots + 2n, whose even anchor does not belong to the odd progression, stays untouched). Anything the gate cannot prove is returned unchanged.
• Polynomial and geometric patterns are recognized too (v2). Successive finite differences identify polynomial general terms — Interpret(1 + 4 + 9 + 16 + \dots + n^2) → \sum_{k=1}^{n} k^2, cubes and triangular numbers likewise — and a constant exact ratio identifies geometric ones: 1 + 2 + 4 + \dots + 2^n → \sum_{k=1}^{n+1} 2^{k-1}, 2 \cdot 4 \cdot 8 \cdot \dots \cdot 2^n → \prod_{k=1}^{n} 2^k. Numeric anchors resolve to concrete bounds (1 + 4 + 9 + \dots + 100 → a sum to 10 that evaluates to 385). An evidence discipline guards against overfitting: a degree-g polynomial needs its constant difference row witnessed twice, or one fewer sample when the anchor structurally confirms the general term — three samples fit any quadratic, so 1 + 2 + 4 + \dots + m stays untouched.
• Linear recurrences are recognized (v3). An exact-rational Berlekamp–Massey pass finds the minimal constant-coefficient recurrence (order ≥ 2) behind the samples, obtains a verified closed form through RSolve, and resolves numeric anchors by iterating the recurrence exactly: Interpret(1 + 1 + 2 + 3 + 5 + 8 + \dots + 55) → \sum_{k=1}^{10} \operatorname{Fibonacci}(k), which evaluates exactly to 143; Pell-number sums likewise (with a Binet-style body). The same evidence discipline applies — a recurrence of order L needs 2L+1 samples (or 2L with a confirming anchor), so primes and factorials stay untouched. Closed forms are verified against every sample before being trusted.
• Subtraction-spelled ellipses are protected too.

Number Theory

• Modular arithmetic reaches common notation. Mod and Congruent now reduce integer-valued expressions in ℤ/mℤ without materializing the (potentially astronomically large) intermediate value. Modular exponentiation, sums, products, negations and factorial reduction are all handled, so 2^{3^{20}} \pmod{100} evaluates to 52 and 2^{3^{20}} \equiv 52 \pmod{100} evaluates to True, where both used to stay inert. The floored-sign convention of Mod (the result follows the divisor) is preserved on the new path.
• New ModularInverse(a, m) returns the modular multiplicative inverse of a modulo m — the integer x in [0, m) with a·x ≡ 1 (mod m) — and stays symbolic when a and m are not coprime.
• Linear congruences and CRT systems solve. solve on a linear congruence returns the parametric residue family with a fresh integer parameter t ∈ ℤ (6n \equiv 4 \pmod 7 → 7t + 3), an empty result when there is no solution (2x \equiv 1 \pmod 4), and reduces gcd-divisible congruences (4x \equiv 2 \pmod 6 → 3t + 2). A system of simultaneous congruences in one unknown is combined via the Chinese Remainder Theorem — including non-coprime moduli — into a single family (x \equiv 2 \pmod 3, x \equiv 3 \pmod 5, x \equiv 2 \pmod 7 → 105t + 23); an inconsistent system reports no solution.
• Huge exact products stay symbolic instead of overflowing. Multiply now applies the same digit-count budget as Power when an exact power term (base^exp) would be folded into a product's numeric coefficient: if materializing that power would exceed the budget, the factor is kept as a symbolic Power term rather than computed eagerly (2 \cdot 3^{5000000} stays 2 \cdot 3^{5000000} instead of building a multi-million-digit bigint). This also lets Mod/Congruent reduce such products — 2 \cdot 3^{5000000} \pmod 7 evaluates to 4 — without ever materializing the giant intermediate value.

Arithmetic

• New operator: PolyLog — the polylogarithm Liₛ(z). Numeric evaluation for integer order s ≥ 2 over the whole complex plane (validated against mpmath to ≈5·10⁻¹⁵; branch cut z ∈ (1, ∞) with the below-the-cut convention), and exact reductions for the elementary orders and special points: Li₁(z) → −ln(1−z), Li₀(z) → z/(1−z), Li₋₁(z) → z/(1−z)², Liₙ(1) → ζ(n), Liₙ(−1) → (2^{1−n}−1)·ζ(n), Liₛ(0) → 0. Parses and serializes as \operatorname{Li}_s(z) (the unsubscripted \operatorname{Li}, conventionally the offset logarithmic integral, is deliberately not claimed).
• LogIntegral now has its standard notation. \operatorname{li}(x) parses to LogIntegral and serializes back (previously the fallback \mathrm{LogIntegral}(x)).
• Repeating decimals box as exact rationals. A LaTeX repeating-decimal literal — vinculum (0.\overline{3}), dots (0.\overset{.}{1}4285\overset{.}{7}), parenthetical (1.54(2345)), or arc (0.\wideparen{142857}) notation — and the MathJSON {num: "0.(3)"} shorthand now box directly to the exact Rational they represent (0.\overline{3} → ["Rational", 1, 3], 1.(2345) → ["Rational", 12344, 9999]) instead of a truncated decimal float carrying a repeating-decimal marker.
• Norm accepts point-like Tuples. \|(-3, 4)\| now evaluates to 5 instead of leaving the expression inert.
• Double-factorial symbolic reductions. Under simplify(), (2n)!! reduces to 2^n \cdot n! and (2n+1)!! reduces to \frac{(2n+1)!}{2^n \cdot n!} when n is integer-typed.

Equation Solving

On a 40-case univariate solving benchmark derived from SymPy's own test suite (graded by substituting the returned roots back into the equation), this release reaches 38/40 — parity with both SymPy and Mathematica — up from 26/40 for the previous release (base engine, without the opt-in solve templates: 33/40, up from 24). The two remaining cases (Dottie-style transcendental fixed points) are unsolved by SymPy and Mathematica as well. What changed:

• Inverse trigonometric and hyperbolic equations solve exactly. \arcsin x = c, \arccos x = c and \arctan x = c return the exact root (\arcsin x = \frac12 → \sin\frac12), with out-of-range constants correctly rejected (\arctan x = 2 has no solution — 2 is outside arctan's range). \sinh x = c and \tanh x = c return their single root, and \cosh x = c returns both roots \pm\operatorname{arcosh}(c).
• Exponential-symmetric equations are recognized. e^x \pm e^{-x} harmonizes to 2\cosh x / 2\sinh x before solving, so e^x + e^{-x} = 4 returns both roots \pm\operatorname{arcosh}(2).
• Two-absolute-value equations solve. a\,\lvert f(x)\rvert = b\,\lvert g(x)\rvert is squared into a^2 f^2 - b^2 g^2 (candidates are validated against the original equation, so no extraneous roots): \lvert x-1\rvert = \lvert x+3\rvert → -1.
• Rational equations cancel correctly before solving. Clearing denominators no longer expands numerators past their common factor (\frac{2x}{x+2} = 1 → 2), and pure-number denominators are no longer multiplied through at all — rational constants stay where the solve patterns expect them.
• LambertW gains the real lower branch W₋₁. The 2-argument form ["LambertW", z, k] selects the branch (k is 0 or -1; other branches stay symbolic): exact evaluation, machine- and arbitrary-precision numerics on the branch domain [-1/e, 0), compilation, and \operatorname{W}_{-1}(x) LaTeX serialization. W(-\frac1{10}, -1) evaluates symbolically and .N()s to -3.5771520639….
• The opt-in solve templates now cover Lambert-type equations on both real branches. With loadIdentities(ce, { solve: true }) (from the identities bundle), equations reducible to W solve exactly and return every real root: x e^x = -\frac1{10} → \{\operatorname{W}(-\frac1{10}), \operatorname{W}_{-1}(-\frac1{10})\}, e^x - x - 2 = 0 → \{-2 - \operatorname{W}(-e^{-2}), -2 - \operatorname{W}_{-1}(-e^{-2})\}, and mixed linear-exponential forms like x + 2^x = 0 → -\operatorname{W}(\ln 2)/\ln 2. Exact rational, float, and integer right-hand sides are all handled. The identities library also simplifies W(x e^x, -1) \to x under assume(x \le -1).

Integration (opt-in Rubi rules)

New rule coverage in the integration-rules bundle (loadIntegrationRules):

• Polynomial × csc²/sec² integrates by parts. \int x\csc^2 x\,dx → -x\cot x + \ln \sin x, and likewise for P(x)\sec^2(ax+b) with any polynomial P (the recursion reduces the polynomial degree).
• Rational × sin/cos of a linear argument reduces to Si/Ci. \int \frac{\sin x}{x+1}\,dx returns the exact \sin(-1)\operatorname{Ci}(x+1) + \cos(-1)\operatorname{Si}(x+1) form via partial fractions over linear factors.
• Secant-family binomials route through the dedicated secant rules. Integrands like \frac{1}{1+\sec x} now resolve (x - \frac{\tan x}{\sec x + 1}) instead of returning unevaluated.
• Cotangent integrands reflect onto the tangent rules, closing forms like \int \cot^3 x\,dx → -\frac{\cot^2 x}{2} - \ln \sin x.

Simplification and Exact Arithmetic (Wester round 1)

• Rational radicands extract perfect-power factors. (1029/1000)^{1/3} now canonicalizes to \frac{7}{10}\sqrt[3]{3} (numerator and denominator factored independently), extending the existing integer-radicand extraction. Also fixed an exactness leak where a higher root of an exact literal could evaluate to a float times Root(1, n) (e.g. Wester 28's 2^{1/3} expressions now stay all-exact under evaluate()).
• Pythagorean factoring in simplify(). \cos^3 x + \cos x\sin^2 x - \cos x now simplifies to 0: a sum with a shared factor times \cos^2 u and \sin^2 u combines (g\cos^2 u + g\sin^2 u \to g), generalizing the bare \sin^2 x + \cos^2 x \to 1 case.
• Rational-function cancellation fires in simplify() (Wester 14): \frac{x^2-4}{x^2+4x+4} simplifies to \frac{x-2}{x+2}. The cancellation machinery existed but its result was destroyed by a subsequent expand-over-sum-denominator rewrite in the same pass; that split is now suppressed (it never reduces complexity).
• Binomial(n, k) and Pochhammer(a, k) expand for small literal k with a symbolic first argument: Binomial(n, 3) evaluates to \frac{n(n-1)(n-2)}{6}, Pochhammer(a, 3) to a(a+1)(a+2) (k ≤ 20). Pochhammer is a newly registered operator (it previously had no definition and was fully inert).
• Six Wester CAS-review tests unskipped in wester.test.ts (the skip ledger drops from 27 to 21).

Linear Algebra

• RowReduce is exact on exact input. Reduction of an integer or rational matrix now uses exact bigint-fraction elimination — the RREF of an integer matrix has exact -1/3 pivots instead of -0.999…/2.999… float artifacts. Float matrices use the numeric path unchanged. (NullSpace/MatrixRank's float elimination is tracked in the ROADMAP for the same treatment.)
• Products of declared matrices type correctly. A product with a matrix/vector/list-typed operand now carries the collection type instead of collapsing to a numeric type: with X and Y declared matrix, 2Y, XY, X - Y and 3X + 2Y all type as matrix (previously finite_number, which made \det(XY) fail validation as an incompatible-type error). Trace of a matrix now types as number. All-scalar products are unchanged. Note: undeclared symbols in a matrix-expecting argument (\det(A+2B) with fresh A, B) still infer as numbers — declare matrix/vector symbols for symbolic matrix algebra (see the ROADMAP "Matrix-operator typing" item for the planned inference-ordering fix).

Units

• Compound units cancel in quantity arithmetic. Multiplying or dividing quantities now cancels units structurally instead of accumulating them: 18 \text{ in} / (12 \text{ in/ft}) evaluates to 1.5 \text{ ft} (previously the inscrutable 1.5 \text{ in/in/ft}). A repeated unit symbol cancels exactly — no conversion factors are introduced — while different units of the same dimension on opposite sides of a fraction bar are converted and folded into the magnitude: \frac{10 \text{ m} \cdot 1 \text{ s}}{5 \text{ in}} → 78.74 \text{ s}. Products of same-dimension units are left as written (2 \text{ in} \cdot 3 \text{ ft} stays 6 \text{ in} \cdot \text{ft}), and simplification to named derived SI units still applies afterwards (2 \text{ N} \cdot 3 \text{ m} → 6 \text{ J}). Works with measurement (uncertainty-carrying) magnitudes as well.
• New units: yd, qt, pt, cup, wk. Yards, quarts, pints, cups (US liquid convention, consistent with the existing US gal) and weeks join the unit registry, with their English word aliases (5 \text{ yards} → 5 \text{ yd}), and convert exactly: 1 \text{ gal} / 1 \text{ qt} evaluates to 4.
• Currency: dollars and cents. A new currency dimension backs the USD and cent units (18 \text{ dollars} → 18 \text{ USD}, 1 \text{ dollar} + 50 \text{ cents} → 1.5 \text{ USD}), and currency participates in unit cancellation (\$6 / (\$2/\text{lb}) → 3 \text{ lb}). Other currencies are deliberately not modeled: exchange rates are not fixed constants, so cross-currency expressions stay inert rather than silently wrong.
• Spaced unit phrases parse. Multi-word unit text such as 60 \text{ miles per hour} now parses to 60 \text{ mi/h} — spaces inside \text{...} unit annotations are preserved and per reads as division — where previously the words ran together and the unit was not recognized.

New Notations

• Base-subscript numerals compute. A numeral with an integer-literal subscript base, e.g. 10111_2 or 2748_{16}, now parses to the numeric BaseForm(value, base) head (10111_2 → ["BaseForm", 23, 2]), so arithmetic on based numerals works: 1011_2 \cdot 101_2 evaluates to 55, and 11_8 - 3_8 = 6_8 evaluates to True. The guard is strict — every digit must be valid for the base (19_2 stays an inert Subscript), subscripted symbols (x_2) are unchanged, and values larger than 2⁵³ stay exact. The BaseForm LaTeX serializer was also fixed (it emitted an unbalanced parenthesis) and now round-trips: BaseForm(23, 2) serializes as 10111_{2}. A numeral with a symbol subscript base, e.g. 161_b or 161_{b}, now parses to BaseForm of the digit polynomial in that base (161_b → ["BaseForm", ["Add", ["Power", "b", 2], ["Multiply", 6, "b"], 1], "b"], i.e. b² + 6b + 1), so arithmetic works symbolically (161_b + 134_b evaluates to 2b² + 9b + 5) and the numeral round-trips back to 161_{b}. Base equations solve: 161_b + 134_b = 315_b reduces to b² − 8b = 0 and solves to b = 8 (and b = 0).
• Sequence-braces notation. \{a_n\}_{n=1}^{\infty} now parses to the new inert IndexedSequence(term, index, lower, upper) head instead of an incompatible-type error. The term uses the operator-call form (["a_", "n"]) so the index binding survives; _{n\in\mathbb{N}} subscripts map to the set's least element as the lower bound; the expression is inert under evaluate() and simplify() and round-trips through LaTeX. Bare \{a_n\} remains a Set, and the parenthesized form (a_n)_{n\in\mathbb{N}} is unchanged.

Ellipsis Expressions

• Sums and products no longer fold numeric terms across an ellipsis. An Add or Multiply containing an ellipsis (\dots, the ContinuationPlaceholder symbol) is a notational pattern, not an arithmetic one: it now keeps its operands in source order with their structure intact, and is returned unchanged by evaluate(), N() and simplify(). Previously 1 + 2 + \dots + n canonicalized to n + 3 + \ldots — folding the sample terms and destroying the pattern — and 2 \cdot 4 \cdot \dots \cdot 2n folded to 16 \cdot \ldots \cdot n, tearing the coefficient out of the 2n anchor. Such products also round-trip through LaTeX now (an explicit \times is emitted around the ellipsis instead of juxtaposition).

LaTeX Parsing

Recovery fixes from the Hendrycks-MATH genre sweep (docs/mathnet/), taking that corpus from 97.09% to 97.38% clean parse:

• Ordinal superscripts devolve to the base number: 13^{\text{th}} now parses as 13 (also 1^{\text{st}}, k^\text{th}, \mbox variants). Only an exact ordinal suffix (st/nd/rd/th, case-insensitive) is dropped; other superscripts are unchanged.
• Empty scripts are dropped: x^{} and x_{} now parse as x instead of producing an error.
• {,} thousands separator: the LaTeX thin-separator idiom 1{,}000 now parses as the number 1000. Only between digits, and a configured decimalSeparator: '{,}' (European convention) takes precedence — 3{,}14 still parses as 3.14 in that mode.
• \cancel, \bcancel, \xcancel unwrap to their body, and \cancelto{4}{72} parses to the replacement value 4 — matching the worked-solution usage the notation comes from.
• \not-prefixed relations compose into the negated relation: \not= → NotEqual, \not\in → NotElement, \not\equiv (incl. a trailing \pmod n) → the negated congruence, \not\subset → NotSubset, and relations without a dedicated negated head wrap in Not(…).
• Standalone \pmod{7} now places the modulus as the second argument of Mod (previously the operands were flipped).
• (2n)!! stays symbolic: Factorial2 accepts symbolic arguments (its signature was integer-only and rejected 2n with an incompatible-type error); numeric double factorials are unchanged (8!! = 384).
• Primed variables type-check as arguments: \sin a' now parses to Sin(Prime(a)) instead of a type error — Prime mirrors the type of its base (a primed value is a value; a primed function is a function). Derivative notation (f'(x) → D(f(x), x)) is unchanged.
• Bare N/D devolve to variables in all argument positions: N \equiv 1 \pmod k now parses as a congruence over the variable N (previously the standard-library N operator's function type failed the relation's numeric parameter check; the existing devolution fallback ran only for arithmetic operators). Applied uses (N(2/3)) still call the operator.
• Congruence chains and fragments: 3^{27}\equiv 3^7\pmod{100}\equiv 87\pmod{100} folds into a conjunction of the adjacent congruence steps; a leading \equiv b \pmod n with an elided left-hand side recovers with a missing-operand placeholder.
• Empty subscripts on multi-letter symbols are dropped: \alpha_{} parses as alpha (completing the earlier x_{}/13^{} fix).
• English unit words in \text{…} parse as quantities. 18 \text{ inches} → ["Quantity", 18, "in"]: common measurement words (singular and plural — inches, feet, miles, gallons, pounds, minutes, hours, meters, liters, degrees, …) are normalized to their canonical unit symbols at the parse boundary, including inside compound units (\text{ inches/foot} → in/ft). An exponent outside the text binds to the trailing unit factor: 7.5 \text{ gallons/ft}^3 → Quantity(7.5, gal/ft³) (gallons per cubic foot, not (gal/ft)³). Strictly gated: the whole text must resolve as a unit, so prose like 9\text{ to }80 is untouched. No ton(s) alias (a US short ton is not the metric tonne t — mapping it would be a silent 10% error).

Restriction Braces

• Comma-separated brace conditions combine as a union (Or). x^2\{x\ge0, x\le3\} now parses to ["When", x², ["Or", 0≤x, x≤3]] — each comma element is piecewise shorthand for cond: 1 evaluated first-match, so the expression is defined where any condition holds. (Previously the condition was a Tuple, which is not boolean and could not compile.) Stacked braces (\{c_1\}\{c_2\}) still AND-combine, unchanged.
• Colon groups parse as piecewise value selectors. x\{x>0:1, x<0:-1\} now parses to ["Multiply", "x", ["Which", 0<x, 1, x<0, −1]]: a brace group is a first-class piecewise value ({cond} ≡ {cond: 1}) attached by juxtaposition — i.e. multiplication, the same convention that makes the bare-condition form a restriction. A trailing bare value is the else branch (\{x>0:1, -1\} → …, "True", −1), and a bare condition inside a colon group means cond: 1. (Previously the cond:val pairs were parsed as a When gate for the body — inverted semantics.)
• When now masks correctly on the interval-js compile target. The interval comparisons return the tri-state string 'true' | 'false' | 'maybe' — all truthy — so the previously-emitted JS ternary could never take its masking branch: an input interval entirely outside the restriction returned a normal interval result. When now compiles to a tri-state-aware runtime helper (_IA.restrict): 'false' masks ({kind: 'empty'}), 'true' passes the value through, and 'maybe' — an input straddling the restriction boundary — reports the value range as domain-clipped ({kind: 'partial', domainClipped: 'both'}) so adaptive samplers see a domain edge rather than a clean interval. Scalar javascript and glsl When emission is unchanged.

Pipelines and Held Operands

• Hold operators reduce transformer heads. Solve, Integrate, and Limit hold their expression operand (so an equation is not collapsed to a boolean before solving) — but a held operand whose head is an expression transformer (Simplify, Expand, ExpandAll, Factor, Together, Distribute, TrigExpand) is a computation step and is now reduced before the algorithm runs. x^2+2x+1 \rhd \operatorname{Simplify} \rhd \operatorname{Solve} now returns [-1] (previously []: the solver found no roots in an expression whose operator was Simplify), and \int \operatorname{Simplify}(x^2)\,dx / \lim of a transformer-wrapped body compute instead of staying inert. Only the curated transformer set is reduced — full evaluation would collapse relations and substitute assigned values into the unknown.
• Unknown-inference defers on the pipe topic placeholder. Operators that infer their variable when omitted (Solve, D, Series, the polynomial operators) no longer run that inference on the pipeline topic placeholder _: ce.box(["Solve", "_"]) stays ["Solve", "_"] instead of canonicalizing to ["Solve", "_", "_"], which baked the placeholder into the unknown slot so a prefix pipeline stage (\rhd \operatorname{Solve}) computed Solve(expr, expr) → [0] where the infix spelling returned [-1]. Solve re-infers the unknown when the applied stage evaluates; the two spellings now agree. Piping an equation through the prefix form works too, now that an undecidable Equal survives the lambda's argument pre-evaluation (see "Undecidable Relations Stay Symbolic" below): Apply(\rhd Solve, x^2 = 4) → [2, -2].

Undecidable Relations Stay Symbolic

• An equation with free variables is a condition, not a falsity. Equal and NotEqual with an undecidable comparison now stay inert under evaluate(): x^2 = 4 evaluates to itself instead of False (and x \ne 4 to itself instead of True). This matches the inequality operators — x^2 < 4 already stayed symbolic — and Mathematica's ==. Decidable comparisons are unchanged (2+2=4 → True, 2=3 → False, x=x → True), list/scalar elementwise comparisons are unchanged, and assumption discharge still applies (assume(z > 0) ⇒ z \ne 0 → True). The previous collapse silently ruined stored equations: a notebook cell holding x^2 = 4 evaluated to False at storage time, breaking every answer-referencing Solve pipe downstream.
• If and Which stay unevaluated on an undecided condition. A condition that is boolean-typed but not yet decidable (e.g. x = 4 with a free x) leaves the conditional inert — it may become decidable once the variables are bound — instead of throwing Condition must evaluate to "True" or "False" (or, previously for Equal conditions, silently taking the else branch). Genuinely non-boolean conditions (a number, a misspelled symbol) still throw with the spell-check hint.

Issues Resolved

• toLatex({ digits: <number> }) no longer throws RangeError: The number NaN cannot be converted to a BigInt on a bignum-precision engine. A bare number — not part of the documented DisplayDigits forms, but the exact shape of a mechanical fractionalDigits: n → digits: n migration — is accepted with the deprecated numeric convention (n ≥ 0 = fractional digits, n < 0 = significant digits), and a genuinely invalid shape reports a clear validation error instead of crashing.
• The engine no longer trips its own `digits` and `fractionalDigits` were both specified deprecation warning. The serializer re-entered the public toMathJson() boundary — which always carries both (resolved) options — for any dictionary-typed expression; for a symbol bound to a dictionary value this also recursed without bound (a warning flood followed by a stack overflow). Dictionary values now serialize inside the serializer proper; the warning fires only for genuine caller mistakes, once.
• BoxedDictionary.toMathJson() called without options no longer throws (Cannot read properties of undefined); it resolves the same defaults as every other expression kind.
• .latex on a dictionary-typed symbol with no value no longer overflows the stack; it serializes as the symbol. (.latex on a dictionary value — which crashed in released builds — now returns an empty string: dictionaries have no LaTeX display form yet.)

Benchmarks

Numeric performance (200-digit precision)

Median time per call, in microseconds — lower is better. — means the tool returned no usable result at that precision.

Expression	CE (current)	CE 0.70.0	SymPy	math.js	Mathematica
\pi^2	18	12	281	275	6.2
\sin 1	34	36	353	945	7.1
\cos 1	40	41	351	1,315	11
\ln 2	27	24	598	8,195	5.8
e^{\pi}	20	22	376	8,984	7.5
\zeta(3)	2,715	2,787	494	—	151
\Gamma(\tfrac13)	1,452	1,424	4,843	—	267
\psi(\tfrac13)	1,269	1,241	3,782	—	235

Symbolic capability & performance

Each cell is how many times faster than Mathematica that engine is on the case (Mathematica ÷ engine, so higher is better; Mathematica itself is 1×). — means the engine can't do the case; ✓ means it solves a case Mathematica can't. Compare the CE (current) and CE 0.70.0 columns to see what is new this release (a — under 0.70.0 next to a number under the current build). The CE + R/F column is the current build with the opt-in Rubi integrator + Fungrim identities loaded (loadIntegrationRules / loadIdentities), on the same minified bundle.

Operation	CE (current)	CE + R/F	CE 0.70.0	SymPy	math.js	Mathematica
Antiderivatives
\int\frac{1}{\sqrt x}\,dx	7.4×	3.0×	5.0×	0.6×	—	1×
\int\frac{x}{\sqrt{1-x^2}}\,dx	8.3×	1.2×	7.7×	0.08×	—	1×
\int\frac{1}{x^3+1}\,dx	4.3×	0.7×	3.4×	0.3×	—	1×
\int\frac{\sqrt x}{1+x}\,dx	—	2.0×	—	0.1×	—	1×
\int\frac{x}{(1+x)^{1/3}}\,dx	—	0.9×	—	0.008×	—	1×
\int\frac{x^2}{(1+x)^{1/3}}\,dx	—	1.2×	—	0.006×	—	1×
Derivatives
\tfrac{d}{dx}\sqrt{1-x^2}	0.01×	0.02×	0.02×	0.0009×	0.002×	1×
Simplification
\sqrt{3+2\sqrt2}	29×	23×	27×	—	—	1×
\sqrt6\,x+\sqrt2\,x	71×	36×	54×	2.8×	9.1×	1×
Evaluation
\lim_{x\to0}\tfrac{\sin x}{x}	43×	20×	38×	2.5×	—	1×
\lim_{x\to\infty}(1+\tfrac1x)^x	6.3×	4.2×	6.0×	2.1×	—	1×
\int_1^2\tfrac1x\,dx	5213×	6049×	5202×	82×	—	1×
\int_{-\infty}^{\infty} e^{-x^2}\,dx	332×	106×	279×	2.3×	—	1×
Solving
x^4+x^2-1=0	0.2×	0.2×	0.2×	0.06×	—	1×
x^3-x-1=0	1.2×	1.4×	1.4×	0.04×	—	1×

Across the cases both solve, Compute Engine is a median 6.3× faster than Mathematica (up to 5213×).

_{Measured 2026-07-10 · Compute Engine 0.72.0 @ 2cf87db4 (current build) · published 0.70.0 · SymPy 1.14.0 · math.js 15.2.0 · Mathematica 14.3.0 for Mac OS X ARM · Node v22.13.1. Correctness is verified numerically against an independent mpmath reference, never another tool. Reproduce with npm run build production && ./venv/bin/python3 benchmarks/gen_cases.py && node benchmarks/report.mjs && node benchmarks/report_changelog.mjs.}

0.72.0 2026-07-09

Angular Units

• Compilation targets honor ce.angularUnit. Compiled code from every built-in target (javascript, interval-js, glsl, wgsl, interval-glsl, python) now reproduces the engine's angular-unit semantics instead of always computing in radians: direct trigonometric arguments (Sin…Csc, Haversine) are scaled by the unit→radian factor and inverse-trigonometric results (Arcsin…Arccsc, Arctan2, InverseHaversine) by its reciprocal, for all units (deg, grad, turn). With ce.angularUnit = 'deg', compile('\\sin(x)') emits Math.sin(0.017453… * x) so run({x: 90}) returns 1, matching evaluate() — previously a degree-mode expression evaluated in degrees but compiled (and therefore plotted) as if radians. Radian mode emits the same code as before.

• Hyperbolic functions are now unit-independent. Their argument (and an inverse hyperbolic's result) is a dimensionless real, not an angle, so sinh, cosh, tanh, coth, sech, csch and arsinh…artanh no longer convert under a non-radian angularUnit. Previously in degree mode \sinh(1) evaluated to \sinh(\pi/180) \approx 0.0175 instead of 1.1752.

• Exact inverse-trigonometric values are returned in the current angular unit. In degree mode \arcsin(1) now evaluates to the exact integer 90 (previously the exact radian value \pi/2, disagreeing with .N(), which returned 90). Similarly 100 in grad mode and the exact rational 1/4 in turn mode; radian mode still returns \pi/2.

• Arctan2 honors angularUnit, consistently with Arctan (it previously always returned radians): in degree mode Arctan2(1, 1) evaluates to the exact 45, with the quadrant corrections applied in the current unit (Arctan2(1, -1) → 135).

  Symbolic calculus (D, Integrate) remains radian-based regardless of angularUnit (no \pi/180 chain-rule factor); this is a known limitation.

Step-by-Step Explanations

• explain('D') handles higher-order and mixed partial derivatives. A new order option requests the n-th derivative (ce.parse('x \\sin x').explain('D', { variable: 'x', order: 2 })), and a receiver that is itself a D expression — including mixed partials such as D(f, x, y) — is traced through its whole differentiation sequence. The explanation differentiates one order at a time: each stage replays the textbook rule applications inside the remaining derivative operators, folds to the simplified derivative, then differentiates again.

• explain('solve') traces systems of equations and alternatives. A List or And of equations is traced through the same solvers solve() runs: Gaussian elimination shows one step per eliminated variable and per back-substituted variable (solve.system.eliminate, solve.system.back-substitute, solve.system.parametric), and nonlinear 2×2 systems show the product–sum or solve-and-substitute strategy (solve.system.product-sum, solve.system.solve-for, solve.system.substitute). An Or of univariate equations is solved case by case (solve.case) with the roots merged. The solutions are identical to solve() — the trace is a pure observation channel. Systems of inequalities and mixed systems are not traced and throw a precise error. To support systems, the variable explain option now also accepts an array of unknowns.

Cortex Language (Experimental)

• Cortex ships as a new entry point @cortex-js/compute-engine/cortex. Cortex is a text-syntax programming language for scientific computing whose intermediate representation is MathJSON, evaluated by the Compute Engine. The entry point exports parseCortex() (Cortex text → MathJSON), serializeCortex() (MathJSON → Cortex text), and executeCortex() (parse and evaluate a program against a host-created engine):

  import { ComputeEngine, executeCortex } from '@cortex-js/compute-engine/cortex';
  const ce = new ComputeEngine();
  const { value } = executeCortex(ce, `
    let x = 1/2
    if (x < 1) { x + 1 } else { 0 }
  `);
  // value.toString() === '3/2'

  This is experimental: the syntax and semantics may change between releases.

Pipeline Operator

• A pipeline operator applies the expression on its left to the function on its right. x \rhd f (also x \triangleright f, x \vartriangleright f, x ⊳ f, or the plain-text shortcut x |> f) parses to f(x). A \square topic marker in the right-hand side names the position the piped value fills, so a stage can be a multi-argument call: x^2 = 4 \rhd \operatorname{Solve}(\square, x) is Solve(x^2 = 4, x). Stages chain left to right (4 \rhd \sqrt \rhd \ln is ln(√4)), a bare function command such as \ln, \lb or \sqrt acts as a function reference (12 \rhd \ln is ln(12)), and the prefix form (\rhd f, with no left-hand side) denotes the anonymous unary function _ ↦ f(_).

• The unknown/variable argument of Solve, D, Series and the polynomial operators may now be omitted. It defaults to the input's single free variable, or to x when there are several free variables and one of them is x; with no inferable default the expression stays unevaluated. This enables point-free pipelines such as x^2 = 4 \rhd \operatorname{Solve} or x^2 \rhd \operatorname{D}. Applies to Solve, D, Series, PolynomialDegree, CoefficientList, PolynomialRoots, Discriminant, PolynomialQuotient, PolynomialRemainder, PolynomialGCD, Resultant, Cancel, PartialFraction and Apart (Factor already inferred its variable). For the two-input polynomial operators the default is inferred from both operands together.

LaTeX Parsing

Notation coverage driven by a cross-genre corpus sweep (Hendrycks MATH, 15,546 fragments across all seven subjects including worked solutions; see docs/mathnet/math-genre-sweep.md), which took the measured clean-parse rate from 95.3% to 97.1%:

• Text-styling commands. \textbf, \textit, \emph, \texttt, \textsf, and \textup parse their argument as a text run and produce an Annotated expression with the matching style (\textbf{Sizes} → ["Annotated", "'Sizes'", {dict: {fontWeight: "bold"}}]) that round-trips back to the same LaTeX. \textrm and \mbox parse like \text. \bold, \boldsymbol, and \bm are synonyms of \mathbf (\bold{v} → the symbol v_bold).

• Vector-norm bars. The \| command is now recognized as a norm delimiter everywhere \Vert is: \|\mathbf{a}\|, \left\| b \right\|, and \|a\|^2 all parse to Norm.

• TeX-primitive binomial. The infix {n \choose k} form parses to Binomial(n, k), joining the already supported \binom, \dbinom and \tbinom.

• Bare mod annotations. x \pmod n with no preceding \equiv parses as Mod(x, n) (-811 \pmod{24} → ["Mod", -811, 24]), matching the existing \bmod behavior. Congruence chains followed by an implication now parse correctly: a+1 \equiv 4 \pmod 7 \implies a \equiv 3 \pmod 7 is Implies(Congruent(…), Congruent(…)) (the congruence previously disintegrated when \implies followed the modulus). \equiv now binds at comparison precedence, tighter than \implies (zero snapshot impact).

• Mixed braced/unbraced fraction and binomial arguments. \frac1{-1}, \frac{900}7, \binom{n}k, \binom n{k+1} parse correctly. Each argument is now independently a group or a single token, per TeX semantics; previously both arguments were forced into the style of the first, and the mixed forms produced a missing error.

Issues Resolved

• Reading .latex (or .toString()) on the canonical, unevaluated form of a scalar×tuple product — ce.parse('3(1,2)').latex, ce.box(['Multiply', 2, ['Tuple', 1, 2]]).latex — no longer throws RangeError: Maximum call stack size exceeded. The pretty-JSON Multiply serializer round-trips through Product.asRationalExpression(), and the tuple-aware branch of canonicalDivide returned an inert Divide(expr, 1) instead of stripping the trivial divisor, sending the serializer into infinite recursion. Trivial /1 and /-1 divisors of tuple-typed expressions are now reduced.

• Juxtaposing a scalar with a tuple-typed symbol now means scaling, not tuple construction: with z declared tuple<number, number>, 3z parses to ["Multiply", 3, "z"] (previously a spurious ["Tuple", 3, "z"]). Literal tuples (3(1,2)) were already handled; heterogeneous tuples such as tuple<string, number> still group as a Tuple.

• Compiled broadcasts over a list operand now compute their values. The generated .map() callback read its element variable from the vars object instead of the callback parameter, so a compiled \sin([x, 2x]) returned [null, null] for every input. Compiled broadcast results now agree with evaluate().

• \operatorname{csch}(x) now parses to the Csch function (previously a free symbol named csch, silently turning the expression into an implicit multiplication), joining the existing \csch command and matching \operatorname{sech}.

• Constructing many ComputeEngine instances in a synchronous loop no longer balloons memory (~430 KB pinned per engine until the task yielded to the event loop, enough to exhaust the default V8 heap after a few thousand engines). Every constant definition subscribed to configuration changes through a new WeakRef(...), and the ECMAScript kept-objects rule pins each WeakRef target until the next microtask checkpoint. The tracker now holds its listeners directly; since it is owned by the engine, the engine and its listeners form a self-contained cycle that is garbage-collected as a unit.

• A bare \ln or \log — with no argument, as in the pipeline 12 \triangleright \ln — now parses to the function symbol ("Ln", "Log"), consistent with \cos, \lg and \lb. It previously parsed to an empty function application, so piping a value into it produced a missing error instead of applying the function: ce.parse('12 \\triangleright \\ln').evaluate() now returns 2\ln 2 + \ln 3. The bare symbols also serialize back to \ln, \log and \lg (previously \ln()).

• A bare \lb (binary log) now parses to the Lb function symbol, so 12 \triangleright \lb computes \log_2 12. It previously parsed to Log, silently computing the base-10 logarithm instead.

• A log with a base but no argument (\log_2) now parses with the pipeline topic marker \square standing in for the argument: 8 \triangleright \log_2 fills the hole and computes \log_2 8 = 3 (composing with inverse superscripts too: 9 \triangleright \log_3^{-1} gives 3^9), and a standalone \log_2 displays as \log_2(\square). It previously parsed as \log_{10} 2 — the base was read as the argument — so piping into it silently discarded the piped value.

• Likewise, a function with a superscript but no argument (\cos^2, \ln^{-1}, \lg^{-1}) holds a topic-marker hole: x \triangleright \cos^2 computes \cos^2 x, 12 \triangleright \ln^{-1} computes e^{12}, and a standalone \cos^2 displays as \cos(\square)^2. These previously produced a Power of the bare function symbol, which failed to type when piped into.

0.71.0 2026-07-08

Differential Equations

• First-order nonlinear equations solve. (contributed by KingArth0r) DSolve now handles four classical first-order classes:

  • Separable equations return an implicit solution when no explicit form is available: y' = x/y gives \frac12 y(x)^2 = \frac12 x^2 + c_1.
  • Bernoulli equations y' = p(x)\,y + q(x)\,y^n reduce via the v = y^{1-n} substitution and return explicit solutions.
  • Homogeneous equations of the form y' = F(y/x) solve by the v = y/x substitution: y' = 1 + y/x gives y(x)/x = \ln x + c_1.
  • Exact equations M(x,y) + N(x,y)\,y' = 0 return the implicit potential: 2xy + y^2 + (x^2 + 2xy)\,y' = 0 gives x^2\,y(x) + x\,y(x)^2 = c_1.

  Implicit solutions are expressed in terms of y(x) itself. Equations outside the supported classes (e.g. the Riccati equation y' = x + y^2) stay inert.

• Initial and boundary conditions are applied. Scalar conditions can be passed in a list alongside the equation: DSolve([y'' = -y, y(0) = 0, y'(0) = 1], y, x) returns y(x) = \sin x. Derivative conditions are recognized in both the Apply(Derivative(y, 1), x0) and flat D(y(x0), x) forms. Conditions also apply to supported implicit solutions (y' = x/y with y(0) = 1 gives \frac12 y(x)^2 = \frac12 x^2 + \frac12), and free parameters survive: y' = kx/y with y(0) = 2 gives \frac12 y(x)^2 = \frac12 k x^2 + 2 with k untouched. If the conditions cannot be applied to the solution class, the equation stays inert rather than silently dropping them.

• Nonhomogeneous constant-coefficient equations of any order. The undetermined-coefficients method now covers polynomial, exponential, and sinusoidal forcing at any order (previously polynomial forcing was second-order only), including resonant cases, which retry the ansatz with powers of x: y'' - y = e^x gives c_1 e^x + c_2 e^{-x} + \frac12 x e^x, and y''' - y = \sin x and resonant y'' + y = \sin x both solve.

• First-order linear homogeneous systems solve. Pass the equations and dependent functions as lists: DSolve([y' = z, z' = y], [y, z], x) returns the general solution built from the eigen-decomposition of the coefficient matrix. Systems with repeated — or numerically indistinguishable — eigenvalues stay inert rather than returning a degenerate basis.

• NDSolve integrates first-order systems. Fixed-step RK4 now handles systems, including nonlinear ones, with the dependent functions and initial values given as lists: NDSolve([y' = z, z' = -y], [y, z], Limits(x, 0, 1), [0, 1], 200) produces samples as [x, [y, z]] pairs. Malformed or unsupported systems stay inert rather than returning partial results.

Recurrence Equations

• New RSolve operator. (contributed by KingArth0r) RSolve(equation, a, n) solves linear homogeneous constant-coefficient recurrences via the characteristic polynomial: geometric (a_{n+1} = 2a_n gives a(n) = c_1\,2^n), Fibonacci-style, repeated roots with n^k r^n modes (a_{n+2} + a_n = 2a_{n+1} gives a(n) = c_1 + c_2\,n), and complex roots (a_{n+2} = -a_n gives a(n) = c_1\,i^n + c_2\,(-i)^n). Initial conditions can be given in list form: RSolve([a(n+1) = 2a(n), a(0) = 3], a, n) gives a(n) = 3 \cdot 2^n. Nonhomogeneous and variable-coefficient recurrences stay inert.

0.70.0 2026-07-08

Breaking Changes

• The published dist/ directory is reorganized into per-variant subdirectories. The flat layout — where the variant was encoded in each filename (compute-engine.min.esm.js, compute-engine.umd.cjs, …) — is replaced by esm/, esm-min/, umd/, umd-min/, and the unchanged types/. The variant marker moves from the filename into the directory, so a bundle is now <dir>/<name>.<ext>. The general mapping is <name>.esm.js → esm/<name>.js, <name>.min.esm.js → esm-min/<name>.js, <name>.umd.cjs → umd/<name>.cjs, and <name>.min.umd.cjs → umd-min/<name>.cjs; for example dist/compute-engine.min.esm.js is now dist/esm-min/compute-engine.js. Consumers importing via the bare package specifier (@cortex-js/compute-engine and its sub-paths such as @cortex-js/compute-engine/identities) are unaffected — the package exports map absorbs the move. Only deep imports that reach into …/dist/… directly, and pinned CDN URLs, need to be updated. Each esm*/ directory is now fully self-contained, with its own chunks/ subdirectory holding only that variant's shared chunks, so vendoring a build is now "copy the directory for the variant you use."

• The non-minified builds are no longer published to npm. The package now ships dist/esm-min/, dist/umd-min/, and dist/types/ only. The non-minified esm/ and umd/ directories — about 60% of the unpacked package, and never referenced by the exports map — are now build-only artifacts: npm run build still produces them locally for development and debugging, but if you need a readable (non-minified) bundle, build from source.

Improvements

• The declaration build and typecheck now run on TypeScript 7 (the native compiler), cutting .d.ts emission from ~31s to ~5s and the full production build from ~45s to ~29s. TS 7.0 ships no programmatic API, so it is installed side-by-side: the module name typescript stays aliased to the TS 6 API (@typescript/typescript6) for ts-jest, typedoc, typescript-eslint and madge, while the native compiler (@typescript/native) drives the CLI. No consumer-facing change — the published declarations are type-identical; only cosmetic emission differences appear (single-quoted string literals, sorted numeric-literal unions, literal non-ASCII property keys instead of \uXXXX escapes).

0.69.1 2026-07-08

Issues Resolved

• #318 Type declarations now resolve correctly in projects using "module": "nodenext"/"node16". The published .d.ts files used extensionless relative imports, which produced TS2834 errors (or collapsed every imported type to any with skipLibCheck). The build now rewrites the emitted declarations with explicit .js extensions and validates them against a nodenext consumer as part of every release build.

0.69.0 2026-07-08

Breaking Changes

• \pm now parses to a Measurement, not PlusMinus. a \pm b parses to ["Measurement", a, b] — a value with an uncertainty (see below) — replacing the previous PlusMinus head that evaluated to the two-branch tuple (a−b, a+b). Consequences: solution sets that previously used \pm (e.g. quadratic roots) are now returned as an explicit List of the branches, and a numeric integral that reports an error estimate now returns ["Measurement", estimate, error] instead of a PlusMinus tuple. Prefix \pm b parses to ["Measurement", 0, b].

• Loop no longer produces a list — comprehensions moved to the new Comprehension operator. ["Loop", body, ["Element", x, coll], …] is now an imperative for-each evaluated for effect: its value is Nothing (or the value carried by a Break/Return), and it no longer collects the body values into a List. The trailing-for comprehension syntax (x^2 \operatorname{for} x = [1...10]) now parses to ["Comprehension", body, ["Element", …], …], which returns exactly what the collecting Loop used to — for consumers of the parse tree this is a head rename. The undocumented arity-2 form ["Loop", body, collection] (body applied as a lambda to each element) has been removed: use Map, or an Element clause; a non-Element iterator argument is now an error.

• scalar + point is now an error. Adding a scalar to a numeric tuple (1 + (2, 3)) previously broadcast the scalar over the components; points are now proper vectors in ℝⁿ (see below) and a scalar term does not broadcast into them. Add a tuple explicitly ((1,1) + (2,3)) instead. Multiplying or dividing a point by a scalar still scales it.

• Comparing a list to a scalar is now elementwise. [1, 4, 4] = 4 previously evaluated to False (whole-list comparison against a scalar); it now broadcasts and evaluates to ["List", "False", "True", "True"], as do <, <=, >, >=, and !=. Comparing two collections is unchanged: Equal(L, M) remains a whole-value comparison ([1,2,3] = [1,2,3] → True).

Measurements and Uncertainty

• New Measurement type — values with a propagated uncertainty. Measurement(value, error) (written value \pm error) represents a measured quantity carrying a 1σ absolute uncertainty, and the uncertainty propagates through arithmetic using standard independent, first-order (quadrature) error propagation:
  • Algebraic and elementary operations propagate the error: (5 \pm 0.2)(3 \pm 0.1) → 15.00 \pm 0.78, \sqrt{4 \pm 0.2} → 2.000 \pm 0.050, \sin(1 \pm 0.1) → 0.841 \pm 0.054 (trig respects the engine's angular unit).
  • Measurements combine with units: (5.1 \pm 0.2)\,\mathrm{cm} is a measured quantity, and the error carries through quantity arithmetic and unit conversion (UnitConvert of (5.1 \pm 0.2)\,\mathrm{cm} to m → (0.0510 \pm 0.0020)\,\mathrm{m}). The bare form 5.1 \pm 0.2\,\mathrm{cm} (no parentheses) parses to the same thing: a unit on only one operand of \pm scopes over the whole measurement (a dimensionless value with a dimensioned error is never meaningful). An error in a different unit than the value (5.1\,\mathrm{cm} \pm 2\,\mathrm{mm}) stays as written.
  • Display follows the physics convention — the uncertainty is shown to two significant figures by default and the value is rounded to the same decimal place (5.134 \pm 0.021, 8.00 \pm 0.22). Controlled by the digits serialization option ({ significant: n }, { fractional: n }, "max"); .toMathJson() stays lossless.
  • Correctness note: propagation is independent — exact when each measured quantity appears once (A = L·W) or in a single operation (x^2), but it over/under-estimates when one measured variable is reused across an expression (x·x, x/(x+1)), which are treated as independent. See the Units guide for details and the simplify workaround.

Points and Tuples

• Numeric tuples are now points/vectors in ℝⁿ, distinct from lists. Arithmetic on tuples is componentwise vector arithmetic and stays a Tuple: (1,2) + (3,4) → (4,6), 3(1,2) → (3,6), (4,2)/2 → (2,1), -(1,2) → (-1,-2). This fixes (1,2)-(3,4), which previously produced a malformed nested list. tuple · tuple is an error (no implicit dot product — use Dot), and scalar + tuple is rejected (see Breaking Changes). Lists keep their existing broadcast semantics.

• Tuple arithmetic and component access work symbolically for typed symbols. A symbol declared tuple<number, number> participates in vector arithmetic without a value, and its components are accessible with the .x/.y/.z member syntax, which parses to First/Second/etc. (P.x → ["First", "P"]). Component access on a point literal ((1,2).x → 1) also works.

• Color functions broadcast over lists, so rgb and hsv applied to list arguments produce a list of colors, matching the other broadcastable numeric operators.

Lists and Collections

• Filtering a list with a condition in index position. L[L > 0] evaluates to the elements of L where the condition holds — the Desmos list-filtering notation. The condition may reference the list itself (L[L>0]), another list (L[d=4] where d is a list), or compute a positional mask from a Range (L[|[1...\operatorname{length}(L)]-i|>0] removes the i-th element). A condition may be combined with integer indexes. The mask applies positionally and truncates to the shorter of list and mask.

• Relational operators broadcast over lists. [-1, 2, -3] > 0 evaluates to ["List", "False", "True", "False"], typed list<boolean>. Scalar and symbolic comparisons are unchanged (x > 0 stays symbolic). For = and != the elementwise form applies only when exactly one operand is a collection — comparing two collections remains a whole-value equality (see Breaking Changes).

• Broadcast results now report an honest list<…> type. A broadcastable numeric operator applied to a list operand produces a list value, and its declared type now says so: Sin([t, 1]) is typed list<finite_number> (previously the scalar finite_number, contradicting the value), and [1,2] \cdot 2 / [1,2] + x report vector<2> rather than a scalar or a number | vector<2> union. Code that inspects .type before evaluating no longer needs to special-case list-broadcast expressions.

• When broadcasts over a list-valued condition. A domain restriction whose condition is a finite list of booleans now masks element by element — the Desmos restriction semantics. x^2\{[1,2,3] > 0\} evaluates to [x^2, x^2, x^2], and with x = 2, x\{x \le [1,2,3]\} evaluates to [Undefined, 2, 2] (one masked branch per element: the value where the element condition is True, Undefined where False, a held When where the element is still symbolic). When the restricted expression is itself a list, the two are zipped elementwise, truncating to the shorter. Scalar restrictions (x^2\{x > 0\}) are unchanged, and the result type is lifted to list<…> only when the condition's type is a list of booleans.

Parsing and Serialization

• Fixed: bracket indexing after a symbol with \left[ delimiters. A\left[1\right] silently dropped the bracket group and parsed as bare A; it now parses to ["At", "A", 1] like A[1] always did. Indexing also works on parenthesized groups and function applications: (3,4)[1] and f(x)[i] parse to At expressions.

• Numbers with a leading or trailing decimal dot parse correctly. .85x parses as 0.85 x, and a trailing-dot literal inside delimiters ((1., 2)) is accepted.

• Scaling a list/vector by juxtaposition is a Multiply, not a Tuple. A scalar written next to a list- or vector-typed operand — including a scaled fraction whose numerator is a list or range, as in Desmos' 2\frac{[0,...,8]}{8} — now canonicalizes to Multiply (element-wise scaling). Previously such juxtapositions produced a spurious Tuple, which raised an incompatible-type error when the result was used in further arithmetic. Genuine tuples (2(3, 4)) and plain list literals ([1,2,3]) are unaffected.

• Restriction braces attach across visual space. A \{...\} domain-restriction suffix now attaches to its base expression even when separated by spacing commands: s(t) = (1-t)^2(1+2t)\ \{t\ge0\}\{t\le1\} parses to a When with both conditions. The space-tolerance is specific to restriction braces: a space before an indexing bracket (x\ \left[1,2\right]) is still not an index access.

• New inert Polygon operator. \operatorname{polygon}((0,0),(1,0),(0,1)) parses to ["Polygon", ...], an opaque geometric primitive like Triangle and Segment, for consumers that render it.

• histogram, pdf, cdf, length, and nCr parse to Histogram, PDF, CDF, Length, and Choose. The lowercase \operatorname{...} forms used by Desmos are now aliases of the existing operators. The member form .length (S.\operatorname{length}) also maps to Length, joining .count, .max, .min, .total, and the .x/.y/.z component accessors. Histogram and BinCounts accept any number as their bin specification (a non-integer bin count is left unevaluated; translate a Desmos bin width to explicit bin edges at the import boundary).

• New digits serialization option for significant-figures and decimal-place display control. Available on expr.toLatex(), expr.toMathJson(), and honored by expr.toString(), digits controls how many digits of a number are displayed (a formatting choice — it does not change the stored value or computation precision):

  • digits: { significant: n } rounds to n significant figures (ce.parse("\\pi").N().toLatex({ digits: { significant: 3 } }) → 3.14). Rounding is independent of notation (1500 at two significant figures stays 1500 in fixed notation; use notation: "scientific" for 1.5 \cdot 10^{3}), and exact integers, rationals, and radicals are shown in full — only inexact values are rounded.
  • digits: { fractional: n } shows n digits after the decimal point (toFixed semantics), and digits: "auto" / "max" behave as before.
  • The fractionalDigits option is deprecated in favor of digits. It continues to work (a numeric n is equivalent to digits: { fractional: n }); if both are provided, digits wins.

• The pipeline operator |> supports a topic marker and a prefix form. A \square in the right-hand side marks where the piped value is substituted, so the right-hand side may be a multi-argument call: x^2 + 2x + 1 |> \operatorname{Solve}(\square, x) parses to Solve(x^2+2x+1, x). Without a marker the value is passed as the sole argument, as before (x |> f → f(x)). A prefix |> f (or |> \operatorname{Solve}(\square, x)) leaves the left-hand side implied and yields an anonymous unary function over the topic (Function(Apply(f, _), _)), which the caller applies to whatever value it wants to pipe in. \rhd, \triangleright, and ⊳ behave identically.

Runtime and Scoping

• Declare now accepts an optional initial value. The three-operand form ["Declare", symbol, type, value] declares the symbol with the given type, sets its initial value, and evaluates to that value (the previous form evaluated to Nothing). This matches the documented signature; earlier the value operand was silently dropped. The one- and two-operand forms are unchanged. A value-carrying Declare also compiles correctly (the initializer is emitted for the JavaScript and GLSL targets), not just when evaluated.

• Declare can attach definition attributes via a trailing dictionary, including declaring constants. An optional final Dictionary operand carries any of type, value, constant, and holdUntil, mirroring the JavaScript ce.declare(name, def) API. For example, ["Declare", "c", "real", 299792458, ["Dictionary", ["KeyValuePair", "constant", "True"]]] declares an immutable constant (a later Assign to it is rejected), and holdUntil controls when the symbol's value is substituted (as for built-in constants such as Pi). A positional type/value takes precedence over the same key in the dictionary. This gives MathJSON a representation for constant declarations (e.g. the target for a const keyword in a surface language).

Control Flow

• Loop is now imperative control flow only (see Breaking Changes above), and ["Loop", body] is a real infinite loop: the body is evaluated repeatedly until it yields a ["Break", value?] (the loop's value) or a ["Return", …] (propagated), guarded by ce.iterationLimit and the evaluation deadline. Previously this form — documented as while(true) — evaluated the body only once. It compiles to while (true) { … } in JavaScript, and Loop with Element clauses compiles to plain for / for…of statement loops with no result array.

• New Comprehension operator: value-producing list comprehensions. ["Comprehension", body, ["Element", x, xs], …] evaluates body for each combination of one or more Element clauses and collects the results into a List. Independent clauses produce a flat Cartesian product; a later clause's collection may reference an earlier binding ([…, ["Element", "x", ["Range", 1, 3]], ["Element", "y", ["Range", 1, "x"]]] iterates the triangle). Bound names do not leak. With a single clause it is equivalent to Map(xs, x ↦ body); unlike Map (lazy) it materializes its result. Compiles to JavaScript as nested array-collecting loops (not available on the GLSL/WGSL targets, which have no dynamic arrays).

• Break and Continue are now registered operators. Break(value?) exits the enclosing loop immediately and its optional value becomes the loop's value; Continue() skips to the next iteration. Outside a loop both are inert.

• Control flow now propagates out of Block statement results. A Break, Continue, or Return produced by a statement's result — e.g. ["If", cond, ["Break"]] — now short-circuits the enclosing Block and propagates to the enclosing loop or function, as the documentation always specified. Previously only a statement that was literally one of those heads short-circuited, so a conditional Break inside a block was silently discarded and the loop ran to the iteration limit. Consequences: the while-loop lowering ["Loop", ["Block", ["If", cond, ["Break"]], …body]] now terminates correctly, and a Block whose value is a Return evaluates to the ["Return", value] expression itself (unwrapped at the function application boundary), where it previously unwrapped eagerly.

• If without an else branch is fixed. ["If", cond, then] — the documented two-operand form — failed to canonicalize (throwing Cannot read properties of undefined) and was left inert. It now canonicalizes and evaluates to Nothing when the condition is false.

• Nested scopes now see the enclosing block's variables (lexical scoping fix). A Block, If branch, or Loop body nested inside a Block resolved symbols against a stale canonicalization-time scope, so it could not read the values of the enclosing block's locals: ["Block", ["Declare", "k", "integer"], ["Assign", "k", 7], ["Block", "k"]] evaluated to symbolic k instead of 7, a while-style ["Loop", ["Block", ["If", cond, …], …]] threw Condition must evaluate to "True" or "False", and an Element-clause loop whose body is a Block left the loop variable symbolic (Loop(Block(Assign(s, s + n)), Element(n, Range(1, 5))) produced 5n instead of accumulating 15). Nested scopes now resolve enclosing block locals, loop variables, and — inside a function body — the function's parameters and locals correctly, so while/for lowerings with block bodies evaluate as expected.

• Re-evaluating a program with Declare statements no longer throws. Evaluating the same Block expression more than once — or a Declare inside a loop body, which re-executes every iteration — threw The symbol "…" is already declared in this scope on the second entry. A Declare statement now resets the binding it created on a previous run of the same scope. Genuine conflicts (redeclaring a function parameter, or ce.declare() on an explicitly declared symbol) still throw.

Benchmarks

The numeric and symbolic state of this release is summarized below against the last packed comparator release (0.66.0), SymPy, math.js, and Mathematica — the reference baseline, since it is the broadest engine in the field. The tables are generated by the harness in benchmarks/ (node benchmarks/report_changelog.mjs); every result is verified numerically against an independent mpmath reference, never another tool. "CE 0.69.0" is this release.

Numeric performance (200-digit precision)

Median time per call, in microseconds — lower is better. — means the tool returned no usable result at that precision.

Expression	CE 0.69.0	CE 0.66.0	SymPy	math.js	Mathematica
\pi^2	5.9	7.9	176	104	3.9
\sin 1	20	20	222	442	5.2
\cos 1	20	20	224	455	7.1
\ln 2	13	81	339	4,315	3.8
e^{\pi}	12	23	213	4,787	4.0
\zeta(3)	1,542	3,395	268	—	49
\Gamma(\tfrac13)	830	—	354	—	214
\psi(\tfrac13)	725	—	2,810	—	172

Biggest gains over 0.66.0: \ln 2 6.1× faster, \zeta(3) 2.2× faster.

Symbolic capability & performance

Each cell is how many times faster than Mathematica that engine is on the case (Mathematica ÷ engine, so higher is better; Mathematica itself is 1×). — means the engine can't do the case; ✓ means it solves a case Mathematica can't. Compare the CE 0.69.0 and CE 0.66.0 columns to see what is new this release (a — under 0.66.0 next to a number under the current build). The CE + R/F column is the current build with the opt-in Rubi integrator + Fungrim identities loaded (loadIntegrationRules / loadIdentities), on the same minified bundle.

Operation	CE 0.69.0	CE + R/F	CE 0.66.0	SymPy	math.js	Mathematica
Antiderivatives
\int\frac{1}{\sqrt x}\,dx	6.8×	3.1×	7.5×	0.5×	—	1×
\int\frac{x}{\sqrt{1-x^2}}\,dx	11×	1.7×	10.0×	0.08×	—	1×
\int\frac{1}{x^3+1}\,dx	6.3×	0.9×	6.7×	0.3×	—	1×
\int\frac{\sqrt x}{1+x}\,dx	—	2.1×	—	0.1×	—	1×
\int\frac{x}{(1+x)^{1/3}}\,dx	—	1.4×	—	0.01×	—	1×
\int\frac{x^2}{(1+x)^{1/3}}\,dx	—	1.3×	—	0.007×	—	1×
Derivatives
\tfrac{d}{dx}\sqrt{1-x^2}	0.03×	0.03×	0.03×	0.001×	0.004×	1×
Simplification
\sqrt{3+2\sqrt2}	46×	30×	41×	—	—	1×
\sqrt6\,x+\sqrt2\,x	98×	58×	97×	3.1×	19×	1×
Evaluation
\lim_{x\to0}\tfrac{\sin x}{x}	55×	25×	56×	3.1×	—	1×
\lim_{x\to\infty}(1+\tfrac1x)^x	9.7×	6.0×	5.1×	2.1×	—	1×
\int_1^2\tfrac1x\,dx	7429×	7782×	7935×	90×	—	1×
\int_{-\infty}^{\infty} e^{-x^2}\,dx	459×	153×	586×	2.5×	—	1×
Solving
x^4+x^2-1=0	0.3×	0.2×	0.1×	0.06×	—	1×
x^3-x-1=0	1.9×	2.0×	0.2×	0.04×	—	1×

Across the cases both solve, Compute Engine is a median 6.8× faster than Mathematica (up to 7429×).

_{Measured 2026-07-08 · Compute Engine 0.68.0 @ 5a2abce1 (current build) · published 0.66.0 · SymPy 1.14.0 · math.js 15.2.0 · Mathematica 14.3.0 for Mac OS X ARM · Node v22.13.1. Correctness is verified numerically against an independent mpmath reference, never another tool. Reproduce with npm run build production && ./venv/bin/python3 benchmarks/gen_cases.py && node benchmarks/report.mjs && node benchmarks/report_changelog.mjs.}

0.68.0 2026-07-05

Breaking Changes

• The ESM builds are no longer single-file: they load a shared chunk from dist/chunks/. compute-engine.esm.js, compute-engine.min.esm.js and the corresponding integration-rules bundles are now built with code splitting, so the engine core is emitted once into a chunks/chunk-*.js file that both entry points import (this fixes instanceof failures when the integration-rules plugin is loaded alongside the main library, which previously carried its own duplicate copy of the engine). If you copy compute-engine.min.esm.js out of the package as a standalone file — for example to vendor it or serve it from your own static assets — you must now copy the chunks/ directory alongside it, preserving the relative layout. Installing the package from npm, importing it from a bundler, or loading it from a CDN that serves the whole package (jsDelivr, unpkg, esm.sh) is unaffected. The .umd.cjs builds remain self-contained single files if you need a copyable artifact.

  Note that the chunk is required even if you don't use the integration-rules plugin: it contains the shared engine core, not the rule data. To vendor the Compute Engine without the optional rule corpora, copy compute-engine.min.esm.js plus the chunks/ directory and omit integration-rules.* (the Rubi corpus) and identities.* (the Fungrim corpus) — neither is loaded unless you import it explicitly. Each entry point imports exactly one chunk, so if you only ship the minified build you only need one of the two chunk files — the smaller one (the minified chunk), or definitively the one named in the entry file's first import statement. Just remember the names contain a content hash that changes between releases. All other sub-path bundles (core, latex-syntax, numerics, compile, interval, identities) remain self-contained. Or.** AB \parallel CD is the parallelism relation, consistent with \perp → Perpendicular. Use \lor or \vee for disjunction (unchanged).

• \rightarrow now parses to the mapping arrow To, not Implies. f: \mathbb{R} \rightarrow \mathbb{R} now parses as a function signature, matching \to. This reverses the mapping introduced for issue #156: \rightarrow-as-implication was far rarer in practice than \rightarrow-as-mapping. Use \Rightarrow, \implies, or \Longrightarrow for implication (unchanged).

New Operators

• Series, BigO, and Normal provide symbolic series expansion. Series(f, x, x0, n) returns the Taylor expansion of f in x about x0 (default x0 = 0) up to and including the power n (default n = 5), plus an explicit remainder term. x0 may be ±∞ for an asymptotic expansion in powers of 1/x.

  • Series(\sin x, x) → x - \tfrac{x^3}{6} + \tfrac{x^5}{120} + O(x^7); Series(\ln(\cos x), x) → -\tfrac{x^2}{2} - \tfrac{x^4}{12} + O(x^6); Series(\arctan x, x, +\infty) → \tfrac{\pi}{2} - \tfrac{1}{x} + \tfrac{1}{3x^3} - \dots. Coefficients are exact (Series(\sin x, x, \frac{\pi}{6}) gives \tfrac12, \tfrac{\sqrt 3}{2}, …), and an undeclared f yields the textbook form f(0) + f'(0)x + \dots.
  • At a pole the result is a Laurent expansion with a finite principal part: Series(\frac{1}{\sin x}, x) → \tfrac{1}{x} + \tfrac{x}{6} + \tfrac{7x^3}{360} + O(x^7), Series(\cot x, x) → \tfrac{1}{x} - \tfrac{x}{3} - \tfrac{x^3}{45} + \dots, and the special functions expand at their poles with exact coefficients — Series(\Gamma(x), x) → \tfrac{1}{x} - \gamma + (\tfrac{\gamma^2}{2} + \tfrac{\pi^2}{12})x + \dots, Series(\zeta(x), x, 1) → \tfrac{1}{x-1} + \gamma + O(x-1). Poles at ±∞ are handled too (Series(\frac{x^2}{x-1}, x, +\infty) → x + 1 + \tfrac1x + \tfrac1{x^2} + \dots). An essential singularity or branch point (e.g. Series(e^{1/x}, x), Series(\ln x, x)) is still left unevaluated rather than expanded incorrectly.
  • BigO(u) is the inert Landau remainder, serialized O\left(u\right) and parsed from \mathcal{O}(u) and \operatorname{O}(u). It is inert under evaluate/simplify; a numeric approximation (.N()) of any expression containing it is NaN.
  • Normal(expr) strips the BigO terms, yielding the compilable/plottable truncated polynomial: Normal(Series(\sin x, x)) → x - \tfrac{x^3}{6} + \tfrac{x^5}{120}.

• TrigExpand, TrigToExp, and TrigReduce rewrite trigonometric and hyperbolic expressions. These are transformation verbs in the spirit of Expand/Factor and preserve exactness.

  • TrigExpand expands functions of sums and integer multiples of angles: TrigExpand(\sin(a+b)) → \sin a\cos b + \cos a\sin b and TrigExpand(\cos(2x)) → \cos^2 x - \sin^2 x (hyperbolic analogs, and \sec/\csc/\cot as reciprocals of the expanded \cos/\sin, are also handled).
  • TrigToExp rewrites trigonometric and hyperbolic functions in terms of the complex exponential, exactly: TrigToExp(\sin x) → -\tfrac{i}{2}e^{ix} + \tfrac{i}{2}e^{-ix}.
  • TrigReduce is the inverse of TrigExpand, rewriting products and integer powers as functions of multiple angles: TrigReduce(\sin^2 x) → \tfrac{1 - \cos 2x}{2} and TrigReduce(\sin x\cos x) → \tfrac{\sin 2x}{2}.

• Probability distributions: NormalDistribution, BinomialDistribution, PoissonDistribution, UniformDistribution, ExponentialDistribution, consumed by the generic PDF, CDF, and Quantile operators. A distribution is a first-class value — assign it, pass it around, query it:

  • PDF(dist, x), CDF(dist, x) and Quantile(dist, p) evaluate to exact closed forms: CDF(NormalDistribution(0, 1), x) → \tfrac12\left(1 + \operatorname{erf}\tfrac{x}{\sqrt2}\right), an ordinary expression that can be simplified, differentiated, compiled and plotted. Exact arguments give exact results — PDF(BinomialDistribution(4, \tfrac12), 2) → \tfrac38 — and .N() numericizes at machine or arbitrary precision.
  • For discrete distributions PDF is the probability mass function, and Quantile (the least k with \operatorname{CDF}(k) \ge p) is computed by exact search: Quantile(PoissonDistribution(9), 0.95) → 14.
  • Mean, Variance, and StandardDeviation now also accept a distribution: Mean(NormalDistribution(\mu, \sigma)) → \mu, Variance(BinomialDistribution(n, p)) → np(1-p).
  • NormalDistribution(\mu, \sigma) takes the standard deviation (not the variance), and ExponentialDistribution(\lambda) the rate — the Mathematica and scipy conventions.

• GammaRegularized and BetaRegularized — the regularized incomplete gamma and beta functions. GammaRegularized(a, z) is Q(a, z) = \Gamma(a, z)/\Gamma(a) and BetaRegularized(x, a, b) is I_x(a, b). They follow the exactness contract (special values fold — GammaRegularized(1, z) → e^{-z} — and exact arguments stay symbolic), evaluate numerically at machine and arbitrary precision, and compile to JavaScript and Python (scipy.special.gammaincc/betainc). The discrete distribution CDFs evaluate to closed forms in these functions, e.g. CDF(PoissonDistribution(\lambda), k) → \operatorname{GammaRegularized}(k+1, \lambda).

• Covariance, PopulationCovariance and Correlation measure the relationship between two data sets. Each accepts either two equal-length collections or a single collection of (x, y) pairs (a scatter of points). Exact data gives exact results — Covariance([1,2,3,4,5], [2,4,5,4,5]) → \tfrac32 and the Pearson correlation of the same data is \tfrac{\sqrt{15}}{5}, exactly. Covariance uses the sample (n-1) convention, PopulationCovariance the population (n) convention, matching Variance/PopulationVariance. Parse aliases: \operatorname{cov} and \operatorname{corr}. Both compile to JavaScript and Python (np.cov/np.corrcoef).

• LinearRegression and PolynomialFit compute least-squares fits. LinearRegression(xs, ys) (or a collection of points) evaluates to (intercept, slope), and PolynomialFit(data, degree) to the list of coefficients, constant term first. Exact data yields exact coefficients: points lying on 1 + x^2 fit at degree 2 to exactly [1, 0, 1], and rational data produces exact rational coefficients rather than floats. With a trailing variable argument the fitted expression is returned directly, ready to plot: PolynomialFit([(0,1), (1,2), (2,5), (3,10)], 2, x) → x^2 + 1.

• Quantile computes empirical quantiles of data. Quantile(collection, p) interpolates the sorted data so that Quantile(xs, 1/4), Quantile(xs, 1/2) and Quantile(xs, 3/4) agree exactly with Quartiles and Median (Moore–McCabe convention), with general p interpolated through the order statistics in rank space. (Combined with the distribution form above, Quantile covers both the theoretical and the empirical case.)

• Divides and NotDivides express divisibility. a \mid b parses to Divides(a, b) and p \nmid ab to NotDivides(...); both evaluate for concrete integers (Divides(3, 12) → True) and stay symbolic otherwise.

• Geometry notation is transcribed as inert heads. \angle ABC → Angle(A, B, C) (also \varangle, ∠), \triangle ABC → Triangle(A, B, C), \square ABCD → Quadrilateral(A, B, C, D), A \perp B → Perpendicular, AB \parallel CD → Parallel, \widehat{ABC} → Arc, \overparen{BC} → OverParen, and \langle a, b \rangle → AngleBracket. These heads have no evaluation semantics — the Compute Engine does not model geometry — but they parse and serialize faithfully so downstream consumers (e.g. graphical clients) get the structure instead of an error. Angle and arc measures are typed as numbers, so \angle A + \angle B + \angle C = 180^\circ composes in arithmetic.

• \sim parses to the generic similarity relation Tilde. It covers triangle similarity (\triangle ABC \sim \triangle DEF), asymptotic equivalence, and "is distributed as" (X \sim N(0, 1)); \nsim negates it, and \simeq now maps to the existing TildeEqual head (it previously had no LaTeX trigger).

Step-by-Step Explanations

• expr.explain() returns a structured, step-by-step explanation of a simplification — the textbook chain expression → step (with a reason) → … → result. Each step carries the expression state after the step, a stable machine id (the localization key for consumers), and a default English description; explain().result is always the same value simplify() returns. ce.parse('\\frac{x^2-1}{x-1}').explain() yields one step, "Cancel the common factors", ending at x + 1.
  • The step chain is curated by default (driver bookkeeping is filtered out); pass {verbosity: 'all'} for the raw trace (rule authoring, debugging). simplify() options (rules, costFunction, strategy) are honored.
  • The most frequently fired simplification rules ship with curated descriptions ("Apply the Pythagorean identity: sin²x + cos²x = 1", "Combine powers with the same base: xⁿ·xᵐ = xⁿ⁺ᵐ", …); other rules get a readable fallback derived from the rule id. registerStepLabels() lets a host application override or extend the descriptions.
  • expr.explain('solve') traces equation solving. Step values are equations — the state after each phase — so the chain reads like textbook working: 2x+1=5 → Move all terms to one side 2x-4=0 → Isolate the unknown x=2. The trace covers the solver's algorithmic phases (clearing denominators, squaring both sides, substitutions like u = eˣ with back-substitution, zero-product factoring, the quadratic formula, checking candidates and rejecting extraneous roots) and the root-template rules, which now carry stable solve.* ids. explain('solve').result is a List of the same roots solve() returns; the unknown is inferred or passed via options.variable. Systems of equations are not traced yet.
  • expr.explain('D') traces differentiation. Steps are whole-expression states in traversal order — each textbook rule (sum, product, quotient, power, chain, exponential, logarithmic differentiation, table lookups) first appears with its unresolved sub-derivatives as inert D(…) terms, which resolve step by step: D(x·sin x, x) → Apply the product rule x·D(sin x, x) + sin x → Differentiate using a known derivative x·cos x + sin x. The variable is inferred when unambiguous (or passed via options.variable), and the result always matches evaluating D(expr, variable).

Solving

• Solve accepts a domain for the unknown. Solve(x^2-5x+6=0,\; x \in 1..1000) restricts solutions to a collection: the equation is solved symbolically and the roots are filtered to the domain (an integer domain also discards non-integer roots up front). When the symbolic solver finds nothing and the domain is finite and reasonably sized, Solve falls back to enumeration with a compiled predicate, confirming every candidate exactly so float rounding never produces a wrong answer (budgeted, interruptible; an unaffordable search returns the expression unevaluated rather than a partial answer). The predicate is not limited to equations — any boolean condition works: Solve(2^n \equiv 1 \pmod{7},\; n \in 1..20) → [3, 6, 9, 12, 15, 18], and an extra condition can ride on the domain (n \in 1..100, n > 5-style, as in Sum indexing sets). The two-argument form is unchanged.

• Multiple unknowns enumerate over the product of their domains. Solve(x^3+y^3=1729,\; x \in 1..12,\; y \in 1..12) → [(1,12), (9,10), (10,9), (12,1)] — a List of Tuples in unknown order, with the same budget, exact-confirmation, and interruption guarantees as the univariate case.

• Integer equations are solved symbolically (diophantine solving). When every unknown ranges over integers, Solve recognizes linear equations in any number of unknowns and Pell-family equations x^2 - Dy^2 = N (including the elliptic case x^2 + |D|y^2 = N) and solves them in closed form — ported from SymPy's diophantine module and validated against its test suite. Over a bounded domain this reaches answers enumeration cannot: Solve(x^2-29y^2=1,\; x \in 1..10^5,\; y \in 1..10^5) → [(9801, 1820)] via continued fractions, where the 10^{10}-candidate sweep would be refused; an unsolvable equation is decided instantly (Solve(6x+9y=4,\; x \in \pm10^6,\; y \in \pm10^6) → []). With integer-typed unknowns and no domain — previously inert — Solve returns the parametric family: Solve(3n+4m=7, n, m) → [(4t-7,\; -3t+7)] with the fresh parameter t ranging over ℤ, and Pell equations yield their exact closed forms \bigl(\tfrac{(3+2\sqrt2)^t + (3-2\sqrt2)^t}{2}, \dots\bigr), and Pythagorean triples return the complete classical parametrization: Solve(x^2+y^2=z^2, x, y, z) → \bigl(t(t_1^2-t_2^2),\; 2t\,t_1 t_2,\; t(t_1^2+t_2^2)\bigr) and its leg-swap — every integer triple, including all signs, lies in one of the two families. Every concrete solution is exact-confirmed by substitution; half-bounded domains (e.g. n \ge 1 alone) are left unevaluated, and forms whose textbook parametrizations are provably incomplete (weighted coefficients, four or more squares) are declined rather than answered partially.

• Periodic equations expand their root families over a bounded domain. Solve(\sin x = \tfrac12,\; x \in [0, 4\pi]) returns all four exact solutions \tfrac{\pi}{6}, \tfrac{5\pi}{6}, \tfrac{13\pi}{6}, \tfrac{17\pi}{6} — not just the principal values. Scaled arguments work too (\sin 2x = 1 over [0, 2\pi] → \tfrac{\pi}{4}, \tfrac{5\pi}{4}). Expansion applies when the unknown appears only inside trigonometric functions of linear arguments; each family member is verified by exact substitution, and unreasonably large expansions degrade gracefully to the principal roots.

• assume() bounds now filter solutions. After assume(n > 0), Solve(n^2 = 16, n) (and expr.solve("n")) returns [4] instead of [4, -4]; assume(n \in 1..10), inequality, and \ne assumptions are honored the same way, conjunctively with any explicit domain. Roots are dropped only when an assumption definitely excludes them — symbolic roots that cannot be decided are kept.

Parsing Resilience

The parser was hardened against a corpus of ~2,300 math fragments extracted from real olympiad problems (the MathNet dataset); the clean-parse rate on that corpus went from 85% to ~96%, and the one crash it exposed is fixed. See docs/mathnet/ for the corpus, the regression checker, and the work plan.

• Ellipsis in a numeric context no longer throws. (1!)^2 + (2!)^2 + \dots + (2018!)^2 crashed with The type of the constant "ContinuationPlaceholder" cannot be changed (type inference attempted to narrow a constant). Inference is now a no-op on constants.

• \cdots, \dotsb, \dotsc, \dotsm, and Unicode … parse as ellipsis. Previously only \dots/\ldots/... did; (2!+2)(3!+3) \cdots (2019!+3) now parses with the placeholder as an inert operand instead of erroring.

• A trailing sentence period no longer breaks an equation. Input copied from prose often ends in ., ; or , (e.g. ... = z^2.). When — and only when — the parse would otherwise contain an error, the trailing punctuation is dropped and the input re-parsed. Valid input is unaffected: 5. still parses as the decimal 5.

• Congruences parse and evaluate. a \equiv b \pmod{n} (also \bmod, the parenthesized (\bmod n), and the ASCII form n ≡ 1 (mod 3)) parse to Congruent(a, b, n), which evaluates for concrete integers (7 \equiv 1 \pmod{3} → True) and now accepts symbolic moduli (2^n \equiv 1 \pmod{p^{k+1}} stays symbolic instead of erroring).

• Common Unicode math symbols are accepted: ≡ (congruence), ∈, ∉, ∪, ∩, ≈, ∠, and … — useful when input comes from plain-text sources rather than LaTeX.

• Alignment environments parse as systems. \begin{aligned} a^2+ab+c=0 \\ b^2+bc+a=0 \end{aligned} (also align, gather, split, multline, eqnarray and their starred variants) parses to a List of the row expressions — the same convention as \begin{cases}, accepted by solve(). Alignment markers are transparent: x &= y is x = y.

• Qualified number sets parse. \mathbb{R}_{>0} → PositiveNumbers, \mathbb{Z}_{\ge0} → NonNegativeIntegers, \mathbb{N}^* → PositiveIntegers, etc., and they round-trip to canonical LaTeX. A qualification with no named set (\mathbb{N}_{>1}) falls back to a faithful set-builder.

• Structural odds and ends: A \backslash B parses as SetMinus (the common alternative spelling of \setminus); a standalone quantified condition \forall n \ge 1 parses instead of erroring; \underbrace mirrors \overbrace.

• A symbol's inferred type narrows instead of erroring. When a free symbol's type was inferred from one use and a later use requires a more specific type, argument validation now narrows the inference (when sound) instead of producing an incompatible-type error. This fixes (A \setminus B) \cup (B \setminus A) — where B was inferred as a value and then rejected as a set — as well as -n!! (double factorial of an undeclared symbol) and a family of similar mixed-use expressions. Declared types are unaffected: passing a declared string where a set is required is still an error.

Packaging

• The integration-rules plugin shares code with the main library. The ESM builds of compute-engine and the opt-in @cortex-js/compute-engine/integration-rules entry point are now emitted with code splitting: the engine core lives in a shared chunk imported by both, instead of being bundled twice. This shrinks the combined download and fixes cross-bundle instanceof failures when a host mixed objects from the two bundles. The UMD builds remain self-contained single files.

Lenient parsing and string helpers

• The string helpers take a strict option. simplify(), evaluate(), N(), expand(), expandAll(), factor(), solve(), and compile() parse string input in lenient (non-strict) mode by default. Note that lenient mode is not a pure superset of strict LaTeX: unbraced multi-digit scripts change meaning — x^23 is x^{23} (not 3x^2), and x_23/a_12 are single multi-digit subscripts. Pass { strict: true } (e.g. N('x^23', { strict: true })) to restore the strict LaTeX grammar.

• Lenient inverse functions, atan2, and letter runs parse correctly. sin^-1 x now means \arcsin x (the inverse function), not 1/\sin x (matching strict \sin^{-1}); sin^-2 x stays 1/\sin^2 x. atan2(1, 2) parses as Arctan2(1, 2), and acot/asec/acsc are recognized. A multi-letter run with an embedded Greek constant is segmented (2pix → 2\pi x, xpi → x\pi) instead of injecting a spurious imaginary unit, and an implicit subscript is accepted on a constant base (alpha2 → \alpha_2).

Differential Equations

• Repeated roots produce correct general solutions. DSolve now clusters numeric characteristic roots by multiplicity: y'''' + 2y'' + y = 0 gives (c_1 + c_2 x)\cos x + (c_3 + c_4 x)\sin x instead of a degenerate basis with spurious e^{\varepsilon x} factors, and repeated real roots keep their x e^{x} modes. A structural self-check returns the equation unevaluated rather than emit a basis with fewer independent solutions than the order.

• No more corrupted solutions. Equations with variable coefficients on higher-order derivatives (e.g. x^2 y'' + x y' = x) previously returned a "solution" containing an internal Error node; they now stay unevaluated when the class is unsupported. Equations whose right-hand side references the dependent function with a transformed argument (e.g. y'(x) = y(2x)) stay unevaluated instead of returning an unevaluated integral as "solved".

• Exponential forcing terms solve. Variation of parameters was silently disabled for exponential bases (an internal Wronskian stayed unsimplified): y'' - y = e^x now returns c_1 e^x + c_2 e^{-x} + \frac12 x e^x - \frac14 e^x, and y'' + y = e^x returns c_1 \cos x + c_2 \sin x + \frac12 e^x, instead of the equation unevaluated. Solutions are returned in collected form (no e^a \cdot e^b products or A\sin^2 u + A\cos^2 u pairs).

• Parsed LaTeX input works end-to-end. ce.parse("y''(x)+y(x)=0") no longer canonicalizes the derivative of an undeclared function into an Error node: a derivative now reports a numeric result type, so prime/dot-notation equations flow from parse() through DSolve (\dot x + \ddot x expressions are likewise no longer corrupted). The implicit first-order form Apply(Derivative(y), x) is also recognized (order defaults to 1).

Evaluation

• Beta is exact and pole-aware. \mathrm{B}(a, m) with a positive integer argument reduces exactly (\mathrm{B}(2,3) = \frac{1}{12}, \mathrm{B}(-2,2) = \frac12), and arguments at gamma-function poles return \tilde\infty instead of a silently wrong finite value (\mathrm{B}(-1,2) previously returned -2.97\times10^{49}).

• Multiplication by infinity respects sign information. x \cdot \infty stays symbolic when the sign of x is unknown, evaluates to -\infty when x is known negative, and to NaN when x is zero — it no longer collapses to +\infty unconditionally.

• Inverse hyperbolic functions have values at their poles. \operatorname{artanh}(\pm 1) and \operatorname{arcoth}(\pm 1) evaluate to \pm\infty, \operatorname{arsech}(0) to +\infty, and \operatorname{arcsch}(0) to \tilde\infty, with result types that no longer claim a finite value at a pole.

• Sum reports incompatible elements. Summing a collection containing a string returns a typed error instead of a silent NaN.

• Sums and products over an infinite domain stay symbolic under evaluate(). An infinite domain has no exact value by truncation, so \sum_{n=1}^{\infty} \frac{1}{n^2} now evaluates to itself; .N() returns the truncated numeric approximation, as before. Previously evaluate() returned a silently truncated partial sum (off by \sim 10^{-4} for this example) — a float where the exactness contract promises an exact value.

• Sums and products with symbolic bounds no longer evaluate to a number. \sum_{k=1}^{n} k with an unbound n evaluated to 50\,015\,001 — the sum truncated at an internal iteration cap of 10\,001 — under both evaluate() and .N(). It now stays symbolic (simplify() still produces the closed form \tfrac{n^2+n}{2}).

• Expand computes constant powers. Expand((2+3i)^{1000}) returns the exact 557-digit Gaussian integer (matching SymPy's expand()), and Expand(2^{1000}) the exact integer; both previously returned unevaluated. Structural expansion of symbolic powers is unchanged, and powers too large for exact computation still stay symbolic.

• Huge exact complex numbers are finite and print in full. An exact Gaussian integer with components beyond float64 range (e.g. (2+3i)^{1000}) reported isInfinity true, serialized as \tilde\infty in plain text, and had a NaN bignumIm — all artifacts of routing through the machine-float projection. It now types as finite_complex, prints its full digits, and bignumIm is exact.

• Perfect-power radicands reduce. (997^3)^{1/6} = \sqrt{997}, 8^{1/6} = \sqrt2, 8^{1/4} = 2^{3/4}: when canonicalization folds a power into an opaque integer, the root now recovers the structure by perfect-power decomposition. In particular the zero-equivalence test \sqrt{997} - (997^3)^{1/6} evaluates to exact 0 (it previously leaked a float residue).

• Logarithms reduce when the argument and base are powers of a common base. \log_8 32768 = 5, \log_8 2 = \tfrac13, \log_4 8 = \tfrac32 — exactly, honoring the exactness contract (\ln 2 and \log_8 10 stay symbolic).

• Xor cancels repeated operands. a \oplus a = \mathrm{False}, so Xor(x, y, y) evaluates to x; cancellation composes with the existing True/False folding.

Linear Algebra

• **3×3 Eigenvalues

  returned wrong values — fixed.** The analytic solver used a sign-flipped term in its depressed cubic, mirroring every eigenvalue about \operatorname{tr}/3: e.g. [[5,-3,-7],[-2,1,2],[2,-3,-4]] returned \{\tfrac{10}{3}, -\tfrac53, \tfrac13\} instead of \{1, -2, 3\}. (Spectra symmetric about their mean — like \{1,2,3\} — were unaffected, which is how it escaped notice.) Additionally, a complex-conjugate eigenvalue pair was returned as its real part twice (\{2, \pm i\} came back \{2, 0, 0\}); complex eigenvalues are now returned as complex numbers.

Rules and Pattern Matching

• Rule conditions must return a boolean. A condition function returning a non-boolean (e.g. the boxed symbol False, which is a truthy JavaScript object) no longer fires the rule; a one-time console warning identifies the malformed condition. Returning the boxed symbol True is accepted.

• e and i work in string rules. Rules such as 'e^2 -> 7' now match: the constants are resolved to ExponentialE and the imaginary unit when the rule is parsed, instead of remaining inert symbols that could never match.

• Explicit wildcards work in LaTeX match patterns. An object-form rule such as {match: '_a + 1', replace: '_a'} now parses _a/__a as wildcards instead of an implicit product.

• A throwing condition no longer discards subexpression rewrites. If a rule condition throws, the rule is skipped at that node but successful rewrites of the operands are kept.

0.67.0 2026-07-03

This release improves correctness and predictability across the public Compute Engine API: exact complex and integer arithmetic stays exact more often, partial derivatives and assumptions are more capable, LaTeX and lenient parsing round-trip more reliably, compiled output agrees more closely with interpreted evaluation, and arbitrary-precision arithmetic is substantially faster. It also fixes many cases where evaluate(), N(), simplify(), isEqual(), assume(), verify(), serialization, or compilation could return a wrong answer, lose exactness, hang, or silently accept invalid input.

Exact and Numeric Evaluation

• Exact complex arithmetic preserves exact values. Gaussian integer and rational complex values now stay exact through arithmetic: (1+i)^3 evaluates to -2+2i, (1+i)^{-2} to -\frac{i}{2}, \frac{1}{1+i} to \frac{1-i}{2}, \sqrt{3+4i} to 2+i, and \sqrt{-4} to 2i. Exact complex numbers also round-trip through MathJSON as ["Complex", re, im] with exact components.

• Integer powers and large integers stay exact. Integer powers such as 2^{127} now evaluate to exact integers, negative integer powers produce exact rationals such as 2^{-2} = \frac14, and powers of Gaussian integers such as (1+i)^2 evaluate exactly. Very large exact integers are no longer rounded when used by IsPrime, IsOdd, IsEven, FactorInteger, Mod, or DigitSum.

• Exact results are preserved more consistently. evaluate() no longer turns exact arguments into floats in cases such as \sqrt{-2}, \operatorname{Fract}(\frac12), \Re(\frac12), |1+i|, \log_2(\pi), Distance, and statistics functions. For example, \operatorname{Mean}([1,2,3,4]) now returns \frac52, and \operatorname{StandardDeviation}([1,2,3,4]) returns \frac{\sqrt{15}}{3}.

• Special functions are more accurate. PolyGamma, Zeta, BesselI, BesselK, Airy functions, logarithms, roots, trigonometric functions near zeros and poles, LambertW, acos, erfInv, Hypergeometric2F1, Gamma, Beta, Fresnel integrals, and complex elementary functions have improved numeric accuracy, including at high precision.

• Negative logarithms and complex logarithms are consistent. Inexact negative arguments now produce the principal complex value under both evaluate() and N(). Exact negative arguments stay symbolic under evaluate() and produce the principal complex value under N(). Logarithms with a complex argument and explicit base now agree between evaluate() and N().

• Roots and radicals are more reliable. Exact perfect powers such as 64^{1/3} and (27/8)^{1/3} evaluate exactly, Root(64, 3).N() returns exactly 4, odd roots of negative numbers keep the real-root convention, and N(\sqrt{4y}) now returns 2\sqrt{y} instead of dropping the radical.

• Sums and infinite sums behave better. Exact sums such as \sum_{k=1}^{5}\sqrt{k} now remain exact, while sums over infinite index sets such as \sum_{n \in \mathbb{Z}^+}\frac{1}{n^2} evaluate numerically when appropriate, remain symbolic when parameters prevent evaluation, and respect timeLimit.

Differentiation, Integration, and Simplification

• Partial derivatives of multivariate functions now work symbolically. D(f(x, y), x) now represents the partial derivative with respect to the first argument, mixed partials accumulate correctly, and multivariate chain, product, power, and quotient rules compose as expected. For example, D(f(x^2, y), x) returns a symbolic chain-rule result proportional to 2x.

• Partial-derivative notation parses and evaluates. Forms such as \partial_x f(x,y), \frac{\partial}{\partial x} f(x,y), \frac{\partial^2}{\partial x \partial y} f(x,y), and \frac{\partial^2}{\partial x^2} f(x,y) now parse to D and evaluate correctly.

• Derivative notation is more robust. Compact derivatives such as d/dx(f(g(x))) preserve unknown-function chain rules, higher-order derivatives round-trip through LaTeX, and \frac{d}{dx}[\sin x] treats the square brackets as grouping rather than a one-element list.

• Several derivative rules are corrected. Variable-degree radicals such as Root(x, x) differentiate as x^{1/x}, \frac{d}{dx}\operatorname{Mod}(x,5) gives 1 almost everywhere, and D(\operatorname{arcoth}(x), x) returns \frac{1}{1-x^2}.

• Definite integrals no longer return fabricated closed forms. If no closed-form antiderivative is found, evaluate() keeps the definite integral symbolic instead of substituting bounds into the integrand. N() still computes a numeric value.

• Default simplification covers more identities. The sine addition identity \sin(x)\cos(y)+\cos(x)\sin(y)=\sin(x+y) now applies in the default simplify() path. Pythagorean identities such as \sin^2 x+\cos^2 x also simplify inside larger sums.

• Simplification is more exact and branch-aware. Combining powers keeps exact exponents, for example x \cdot x^{\sqrt2} becomes x^{1+\sqrt2}. The simplification of \ln(x^2) now produces 2\ln(|x|) for real x, and identities that require real arguments no longer apply to symbols declared as complex.

• Some unsafe rewrites were removed. simplify() no longer rewrites |\sin x| as \sin|x|, Arctan2 preserves the correct quadrant, rule conditions such as x \ne 0 require proof rather than assuming unknown symbols satisfy them, and alternating-binomial sum simplifications now check their validity bounds.

• Differential equation solvers handle higher-order equations. (contributed by KingArth0r) DSolve now solves linear constant-coefficient homogeneous equations of any order via the characteristic polynomial — distinct real, repeated, and complex roots — for example y''(x) = y(x) → [y(x) = c_1·e^x + c_2·e^{-x}] and y''(x) + y(x) = 0 → [y(x) = c_1·cos(x) + c_2·sin(x)]. Roots are kept exact when the characteristic polynomial factors and fall back to numeric roots otherwise. It also solves second-order constant-coefficient nonhomogeneous equations (undetermined coefficients for polynomial forcing, variation of parameters otherwise) and second-order Cauchy–Euler equations. Integration constants are now named c_1, c_2, … (fresh names are chosen if those are already in use). Correspondingly, NDSolve now solves explicit higher-order initial value problems y⁽ⁿ⁾(x) = f(x, y, y', …, y⁽ⁿ⁻¹⁾) by reducing them to a first-order RK4 system, with the initial condition given as a list [y(x0), y'(x0), …]. Equations outside these classes remain inert.

Parsing and Serialization

• \binom is supported. \binom{n}{k}, \dbinom{n}{k}, and \tbinom{n}{k} parse to Binomial(n, k), and Binomial serializes back to \binom.

• LaTeX parsing accepts more common notation. N(...) and D(...) parse as numeric evaluation and differentiation outside quantifier scopes, superscripts on \log, \ln, \lg, and \exp bind to the applied function, and ==, !=, chained \ne, mixed-direction inequality chains, parenthesized relations, and double negation such as x--y now parse with the expected meaning.

• Lenient parsing is more useful. In lenient mode, digit suffixes parse as subscripts (x2 as x_2), bare known function names apply to the following factor (sin x), log2(8) means \log_2 8, [1,...,10] parses as a range, and \mathbb{Z}^+ parses as PositiveIntegers.

• The public string helpers accept their documented syntax. simplify(), evaluate(), N(), expand(), expandAll(), factor(), solve(), and compile() now parse string input in non-strict mode, so expressions such as sqrt(5), sin(alpha), and x**2 work as documented.

• verify() and assume() accept strings. ce.verify('x > 0') and ce.assume('$x > 0$') parse the predicate and report clear errors for unparseable input.

• MathJSON .json serialization is lossless for more numbers. Exact large integers, 16- and 17-digit values, high-precision complex numbers, exact rational/radical values such as \frac{\sqrt3}{2}, and repeating decimals now round-trip without silently changing value.

• LaTeX round-trips are improved. Repeating decimals serialize with an overline, sequence expressions no longer serialize as ambiguous adjacent numbers, set-builder notation attaches conditions to the comprehension, and toMathJson({exclude: ...}) honors exclusions for number literals.

Assumptions, Types, and Equality

• The numeric type hierarchy is more natural. real is now a subtype of complex, so real-typed symbols satisfy complex-typed signatures and guards. Union types flatten and canonicalize their member order, and type negation now distinguishes never from nothing.

• Complex and non-finite type inference is more precise. Expressions such as \sqrt2 i, i^2, i/2, i^3, e^i, and \ln(-1) infer more accurate types. Non-finite values such as \tan(\frac\pi2), \Gamma(0), \zeta(1), \ln(0), 0 \cdot \infty, and k/0 no longer claim finite types when that is unsound.

• Assumptions can prove more facts. Assumptions over signed integer and real sets refine both type and sign. Inequality bounds now affect equality checks, comparisons between bounded symbols, and verify(). Chained inequalities and equations with multiple roots are recorded correctly, contradictory assumptions are rejected atomically, and forget() clears values introduced by assume('x = 5') while preserving values set with assign().

• Assumptions respect scope. Assumptions made inside a pushed scope no longer leak into parent scopes or continue to affect expression results after popScope().

• Equality and ordering are more coherent. isSame is now an equivalence relation, isEqual returns undefined for indeterminate equality with free variables, equality and ordering share one tolerance, collection equality uses scalar tolerance semantics, and complex numbers are no longer ordered against real numbers.

• Set, collection, and statistics behavior is corrected. Intersection, SymmetricDifference, Union, and SetMinus produce correct finite-set results, Reverse([1,2,3]) returns [3,2,1], Quartiles consistently uses the Moore-McCabe convention, and single-argument \operatorname{KroneckerDelta}(0) returns 1.

Compilation

• Compiled JavaScript, Python, GLSL, WGSL, and interval output now match the interpreter more closely. Equality uses the engine tolerance, Mod and Remainder use consistent conventions, chained relations evaluate middle operands once, dynamic 0^0 returns NaN, non-boolean Which and When conditions throw, and interval arithmetic matches interpreter conventions for branches, rounding, modulus, and odd roots of negative numbers.

• Compilation fails closed when a target cannot represent an expression correctly. Unsupported or unsafe cases such as invalid shader constructs, reserved shader variable names, non-real values in real-only target helpers, multi-index sums or products that a target cannot express, and invalid constant folds now fail at compile time instead of emitting wrong code.

• The Python target emits valid Python for more expressions. Conditional expressions, NaN, logical operators, chained relations, assigned symbols, vars, and target options now compile consistently with the JavaScript target and the interpreter.

Additional Resolved Issues

• Evaluation limits are honored more reliably. Hard limits such as nested-exponential limits, divergent infinite sums, and very large special function inputs now return promptly, remain symbolic, or throw a CancellationError when timeLimit or the recursion limit is exceeded. Examples include \lim_{x\to\infty} e^{e^{e^x}}/e^{e^{e^{x-1}}}, \Gamma(10^{300}), \zeta(\pm 10^{300}), \operatorname{Fib}(10^9), \binom{2 \times 10^9}{10^9}, and \operatorname{Subfactorial}(10^6).

• simplify() honors more of its public contract. simplify({rules: null}) now applies no rewrite rules, as documented, and logarithmic simplifications such as \ln(a)/\ln(b) no longer reduce to an integer unless the identity can be verified exactly. simplify() also preserves exact exponents when combining powers, so x \cdot x^{\sqrt2} becomes x^{1+\sqrt2} rather than a decimal exponent.

• Numeric comparison and formatting edge cases are fixed. Two large 15-digit values that previously compared in the wrong order now compare correctly, toPrecision(15) no longer corrupts 999999999999999, NaN has a deterministic place in canonical ordering, and high-precision toString(), .json, and toFixed() avoid long stalls on enormous exponents.

• Substitution and collection operations are more complete. subs() now reaches into lists and tensors, for example Median([a,b,c]).subs({a: 1}). Finite set operations such as Intersection({1,2}, {2}) and SymmetricDifference now evaluate correctly, and Reverse([1,2,3]) returns [3,2,1] instead of throwing.

• Strict and non-strict validation are more predictable. In strict mode, user-declared function signatures are enforced for closed arguments, numeric operators reject provably non-numeric operands such as Sin("hello"), big-op bounds are type-checked, and Map([1,2,3], "nf") is rejected. In non-strict mode, missing required arguments such as Sqrt() or Power(2) no longer crash.

• Rule replacement is safer. Rule guards such as x \ne 0 must now be provable before they match, wildcard conditions such as :notzero no longer assume unknowns satisfy the condition, and failed sequence-wildcard matches no longer drop operands from the expression being transformed.

• Special values and combinatorics are corrected. Choose and Binomial now share standard conventions, including Choose(2,3) = 0 and negative upper indices such as Binomial(-2,3) = -4. Argument(1+i) evaluates to \pi/4, several Digamma special values simplify when the Fungrim pack is loaded, and integer-domain functions such as Fibonacci(+Infinity) and MoebiusMu(Infinity) stay symbolic instead of throwing.

• Modular arithmetic is consistent. Mod is floored everywhere, so Mod(-7, 3) returns 2, while Remainder uses round-to-nearest semantics. Exact rational inputs stay exact, for example Mod(\frac12, \frac13) returns \frac16.

• Complex and matrix products no longer lose meaning. Multiplying a scalar by a complex literal such as ["Complex", 1, 1] preserves both real and imaginary parts, and symbolic matrix products preserve their written order, so a commutator such as MP - PM no longer collapses to 0 for declared matrix symbols.

• Parsing rejects or preserves ambiguous forms more reliably. x^2^3 is now a parse error instead of an unintended list power, Sequence(1,2) no longer serializes as 1 2, parenthesized relations are treated as atomic operands inside larger chains, and a scalar or matrix next to a function- or matrix-valued symbol is parsed as multiplication rather than a tuple.

• 0^0 and non-finite values are consistent across paths. evaluate(), N(), and compiled JavaScript now agree that 0^0 is NaN. Trigonometric poles such as N(\cot \pi) and N(\csc \pi) now return complex infinity rather than huge finite artifacts.

Performance

• LaTeX parsing is 15-28% faster. Parsing is faster on derivative, polynomial, matrix, and definite-integral inputs, with the same parse results as before.

• Arbitrary-precision arithmetic is substantially faster. At 100 significant digits, addition, subtraction, multiplication, division, and comparison are now much faster than in 0.66.0, and high-precision ln, exp, Gamma, and related operations also benefit. The improvements are visible in both direct numeric work and symbolic operations that depend on arbitrary-precision arithmetic.

  Arbitrary-precision arithmetic at 100 significant digits (ns per operation, lower is better; warm median, distinct operands per call):

  op	CE 0.67.0	CE 0.66.0	math.js¹	Mathematica²
  add	75	152	278	1,023
  sub	91	167	329	1,212
  mul	202	319	7,984	1,025
  div	501	1,748	11,890	1,366
  cmp	29	198	61	984
  sqrt	3,163	4,018	54,696	1,055
  exp	4,795	8,698	728,876	1,682
  ln	5,887	32,398	670,206	1,353
  cos	6,914	7,467	1,666,292	2,059

  ¹ math.js BigNumber (decimal.js) at precision 100. ² Mathematica 14.3 timed inside the kernel with result caches disabled; its ~1 µs per-call dispatch floor dominates its small-op rows. CE and 0.66.0 from benchmarks/big-decimal/ops-results.json; reproduce with node benchmarks/big-decimal/run-ops.mjs and wolframscript -file benchmarks/big-decimal/ops-bench.wls.

  Symbolic operations (ms per call, lower is better; warm median, from the cross-library suite in benchmarks/REPORT.md):

  case	CE 0.67.0	CE 0.66.0	math.js	SymPy	Mathematica
  simplify √(3+2√2)	0.07	0.09	🟡 0.92	🟡 3.56	3.28
  simplify √6·x + √2·x	0.16	0.19	1.13	5.69	18.0
  simplify (x²−1)/(x−1)	0.10	0.15	🟡 0.99	8.53	0.17
  d/dx √(1−x²)	0.22	0.21	2.13	5.70	0.008
  d/dx xˣ	0.04	0.04	1.83	1.80	0.005
  ∫ x eˣ dx	0.08	0.09	—	6.53	0.57
  ∫ x/(x²+1) dx	0.16	0.18	—	7.23	0.60
  lim sin(x)/x	0.03	0.04	—	0.62	1.93
  lim (1+1/x)ˣ	0.55	1.13	—	2.76	5.81
  solve x⁴+x²−1 = 0	1.88	4.65	—	8.56	0.55
  solve x³−x−1 = 0	0.11	1.18	—	5.73	0.23

  🟡 = value-correct but not fully simplified. — = not supported. SymPy 1.14 via sympify/evalf (per-call parse included, as for every string-based tool). All engines measured warm, per-call from source, same protocol (benchmarks/REPORT.md, "Methodology").

• Integration, assumptions, polynomial solving, and factoring are faster. Rubi-backed integration spends less time on integrals it cannot solve, sign-related assumption queries respond faster, polynomial equations solve faster, and Factor handles common square-pattern cases more efficiently.

0.66.0 2026-06-28

New Features

• Multiply now operates on vectors and matrices. Previously a product with any list/matrix operand was left unevaluated — even 2 * [1, 2, 3]. Multiply (i.e. *, \cdot, \times, and implicit products) now follows matrix-product / scalar-scaling semantics, matching Add's existing element-wise threading:

  • Scalar × tensor scales every element: 2 * [1, 2, 3] → [2, 4, 6], 2 * \begin{pmatrix}1&2\\3&4\end{pmatrix} → \begin{pmatrix}2&4\\6&8\end{pmatrix} (exact values are preserved, e.g. \frac12 [2, 4, 6] → [1, 2, 3]).
  • Two or more matrices/vectors form the matrix product, folded left-to-right in the written order: \begin{pmatrix}1&2\\3&4\end{pmatrix}\begin{pmatrix}5&6\\7&8\end{pmatrix} → \begin{pmatrix}19&22\\43&50\end{pmatrix}. The product is not commutative — operand order is preserved (including for matrix·vector vs vector·matrix), and vector·vector reduces to the dot product. This reuses the existing MatrixMultiply implementation.

  Element-wise (Hadamard) multiplication of two same-shape tensors is therefore not what * does; tensors of incompatible dimensions are left unevaluated, and symbolic operands of unknown shape are unaffected.

• Hadamard (element-wise) product \odot. A new HadamardProduct operator, written \odot, multiplies two vectors or matrices of the same shape entry by entry: [1,2,3] \odot [4,5,6] → [4,10,18] and \begin{pmatrix}1&2\\3&4\end{pmatrix} \odot \begin{pmatrix}5&6\\7&8\end{pmatrix} → \begin{pmatrix}5&12\\21&32\end{pmatrix} (compare the matrix product *, which gives \begin{pmatrix}19&22\\43&50\end{pmatrix}). Operands of incompatible shape report an incompatible-dimensions error. It binds like multiplication and round-trips through LaTeX as \odot.

Resolved Issues

• Mixed chained inequalities keep their middle term. A chain combining different operators — e.g. 5 \le b \lt 7 — canonicalized to And(5 \le 7, b \lt 7), dropping b from the first link (so 3 \le 2 \lt 7 wrongly evaluated to True). It now canonicalizes to And(5 \le b, b \lt 7). Uniform chains (5 \le b \le 7) and the already-correct a \lt b \le c form are unchanged.

• A transcendental of an exact constant expression stays symbolic. Per the exactness contract, evaluate() of a transcendental of an exact argument returns a symbolic result and only .N() numericizes. This held for number literals (sin(2) → sin(2)) but not for exact constant expressions: sin(\pi^2) numericized to -0.4303… instead of staying sin(π²) (and likewise cos(√2), etc.). These now stay symbolic under evaluate(); an inexact (float) argument such as sin(2.5) still numericizes.

• An exact real added to the imaginary unit keeps its exact real part. \frac12 + i evaluated to 0.5 + i, and \frac34\sqrt3 + i to 1.299… + i — the exact real part was floatified when folded with i. Exact reals (rationals, radicals) are now preserved alongside the imaginary unit (1/2 + i, 3/4·√3 + i); .N() still numericizes, and inexact reals (1.5 + i) are unchanged.

• Matrix/vector arithmetic preserves exact entries. A tensor with exact rational or radical entries was stored with a float64 element type, so element-wise operations silently produced floats — e.g. \begin{pmatrix}½&⅓\end{pmatrix} + \begin{pmatrix}½&⅓\end{pmatrix} returned [1, 0.666…] instead of [1, ⅔], and a matrix of √2 entries decayed to decimals. Exact entries now use the expression element type and stay exact; inexact (machine/decimal) values continue to use float64.

• A^n is now the matrix power for an integer exponent. A power of a matrix was element-wise for non-negative exponents (A^2 squared each entry, A^0 gave a matrix of ones) yet A^{-1} already returned the inverse, and \begin{pmatrix}…\end{pmatrix}^2 did not evaluate at all. A^n is now the matrix power — repeated matrix multiplication — consistent with * being the matrix product: A^2 = A·A, A^0 is the identity, A^{-1} the inverse, and A^{-n} = (A^n)^{-1}. A non-square base reports expected-square-matrix. (Also fixes MatrixPower(A, n) for n < -1, which previously collapsed to A^{-1}.)

• Element-wise functions now distribute over matrix/vector-valued sub-expressions. A broadcastable unary function applied to an operand that only becomes a collection after evaluation — e.g. \sqrt{AB}, \sin(AB), |AB| where AB is a matrix product — was left unevaluated, because broadcasting was decided from the raw (un-evaluated) operand. It now also broadcasts over the evaluated operand, so these distribute element-wise like \sqrt{M} on a literal matrix already did. (Add/Multiply keep their dedicated tensor handling.)

• Juxtaposed matrices now form the matrix product. Writing two matrices next to each other (\begin{pmatrix}…\end{pmatrix}\begin{pmatrix}…\end{pmatrix}), or a scalar next to a matrix (2\begin{pmatrix}…\end{pmatrix}), previously produced a Tuple instead of a product, because the Matrix(…) wrapper is not reported as an indexed collection. The invisible (implicit) operator now treats matrix operands as multiplication, consistent with */\cdot/\times.

• Negate (and hence Subtract) of a matrix-valued product is distributed correctly. A negation whose operand only became a vector/matrix after evaluation — e.g. Negate(Multiply(A, B)) from A B - A B — was left undistributed, so the following Add/Subtract misclassified it as a scalar and broadcast it over the other matrix, yielding a bogus higher-rank result. Matrix subtraction (e.g. the commutator AB - BA) now evaluates correctly.

• A \textcolor wrapping a bare operator now parses as that operator. Input such as x \textcolor{red}{=} y previously failed — the = could not be parsed as a standalone group, producing a Tuple around an expected-closing-delimiter error. The color command is now transparent in operator position, so x \textcolor{red}{=} y parses as Equal(x, y) (and likewise for +, <, \le, \times, …). Because MathJSON has no way to annotate a lone operator glyph, the operator's color is dropped; coloring an operand (\textcolor{red}{y}, \textcolor{red}{x+1}) is unchanged and still yields an Annotated.

• One-sided \left( … \right. enclosures now parse. \right. (and the \bigr./\Bigr./… variants) is a TeX null delimiter: a fence with no visible closing glyph. Previously a one-sided group such as \sin\left(x\right. was rejected, leaking the \left out as an unexpected-command error; it now parses the same as \sin\left(x\right) (→ Sin(x)). The null open form (\left.…\right|, used by EvaluateAt) and ordinary two-sided delimiters are unchanged.

• Summation/product indices written as a \le range are now recognized. An index set of the form \sum_{1 \le i \le 10} i^2 (and the one-sided \sum_{i \le 10}) is now turned into the expected Limits, so the index i is bound by the sum instead of falling through to the imaginary unit. The example above now evaluates to 385 rather than staying symbolic with i → Complex(0, 1). This mirrors the existing handling of i \ge 1 and i = 1; strict < chains are not yet treated as index sets.

0.65.0 2026-06-28

New Features

• Differential equation solvers. (contributed by KingArth0r) Two new functions in the calculus library provide an initial slice of ordinary differential equation (ODE) support:

  • DSolve(eq, y, x) — symbolic solver for first-order linear scalar equations of the form y'(x) + p(x)·y(x) = q(x). It returns a List of solutions, each an Equal expression for y(x), introducing an integration constant C (a fresh name is chosen if C is already in use). For example, DSolve(y'(x) = y(x), y, x) → [y(x) = C·e^x] and DSolve(y'(x) + y(x) = x, y, x) → [y(x) = x - 1 + C·e^{-x}]. Nonlinear or higher-order equations are left unevaluated (inert).

  • NDSolve(eq, y, limits, y0, steps?) — numerical solver for explicit scalar first-order initial value problems y'(x) = f(x, y), y(x0) = y0, using a fixed-step fourth-order Runge–Kutta (RK4) method. It returns a List of [x, y] sample pairs over the interval given by limits (a Limits or Tuple of (x, x0, x1)); the number of steps defaults to 100. It handles integrands with no elementary antiderivative (e.g. a Gaussian IVP whose solution is expressed with Erf).

  This slice is intentionally narrow so the API and result shape can get feedback before broader ODE support (adaptive RK45, systems, higher-order reductions, stiff and implicit solvers) is added.

• \keyword{…} command for control-flow and logic keywords. Keyword constructs — if/then/else, for/from/to/do, where, such that, and, or, iff, for all, there exists, break, continue, return — can now be written with a dedicated \keyword{…} command, for example:

  \keyword{if} x > 0 \keyword{then} 1 \keyword{else} 0

  Unlike \text{…}, \keyword{…} keeps the input in math mode, and unlike \operatorname{…} it is rendered with symmetric keyword spacing. The existing \text{…} and \operatorname{…} spellings continue to work, and all three parse to the same expression. Multi-word keywords are written as a single token (e.g. \keyword{for all}). \keyword{otherwise} / \keyword{else} also serve as the default-branch marker inside a cases environment.

  A new keywordStyle serialization option — "text" (default), "keyword", or "operatorname" — selects which spelling is emitted when serializing If, Loop, Break, Continue, and Return back to LaTeX. The default preserves the previous \text{…} output.

0.64.0 2026-06-27

New Features

• Expanded number-theory library. A set of standard number-theoretic functions has been added to the number-theory library. Integer arguments use arbitrary-precision (bigint) arithmetic, and long-running cases honor the evaluation deadline.

  Factorization & divisors:

  • FactorInteger(n) — prime factorization as a list of [prime, exponent] tuples ordered by ascending prime: FactorInteger(360) → [(2, 3), (3, 2), (5, 1)]. Following Mathematica's conventions, FactorInteger(0) → [(0, 1)], FactorInteger(1) → [(1, 1)], and a negative integer carries its sign in a leading [-1, 1] tuple.
  • PrimeFactors(n) — the sorted distinct prime factors: PrimeFactors(360) → [2, 3, 5].
  • Divisors(n) — the sorted positive divisors: Divisors(12) → [1, 2, 3, 4, 6, 12]. Divisors(0) is left unevaluated.
  • Radical(n) — the square-free kernel (product of distinct primes): Radical(360) → 30.
  • PrimeNu(n) / PrimeOmega(n) — the number of prime factors without / with multiplicity (ω and Ω).
  • MoebiusMu(n) — the Möbius function μ(n).
  • DivisorSigma(k, n) — the divisor function σ_k(n) (generalizes the existing Sigma0/Sigma1).
  • IsSquareFree(n) — whether n is square-free.
  • IsPerfectPower(n) — whether n = a^b for integers a, b ≥ 2.

  Primes:

  • NthPrime(n) — the nth prime (1-based): NthPrime(10) → 29. (Mathematica names this Prime, but in the Compute Engine Prime denotes derivative notation, so the prime-number function is NthPrime.)

  • NextPrime(n) / NextPrime(n, k) — the smallest prime greater than n; with k, the kth prime after n (or the |k|th before it when k < 0).

  • PrimePi(n) — the prime-counting function π(n): PrimePi(10) → 4.

  • RandomPrime(n) / RandomPrime(m, n) — a random prime in the range.

    Primality for these uses exact 6k±1 trial division for small n and switches to Miller–Rabin above 2³² (deterministic for the supported range), so NextPrime and RandomPrime are fast even for very large arguments.

  Modular arithmetic & GCD:

  • PowerMod(a, b, m) — modular exponentiation a^b mod m; a negative b uses the modular inverse (undefined when a and m are not coprime).
  • ExtendedGCD(a, b) — the GCD with Bézout coefficients, as (g, x, y).
  • ChineseRemainder(residues, moduli) — solves a system of simultaneous congruences (moduli need not be coprime).
  • MultiplicativeOrder(a, n) — the order of a modulo n; PrimitiveRoot(n) — the smallest primitive root mod n.
  • JacobiSymbol(a, n) / LegendreSymbol(a, p) — the Jacobi and Legendre symbols.

  Other primitives:

  • IntegerSqrt(n) — the integer (floor) square root.
  • CarmichaelLambda(n) — the reduced totient λ(n).
  • LucasL(n) — the nth Lucas number; CatalanNumber(n) — the nth Catalan number.
  • BernoulliB(n) — the nth Bernoulli number as an exact rational, with the convention B₁ = -1/2.
  • ContinuedFraction(x, n?) / FromContinuedFraction(list) — the continued-fraction expansion of a number (exact for rationals) and its inverse.
  • IntegerDigits(n, base?, length?) / FromDigits(list, base?) — the digits of n in a given base, and its inverse. DigitCount(n, base?, digit?) — digit-occurrence counts; DigitSum(n, base?) — the digit sum.

• IsPrime is now reliable for large integers. Primality was previously left unevaluated above ~10¹⁵ and could silently round integers beyond 2⁵³ to a wrong machine value. IsPrime (and IsComposite) now route through a single deterministic Miller–Rabin implementation shared with the number-theory library, so e.g. IsPrime(2^61 - 1) correctly returns True. (The previous duplicate Miller–Rabin code, which used random bases and overflowed for large inputs, has been removed.) Relatedly, the internal toInteger helper now returns null instead of a precision-lost value for integers beyond the safe-integer range, so this class of silent-rounding bug cannot recur in the operators that use it for counts and indices.

• Factorial2, Subfactorial, and BellNumber no longer round a non-integer argument. These are defined only on integers; in non-strict mode they previously rounded a non-integer (e.g. Factorial2(5.5) returned 6!!). They now stay symbolic for non-integer arguments. (In strict mode the (integer) signature already rejected such inputs.)

• N(expr, precision) evaluates to a requested number of significant digits. The N function (and the ["N", expr] MathJSON form) now accepts an optional precision argument: ["N", "Pi", 50] returns π to 50 significant digits. When the requested precision exceeds the engine's working precision, the working precision is raised to match — and kept, since display precision is a global setting. When it is at or below the working precision, the result is rounded to that many significant digits without changing the global precision (N(1/3, 4) → 0.3333).

• New linear-algebra operators.

  • Dot(a, b) — vector inner product / matrix product (Mathematica's .): Dot([1,2,3], [4,5,6]) → 32.
  • Cross(a, b) — cross product of two 3-vectors.
  • MatrixRank(m) — the rank (number of linearly independent rows/columns) via the rank–nullity theorem.
  • MatrixPower(m, n) — a square matrix raised to an integer power (the repeated matrix product A·A·…, with negative powers using the inverse). Distinct from ["Power", m, n], which threads element-wise.
  • CharacteristicPolynomial(m, x?) — the monic characteristic polynomial det(x·I − A) (variable defaults to x): [[1,2],[3,4]] → x² − 5x − 2.
  • RowReduce(m) — the reduced row echelon form (RREF) of a matrix.
  • IsSymmetric(m) / IsDiagonal(m) / IsSquareMatrix(m) — matrix-shape predicates returning True/False.

Resolved Issues

• ["N", expr] now numerically evaluates its operand. The N operator holds its argument unevaluated and previously called .N() on the still unbound operand — a no-op for symbolic constants — so ["N", "Pi"] returned Pi unchanged (and ["N", ["Sqrt", 2]] returned Sqrt(2)) instead of a numeric value. The operand is now bound before evaluation, making ["N", expr] equivalent to expr.N().

0.63.0 2026-06-26

New Features

• LaTeX parse errors carry their source location. (contributed by zojize) The Error expressions produced by the LaTeX parser now include a sourceOffsets: [start, end] character range identifying where in the input the error occurred, so a consumer can map a parse error back to the offending span — e.g. to highlight an invalid token in a mathfield. Offsets are zero-based and end-exclusive into the serialized LaTeX (tokensToString); for input that round-trips through the tokenizer unchanged — editor-generated LaTeX, with no comments, Unicode normalization, or macro expansion — they match the original input string. Missing-operand errors (an empty \sqrt{} or \frac{}{}) use a zero-width range at the position where the token was expected. The new Parser.sourceOffsets(startToken, endToken?) helper lets custom dictionary entries attach a range to errors they raise. The raw parser output (LatexSyntax().parse()) always carries these offsets, so an Error node is now emitted in object form ({ fn: ["Error", …], sourceOffsets }) rather than the bare ["Error", …] array whenever a range is available — a consumer matching expr[0] === "Error" should also handle expr.fn?.[0] === "Error". Through the boxed path (ce.parse(latex).toMathJson()), source offsets are opt-in metadata like latex and wikidata: included with metadata: ['sourceOffsets'] or metadata: 'all', and omitted from the default serialization.

• Long numerators over a single power serialize with an inline solidus. When prettifying, a large numerator divided by a single power of a small base now serializes as (3x^4+2x^3+x+5)/x^{23} instead of the tall, lopsided fraction \frac{3x^4+2x^3+x+5}{x^{23}}. This rounds out the existing prettify heuristics, which already factor a small denominator out of a large numerator (\frac{1}{x}(…)) and write a small numerator over a large denominator with a negative exponent ((a)(…)^{-1}). The new form applies when the numerator is large and the denominator is a single power of a small base — base^{k} with an integer exponent k ≥ 2 (/x^{23}), a square (/x^2), or a square root (/\sqrt{x}). Lone powers (\frac{1}{x^{23}}), products in the denominator (a·x^n), compound bases ((x+1)^{23}), and all other shapes are unchanged. As with the other rewrites, it is disabled by prettify: false.

• Double-quoted string literals in LaTeX. "hello" now parses to a string (previously " was an unexpected-token). Content is read verbatim up to the closing quote, with LaTeX commands normalized to Unicode like \text{…} ("\alpha" → α); there is no escaping (use \text{…} for a string that must contain a "). Strings still serialize back to \text{…}. A " inside \unicode{…}/\char remains a hex prefix and is unaffected.

• Dictionary values can be read by key with At. ["At", dict, "key"] (string key) now returns the value of that entry in a dictionary — e.g. ["At", { dict: { height: 42 } }, "height"] → 42. A missing key yields Nothing. Previously At was restricted to indexed (positional) collections and rejected dictionaries with an incompatible-type error; its value type is now indexed_collection | dictionary. In LaTeX, the postfix bracket form accepts a string key, so \mathrm{data}["height"] (or \mathrm{data}[\text{height}]) parses to ["At", "data", "height"]. Dot-notation also works when the base is a symbol declared as a dictionary: \mathrm{data}.height → ["At", "data", "height"] (the key is an alphabetic, space-free name; for a dictionary base, .x / .real are key lookups, not First / Real component access). Positional indexing of indexed collections is unchanged.

• BoxedExpression.referencedFunctions and BoxedExpression.references. Two accessors aimed at dependency graphs (e.g. notebooks). The operator head of a function application — the f in f(x) or g(x) := f(x) + 1 — is not a symbol of the expression, so it appears in neither symbols nor freeVariables; referencedFunctions recovers those applied user-function names (excluding built-in operators, constants, and names bound by an enclosing scope, using the same predicate freeVariables applies to ordinary symbols). references is the complete in-edge set — freeVariables ∪ referencedFunctions, minus defines — so it pairs with defines (the out-edges) to build a use/def graph in one call. Subtracting defines drops self-references, so a recursive g(x) := g(x - 1) reports no dependency on itself.

• ce.declare() refines an auto-declared binding instead of throwing. Parsing auto-declares the names it encounters (a free variable a in a + 1, a called function f in f(x)), recording an inferred binding. Calling ce.declare(name, …) for such a name now refines that inferred binding rather than throwing "… already declared in this scope" — which is exactly what the inferred flag is for. This lets a declare-first workflow parse cells to discover names and then declare them on the same engine. Re-declaring an explicit binding still throws, and a name bound to a value (e.g. a function argument) is still a genuine conflict.

Resolved Issues

• canonical and structural options are now honored by parse(), expr(), and function(). These methods only consulted the form option when deciding how to box their result, so the documented canonical / structural shortcuts were silently ignored: ce.parse(latex, { canonical: false }) returned a canonical expression (and, as a side effect of canonicalization, auto-declared its symbols), and ce.function('Power', ops, { structural: true }) returned canonical Root instead of a structural Power. The keys now resolve the same way form does, with an explicit form taking precedence. As part of this, ce.assume() now canonicalizes its predicate so the assumption machinery always sees a normalized form (e.g. Negate(ImaginaryUnit) folded to the complex literal -i) regardless of how the caller boxed it.

0.62.1 2026-06-22

New Features

• indexStyle serialization option for collection indexing. The At operator (e.g. ["At", v, 1]) can now be serialized either as a subscript (v_1, M_{i,j}) or with programming-style brackets (v[1], M[i,j]). Like the other style options (fractionStyle, rootStyle, …) it is a callback (expr, level) => 'subscript' | 'bracket', settable engine-wide via ce.latexOptions.indexStyle or per-call via expr.toLatex({ indexStyle }). The default is 'subscript'.

Resolved Issues

• Collection indexing (At) now serializes to valid, round-tripping LaTeX. ["At", v, 1] previously serialized to \lbrack v, 1\rbrack — i.e. the list [v, 1], which re-parsed as ["List", v, 1], silently changing the meaning on a serialize→parse cycle. It now serializes as v_1 (or v[1] with indexStyle: 'bracket'), both of which parse back to At.

• Accents and decorations serialize with brace notation and round-trip. OverHat, OverVector, OverTilde, OverBar, UnderBar, the over-arrows, OverBrace, etc. had no serializer and fell back to function-call notation — \hat{x} came back out as \hat(x), which re-parsed to ["Multiply", x, ["OverHat"]] instead of ["OverHat", x]. They now serialize as \hat{x}, \vec{v}, \overline{x}, … and round-trip correctly, including when subscripted (\hat{x}_0).

• Subscripted single-letter symbols serialize with an italic base instead of an upright one. When a symbol name carried a subscript (e.g. a_1, x_n, S_t), the serializer chose its font style from the decorated string rather than the base: the subscript inflated the token count, so the multi-character rule wrapped the whole thing in \mathrm{…} and rendered the base letter upright (\mathrm{a_1}). A single-letter variable with a subscript is now rendered italic, as a variable should be — a_1 serializes to a_1, not \mathrm{a_1}. The font style is now decided from the base alone: multi-letter bases are still upright with the wrapper enclosing the whole symbol, so descriptive subscripts stay roman (speed_max → \mathrm{speed_{max}}), and explicit style modifiers (\mathbf, \mathbb, …) are unchanged. Greek single-letter bases are likewise rendered with their default (italic) style.

0.62.0 2026-06-20

Resolved Issues

• Arbitrary-precision sums of three or more terms no longer collapse to machine precision. BigNumericValue.add had a fast path that, when adding to a zero value, cloned the other operand through a constructor that reads its machine real part (decimal.toNumber()), silently truncating a full-precision bignum to ~16 significant digits. The exact (rational/radical) arithmetic path was unaffected, and two-term sums were unaffected, so this only surfaced when summing three or more inexact values at a precision above machine: ExactNumericValue.sum folds those starting from a zero accumulator, and the very first 0 + xᵢ step lost all extra precision. The degradation was invisible when the terms were of similar magnitude (the result was merely capped at ~16 digits), but became a wrong answer under cancellation — e.g. numerically evaluating a high-order symbolic derivative at a point (large factorial-scale terms cancelling to a small value) returned garbage at any working precision. The zero-accumulator path now reads the full-precision real part, matching the non-zero path. Coefficients were always computed exactly; only the final numeric summation was affected.

• High-order derivatives are reduced instead of blowing up. The Derivative operator applies the differentiation rules iteratively, and the quotient and product rules square the denominator at each step, so the r-th derivative of a quotient carried an x^(2ʳ)-scale denominator — e.g. the 75th derivative of sin(x)/x came back over x^(2⁷⁵). The result was mathematically exact (the integer coefficients are computed exactly), but the enormous exponent made it unusable and overflowed to NaN when evaluated at a point. Derivative of order ≥ 2 now runs a single simplification at the end, cancelling the common factors back to a linear-degree denominator (x^(2⁷⁵) → x⁷⁶). It is applied once, not per step, so it is cheap (~30 ms at order 75) and leaves first derivatives and the existing low-order results unchanged.

• interval-glsl is now outward-rounded, making it a sound standalone exclusion oracle in float32 (preview). As shipped in 0.61.0 the _iv_* ops clamped to the sentinel range but rounded to nearest, so an operation — or the cell box itself — could come back slightly narrower than the true range. At a boundary that is enough to flip the exclusion verdict for a box the curve only grazes (e.g. the unit circle's tangent corner at (1, 0)), violating the containment contract that the GLSL interval must contain the interval-js (float64) result — a spuriously narrow interval can exclude a box the curve actually passes through. Every inexact operation now widens its result outward (lo toward −∞, hi toward +∞) before the clamp: by ~1 ulp for the correctly-rounded ops (+ − ×, Square), and by a larger relative margin for the GLSL ES built-ins that are not correctly rounded — 8 ulp for /, Sqrt, Exp/Ln/Log, and inverse trigonometry, and 32 ulp for Power (x^n with n ≥ 3, and fractional powers such as the astroid x^{2/3}). Crucially, the cell box that compileExclusionShader's main() builds is itself outward-rounded (via the new _iv_widen_box): the float32 mix that constructs it rounds to nearest and is the actual source of the grazing miss, which per-op widening alone cannot fix (with exact endpoints the op chain is exact). That box pad is scaled to the domain extent, not the edge value, since that is what bounds the mix error — a value-relative pad would vanish for a box edge near 0 in a wide domain. Widening only ever moves a bound outward, so it cannot break soundness; the empty (lo > hi) / entire (±IV_INF) encodings, the finite IV_INF sentinel, the per-op clamp, and exact empty-propagation are all preserved. Sin/Cos remain best-effort (see below).

• freeVariables / unknowns no longer report the bound variables of Function literals and integrals. A function literal leaked its own parameters, and Integrate / Limit leaked their variable — e.g. freeVariables of f(x) := x^2 + b wrongly included the parameter x, and a definite integral leaked its integration variable. They now return only genuinely free symbols ([b, f] for that definition, [] for ∫ sin(x) dx), while a free coefficient is still reported (∫ a·sin(x) dx → [a]). Sum / Product were already correct, and symbols is unchanged (it still includes bound variables). This is a behavior change for code that relied on the previous, over-inclusive result.

• Runaway user-function recursion now throws a catchable CancellationError instead of a native RangeError. A recursive definition with no reachable base case (e.g. f(x) := f(x-1) + 1) previously overflowed the JavaScript call stack with an uninformative RangeError. recursionLimit — previously defined but never enforced — is now applied to user-function application: exceeding it throws a CancellationError with cause: 'recursion-depth-exceeded', consistent with how timeLimit and iterationLimit are surfaced. The default recursionLimit is now 256 (was a nominal, unenforced 1024), chosen to fire below the native stack limit on typical engines; raise ce.recursionLimit for legitimately deep recursion. Iterating a user function (e.g. \sum f(i)) is not counted as recursion. (A sufficiently complex single call can still exceed the native stack before the limit is reached, so a robust caller catches RangeError as a backstop.)

• Integrate binds only the integration variable in its canonical integrand. ∫ a·sin(x) dx previously canonicalized to Integrate(Function(body, a, x), …), listing the free coefficient a as a spurious integrand parameter; it is now Integrate(Function(body, x), …). Introspecting the integrand (expr.op1) therefore reports a as free, and the integrand is a proper single-variable function. Evaluation is unchanged.

• Nested (multivariate) integrals now parse and evaluate correctly. \int_1^2\int_3^4 x y \, dx \, dy previously attached all the trailing differentials to the innermost integral, leaving the outer integrals with a Nothing integration variable — so the expression could not evaluate. Each \int now consumes only its own differential (the innermost dx pairs with the innermost \int, the next dy with the next), producing a properly nested Integrate where every level carries its own variable and limits (\iint / \iiint still bind 2 / 3 variables at one level). Combined with the definite-integral evaluator now applying the limits to a parametric antiderivative (e.g. ∫_3^4 k·x dx → 7/2·k; the symbolic f(b) - f(a) was previously left as an unevaluated EvaluateAt), nested definite integrals evaluate to a value: ∫_1^2∫_3^4 x·y dx dy → 21/4.

• Multiple-integral and contour-integral serialization round-trips. \iint / \iiint (and \oiint / \oiiint) now serialize back to the compact sign with a single region subscript (\iint_{D}\!…) instead of a stack of \ints, so a flat multiple integral round-trips to the same structure. A separate long-standing bug that emitted the literal text \ointundefined for any \oint with a region (its limit is a 3-element Tuple, serialized to MathJSON as Triple, which the serializer did not recognize) is also fixed: \oint_V f(s)\,ds now serializes as \oint_{V}\!f(s)\, \mathrm{d}s.

• 1^x simplifies to 1 for any finite exponent. A symbolic or function exponent (e.g. 1^{n+1}, 1^{\sin x}) previously left Power(1, x) un-reduced because the canonicalizer bailed before its base-1 rule. 1^x → 1 now (matching SymPy / Mathematica); only a genuinely infinite or NaN exponent stays indeterminate (1^∞ → NaN, unchanged).

New Features

• interval-glsl: public outward-rounding helpers and an opt-in absolute trig pad (preview). The widen helpers _iv_widen / _iv_widen_t / _iv_widen_pow / _iv_widen_sc / _iv_widen_box, and their epsilons IV_EPS / IV_EPS_FN / IV_EPS_POW / IV_BOX_EPS, are a stable, public part of the emitted preamble: a renderer that builds its own cell box (instead of using compileExclusionShader) outward-rounds it by calling _iv_widen_box(vec2(lo, hi), extent) per axis, where extent is the domain extent for that axis (the box pad is domain-scaled, not value-relative). The preamble is now emitted for any expression with free variables (not only ones that reference an _iv_* op), so those helpers are always available — e.g. for an axis line f = x. GLSL ES Sin/Cos carry an absolute, implementation-defined error (≈2⁻¹¹ in the worst case; macOS ANGLE→Metal differs) that no relative pad can cover. A new trigAbsPad option (default 0, off) on compile(), IntervalGLSLTarget.compileExclusionShader(), and the new IntervalGLSLTarget.getPreamble() adds an absolute Sin/Cos pad, so a trigonometric implicit curve can be a strictly-sound standalone oracle at the cost of fatter trig intervals.

• BoxedExpression.defines. A new accessor returning the symbols an expression defines: the target of a top-level Assign / Declare (a in a := 3, f in f(x) := …), recursing through Block. It complements freeVariables (the symbols an expression references) — together they let tooling build a definition/use dependency graph, with references = freeVariables minus defines.

• ComputeEngine.appliedNonFunctions(latex). Returns the symbols written in function-application syntax f(…) in latex that are not functions in the current scope, and so parse as implicit multiplication (f·x) or are left unresolved. The check is scope-aware (a symbol declared as a function is not reported) and has no side effects. Useful for flagging a likely call to an undefined function — e.g. warning that f(x) was read as f·x.

0.61.0 2026-06-17

New Features

• interval-glsl compilation target (preview). A GPU compilation target that evaluates an expression with interval arithmetic in GLSL — each value is a vec2 (lo, hi) — so a robust implicit-curve renderer can run its per-cell exclusion test (lo > 0 || hi < 0) on the GPU instead of CPU-side via interval-js. (Reinstates the interval-glsl target removed in 0.52, with a simpler vec2-only representation — the GPU acts as an exclusion oracle and the CPU keeps curve extraction — instead of the former status-flag struct.) compile(expr, { to: 'interval-glsl' }) emits _iv_* helper calls plus a preamble library. Coverage: arithmetic, integer and positive rational powers, Abs, Sqrt, Exp, Ln/Log/Lb, trigonometry / inverse trigonometry (Sin, Cos, Tan, Arcsin, Arccos, Arctan, with interval range reduction), and the step / rounding family (Floor, Ceil, Round, Truncate, Fract, Sign, Heaviside, Mod, Min, Max) — covering polynomial, rational, algebraic, trigonometric, and lattice/periodic implicit curves (conics, lemniscate, astroid, superellipse, trig lattices, floor/mod grids, …). Jump-discontinuity functions return a tight, sound value-range enclosure (so cells can still be excluded), with discontinuity classification left to the CPU; only genuine poles widen to the full range. A head that is not yet supported (e.g. hyperbolic functions) is reported in the result's unsupported field, so a caller can fall back to another target per-expression. Values use a finite ±∞ sentinel and a lo > hi encoding for the empty (domain-undefined) interval, propagated through every operation; domain-restricted functions (sqrt/ln/asin/rational pow of an out-of-domain argument) yield empty, and a pole (zero-spanning denominator, tan asymptote) yields the full range. Parity with the interval-js target is verified against a shared corpus. IntervalGLSLTarget.compileExclusionShader() emits a complete, self-contained fragment shader (preamble + an _implicit interval evaluator + a reference main that derives each fragment's cell box and applies the exclusion test) ready to drop into a WebGL2 renderer.

Resolved Issues

• A function parameter now shadows a same-named constant. A parameter named like a constant (i, e, Pi/\pi, …) was rewritten to the constant while the function body was canonicalized, so the binding was lost — λi. 2i applied to 5 returned 2i (the imaginary unit doubled) instead of 10. Parameters now shadow whatever their name means in the enclosing scope — a constant, an assigned variable, or nothing — which is standard lexical scoping. A free symbol that is not a parameter is unchanged (i outside a parameter is still the imaginary unit), and closure capture is preserved (λi. λz. (z + i) captures i correctly).

• compile() no longer emits a dangling reference to a symbol that has an assigned value (GLSL, WGSL, JavaScript, and interval-JS targets). When an expression referenced a symbol with an assigned value in the engine (ce.assign("a", 1.5)), compile() emitted a bare a — an undeclared GLSL identifier (a shader that silently fails to compile) or a bare JS global (a ReferenceError when the compiled function is called) — even though the symbol is omitted from expr.unknowns and folded by evaluate(). The value is now folded into the generated code (sin(a·x) → sin(1.5 * x)), making compile(), evaluate(), and unknowns consistent. This also folds user-declared constants (ce.declare("c", { value: 3 })), and applies on the direct-target compile(expr, { target }) path as well. A symbol supplied through the compile() vars option is never folded — the mapping always wins, so a per-frame GLSL uniform / JS argument keeps updating the result without recompiling — and a genuinely free symbol is unchanged.

• compile() folds a symbolic assigned value correctly, parenthesizing it and resolving the free symbols it references. When a symbol was assigned an expression rather than a number (ce.assign("b", ce.parse("c + 1"))), folding b into a larger expression had two bugs: the compound value was spliced in without parentheses, so b · x compiled to c + 1 * x (i.e. c + x) instead of (c + 1) * x — a silently wrong result (2·b → 2 * c + 1, b² → (c + 1 * c + 1)); and the inner free symbol c, hidden behind b's value and therefore absent from expr.unknowns, was emitted as a bare global (ReferenceError on the JS target). The folded value is now parenthesized for its context, and a free symbol reachable only through a folded value routes through the normal free-symbol plumbing (_.c on the JS / interval-JS targets; a uniform on GPU) and is reported in the result's freeSymbols.

• GPU compilation rejects non-finite numbers instead of emitting a non-compilable shader. GLSL and WGSL have no infinity or NaN literals, but compile() emitted Infinity.0 / NaN.0 for a ±∞ or NaN value (e.g. from a literal \infty or a constant-folded 1/0) and reported success: true — a shader that silently fails to compile on the GPU. Such values now throw a clear error from the GLSL/WGSL targets (so the free compile() falls back to success: false with a diagnostic), consistent with how other GPU-unsupported constructs are handled. The JavaScript target is unchanged (Infinity / NaN are valid there).

• The JavaScript compilation target now lowers the exponential, trigonometric, and logarithmic integrals. SinIntegral (Si), CosIntegral (Ci), ExpIntegralEi (Ei), and LogIntegral (li) compile to _SYS runtime helpers, matching the existing support for Erf, FresnelS, Gamma, BesselJ, etc. These are the closed forms the antiderivative engine emits (e.g. ∫ sin x / x dx = SinIntegral(x)), so an "evaluate then compile" pipeline — such as plotting ∫ f dx from its closed form — no longer throws Unknown operator and falls back to numeric sampling. (GLSL/WGSL shader approximations of these are not yet provided.)

• The JavaScript compilation target now lowers the elliptic, AGM, and hypergeometric kernels. AGM, EllipticK, EllipticE, EllipticF, EllipticPi, Hypergeometric2F1, Hypergeometric1F1, Erfi, and Choose compile to _SYS runtime helpers. Like the integral functions above, these are closed forms evaluate()/.N() produces (e.g. a pendulum period or an arc length reduces to an elliptic integral), so they can now be plotted from the closed form rather than re-sampled numerically. EllipticE and EllipticPi keep their arity-overloaded complete/incomplete forms, and AGM accepts the one-argument AGM(z) = AGM(1, z) shorthand. (Real-valued like the other special functions on this target; GLSL/WGSL not provided.)

Improvements

• compile() results now report their external references. A CompilationResult carries two new fields so a caller can check that a result is self-contained declaratively, instead of executing or GPU-compiling the code to discover a dangling reference:

  • freeSymbols — the identifiers the generated code references that the caller must supply at run time (JS vars-object keys / GLSL uniforms). These are the free symbols as codegen sees them: assigned values and constants are folded out, bound variables (lambda parameters, Sum/Product/ Integrate/Loop indices, Block locals) are excluded, and vars-mapped symbols are always included. Unlike expr.unknowns, it also surfaces a free symbol reachable only through a folded value (e.g. b assigned c + 1 exposes c). Use it to build a uniforms / vars mapping that is guaranteed consistent with the emitted code.
  • unsupported — operator heads the target cannot lower (no operator/function mapping, not a structural form). On a failed compile() this is populated alongside a human-readable error, so an unlowerable operator (e.g. SinIntegral on the GLSL target) surfaces as success: false with a machine-readable list rather than only a thrown exception.

  Built-in targets populate freeSymbols (and an empty unsupported) on every successful compile. The direct getCompilationTarget(name).compile(expr) path still throws on a genuinely unsupported operator (so the engine-level compile() can fall back to interpretation); the unsupported / error fields are how the engine-level compile() reports that condition without a throw.

0.60.0 2026-06-16

Behavior Changes

• isFinite is now known for finite symbolic constants. Expressions such as √π, 1/π, and π^π report expr.isFinite === true (previously undefined), because finiteness is propagated through Sqrt, Root, Power, and Divide of finite operands. Cases that are genuinely indeterminate (e.g. 1/x for an unconstrained x) still report undefined.

• Exact transcendental expressions now remain symbolic under evaluate(). For example, ln(2) remains ln(2) instead of becoming 0.693…. Use .N() or { numericApproximation: true } when a numeric approximation is wanted. Inexact inputs still evaluate numerically, and known exact values such as cos(π) = -1 and arctan(1) = π/4 still simplify. As a result, definite integrals also preserve exact results, such as ∫₁² 1/x dx = ln(2) and ∫₀¹ 1/(1+x²) dx = π/4.

• (aⁿ)ᵐ no longer folds to aⁿᵐ based solely on an odd inner exponent. This combine was unsound on the principal branch: when a < 0 and m is not an integer, the two sides differ by a phase. For example (x³)^{1/2} now stays √(x³) (which is 8i at x = -4) instead of becoming the inequivalent x^{3/2} (-8i), and it is again confluent with the √(x³) form. The fold still applies when the base is non-negative or the outer exponent is an integer. (Roots are unaffected: (x³)^{1/3} = x still holds, since odd-index roots use the real-root convention.)

• Logarithms are no longer combined across a branch cut. ln(a) + ln(b) → ln(ab) (and the log and subtraction variants) is only valid on the principal branch; for arguments on the negative real axis the two sides differ by a multiple of 2πi. For example ln(-2) + ln(-3) no longer simplifies to the inequivalent ln(6) (its true value is ln(6) + 2πi). The combine still applies to positive and unconstrained-symbolic arguments. The guard consults the analytic-property store's branch-cut records (see Special Functions).

• e^{iθ} stays in exponential form under evaluate() for a symbolic angle. Euler's formula e^{iθ} → cos θ + i·sin θ is now applied only when θ is a constant that reduces to a closed form (e^{iπ/2} = i, e^{iπ} = -1, e^{ln y} = y are unchanged); for a symbolic angle, e^{ix} stays e^{ix} — a basis change is not an evaluation, and it no longer differs from the previous inconsistency where (e^{ix})² expanded while e^{ix} did not. Convert to trigonometric form on demand with the new strategy expr.simplify({ strategy: 'trig' }).

• N() at a known pole now returns ComplexInfinity instead of NaN. When a function is evaluated numerically at a pole recorded in the new analytic-property metadata store (see Special Functions), the result is ComplexInfinity rather than NaN or an unevaluated expression — for example Digamma(0).N() and Digamma(-2).N(). Functions whose kernels already returned an infinity at their poles (such as Gamma) are unchanged.

Benchmarks

The numeric and symbolic gains in this release are summarized below against the last release (0.59.0), SymPy, math.js, and Mathematica — the reference baseline, since it is the broadest engine in the field. The tables are generated by the harness in benchmarks/ (node benchmarks/report_changelog.mjs); every result is verified numerically against an independent mpmath reference, never another tool. "CE 0.60.0" is this release.

Numeric performance (200-digit precision)

Median time per call, in microseconds — lower is better. — means the tool returned no usable result at that precision.

Expression	CE 0.60.0	CE 0.59.0	SymPy	math.js	Mathematica
\pi^2	15	20	174	107	3.9
\sin 1	25	61	220	429	5.2
\cos 1	24	60	222	455	7.1
\ln 2	87	302	339	4,374	3.7
e^{\pi}	31	398	214	4,771	4.6
\zeta(3)	3,419	—	264	—	49
\Gamma(\tfrac13)	1,867	427,938	341	—	212
\psi(\tfrac13)	1,689	404,300	2,831	—	169

Biggest gains over 0.59.0: \psi(\tfrac13) 239× faster, \Gamma(\tfrac13) 229× faster, e^{\pi} 13× faster (it no longer recomputes \ln e on every call), \ln 2 3.5× faster, \sin 1 / \cos 1 ~2.5× faster. The elementary functions widen further at 1000+ digits (e.g. \ln 2 ≈ 21× faster, where it now also leads SymPy and mpmath). 0.59.0 could not reach 200 digits for \zeta(3) (it was capped near machine precision); math.js has no arbitrary-precision ζ/Γ/ψ. Mathematica's native bignum kernel is faster still on these constants.

Symbolic capability & performance

Each cell is how many times faster than Mathematica that engine is on the case (Mathematica ÷ engine, so higher is better; Mathematica itself is 1×). — means the engine can't do the case. Compare the CE 0.60.0 and CE 0.59.0 columns to see what is new this release (a — under 0.59.0 next to a number under CE 0.60.0). The CE + R/F column is CE 0.60.0 with the opt-in Rubi integrator and Fungrim identities loaded (loadIntegrationRules / loadIdentities), on the same minified bundle: sometimes it improves performance, sometimes it hurts it, but the overall effect is improved coverage.

Operation	CE 0.60.0	CE + R/F	CE 0.59.0	SymPy	math.js	Mathematica
Antiderivatives
\int\frac{1}{\sqrt x}\,dx	1.5×	3.7×	—	0.5×	—	1×
\int\frac{x}{\sqrt{1-x^2}}\,dx	2.5×	2.6×	—	0.09×	—	1×
\int\frac{1}{x^3+1}\,dx	2.2×	11×	—	0.3×	—	1×
\int\frac{\sqrt x}{1+x}\,dx	—	3.7×	—	0.1×	—	1×
\int\frac{x}{(1+x)^{1/3}}\,dx	—	3.9×	—	0.01×	—	1×
\int\frac{x^2}{(1+x)^{1/3}}\,dx	—	4.1×	—	0.007×	—	1×
Derivatives
\tfrac{d}{dx}\sqrt{1-x^2}	0.01×	0.03×	0.01×	0.001×	0.004×	1×
Simplification
\sqrt{3+2\sqrt2}	11×	20×	—	—	—	1×
\sqrt6\,x+\sqrt2\,x	28×	65×	30×	3.3×	18×	1×
Evaluation
\lim_{x\to0}\tfrac{\sin x}{x}	9.2×	23×	—	3.1×	—	1×
\lim_{x\to\infty}(1+\tfrac1x)^x	1.6×	1.6×	—	2.1×	—	1×
\int_1^2\tfrac1x\,dx	1996×	1907×	—	92×	—	1×
\int_{-\infty}^{\infty} e^{-x^2}\,dx	106×	428×	—	2.5×	—	1×
Solving
x^4+x^2-1=0	0.07×	0.08×	—	0.06×	—	1×
x^3-x-1=0	0.08×	0.1×	—	0.04×	—	1×

Across the cases both solve, Compute Engine is a median 3.7× faster than Mathematica (up to 1996×). The — entries under 0.59.0 show what is new this release: limits, exact definite/improper integrals, and polynomial solving. The bottom three antiderivative rows are integrals the base engine still leaves unevaluated but the opt-in Rubi rules solve. Mathematica still leads on raw derivative and root-finding latency (the <1× rows), where its native kernel is hard to beat.

_{Measured 2026-06-16 · SymPy 1.14.0 · math.js 15.2.0 · Mathematica 14.3.0 · Node 22 · verified against mpmath. Reproduce: npm run build production && ./venv/bin/python3 benchmarks/gen_cases.py && node benchmarks/report.mjs && node benchmarks/report_changelog.mjs.}

Calculus

• Limit can now return exact symbolic results. This includes direct substitution, indeterminate quotients, rational functions at infinity, dominant-term analysis, and exponential forms. Examples include lim(x→0) sin(x)/x = 1, lim(x→∞) (1+1/x)^x = e, and lim(x→∞) arctan(x) = π/2. Limits that cannot be determined reliably fall back to numeric evaluation or remain unevaluated. NLimit remains numeric.

• Limits no longer return a wrong value at a special-function pole. A limit whose expression contains a special function (Gamma, Digamma, PolyGamma, Zeta, …) evaluated at one of its poles — e.g. lim(x→-1) (x+1)·Digamma(x) — previously substituted the pole as a finite value and returned a confident wrong result (0). Such limits now stay unevaluated (or are recovered numerically where sampling allows) rather than reporting a false value.

• Symbolic integration supports many more integrands, including:

  • Gaussian integrals and quadratic exponentials using Erf and Erfi
  • Fresnel integrals
  • Sine, cosine, exponential, and logarithmic integrals
  • Products of polynomials, exponentials, and trigonometric functions
  • More radical and quadratic-root integrands
  • Powers of secant, cosecant, tangent, and cotangent
  • Reverse power-chain forms such as ∫ln(x)/x dx = ½ln²(x)
  • Products with symbolic exponents that previously failed or timed out
  • Powers and radicals of a linear function, e.g. ∫√(1+x) dx, ∫x√(1+2x) dx, and ∫(a+bx)^p dx
  • Radical powers of a polynomial via the reverse chain rule, e.g. ∫x√(1−x²) dx = −⅓(1−x²)^{3/2}
  • Quotients by a sum of two square roots, e.g. ∫1/(√(a+bx)+√(c+bx)) dx, by conjugate rationalization
  • Absolute value of a linear argument, e.g. ∫|x| dx = x|x|/2 and ∫|ax+b| dx = (ax+b)|ax+b|/(2a) (valid for all x)

• Rational-function integration is more exact and complete. Partial fractions now preserve rational and radical coefficients for a wider range of denominators, including x³+1, x⁴+1, x⁴-1, and biquadratic polynomials. Several cases that previously returned incomplete results, floating-point coefficients, or no result now return exact antiderivatives.

• More improper integrals evaluate correctly. Exact results now include Gaussian, rational, and Fresnel integrals over infinite intervals. Numeric integration of convergent oscillatory integrals is also more reliable, while divergent or low-confidence cases remain unevaluated instead of returning a misleading finite value.

• Fixed incorrect or missing antiderivatives for sin²(ax+b), cos²(ax+b), √x, 1/√x, 1/√(1-x²), and related forms.

• New Residue(f, x, a) operator computes the residue of f at x = a (the coefficient of (x-a)⁻¹ in its Laurent expansion). It detects the pole order and evaluates exactly via the symbolic limit engine, e.g. Residue(1/(x²-1), x, 1) → 1/2, Residue(eˣ/(x-1)², x, 1) → e, and Residue(cot(x), x, 0) → 1. Residues of Gamma, Digamma, and Zeta at their poles use closed forms gated by the analytic-property store, e.g. Residue(Gamma(x), x, -2) → 1/2 and Residue(Zeta(s), s, 1) → 1 — including in a product or quotient with an analytic cofactor, such as Residue(Gamma(x)/(x-5), x, -2) → -1/14.

Algebra and Solving

• solve handles equations between two different inverse-trigonometric functions by applying tan to both sides to clear them, then solving the resulting algebraic equation. For example arcsin(x) = arctan(x) → 0 and arccos(x) = arctan(x) → √((√5−1)/2). As part of this, √(f(x)) = g(x) with a non-linear right-hand side now solves too (e.g. √(1−x²) = x²).

• New Solve operator. Solve(equation, unknown) returns the list of solutions of an equation for an unknown, using the same solver as the expr.solve() method — for example ["Solve", ["Equal", "x^2", 1], "x"] returns ["List", 1, -1]. The equation may be an Equal expression or a bare expression read as = 0; the arguments are held, so the equation is no longer prematurely reduced to a boolean.

• solve now handles general cubic, quartic, and higher-degree polynomials. Exact roots are still preferred; when no supported exact form is available, real roots are returned as numeric approximations.

• Absolute-value equations solve more reliably. This includes equations such as |x| = 2, |x-1| = 2, non-linear arguments such as |x²-3| = 1, and equations with an absolute value on both sides.

• solve handles more transcendental and substitution equations. Equations with equal exponential bases reduce by their exponents (e^{2-x²} = e^{-x} → -1, 2; 2^x = 2^3 → 3); a·sin(x) + b·cos(x) = 0 solves via the tangent (sin x = cos x → π/4); equations that are polynomials in a root of the unknown solve by substitution (2√x + 3·⁴√x = 2 → 1/16); and a single square root with a non-constant coefficient is eliminated by squaring (x = 1/√(x²+1)).

• Biquadratic and sparse-power equations return exact roots. Polynomials whose exponents share a common factor — such as x⁴ + x² − 1 — are solved by substituting u = x² (or x³, …), so the roots are exact radicals (±√((√5−1)/2)) instead of numeric approximations.

• solve handles equations that are polynomials in a single nonlinear generator, by substituting u = g(x) for a logarithmic, exponential, trigonometric, or radical generator g, solving for u, and inverting. For example (ln x)² = 4 → e², e⁻², e^{2x} − 3eˣ + 2 = 0 → 0, ln 2, and √(ln x) = ln√x → 1, e⁴.

• solve factors a zero product. When an equation is a product whose factors each involve the unknown — such as ln(x)·(x − 1) = 0, or an already-factored (x + 1)·cos³(3x) = 0 — its roots are the union of the roots of each factor.

• GCD now finds common polynomial factors for univariate and multivariate polynomials. Integer operands retain their existing behavior; use PolynomialGCD() when an explicit polynomial result of 1 is needed for coprime inputs.

• New Resultant(a, b, x) operator computes the resultant of two polynomials with respect to a variable (the Sylvester-matrix determinant). It is zero exactly when the polynomials share a common factor, e.g. Resultant(x² - 1, x - 1, x) → 0 and Resultant(x² + 1, x² - 1, x) → 4. Symbolic coefficients are supported: Resultant(x² + a, x + b, x) → a + b².

• Polynomial factorization is more complete and reliable. In particular, Factor(xⁿ-1) now returns polynomial factors without introducing branch-dependent radicals, and the public factor() function once again factors expressions such as x²+5x+6.

• Nested radicals are simplified when possible, for example √(3+2√2) = 1+√2.

Special Functions

• Added numeric evaluation for:

  • Complete and incomplete elliptic integrals: EllipticK, EllipticE, EllipticF, and EllipticPi
  • The arithmetic-geometric mean AGM
  • Hypergeometric2F1, Hypergeometric1F1, and AppellF1
  • Jacobi theta functions and the Dedekind eta function
  • Erfi, SinIntegral, CosIntegral, ExpIntegralEi, and LogIntegral

• Gamma now accepts a second argument, the upper incomplete gamma function Γ(s, z) = ∫_z^∞ tˢ⁻¹ e⁻ᵗ dt (e.g. ["Gamma", s, z]). It is evaluated numerically for real and complex arguments, including negative and fractional orders s (Gamma(-4, 2), Gamma(1/2, -1)), and honors the exactness contract: it stays symbolic under evaluate() and reduces Γ(s, 0) to the ordinary Γ(s). Use .N() for a numeric value. The one-argument Γ(z) is unchanged.

• Hypergeometric2F1 now supports analytic continuation across most of the complex plane, rather than being limited to its defining power series.

• Zeta and Gamma now honor the requested precision. At high ce.precision, numeric evaluation of Zeta, Gamma, GammaLn, Beta, Digamma, Trigamma, and PolyGamma previously stalled near machine precision (e.g. Zeta(3) was correct to only ~16 digits regardless of precision). They now return the full requested precision — Zeta uses the Cohen–Villegas–Zagier acceleration, and all of these kernels compute with guard digits.

• EulerGamma (γ) now honors the requested precision. It was previously a fixed ~858-digit constant, so at higher ce.precision it silently stopped at ~858 correct digits (making identities such as Digamma(1) = -γ appear wrong past that point). It is now computed on demand to the full working precision.

• Gamma and the polygamma family are dramatically faster at high precision (~340× at 300 digits — Gamma(1/3) ≈1.9 s → ≈5 ms; ~130× at 1000 digits). The Stirling-series kernels (Gamma, GammaLn, Digamma, Trigamma, PolyGamma) were both shifting their argument just short of where the series converges (running far more terms than needed) and letting intermediate products grow in size without bound; the shift, term count, and per-step rounding are now chosen so the series converges quickly with bounded-size arithmetic. Results are unchanged to full precision.

• The Identities Library has been updated from 1,350 to 1,376 verified rules, including corrected Jacobi theta identities.

• Modular and theta-function identities now discharge under Im(τ) > 0. The upper-half-plane condition guarding these identities is expressed as the part inequality Im(τ) > 0, so they apply once you assume(Im(τ) > 0) (previously an opaque τ ∈ HH set membership was required). A new LaTeX shorthand, \mathbb{C}^+ (also \C^+), denotes the open upper half-plane: z \in \mathbb{C}^+ canonicalizes to Im(z) > 0. As a side effect three further identities became available — the derivative of the modular j-function and the θ₁/θ₂ logarithmic derivatives — recovered because the inequality form is verifiable where the opaque set was not.

• EisensteinE(s, τ) now evaluates numerically. The normalized Eisenstein series of even weight s ≥ 2 gets a numeric kernel (Lambert-series q-expansion in the upper half-plane), joining JacobiTheta/DedekindEta. For example EisensteinE(4, i).N() is 1.45576…, EisensteinE(2, i).N() is 3/π, and EisensteinE(6, i).N() is 0 (an elliptic fixed point). Exact arguments stay symbolic under evaluate(); the kernel requires Im(τ) > 0.

• New analytic-property metadata store. ce.functionProperties(name) exposes per-operator analytic properties drawn from the Fungrim corpus — poles, zeros, branch points and cuts, residues, and holomorphic/meromorphic domains. For example ce.functionProperties('Gamma')?.poles is the set NonPositiveIntegers. Convenience accessors (poles, zeros, branchCuts, holomorphicDomain, …) return the unconditional record of each kind; parametric records (such as residues that depend on parameters) are available via entries. This also powers pole-aware N() (see Behavior Changes).

Numeric Evaluation

• Arbitrary-precision elementary and transcendental functions are substantially faster, especially at hundreds or thousands of digits. High-precision π and trigonometric functions are no longer limited to about 2,350 digits. Square root is roughly twice as fast at 1,000+ digits (a giant-steps integer square root), the natural logarithm switches to the faster arithmetic–geometric-mean method from around 700 digits (previously ~1,250), and a power no longer recomputes the logarithm of its base on every call — at 1,000 digits Exp(x).N() is about three times faster, and a repeated base such as 2^x or 10^x about 2.8 times faster. Results are unchanged.

• Odd roots of negative real numbers now use the real-root convention, so Root(-8, 3) and (-8)^(1/3) evaluate to -2.

• N() of a non-unit rational power of a negative base no longer returns NaN. Previously only unit fractions worked (they route through Sqrt/Root); (-4)^{3/2}, (-8)^{2/3}, and similar fell through to Math.pow(negative, non-integer) = NaN. They now follow the same branch conventions as the roots above: an even denominator takes the principal complex value ((-4)^{3/2} = -8i, consistent with Sqrt(-4) = 2i), and an odd denominator the real root ((-8)^{2/3} = 4, (-8)^{5/3} = -32, consistent with (-8)^{1/3} = -2).

• Exact evaluate() of a non-unit rational power of a perfect power now reduces. When x^{p/q} has a real base and its q-th root is an exact perfect power, it reduces to an exact value (8^{2/3} = 4, 27^{2/3} = 9, (-8)^{5/3} = -32), extending the unit-fraction behavior (8^{1/3} = 2) to non-unit numerators and matching N(). Non-perfect powers (2^{2/3}) and the negative even-root branch ((-4)^{3/2}, complex) stay symbolic under evaluate().

• N() now fully evaluates applied functions and constants such as e, i, and expressions in Euler form.

• Complex equality and arbitrary-precision complex square roots are more robust in the presence of small rounding errors.

Collections and Matrices

• Take, Drop, Slice, and Count now operate on matrix rows consistently. For example, Count(matrix) returns the number of rows.

• Join now preserves list order, duplicates, and all elements when joining lists. Joining sets continues to produce a deduplicated set.

• Sums and products over ranges from -∞ to a finite bound, or from -∞ to ∞, now iterate over an appropriate finite approximation instead of an empty range.

Resolved Issues

• Significant performance boost when many boxed expressions are involved in computations, due to improved handling of configuration changes and listener management.

• Long-running evaluation is interruptible. Collection operations, number-theory functions, limits, differentiation, simplification, and integration now respect ce.timeLimit more consistently. Operations that cannot finish in time either throw CancellationError or return the best numeric estimate available, as appropriate.

• Fractional powers and radicals now preserve the correct principal complex branch. This fixes several unsafe transformations involving negative or unknown-sign values, including x/√(x²), negative factors under roots, products and quotients raised to fractional powers, and 1/√u.

• Infinity arithmetic is more reliable for finite symbolic denominators, while indeterminate forms such as ∞/∞ remain indeterminate.

• Numeric limits now reject overflow, catastrophic cancellation, oscillation, and other low-confidence results instead of returning spurious values.

• Fixed hangs and crashes when factoring certain sums, simplifying expressions with radical coefficients, or mixing non-finite rational values with arbitrary-precision integers.

• ce.number() now throws a helpful error when passed a MathJSON expression array; use ce.expr() for expressions.

• Fixed incorrect simplification or evaluation of 2^i, division by a floating-point zero coefficient, and several exact expressions involving negative radicals.

• Fixed a rational function such as 1/(x(x²+x)) wrongly simplifying (and integrating) to 0 when its factored denominator contained factors sharing a common root. The partial-fraction solver now detects the inconsistent system instead of returning a spurious all-zero decomposition.

• Factor is more complete: it now extracts a common monomial factor (e.g. x³+x² → x²(x+1), 3x⁴+2x³ → x³(3x+2)) and fully factors already-factored products and powers, so partial-fraction decomposition sees irreducible factors with correct multiplicities.

• Partial-fraction decomposition now uses exact arbitrary-precision integer arithmetic, so decompositions of higher-degree denominators no longer lose precision (the previous machine-integer solver overflowed past 2⁵³ and could return wrong coefficients).

• Rational functions with repeated linear or irreducible-quadratic factors now integrate to a closed form via full partial-fraction decomposition — e.g. ∫1/(x²(x+1)) dx and ∫1/(x(1+x²)²) dx, which previously returned an unevaluated integral.

• Nested powers serialize to LaTeX and round-trip correctly. A Power whose base is itself a Power — i.e. (aᵇ)ᶜ — was serialized as a^{bᶜ}, which re-parses as a^(bᶜ), a different expression. It now serializes as {aᵇ}^ᶜ, so e.g. (x³)^{2/5} round-trips instead of becoming x^{3^{2/5}}.

• GLSL/WGSL compilation no longer declares int/i32 for a Block's local bindings. An integer-valued local (e.g. ["Assign", "r", 3]) was declared as int r; while its value was emitted as a float literal (r = 3.0;), producing non-compilable shader code that also poisoned downstream float arithmetic. Scalar locals are now declared as float/f32 — consistent with the always-float number literals and scalar shader math — and an explicit ["Declare", "r", "complex"] type is honored. Complex locals still declare as vec2/vec2f.

• Loop now compiles to JavaScript that returns its collected values. A value loop such as Loop(i², Element(i, Range(1, 5))) compiled to a for-loop IIFE with no return, so it evaluated to undefined at runtime instead of the [1, 4, 9, 16, 25] the interpreter produces. The compiled loop now collects each iteration's value and returns the array. Imperative loops that mutate an outer accumulator or use Break/Continue/Return are unchanged.

• Integrate now compiles to JavaScript that returns a numeric estimate. For the common \int x^2 dx parse shape (where the integrand is a Function expression), the integrand was wrapped in a double lambda ((x) => ((x) => x*x)), so the Monte-Carlo estimator never called the inner function and returned NaN; it now compiles to a single lambda and returns the estimate (e.g. ∫₀¹ x² dx ≈ 0.333). Integration bounds are also no longer floored, so non-integer limits such as ∫₀^0.5 integrate over the correct interval.

0.59.0 2026-06-10

This is a significant update to the Compute Engine.

The headline feature of this release is a large collection of curated mathematical identities, the Identities Library:

// When the Identities Library is loaded, CE can prove that...
console.log(parse("\\arctan(2-\\sqrt{3})").simplify().latex);
// ➔ "\frac{\pi}{12}"

// Declare that n is a positive integer...
ce.declare("n", "integer");
ce.assume(parse("n > 0"));

// ...and the parity identity applies:
console.log(parse("\\sin(\\pi n + \\frac{\\pi}{2})").simplify().latex);
// ➔ "(-1)^n"

Read more about the Identities Library in the dedicated guide.

This release also includes a large collection of performance improvements and Resolved Issues across the library.

This release includes some breaking changes.

Breaking Changes

• replace() no longer eagerly canonicalizes the complete result. The requested form, or the form produced by the rule, applies to replaced subexpressions. Call .canonical on the result to restore the previous behavior.

• Fixed-size numeric collections now infer dimensioned types. For example, [1, 2, 3] is now vector<3> instead of list<number>, and a 3×3 numeric collection is matrix<3x3>.

Features

• Curated mathematical identities: the new opt-in loadIdentities() API loads over 1,300 guarded simplification rules and special values derived from Fungrim. Identities can be selected by topic, class, or purpose, and rules apply only when their side conditions can be proven.

  import { ComputeEngine } from '@cortex-js/compute-engine';
  import { loadIdentities } from '@cortex-js/compute-engine/identities';
  
  const ce = new ComputeEngine();
  loadIdentities(ce); // Or: loadIdentities(ce, { topics: ['gamma'] })
  ce.parse('\\Gamma(\\frac12)').simplify(); // → √π

  The loader is synchronous and idempotent per engine. Importing the identities subpath is required, so applications that do not use it incur no bundle cost.

  Simplifying with the full Identities Library loaded is now substantially faster: simplify() runs at roughly 1.2–1.3× the unloaded baseline (previously ~1.6×). The many guarded rules that share a common arithmetic head — Multiply, Add, Divide, … — are dispatched together per head instead of one at a time, so the per-rule overhead on every arithmetic node is paid once per head rather than once per rule. Results are unchanged.

• More control over replacements:

  • ReplaceOptions.form controls the form of replacement expressions: 'canonical', 'structural', 'raw', or a specific canonical transform. The previous canonical option is deprecated and remains available as an alias for this release.
  • ReplaceOptions.direction selects left-to-right or right-to-left traversal for order-sensitive rules.
  • Custom rules can now match user-defined function operators in replace() and simplify({ rules }).

• Improved algebra:

  • solve() now handles quadratics with symbolic coefficients, including x^2 - a x + 1 = 0 and the general a x^2 + b x + c = 0. (#300)
  • Factor infers the variable of a univariate polynomial and preserves extracted numeric content: Factor(x^2 + 5x + 6) returns (x+2)(x+3), and Factor(6x + 9) returns 3(2x + 3). (#309)

• Parsing improvements:

  • Two-argument \arctan(y, x) and \tan^{-1}(y, x) now parse as Arctan2.
  • LaTeX input is normalized to Unicode NFC, so decomposed identifiers parse like their precomposed equivalents.
  • A trailing bare \ and trailing visual spacing commands are tolerated.
  • Multi-character subscripted identifiers such as D_{etectsize} no longer collide with Euler derivative notation.

Resolved Issues

• Numeric evaluation and arithmetic:

  • Corrected complex powers, reciprocals, roots, and logarithms, including i^2, i^i, negative complex exponents, and even roots of negative reals.
  • Restored arbitrary-precision accuracy for roots, exp(), ln(), mod(), gammaln(), and large integer conversion. Very small real results such as Power(10, -100).N() are no longer rounded to zero.
  • Exact floor(), ceil(), and round() no longer lose digits beyond 2^53. Large decimal powers no longer report false overflow.
  • Division by zero, NaN * 0, infinity comparisons, and signed infinities now behave consistently across numeric representations.
  • Corrected Arctan2 quadrants, ln(Root(a, b)), non-integer logarithm bases, exact radicals such as sqrt(8), and division of Gaussian integers.

• Special functions and statistics:

  • Added complex Gamma and GammaLn evaluation. Gamma and factorial poles at non-positive integers now return ComplexInfinity, while factorials of positive non-integers evaluate through Gamma(x + 1).
  • Improved Erf/Erfc to machine precision and corrected small-argument gammaln().
  • Corrected GCD, LCM, Congruent, Subfactorial, negative-index Fibonacci, IsOctahedral, Multinomial, and BellNumber.
  • Corrected skewness, kurtosis, interquartile range, histogram/bin endpoints, and exact combinatorial calculations.

• Simplification, comparison, and assumptions:

  • Indeterminate comparisons now remain unknown instead of becoming false; this also improves sign inference, Boole, and KroneckerDelta.
  • Fixed equality handling for unordered expressions and multi-variable equation equivalence.
  • Prevented invalid simplification of rational powers such as (-x)^(3/4).
  • Set membership now remains undecided when a symbol's type is unknown, and Subset, SubsetEqual, Superset, and empty-set relations use the correct direction.
  • Symbolic common factors are now recognized, and unresolved derivatives remain symbolic instead of recursing indefinitely.

• Collections, matrices, and tensors:

  • Corrected Rest, Slice, Drop, Cycle, Position, SetFrom, TupleFrom, Filter, Zip, and compiled Reduce behavior.
  • Determinants now work for matrices of any supported size, with exact integer results; inverses work beyond 2×2.
  • Corrected matrix row access and the isUpperTriangular, isDiagonal, and isTriangular predicates.
  • Incompatible tensor broadcasts now throw instead of producing invalid data; diagonal() respects its axis arguments, and mixed real/complex dtype joins preserve precision.

• Types and serialization:

  • Dimensioned list and matrix type strings now parse and round-trip, including unknown dimensions, spaces, parenthesized element types, and single ^N dimensions.
  • Corrected union reduction, never subtyping, narrowing of disjoint types, bare matrix handling, numeric literal subtyping, and invalid range validation.
  • String literals now remain strings after MathJSON round-trips, dictionary conversion retains every entry, and function literals can be applied directly.
  • Plain symbols no longer report themselves as empty finite collections.

• LaTeX parsing and serialization:

  • Corrected scaled/big delimiters, nested \text{...}, repeating decimals beginning with ., digit-like symbol names, prefixed-symbol errors, and unbalanced environment names.
  • Multiplication signs are now emitted where juxtaposition would merge numeric factors, for example 3 \times 2^2 instead of 32^2. (#302)
  • Re-declaring a parser symbol with the same type no longer reports a conflict.

• Compilation:

  • Corrected JavaScript compilation of symbolic Range, compound-bounded interval Sum/Product, and interpreted fallback for multi-argument lambdas.
  • Corrected Python parentheses for (a^b)^c.
  • Corrected GLSL/WGSL output for Degrees, complex multiplication, Gamma/Factorial/Beta/Erf, and If/Which/When.
  • Corrected symbolic derivatives of Arcsec and Arccsc.

• Interval arithmetic:

  • Restored conservative enclosures for multiplication involving zero and infinity, negative-modulus mod, clamp, binomial, gcd, lcm, gamma, gammaln, sinc, and Fresnel integrals.

0.58.0 2026-05-12

Added

• \operatorname{count}(L) lowercase alias — function-call form now parses to ["Length", L], matching the existing dot-notation form (L.\operatorname{count}) and the other lowercase aliases (mod, var, shuffle, repeat, join).

• Repeat(value, count) 2-arg form — Repeat now accepts an optional integer count and evaluates to a finite list of count copies of value. The 1-arg Repeat(value) keeps its existing infinite-sequence semantics. Materialization is gated by ce.maxCollectionSize; larger values stay lazy (still accessible via .at() / iterator).

• ce.maxCollectionSize — new configurable cap (default 10_000) on the number of elements a collection may have when materialized into a concrete List. Assigning <= 0 or Infinity disables the cap (matching iterationLimit and recursionLimit).

• Sum(L) collection-reducer form — Sum now accepts a single collection argument and reduces to the sum of its elements: ["Sum", ["List", 1, 2, 3, 4, 5]] // ➔ 15. The big-op form Sum(body, [i, a, b], …) is unchanged. The Sum head is now preserved through canonicalization (previously rewritten to Reduce(L, "Add", 0)), so L.\operatorname{total} round-trips cleanly with latexOptions.dotNotation = true. The async path throws CancellationError on signal abort.

• At extended with boolean-mask and integer-list indices — At(L, mask) where mask is a finite collection of True/False returns the elements of L where the mask is True. At(L, indices) where indices is a finite collection of integers returns a sublist picked at those positions; out-of-range positions are filtered. Integer indices (At(L, 2)) and string keys (At(d, "key")) work as before.

• Function-application broadcasting for user-defined lambdas — when a user function with scalar-typed parameters is applied to a finite indexed collection, CE now broadcasts the call elementwise. For ce.assign('f', ce.parse('x \\mapsto x^2 + 1')), the expression ["f", ["List", 1, 2, 3]] evaluates to ["List", 2, 5, 10]. Multi-arg functions broadcast with zip semantics, mixing scalars and lists naturally. The inferred default for \mapsto lambdas is scalar parameters, so most user functions broadcast by default. To opt out, declare an explicit list parameter type via ce.declare(name, '(list<X>) -> Y').

• List type for mixed-kind and mixed-dimension elements — widen() now builds a structural union when the common supertype would otherwise collapse to a lossy generic category (scalar, value, list, tuple, dictionary, …). Consumers can detect heterogeneous lists by inspecting expr.type.toString():

  • [1, 2, 3] → list<number> (precise)
  • [1, "hello", 3] → list<finite_integer | string> (union)
  • [(1,2), (1,2,3)] → list<tuple<finite_integer, finite_integer> | tuple<finite_integer, finite_integer, finite_integer>> (mixed dimension)
  • [] → list<nothing> (empty)

• ce.expr(true) / ce.expr(false) — JS boolean primitives now box to the True / False symbols (previously fell through to Undefined).

• Length operator definition — ce.operatorInfo('Length') now returns a valid entry. The evaluator returns an integer count for finite collections and leaves the expression unevaluated for non-collection or infinite inputs.

• Library entries for Complex, Colon, Prime — ce.operatorInfo() now returns introspection data for these heads (previously undefined). Complex boxing is unchanged — ["Complex", re, im] still produces a BoxedNumber.

• ce.symbolInfo(name) — new public API parallel to ce.operatorInfo(), for introspecting constants and declared variables. Returns { kind: 'constant' | 'variable', type: BoxedType } for symbols like Pi, True, ExponentialE, ImaginaryUnit. Returns undefined for unknown names and for operator heads. Added SymbolInfo type to the public type surface.

  • Note: Infinity is registered as PositiveInfinity / NegativeInfinity; Undefined has no value definition.

• ce.normalizeIdentifier(latex) — new public helper that converts a LaTeX identifier string to its canonical MathJSON name without side effects. Examples: R_{3} → R_3, f_{Bm} → f_Bm, \theta_x → theta_x. Inputs that aren't identifiers ('1 + 2', empty string) return ''. Useful in importer pipelines that need to call ce.declare() with normalized names before parsing referencing rows.

• First/Second/Third compile entries — component access (p.x, p.y, p.z) now compiles cleanly. JS uses [0]/[1]/[2] index access; GLSL/WGSL use .x/.y/.z swizzles, assuming the argument compiles to a vec2/vec3/vec4. 5+-element tuples (which compile to float[N] arrays) aren't supported.

• Range GPU compile entry — Range(lo, hi[, step]) with compile-time-constant bounds emits an inline float[N](...) (GLSL) or array<f32, N>(...) (WGSL) literal. Non-constant bounds throw a clear error directing the caller to materialize on the JS host and upload as a uniform. Sequence count is capped at 256 elements per call site.

• Variance/GCD/Median GPU compile entries — GLSL+WGSL parity with their JS counterparts.

  • Variance is inlined (no size limit).
  • GCD uses a preamble function implementing the Euclidean algorithm.
  • Median is supported for list sizes 2–8; lists with 9+ elements throw.

• Random GPU compile with deterministic seed — Random(seed) in GLSL/WGSL compiles to a hash-based pseudorandom. Random() (no args) in GLSL falls back to a gl_FragCoord-derived seed (fragment-shader only); in WGSL it throws — callers must provide an explicit seed.

  • The fract-sin hash exhibits banding near seed ≈ kπ. For high-quality shader random, use a more robust hash (e.g. PCG or xxHash).
  • JS-side Random is unchanged (still Math.random, non-seeded). A seeded JS form will land in a future release.

• toSignedFunction() — new method on BoxedExpression for implicit-surface rendering and region classification:

  • Equal(a, b) → a - b (zero on the surface)
  • Less(a, b) / LessEqual(a, b) → a - b (negative when relation holds)
  • Greater(a, b) / GreaterEqual(a, b) → b - a (negative when relation holds)
  • NotEqual(a, b) → a - b
  • Non-relation expressions return undefined.

  Strictness and direction are encoded in expr.operator. Note that CE canonical form normalizes GreaterEqual to LessEqual(b, a) (and similarly Greater to Less), so callers will typically see the Less/LessEqual operator on parsed expressions — the signed-function semantics are preserved.

• BoxedExpression.getInterval(symbol) — new method for extracting domain bounds from restriction expressions. Returns IntervalBounds with lower/upper/lowerStrict/upperStrict for When(e, cond), And(c1, c2, …), and bare comparison expressions; returns undefined for unsupported shapes. Useful for 2D-plot domain derivation (e.g. clipping y = f(x)\{0 < x < 5\} to [0, 5]). Added IntervalBounds type to the public type surface.

• Compact piecewise \{cond_1 : val_1, …, default\} — now parses to Which(c_1, v_1, …, True, default), the same head CE produces for \begin{cases}…\end{cases}. Disambiguated from set-builder \{x : type\} by inspecting the LHS of the top-level Colon: comparison/boolean heads (Less, Greater, Equal, And, Or, Not, …) → piecewise branch; otherwise → set-builder. Normal set literals (\{1, 2, 3\}) and set-builder via \mid are unchanged.

Fixed

• Linspace endpoint inclusion — Linspace(a, b, n) now produces n points evenly spanning [a, b] inclusive of both endpoints (matching NumPy, Julia, and MATLAB). Previously the last sample fell short of b (e.g. Linspace(0, 1, 5) yielded 0, 0.2, 0.4, 0.6, 0.8 instead of 0, 0.25, 0.5, 0.75, 1). Linspace(a, b, 1) is the degenerate case and returns just a. The contains check is now tolerance-based (was an exact % test that failed for typical floating-point values).

• Heterogeneous-list type rendering — lists containing mixed kinds or mixed-dimension tuples previously rendered their type as "[object Object]" in some paths (BoxedDictionary.type, collectionElementType). Types are now constructed programmatically. Lists containing tuples, sets, dictionaries, records, or strings are no longer misclassified as numeric BoxedTensors.

Known issues

• JS Loop compile produces undefined — the imperative for-loop IIFE generated for Loop(body, Element(i, Range(lo, hi))) has no return statement, so the compiled function returns undefined rather than the list of body values. Tracked for a future release.

• JS Integrate compile produces NaN — when args[0] is a Function expression (the common \int x^2 dx parse shape), compileIntegrate produces a double-lambda, so _SYS.integrate receives a function-returning function. Tracked for a future release.

0.57.0 2026-05-10

Added

• verbatim opt-in for toLatex() — expr.toLatex({ verbatim: true }) returns the original LaTeX source captured at parse time when the expression was parsed with preserveLatex: true. Falls back to normal re-serialization if no verbatim is available (e.g. for synthetic or transformed expressions). The default behavior of expr.latex and expr.toLatex() is unchanged — verbatim is strictly opt-in. Useful for round-tripping authored LaTeX (e.g. p.x, \sin(x)) without rewriting it to canonical form.

  • Verbatim is set only on the top-level boxed expression produced directly by ce.parse(..., { preserveLatex: true }). Canonicalization, simplify(), evaluate(), subs(), and ce._fn() produce fresh expressions with verbatimLatex === undefined.
  • Function expressions whose operator has a custom canonical handler (e.g. Sin, Add) currently do not preserve top-level verbatim through canonicalization — the handler reconstructs the result without threading metadata. Atoms (symbols, numbers) and functions without custom canonical handlers (e.g. First) do preserve it. Use form: 'structural' to skip canonical handlers when verbatim preservation matters.

• dotNotation serialization option — when enabled (default off), member-access heads serialize to dot notation rather than function-call form: First(p) → p.x, Length(L) → L.\operatorname{count}, etc. Useful for round-tripping editor-authored dot-notation back to its source form. Set via ce.latexOptions.dotNotation = true or per-call expr.toLatex({ dotNotation: true }). Only applies to arity-1 forms; multi-operand forms (e.g. Sum with an index range) keep their standard serialization.

  • Serializer-only. The flag lives in SerializeLatexOptions and has no effect on parsing. All input forms continue to parse as before regardless of the flag: |L|, \operatorname{count}(L), L.\operatorname{count}, \operatorname{length}(L) all still parse to ["Length", L] whether dotNotation is on or off. The flag only decides which form the serializer emits.

• Component access (p.x, L.\operatorname{count}, z.\operatorname{re}) — dot notation now parses to existing semantic heads at parse time. No generic accessor head was introduced.

  • Recognized members and their AST mapping: x/y/z → First/Second/Third; real/re → Real; imag/im → Imaginary; count → Length; total → Sum; max → Max; min → Min.
  • Disambiguation: after a terminated integer or decimal, . followed by a letter or \operatorname{...} is component access, not a decimal point. Examples: 1.x parses as ["First", 1] (not a malformed decimal); 1.5.x parses as ["First", 1.5].
  • Only \operatorname{...} and bare-letter identifiers are recognized after .. \mathrm{...} is not accepted (deliberately tight).
  • Third is a new operator (parallels First/Second) with signature (any) -> any. First/Second were widened from (collection) -> any to (any) -> any so component access on a non-collection (e.g. 1.x) defers type-checking to evaluation; evaluation returns an Error expression for incompatible types.

• Restriction braces (expr\{cond\}) — trailing brace predicates parse to a new When head.

  • f(x)\{0 < x < 2\} → ["When", ["f", "x"], ["Less", 0, "x", 2]].
  • Stacked restrictions canonicalize: expr\{c_1\}\{c_2\} → ["When", expr, ["And", c_1, c_2]]. Downstream simplification, evaluation, interval intersection, and compilation see a single canonical shape regardless of source form.
  • Disambiguation from set literals is positional: standalone \{1, 2, 3\} continues to parse as a Set; <expr>\{cond\} parses as a When restriction. Allowed left operands include function calls, tuples, list/set literals, bare symbols, subscripted symbols, member access, power expressions, and chained restrictions.
  • Evaluator semantics: When(e, True) evaluates e; When(e, False) returns Undefined; indeterminate cond holds the form.
  • Serializer round-trips to the stacked-brace form (not \wedge inside one set of braces) so authored source and re-serialized output stay visually consistent.
  • JS and GLSL compilation: ternary (cond ? e : NaN).

• List-range ellipsis ([1...9], [0, 0.1, ..., 1]) — ranges inside list literals parse to the existing Range head.

  • Endpoint-only form: [a...b] → ["Range", a, b]. Triggers ..., \ldots, and \dots are all accepted.
  • Inferred-step form: [a_0, a_1, ..., a_n] → ["Range", a_0, a_n, step] where step = a_1 - a_0 is inferred from the first sample pair. Intermediate samples are validated against a_0 + k·step within ce.tolerance; inconsistent samples produce a parse error.
  • The float idiom [0, 0.1, 0.2, ..., 1] is supported (tolerance-aware comparison; 0.1 + 0.1 ≠ 0.2 exactly but is accepted within tolerance).
  • Outside [...] brackets, \ldots/\dots/... continue to parse as the ContinuationPlaceholder symbol. The trigger is bracket context.

• For-comprehensions ((x, y) \operatorname{for} x=L_1, y=L_2) — the Loop head now accepts multiple Element clauses, evaluated as nested loops with later bindings seeing earlier ones in scope.

  • Loop(body, Element(x, L_1), Element(y, L_2), ...) produces an indexed_collection<T> of body evaluations, in row-major order.
  • For independent bindings this is the Cartesian product: (x, y) \operatorname{for} x = [1...2], y = [1...2] → 4 tuples.
  • For dependent bindings later clauses see earlier: (x, y) \operatorname{for} x = [1...3], y = [1...x] → 6 tuples (triangle, not Cartesian).
  • Precedence: \operatorname{for} binds looser than , and =, tighter than ;. So (x + y) \operatorname{for} x = L_1, y = L_2 parses with body x + y and two bindings.
  • Bound names do not leak into the enclosing scope (uses Scope.noAutoDeclare).
  • Legacy single-Element form continues to round-trip via the existing \text{for } i \text{ from } a \text{ to } b \text{ do } body syntax. Multi-Element comprehensions serialize to the \operatorname{for} form.

• Range type is now dynamic — element type narrows based on the step argument: integer step (or no step) yields indexed_collection<integer>; non-integer step yields indexed_collection<number>. Previously the type was always indexed_collection<integer>, which was incorrect for float-step ranges.

• When head — new conditional-value operator. When(expr, cond) returns expr when cond is true, Undefined when cond is false, and holds when cond is indeterminate. Used by restriction-brace parsing (see above) but also usable directly.

• ce.operatorInfo(head) — new method on ComputeEngine for introspecting registered operator heads. Returns { kind: 'function' | 'opaque', signature?: BoxedType } or undefined.

  • 'function' — head has an evaluate handler or a collection handler (lazy producers like Range, Linspace, Tuple work via the latter).
  • 'opaque' — head is declared with a signature but has neither (e.g., Triangle, Sphere, GeometricVector).
  • undefined — no operator definition (constants like Pi and unknown heads).
  • Lets external tooling classify heads by capability without maintaining a parallel list of supported operators.

• tolerance in ParseLatexOptions — populated automatically from ce.tolerance when parsing through ce.parse(). Used by list-range sample validation; available to other parse handlers that need tolerance-aware comparison.

Fixed

• Loop with Element clause — single-Element Loop(body, Element(i, range)) previously did not produce a list of body evaluations (the iteration path for Element form had a bug). The new variadic evaluator correctly yields a List of body values for each iteration.

0.56.0 2026-03-10

Added

• First-class color values — colors are now typed values with a dedicated color primitive type and per-colorspace constructor heads, rather than anonymous tuples.

  • Constructor heads: Rgb, Hsv, Hsl, Oklab, Oklch. Each takes 3 components plus an optional alpha. Channels follow each colorspace's own conventions (RGB: 0–1 sRGB; HSV/HSL: hue in degrees, S/V/L 0–1; Oklab/Oklch: standard ranges).
  • LaTeX: \operatorname{rgb}(...), \operatorname{hsv}(...), \operatorname{hsl}(...), \operatorname{oklab}(...), \operatorname{oklch}(...), parsing and serialization both directions.
  • Conversions: AsRgb, AsHsv, AsHsl, AsOklab, AsOklch convert any color to the named space (identity if already there).
  • ColorDelta(a, b) — perceptual color difference (ΔE_OK, Euclidean distance in OKLab). Wide-gamut inputs are not clipped before measurement.

• JavaScript compile-target support for color values — all color constructors, the As* converters, ColorDelta, and Distance are supported. At runtime a color is a 3- or 4-element OKLCh array ([L, C, H] or [L, C, H, alpha]), matching the GPU target's vec3/vec4 representation, so values move between JS, GLSL, and WGSL without conversion.

• Distance(p1, p2) — Euclidean distance between two points represented as tuples. Accepts any positive dimension; mismatched dimensions return a typed error. LaTeX trigger \operatorname{distance}(p1, p2).

• Geometric primitive heads — Triangle, Sphere, Segment, and GeometricVector are now recognized as typed function heads (no evaluator, preserved structurally for downstream consumers). LaTeX triggers \operatorname{triangle}, \operatorname{sphere}, \operatorname{segment}, \operatorname{vector}(p1, p2). GeometricVector is distinct from the existing Vector (column-vector construction).

• To head registered — \to already parsed to ["To", a, b] but was classified as unsupported-operator; it is now a known typed head.

• Function-style aliases — lowercase \operatorname{...} forms common in Desmos-style notation now parse to their existing capitalized operators: \operatorname{mod} → Mod, \operatorname{var} → Variance, \operatorname{shuffle} → Shuffle, \operatorname{random} → Random, \operatorname{repeat} → Repeat, \operatorname{join} → Join.

• ce.latexOptions — new mutable, engine-wide bag of LaTeX parse/serialize options (e.g. decimalSeparator, digitGroupSeparator). Available as a constructor option and as a read/write property:

  const ce = new ComputeEngine({ latexOptions: { decimalSeparator: '{,}' } });
  // or post-construction:
  ce.latexOptions = { decimalSeparator: '{,}' };

  These options are merged into every ce.parse() and expr.toLatex() call. Precedence (most-specific wins): LatexSyntax instance defaults < ce.latexOptions < per-call options. Previously, options like decimalSeparator could only be changed per call post-construction (and expr.latex could not be customized at all).

Changed

• Color('...') now returns an Oklch head instead of a 0–1 sRGB Tuple. The string parser still accepts the same set of CSS-style inputs.
• ColorMix now returns an Oklch head and mixes in OKLCh directly, preserving out-of-gamut chroma. Hue interpolation takes the shortest path around the wheel; mixing with an achromatic endpoint carries the other endpoint's hue (matches CSS Color 4 color-mix).
• ContrastingColor now returns an Rgb head (was: 0–1 sRGB Tuple).
• Colormap now returns Oklch heads — either a List(Oklch, ...) or a single Oklch for position-sampling.
• ColorToString with 'oklch' format serializes typed color inputs without an sRGB round-trip; out-of-gamut chroma serializes losslessly. 'hex'/'rgb'/'hsl' paths are unchanged.
• Color-consuming signatures tightened — (any, any) → (color | string | tuple, color | string | tuple) for ColorDelta, ColorContrast, ColorMix, ContrastingColor, ColorToString, ColorToColorspace. The As* converters take (color) -> color.

Migration notes

Code that consumed the tuple output of Color('...'), ColorMix, ContrastingColor, or Colormap now sees a typed color head. To get the previous 0–1 sRGB shape, wrap with AsRgb:

// Before: const tuple = ce.expr(['Color', "'red'"]).evaluate();  // [r, g, b] in 0-1
// Now (equivalent 0-1 sRGB):
const rgb = ce.expr(['AsRgb', ['Color', "'red'"]]).evaluate();
// rgb is ['Rgb', r, g, b] with channels 0-1

Rgb head components are 0–1 sRGB across all layers (engine, JS compile, GPU compile).

Fixed

• Super-linear parse time on deeply-nested parametric expressions — ce.parse() could exhibit exponential blowup on inputs like nested rotation matrices \left(\cos(\theta)\cdot S+\sin(\theta)\right) (depth 6 took ~44s). Two underlying causes were addressed: the type/sign cache on BoxedFunction was effectively disabled (causing every .type access to recurse through all operands), and parseEnclosure was speculatively trying matchfix definitions whose close-delimiter token wasn't even present in the input. Parse time on the affected inputs is now linear.

• ce.parse() ignored the injected LatexSyntax instance's decimalSeparator — ce.parse() hardcoded decimalSeparator: '.', silently overriding any value configured on a LatexSyntax passed via the constructor's latexSyntax option. The injected instance's configured separator now takes effect end-to-end.

• expr.toMathJson({ metadata: ['latex'] }) was silently dropped — passing a metadata array of specific fields (e.g. ['latex'] or ['wikidata']) was ignored; only metadata: 'all' worked. The array form now correctly populates the requested fields.

• expr.toMathJson({ shorthands: ['all'] }) disabled all shorthands — the ['all'] array form had the opposite of its intended effect. The string form 'all' and explicit lists like ['function'] were unaffected.

0.55.6 2026-03-08

Resolved Issues

• LaTeX parsing: \lim with postfix operators — \lim_{x\to 0}\left(x\right)^x now correctly parses as Limit(x^x) instead of Power(Limit(x), x). The \lim parser was using parseArguments('implicit') which stripped the delimiters and left the ^x unconsumed; it now uses parseExpression so postfix operators are included in the limit body.

• LaTeX parsing: style, size, and color switch commands — \displaystyle, \textstyle, \scriptstyle, \scriptscriptstyle, \tiny..\Huge (10 size commands), and \color{...} were silently discarded during parsing. They now produce Annotated expressions that preserve the styling information and round-trip correctly through serialization. Added \scriptstyle / \scriptscriptstyle serialization support (previously only \displaystyle and \textstyle were handled).

• LaTeX parsing: set-builder notation — \{x \in \R \mid x > 0\} now parses to ["Set", expr, ["Condition", cond]]. Registered \mid as an infix operator (Divides, precedence 160). The serializer round-trips set-builder notation correctly.

• LaTeX serialization: Complement — ["Complement", "A"] now serializes to A^\complement instead of falling back to the generic function form. Removed stale @todo comments about a non-existent multi-argument case.

• LaTeX parsing: spacing commands — \hspace{dim}, \hspace*{dim}, \hskip, and \kern are now consumed during parsing (previously caused "unexpected token" errors). These are treated as visual spacing and skipped.

• LaTeX serialization: HorizontalSpacing math classes — the 2-argument form ["HorizontalSpacing", expr, "'bin'"] now serializes to \mathbin{expr} (and similarly for rel, op, ord, open, close, punct, inner). Previously the second argument was silently dropped.

• LaTeX serialization: redundant parens on matchfix operators — wrap() no longer adds parentheses around Abs, Floor, Ceil, Norm, and other matchfix expressions that already have visible delimiters.

• LaTeX serialization: tabular environments — default environment serializer now renders matrix bodies (List of Lists) with & column separators and \\ row separators instead of nested function calls.

• LaTeX serialization: matchfix delimiter scaling — default matchfix serializer now respects groupStyle to choose between bare delimiters, \left..\right, or \bigl..\bigr scaling.

• LaTeX parsing: Greek symbols in string groups — \alpha, \beta, etc. in parseStringGroupContent() (used by \begin/\end, color arguments) are now interpreted as their Unicode equivalents instead of passing through as raw LaTeX commands.

0.55.5 2026-03-06

Resolved Issues

• Deep-zoom fractal precision — emulated-double (dp) and perturbation (pt) shaders now compute per-pixel coordinates from v_uv and viewport uniforms instead of the shader template's single-precision mix(), which lost distinguishability at high zoom levels.
• Perturbation theory: absolute vs delta coordinates — the perturbation Mandelbrot/Julia handlers were passing absolute single-precision coordinates to the shader instead of the small delta from the reference center. Fixed by introducing _pt_delta() which computes the per-pixel offset from viewport uniforms.
• compile() free function dropped hints — the hints option (viewport center/radius) was accepted but silently not forwarded to the language target. Fixed in compile-expression.ts.

New Features

• BigDecimal export — the arbitrary-precision decimal class is now exported from the public API for use by plot engines and other consumers that need precision beyond float64.
• HighPrecisionCoord type — new union type (number | string | { hi: number; lo: number }) for passing extended-precision viewport coordinates through the compile API. The viewport.center option now accepts this type instead of plain [number, number].

0.55.4 2026-03-06

Resolved Issues

• #254 LaTeX parsing: interval notation with \lbrack/\lparen — parsing \lbrack5,7) or \left\lbrack5,7\right) now correctly produces an Interval expression. Previously, when the open delimiter was a LaTeX command (e.g., \lbrack), the parser incorrectly required the close delimiter to also be a LaTeX command (e.g., \rparen instead of )), causing mismatched-delimiter intervals to fail.
• LaTeX parsing: invalid symbols in \mathrm{} and related prefixes — invalid content inside \mathrm{}, \operatorname{}, etc. (e.g., \mathrm{=} or \mathrm{DavidBowie👨🏻‍🎤}) now produces the correct invalid-symbol error instead of cascading parse errors. Also fixed matchPrefixedSymbol leaking parser state on failure, and emoji sequences are now properly recognized inside symbol prefixes (e.g., \operatorname{😎🤏😳🕶🤏}).

New Features

• High-precision Mandelbrot/Julia compilation — the GPU compilation targets (GLSL, WGSL) now support three precision tiers for fractal rendering, selected automatically based on viewport hints:
  • Single float (zoom < 10^6x): existing implementation, no overhead
  • Emulated double (zoom 10^6x–10^14x): double-single (float-float) arithmetic using Dekker/Knuth algorithms, ~48-bit mantissa from two 32-bit floats
  • Perturbation theory (zoom > 10^14x): reference orbit computed on CPU at arbitrary precision via BigDecimal, GPU iterates only the small delta from the reference, with glitch detection and single-float rebase fallback
• Viewport-aware compile API — compile() accepts optional hints: { viewport: { center, radius } }. The compiler auto-selects the precision strategy and returns staleWhen thresholds for cheap staleness checking by the plot engine.
• CompilationResult extensions — new optional fields: staleWhen (plain data staleness predicate), uniforms (scalar shader uniforms), textures (typed texture data with format/dimensions for GPU upload).

0.55.3 2026-03-05

Improved

• Compilation: constant folding — Add, Multiply, Subtract, Negate, Divide, Power, Sqrt, and Root handlers now fold numeric literals at compile time and eliminate identity values.
  • x + yi compiles to vec2(x, y) instead of vec2(x, 0.0) + (y * vec2(0.0, 1.0))
  • 2 + 3 → 5.0, x + 0 → x, x * 1 → x, x * 0 → 0.0
  • Power(x, 2) → (x * x) for simple operands, pow(f(x), 2.0) for complex expressions to avoid duplicate computation
  • Power(x, 0.5) → sqrt(x), Power(x, 0) → 1.0, Power(x, -1) → (1.0 / x)
  • Sqrt(4) → 2.0, Root(x, 2) → sqrt(x)
• isComplexValued uses expression type system instead of hard-coded operator list.
• Integer arguments in GPU fractal functions emit as 200 instead of int(200.0).
• Type-based optimizations — compilation handlers now use expression type information for better code generation:
  • Floor/Ceil/Round/Truncate are no-ops when the operand is integer-typed
  • Abs is a no-op when the operand is provably non-negative
  • Power(x, 2) only expands to (x * x) for simple operands (symbols, literals) — function calls like Power(Sin(x), 2) use pow/Math.pow to avoid duplicate evaluation
  • Integer Mod with non-negative dividend uses plain % instead of the Euclidean double-mod formula
  • GPU variable declarations infer i32/int type for integer-typed locals

Resolved Issues

• Abs signature: return type is now real instead of propagating the input type (which incorrectly returned complex for complex inputs).
• Compilation fallback: uses pushScope/assign pattern instead of crashing when receiving a vars object.

New Features

• Mandelbrot and Julia operators in JavaScript and GPU compilation targets.

0.55.2 2026-03-04

Resolved Issues

• \text{} flush bug: \text{a$x$b} now correctly produces ["Text", "'a'", "x", "'b'"]. Previously the text before and after inline math were merged due to a missing flush() call in parseTextRun.
• # / * parsed as valid symbols: Bare # and * tokens were incorrectly accepted as valid symbol names because they match the Unicode Emoji property (keycap base characters). They now produce unexpected-token errors as expected. The fix excludes ASCII characters from the emoji regex in symbol validation.
• Text operator type: The Text operator now has return type string instead of expression.
• \textcolor inside \text{}: \textcolor{red}{RED} inside \text{} now correctly parses the body as text ('RED') instead of switching to math mode and treating each letter as a separate symbol.
• parseSyntaxError token consumption: Non-command tokens (like #, &) are now consumed when producing errors, preventing potential parser loops.
• parseSymbolToken hardening: Raw tokens are pre-validated against \p{XIDC} before being consumed as symbols, providing defense-in-depth against future isValidSymbol regressions.

New Features

• Text promotion: When InvisibleOperator canonicalization encounters a Text expression or a string operand, it now absorbs all operands into a single Text expression. For example, a\text{ in $x$ }b canonicalizes to ["Text", "a", " in ", "x", " ", "b"] instead of producing a Tuple.
• Text infix keywords: \text{and}, \text{or}, \text{iff}, and \text{if and only if} are now recognized as infix operators that produce And, Or, and Equivalent expressions respectively, following the existing \text{where} pattern.
• Additional text keywords: \text{such that} (maps to Colon), \text{for all} (maps to ForAll), and \text{there exists} (maps to Exists) are now recognized as operators.
• Text serializer: Text expressions now round-trip back to proper \text{...} LaTeX with inline $...$ for math sub-expressions, instead of falling through to the default \mathrm{Text}(...) output.
• Text evaluate handler: Evaluating a Text expression now concatenates all operands into a single string.

0.55.1 2026-03-04

Resolved Issues

• After parse('f(x):=\\sin(x)'), the symbol f is now immediately recognized as having type function. Previously its type remained unknown until the Assign expression was explicitly evaluated.
• 2f(x) and 2f \left(x\right) now both correctly parse as ["Multiply", 2, ["f", "x"]] when f is a known function symbol. Previously, a space before \left caused the parser to produce a Tuple instead of Multiply, and expressions whose return type was any (e.g., calls to generically-typed functions) were also misclassified as Tuple.
• Expressions involving operators that return expression type (such as D, Simplify, Annotated) are now correctly treated as multiplicable in juxtaposition contexts. For example, 2f'(x) now produces ["Multiply", 2, ["D", ...]] instead of Tuple.
• The D (derivative) operator now returns a numeric type when its body is numeric, instead of always returning the generic expression type.
• Undeclared symbols followed by parenthesized multi-argument expressions (e.g., 2g(x,y)) are now auto-declared as functions in all invisible operator paths, not just the two-operand path.

0.55.0 2026-03-04

Breaking Changes

• ce.box()/box() renamed to ce.expr()/expr() (ce.box() remains as a deprecated wrapper).
• Removed ce.latexDictionary getter/setter; configure dictionaries through new LatexSyntax({ dictionary: [...] }).
• Removed ComputeEngine.getLatexDictionary(); import dictionary constants from package exports.
• Removed deprecated type guard aliases: isBoxedExpression, isBoxedNumber, isBoxedSymbol, isBoxedFunction, isBoxedString, isBoxedTensor (use isExpression, isNumber, isSymbol, isFunction, isString, isTensor).
• Removed LibraryDefinition.latexDictionary; LaTeX dictionaries now live in the latex-syntax module.

Resolved Issues

• #295 The parse() free function now accepts the form options object, so parse("\\frac{10}{2}", { form: "raw" }) return ["Divide", "10", "2"].
• Undeclared symbols followed by parenthesized numeric expressions are now interpreted as multiplication, not implicit function calls (for example, q(2q) -> 2q^2). Function-call behavior remains for explicitly declared function symbols and non-numeric argument forms.

New Features

• Modular package exports for smaller bundles: @cortex-js/compute-engine/core, @cortex-js/compute-engine/compile, @cortex-js/compute-engine/latex-syntax, @cortex-js/compute-engine/numerics, and @cortex-js/compute-engine/interval (with existing sub-paths still available, including math-json).
• New standalone LatexSyntax API (class + parse()/serialize() helpers) for LaTeX ↔ MathJSON without a ComputeEngine instance.
• New ILatexSyntax interface exposed via IComputeEngine.latexSyntax to allow custom LaTeX parser/serializer implementations.
• All 16 LaTeX domain dictionaries are now exported individually, plus the combined LATEX_DICTIONARY.
• Parser type is now exported from the main package for typed custom LatexDictionaryEntry parse handlers.

Changed

• ComputeEngine now accepts an injectable latexSyntax dependency.
  • Full package imports still auto-create a LaTeX syntax instance.
  • Core-only imports do not bundle LaTeX support; parse(), .latex, and toLatex() require an injected LatexSyntax.
  • MathJSON serialization omits optional LaTeX metadata when no LaTeX syntax is present.
• decimal.js has been replaced with a native bigint-backed BigDecimal implementation, reducing dependency surface and bundle size.
• BigDecimal add(), sub(), and mul() are now exact; rounding is limited to operations that require it (div(), non-integer pow(), transcendentals).
• Numeric string/LaTeX serialization now respects precision settings: .latex/.toString() round to ce.precision, while .json/toJSON() remain lossless.
• High-precision special functions (bigGamma, bigGammaln, bigDigamma, bigTrigamma, bigPolygamma, bigZeta) now scale with BigDecimal.precision; integer Gamma values are exact.

0.54.0 2026-02-26

• New expr.polynomialCoefficients() method: Returns the coefficients of a polynomial expression in descending order of degree, or undefined if the expression is not a polynomial. Auto-detects the variable when the expression has exactly one unknown. Subsumes isPolynomial (check !== undefined) and degree computation (length - 1).

• polynomialCoefficients() now accepts an array of variables: Pass ['x', 'y'] to verify the expression is polynomial in all listed variables. Coefficients are decomposed by the first variable.

• New expr.polynomialRoots() method: Returns the roots of a polynomial expression, or undefined if not a polynomial. Handles degree 3+ polynomials with rational roots via the Rational Root Theorem.

• New Polynomial CAS function: Constructs a polynomial from a coefficient list (descending order) and a variable. Inverse of CoefficientList: Polynomial([1, 0, 2, 1], x) evaluates to x³ + 2x + 1.

• Improved Factor for degree 3+ polynomials: Factor now uses the Rational Root Theorem to factor polynomials with integer coefficients and rational roots. Previously only handled degree ≤ 2.

• Improved Factor with content extraction: Factor now extracts the GCD of integer coefficients before applying other strategies. For example, Factor(6x² + 12x + 6, x) now produces 6(x+1)².

• New PartialFraction CAS function: Decomposes rational expressions into partial fractions. Supports distinct and repeated linear factors, irreducible quadratic factors, and improper fractions (polynomial division performed first). Example: PartialFraction(1/((x+1)(x+2)), x) → 1/(x+1) - 1/(x+2).

• New Apart CAS function: Alias for PartialFraction.

• New PolynomialRoots CAS function: Returns the roots of a polynomial as a set. Example: PolynomialRoots(x² - 5x + 6, x) → {2, 3}.

• New Discriminant CAS function: Returns the discriminant of a polynomial of degree 2, 3, or 4. Supports symbolic coefficients. Example: Discriminant(x² - 5x + 6, x) → 1.

• simplify() auto-decomposes partial fractions: When a Divide expression has a denominator already in factored form (product or power) and the decomposition is simpler, simplify() automatically applies partial fraction decomposition.

• Breaking: CoefficientList now returns descending order: The CAS function CoefficientList now returns coefficients from highest to lowest degree (e.g., [1, 0, 2, 1] for x^3 + 2x + 1), matching the new polynomialCoefficients() method and common external conventions. Previously it returned ascending order.

• expr.match() now accepts string patterns with auto-wildcarding: Pass a LaTeX string like 'ax^2+bx+c' and single-character symbols are automatically treated as wildcards. Results use clean unprefixed keys ({a: 3, b: 2, c: 5}) with self-matches filtered out. useVariations and matchMissingTerms default to true for string patterns.

• expr.match() now accepts MathJSON arrays directly: Pass a raw MathJSON pattern like ['Add', '_a', '_b'] without calling ce.box() first.

• New matchMissingTerms option for match(): When enabled, expressions with fewer operands than the pattern can still match by treating missing terms as identity elements (0 for Add, 1 for Multiply). For example, 3x^2+5 matches the pattern ax^2+bx+c with b = 0. Enabled by default for string patterns.

• Non-strict parsing: implicit superscript for letter+digit: In non-strict mode, a single letter immediately followed by a digit 2–9 is parsed as an exponent: x2 + y2 → x^2 + y^2. Handles common copy-paste from web pages. Only digits 2–9, only single ASCII letters, and only when adjacent (no space).

0.53.1 2026-02-25

• timeLimit now reliably interrupts long-running evaluations: Factorial, Sum, Product, Loop, and Reduce all respect the timeLimit property and throw CancellationError when the deadline is exceeded. Previously, generators yielded too infrequently (every 1,000–50,000 iterations), allowing a single gen.next() call to block for longer than the timeout. All generators now yield every iteration. The Factorial handler no longer silently swallows CancellationError, and withDeadline/withDeadlineAsync now use try/finally to always reset the engine deadline.

• Fixed GPU compilation of Sum, Product, Loop, and Function: These constructs no longer leak JavaScript-specific syntax (IIFEs, let, while, arrow functions, { re, im } objects) into GLSL/WGSL output. Sum/Product with small constant bounds are unrolled inline; larger ranges emit native for loops. Loop emits a GPU for loop with int/i32 index. Function (lambda) now throws a clear error for GPU targets. Block-level Declare statements infer vec2/vec2f type from subsequent complex-valued assignments.

• Added GLSL/WGSL compilation for Heaviside, Sinc, FresnelC, FresnelS, BesselJ: These five special functions now compile to GPU shader targets. FresnelC/FresnelS use a three-region rational Chebyshev approximation (ported from Cephes/scipy) with a shared _gpu_polevl helper. BesselJ uses power series, Hankel asymptotic, and Miller's backward recurrence depending on the argument range. Both GLSL and WGSL preambles are emitted on demand.

• Fixed GLSL/WGSL block expression compilation: Block expressions (produced by \coloneq / semicolon blocks) now emit valid GPU shader code instead of JavaScript syntax. Variable declarations use float x (GLSL) or var x: f32 (WGSL) instead of let x, and blocks are emitted as plain statements instead of JavaScript IIFEs. compileFunction correctly formats multi-statement bodies.

• Fixed \; in \text{where} clauses: Visual spacing commands like \;, \,, \quad, etc. between comma-separated bindings in where-clauses are now correctly skipped instead of being parsed as HorizontalSpacing expressions wrapped in InvisibleOperator.

• Fixed require() returning empty exports on Node 22+ (#292): Because the package sets "type": "module", Node treated the UMD .js files as ESM, breaking the UMD factory pattern. The UMD builds now use a .cjs extension so Node always treats them as CommonJS.

0.53.0 2026-02-21

Runtime and Scoping

• True lexical scoping for Function expressions: Functions now capture their defining scope and resolve free variables from that scope chain (not the call site), with a fresh child scope on each call.

• BigOp scope pollution fixed: Sum, Product, and other big operators now only declare their index variable locally. Other names are declared in the enclosing scope via noAutoDeclare.

• Closure capture for nested functions: Returned functions now correctly capture outer parameters across multiple nesting levels.

• EvalContext.values removed: Symbol values now live only in BoxedValueDefinition.value. The per-frame shadow map and withArguments option were removed.

• forget() now resets values set by assume(): forget('x') now clears values introduced by assume('x = ...') (value reset to undefined), in addition to clearing assumptions.

Expressions and Equality

• expand() now returns the input expression instead of null: Both the free function and internal expand()/expandAll() now return the original expression when no expansion is possible.

• New .toRational() method: Returns [numerator, denominator] integers for rational expressions, or null otherwise.

• New .factors() method: Returns multiplicative factors as a flat array by decomposing Multiply and Negate structurally.

• .is() now tries expansion: After structural comparison, .is() expands both sides before numeric fallback, catching forms like (x+1)^2 and x^2+2x+1.

• .is() is now symmetric: a.is(b) === b.is(a) now holds across all expression types.

LaTeX Parsing

• Parse \mleft/\mright delimiters: Alternative delimiters from the mleftright package are now treated like \left/\right.

• Parse \color in math mode: \color{...} is now recognized in math mode; the color argument is consumed so the following math parses normally.

• Parse : and \colon as infix operators: Outside quantifier contexts, a bare :/\colon now parses as Colon (e.g. f:[a,b]\to\R), without affecting := assignment or quantifier syntax.

• Parse \dfrac, \tfrac, and \cfrac as fractions: These variants now parse the same as \frac.

Fractals

• New Mandelbrot and Julia functions: Added built-in escape-time fractal operators. Mandelbrot(c, maxIter) and Julia(z, c, maxIter) return a smooth, normalized value in [0, 1] (1 for interior points, fractional for escaping points via log₂(log₂(|z|²)) smoothing). Both evaluate in JavaScript and compile to GLSL/WGSL.

0.52.1 2026-02-19

Expressions

• Exact number literal check: Use isNumber(expr) && expr.isExact to test for exact numeric literals.

• raw form preserves subtraction: x-1 now parses as ["Subtract", "x", "1"] (instead of ["Add", "x", -1]) when using raw form.

Parsing and Blocks

• Fix ;\; parsing in semicolon blocks: Spacing commands after semicolons (\;, \,, \quad, etc.) no longer create spurious Nothing operands.

• Fix \text{if} parsing with \; spacing: \text{if}\;...\;\text{then}\;...\;\text{else}\;... now parses correctly as If.

• Block serializer now uses ; : Serialization emits ; (not ;\; ) to avoid reintroducing spacing-related parse issues on round-trip.

• Block compiler filters Nothing operands: The Block compiler now removes Nothing symbols and empty compile results before generating code.

• Subscripted variable names in blocks: Names like r_1 are treated as compound symbols (not Subscript) when the base is not a known collection.

• Non-strict parser supports exponents on bare functions: In strict: false mode, forms like sin^2(x) and cos^{10}(x) now parse correctly as powers.

• Unicode superscript/subscript digits supported: Superscript and subscript Unicode digits now normalize to ^{...} / _{...} in parsing.

Compilation

• Selective GLSL interval preamble: interval-glsl now emits only used helper functions (plus dependencies), typically reducing preamble size by 60-80%.

• Selective WGSL interval preamble: interval-wgsl now applies the same used-only preamble strategy.

• Fix recursive GLSL gamma helper: Replaced recursive _gpu_gamma() reflection logic (illegal in GLSL) with a non-recursive implementation.

Equality

• .is() now works with assigned variables: Numeric fallback now applies to expressions with no free variables, including variables with assigned values.

• .is() now accepts an optional tolerance: A per-call tolerance can override engine.tolerance for numeric comparison.

0.52.0 2026-02-18

New Features

• Smart .is() / exact .isSame() separation: The .is() and .isSame() methods on expressions now have distinct roles:

  • .isSame(v) — Fast exact structural check. No evaluation, no tolerance. Now accepts primitives (number, bigint, boolean, string) in addition to Expression. This is the method used internally throughout the engine.

  • .is(v) — Smart check with numeric evaluation fallback. Tries .isSame() first; if that fails and the expression is constant (no free variables), evaluates numerically and compares within engine.tolerance. For literal numbers, behaves identically to .isSame() — tolerance only applies to expressions that require evaluation.

  This resolves a common pain point where ce.parse('\\cos(\\pi/2)').is(0) returned false because .is() was purely structural. Now it returns true:

  ce.parse('\\sin(\\pi)').is(0);            // true  (evaluates, within tolerance)
  ce.parse('\\cos(\\frac{\\pi}{2})').is(0); // true
  ce.number(1e-17).is(0);                   // false (literal number, no tolerance)
  ce.parse('x + 1').is(1);                  // false (not constant, no fallback)

• numericValue() convenience helper: New standalone function that combines the isNumber() guard with .numericValue access. Returns the numeric value if the expression is a number literal, or undefined otherwise. Useful for safely extracting numeric values without verbose ternary patterns:

  import { numericValue } from '@cortex-js/compute-engine';
  
  // Before
  const val = isNumber(expr) ? expr.numericValue : undefined;
  
  // After
  const val = numericValue(expr);

• Stochastic equality check for expressions with unknowns: expr.isEqual() now uses a stochastic fallback when symbolic methods (expand + simplify) can't prove equality. Both expressions are evaluated at 50 sample points (9 well-known values + 41 random) and compared with relative+absolute tolerance. This detects equivalences like sin²(x) + cos²(x) = 1, (x+y)² = x²+2xy+y², and sin(2x) = 2sin(x)cos(x) that were previously returned as undefined. Singularities (NaN at a sample point) are skipped rather than treated as disagreements. The check also works when the two expressions have different unknowns (e.g. x - x + y vs y).

• expr.freeVariables property: New property on BoxedExpression that returns the free variables of an expression — symbols that are not constants, not operators, not bound to a value, and not locally scoped by constructs like Sum or Product. Semantically identical to expr.unknowns.

• New interval-js compilation functions: Added Binomial, GCD, LCM, Chop, Erf, Erfc, Exp2, Arctan2, and Hypot to the interval-js compilation target, with corresponding interval arithmetic implementations.

• GLSL/WGSL variable exponent support: The interval GLSL and WGSL targets now support Power with variable exponents (e.g. (-1)^k, x^n). Previously these threw at compile time. Added ia_pow_interval() to both GPU library preambles using four-corner exp(exp * ln(base)) evaluation with special cases for point-integer exponents and (-1)^n.

• Factorial, Gamma, GammaLn for GLSL/WGSL interval targets: Added ia_factorial (via ia_gamma(x+1)) to both GPU targets. Added ia_gamma (Lanczos approximation) and ia_gammaln (Stirling asymptotic) to the WGSL target, matching existing GLSL implementations.

Resolved Issues

• parse() with form: 'structural' ignored the structural flag: The structural option from formToInternal() was dropped in parseLatexEntrypoint(), making ce.parse(s, { form: 'structural' }) behave identically to { form: 'raw' } (unbound, unsorted). Now correctly produces a bound, structural expression.

• Partial canonicalization with 'Flatten' form folded numerics: Using ce.parse(s, { form: ['Flatten', 'Order'] }) unexpectedly evaluated numeric operands (e.g. 3×2+1 became 7) because flattenForm() used ce.function() which defaults to full canonical mode. Now uses ce._fn() to preserve operand structure. This enables structural comparison of expressions modulo commutativity and associativity without numeric evaluation — useful for checking the method used to solve a problem rather than just the numeric result:

  const a = ce.parse('3\\times2+1', { form: ['Flatten', 'Order'] });
  const b = ce.parse('1+2\\times3', { form: ['Flatten', 'Order'] });
  a.isSame(b);  // ➔ true  (same structure, different order)
  
  const c = ce.parse('7', { form: ['Flatten', 'Order'] });
  a.isSame(c);  // ➔ false (different structure)

• Sum/Product with symbolic bounds compiled incorrectly: Expressions like \sum_{k=0}^{n} f(k, x) where the upper bound is a variable produced loops that iterated 10001 times instead of using the variable n. The compilation extracted bounds via normalizeIndexingSet() which converted symbolic bounds to NaN and fell back to a hardcoded limit. Now bounds are extracted as expressions and compiled to code (e.g. Math.floor(_.n) for JS, Math.floor((_.n).hi) for interval-js). This fixes Taylor series patterns like \sum_{k=0}^{n} \frac{(-1)^k x^{2k+1}}{(2k+1)!} for both JS and interval-js targets.

• Interval (-1)^k returned empty instead of correct value: The powInterval() function required positive bases for variable exponents, causing (-1)^k patterns in summations (e.g. Taylor series) to fail at runtime. Now correctly delegates to intPow() when the exponent is a point interval with an integer value, preserving even/odd parity. Also handles the case where base is -1 and the exponent spans multiple integers by returning the conservative interval [-1, 1].

• Factorial missing from interval-js compilation target: Expressions containing n! (e.g. \frac{(-1)^k x^{2k+1}}{(2k+1)!}) failed interval-js compilation with success: false. Added Factorial and Factorial2 interval functions and compilation handlers.

• expr.unknowns included bound variables: Scoped constructs like Sum, Product, Integrate, and Block bind index variables in a local scope, but expr.unknowns was reporting them as free unknowns. For example, \sum_{k=0}^{10} k \cdot x returned ["k", "x"] instead of ["x"]. Now correctly excludes locally bound variables from the result.

• Symbolic upper bounds missing from expr.unknowns: In expressions like \sum_{k=0}^{M} k \cdot x, the symbolic upper bound M was incorrectly excluded from unknowns because the scope's bindings map captured all symbols referenced during canonicalization. Now extracts bound variables structurally from Limits/Element/Assign/Declare expressions, so only true bound variables are excluded. This also fixes Block expressions where locally assigned variables (via Assign or Declare) were reported as unknowns.

• Integrate with symbolic bounds compiled incorrectly: Same issue as Sum/Product — compileIntegrate() used normalizeIndexingSet() which converted symbolic bounds to NaN. Now uses extractLimits() and compiles bounds as expressions.

• Interval piecewise test fix: Fixed test that incorrectly accessed result.lo directly instead of unwrapping the IntervalResult envelope (result.value.lo). The piecewise() function correctly returns IntervalResult objects.

0.51.1 2026-02-15

Features

• #172 Degrees-Minutes-Seconds (DMS) notation: Parse and serialize geographic angle notation such as 9°30'15". The LaTeX parser now recognizes arc-minute (', \prime) and arc-second (", \doubleprime) symbols when they follow a degree symbol, producing Add(Quantity(…, deg), Quantity(…, arcmin), …) expressions that evaluate and simplify through the existing unit system. Negative angles (e.g. -45°30') are fully supported for latitude/longitude coordinates.
• dmsFormat serialization option: Set dmsFormat: true in SerializeLatexOptions to serialize angle quantities as DMS notation (e.g. Quantity(9.5, deg) → 9°30').
• angleNormalization serialization option: Normalize angles during serialization with '0...360' (useful for bearings) or '-180...180' (useful for longitude). Default is 'none'.
• realOnly compilation option: Pass { realOnly: true } to compile() to automatically convert complex { re, im } results to real numbers — returns re when im === 0, NaN otherwise. Useful for plotting and other contexts that only need real-valued output.
• Sinc function: Unnormalized cardinal sine sinc(x) = sin(x)/x with sinc(0) = 1. Includes LaTeX parsing via \operatorname{sinc}, JavaScript and interval-arithmetic compilation targets.
• Fresnel integrals (FresnelS, FresnelC): Numeric evaluation using Cephes rational Chebyshev approximation, LaTeX parsing via \operatorname{FresnelS} / \operatorname{FresnelC}, JavaScript and interval-arithmetic compilation targets.
• Heaviside step function: H(x) = 0 for x < 0, 1/2 for x = 0, 1 for x > 0. LaTeX parsing via \operatorname{Heaviside}, JavaScript and interval-arithmetic compilation with singularity detection at zero.

LaTeX Syntax

• Which compilation: \begin{cases} expressions now compile to JavaScript and interval-js targets as chained ternary operators with NaN fallback when no condition matches.
• Sum/Product compilation: \sum_{k=a}^{b} and \prod_{k=a}^{b} expressions with numeric bounds now compile to JavaScript loops with accumulator variables, including complex number support.
• Loop compilation: Loop, Break, Continue, and Return operators compile to JavaScript for loops wrapped in IIFEs with standard control flow keywords.
• Inline If syntax: Parse \text{if } C \text{ then } A \text{ else } B (or \operatorname{if}) to ["If", C, A, B] expressions.
• where syntax: Parse E \text{ where } x \coloneq V to Block expressions with implicit variable declarations.
• Semicolon block syntax: Semicolons (;, \;) act as statement separators, building Block expressions with auto-declared variables when assignments are present.
• for loop syntax: Parse \text{for } i \text{ from } a \text{ to } b \text{ do } body to ["Loop", body, ["Element", "i", ["Range", a, b]]].

Resolved Issues

• Interval-JS compilation for Gamma functions: Added missing gamma and gammaln exports and implementations in the interval-arithmetic library.
• Interval-JS graceful fallback: The interval-js target no longer throws when encountering unsupported functions. Unsupported operators now produce { success: false } at compile time, and runtime errors return { kind: "entire" } instead of propagating.
• CompilationResult.run type signature: The TypeScript type for run now correctly reflects the actual calling convention ((...args: unknown[])) instead of the previous misleading (...args: (number | {re, im})[]).
• Loop compilation for interval-js target: Loop counter now uses raw numbers (not _IA.point()) for the for statement, with loop index references properly wrapped in the body. Conditions in if/break/ continue statements inside loops use scalar comparisons instead of interval comparison functions.

Other Changes

• Updated color palettes
• Deduplicated runtime helper object (SYS_HELPERS) shared between ComputeEngineFunction and ComputeEngineFunctionLiteral in compilation target
• Centralized sinc implementation in numerics/special-functions.ts (shared by library evaluation and JS compilation runtime)
• Removed dead args === null checks in compilation base class

0.51.0 2026-02-14

Colors

• New colors library: Four MathJSON operators for color manipulation and color space conversion, available as the "colors" library category.
• Color: Parse a color string (hex 3/6/8-digit, rgb(), hsl(), named CSS color, transparent) into a canonical sRGB Tuple with components normalized to 0-1. Alpha is included as a fourth component when not equal to 1.
• Colormap: Sample named visualization palettes. Three variants: no second argument returns the full palette as a List; integer n >= 2 resamples to n evenly spaced colors; real t in [0, 1] interpolates at position t using OKLCh color space with shorter-arc hue interpolation. Includes 8 sequential palettes (viridis, inferno, magma, plasma, cividis, turbo, rocket, mako), 6 categorical palettes (graph6, spectrum6, spectrum12, tableau10, tycho11, kelly22), and 12 diverging palettes (roma, vik, broc, rdbu, coolwarm, ocean-balance, plus reversed variants).
• ColorToColorspace: Convert an sRGB color (string or Tuple) to components in "rgb", "hsl", "oklch", or "oklab" (alias "lab"). Preserves alpha when present.
• ColorFromColorspace: Convert color space components back to a canonical sRGB Tuple. Accepts the same color space names as ColorToColorspace.
• ColorToString: Convert a color (string or sRGB Tuple) to a formatted string. Supports optional format argument: "hex" (default), "rgb", "hsl", or "oklch" for CSS-style output. Alpha is included when not equal to 1.
• ColorMix: Blend two colors in OKLCh space with an optional ratio (default 0.5). Accepts color strings or sRGB Tuple values. Interpolates lightness and chroma linearly, hue with shorter-arc interpolation.
• ColorContrast: Compute the APCA contrast ratio between a background and foreground color. Returns a positive value for dark-on-light and negative for light-on-dark.
• ContrastingColor: Choose the foreground color with better APCA contrast against a background. With one argument, picks between white and black. With three arguments, picks the better of two foreground candidates.
• LaTeX color support: \textcolor{color}{body}, \colorbox{color}{body}, and \boxed{body} now roundtrip through Annotated expressions. Parsing and serialization are handled in the core Annotated infrastructure.
• LaTeX font annotations: \textbf, \textit, \texttt, \textsf, \textup now serialize correctly from Annotated expressions via fontWeight, fontStyle, and fontFamily dict keys.
• JavaScript compilation: All color operators (Color, ColorToString, ColorMix, ColorContrast, ContrastingColor, ColorToColorspace, ColorFromColorspace, Colormap) now compile to JavaScript.
• oklab() CSS parsing: parseColor() now accepts oklab(L a b) and oklab(L a b / alpha) syntax, matching the existing oklch() support.
• GPU compilation: ColorMix, ColorContrast, ContrastingColor, ColorToColorspace, and ColorFromColorspace now compile to GLSL and WGSL. Preamble functions provide sRGB ↔ OKLab ↔ OKLCh conversion, color mixing with shorter-arc hue interpolation, and APCA contrast on the GPU.
• Added rgbToHsl() conversion function. Exported hslToRgb() (previously private).

Resolved Issues

• (#290) Derivatives of user-defined functions: \frac{d}{dx} f and f'(x) now correctly evaluate when f is a user-defined function (e.g., f(x) := 2x). Previously \frac{d}{dx} f returned 0 and f'(x) returned a symbolic Apply(Derivative(...)).
• Cleaner D canonical form: f'(x) now canonicalizes to ["D", ["f", "x"], "x"] instead of the verbose ["D", ["Function", ["Block", ["f", "x"]], "x"], "x"]. Function calls are no longer redundantly wrapped in Function(Block(...)). Similarly, \frac{d}{dx} f where f is a known function symbol canonicalizes to ["D", ["f", "x"], "x"] by applying the function to the differentiation variable.

Free Functions

• Free functions (simplify, evaluate, N, expand, expandAll, factor, solve, compile) now accept ExpressionInput in addition to LatexString and Expression. This means you can pass numbers, MathJSON objects, or tuple arrays directly — e.g., evaluate(["Add", 1, 2]) or simplify(["Power", "x", 2]).
• Added declare() free function to declare symbols without instantiating a ComputeEngine explicitly — e.g., declare('x', 'integer') or declare({ x: 'integer', y: 'real' }).

Units and Quantities

• New units library: A comprehensive unit system for physical quantities, available as the "units" library category. Supports SI base units, 18 named derived units, SI prefixes (quetta through quecto), and common non-SI units (imperial, angles, logarithmic).
• Quantity expression: Pairs a numeric value with a unit: ["Quantity", 9.8, ["Divide", "m", ["Power", "s", 2]]]. Accessors QuantityMagnitude and QuantityUnit extract the parts.
• Quantity arithmetic: Add, Subtract, Multiply, Divide, and Power are unit-aware. Addition and subtraction automatically convert compatible units and express the result in the unit with the largest scale factor (e.g., 12 cm + 1 m evaluates to 1.12 m). Incompatible dimensions remain unevaluated.
• Unit conversion: UnitConvert converts between compatible units, including compound units like m/s to km/h. Supports affine temperature conversions (degC, degF, K). Returns an error for incompatible units. UnitSimplify reduces compound units to named derived units when possible (e.g., kg*m/s^2 to N).
• Dimensional analysis: IsCompatibleUnit tests dimensional compatibility. UnitDimension returns the 7-element SI dimension vector. Both support compound unit expressions.
• LaTeX parsing: \mathrm{...} and \text{...} containing recognized units produce Quantity expressions when juxtaposed with numbers. Compound units with /, ^, and \cdot are supported (e.g., 5\,\mathrm{m/s^{2}}).
• siunitx commands: \qty{value}{unit}, \SI{value}{unit}, \unit{unit}, and \si{unit} are parsed.
• LaTeX serialization: Quantity expressions serialize to value\,\mathrm{unit} notation.
• DSL string sugar: Compound units can be specified as strings in MathJSON: ["Quantity", 9.8, "m/s^2"] is canonicalized to the structured form. Parentheses are supported for grouping: "kg/(m*s^2)".
• Temperature units: degC and degF with affine offset conversions.
• Angular unit unification: Trigonometric functions (Sin, Cos, Tan, etc.) accept Quantity arguments with angular units (deg, rad, grad, arcmin, arcsec) and convert to radians automatically.
• Physics constants: 11 CODATA 2018 constants defined as Quantity expressions: SpeedOfLight, PlanckConstant, Mu0, StandardGravity, ElementaryCharge, BoltzmannConstant, AvogadroConstant, VacuumPermittivity, GravitationalConstant, StefanBoltzmannConstant, and GasConstant.

Compilation

• Tuple and Matrix compilation: Tuple and Matrix expressions can now be compiled across all targets. compile('(\\sin(t), \\cos(t))') produces [Math.sin(t), Math.cos(t)] in JavaScript, vec2(sin(t), cos(t)) in GLSL, vec2f(sin(t), cos(t)) in WGSL, and (np.sin(t), np.cos(t)) in Python.
• GPU-native matrix types: Square matrices (2x2, 3x3, 4x4) compile to native GPU matrix constructors (mat2/mat3/mat4 in GLSL, mat2x2f/mat3x3f/mat4x4f in WGSL) with proper column-major transposition. Column vectors are flattened to vecN/vecNf instead of nested single-element arrays.
• Complex number compilation: The JavaScript compilation target now supports complex-valued expressions. The compiler performs static type analysis at compile time to determine whether each subexpression is real or complex, and emits the appropriate code path. Simple arithmetic (Add, Subtract, Multiply, Divide, Negate) uses inline {re, im} field math to avoid allocation. Transcendental functions (Sin, Cos, Exp, Ln, Sqrt, Power, and others) delegate to runtime helpers backed by the complex-esm library. Mixed real/complex operands are promoted inline. ImaginaryUnit compiles to {re: 0, im: 1}. Symbols with unknown type are assumed real. Complex-aware Sum and Product loops emit {re, im} accumulators when the loop body is complex-valued. Reciprocal trig/hyperbolic functions (Cot, Sec, Csc, Coth, Sech, Csch) and their inverses dispatch to complex helpers when operands are complex.
• Python complex compilation: The Python target now supports complex-valued expressions using Python's native complex() constructor and the cmath module for transcendental functions. Real-valued expressions continue to use NumPy.
• Gamma function compilation: Gamma and GammaLn can now be compiled to interval-js, glsl, wgsl, and interval-glsl targets. The interval targets include pole detection at non-positive integers and correct monotonicity handling around the minimum at x ≈ 1.46.
• Special function compilation: 27 additional functions can now be compiled to JavaScript: Erf, Erfc, ErfInv, Beta, Digamma, Trigamma, PolyGamma, Zeta, LambertW, BesselJ, BesselY, BesselI, BesselK, AiryAi, AiryBi, Factorial, Factorial2, Exp2, Log2, Log10, Lg, Arctan2, Hypot, Degrees, Haversine, InverseHaversine, Binomial, and Fibonacci.
• GPU special functions: Erf, Erfc, ErfInv, Beta, Factorial, Arctan2, Hypot, Haversine, InverseHaversine, Log10, and Lg can now be compiled to GLSL and WGSL targets. Erf/ErfInv use Abramowitz & Stegun polynomial approximations; Beta and Factorial leverage the existing GPU Gamma preamble.

Simplification

• Factorial quotient simplification: n!/k! is now simplified to a partial product for both concrete integers (e.g., 10!/7! → 720) and symbolic expressions with small constant difference (e.g., n!/(n-2)! → n(n-1)).
• Binomial detection: Expressions of the form n!/(k!(n-k)!) are automatically recognized and simplified to Binomial(n, k).
• Binomial identity simplification: C(n,0) → 1, C(n,1) → n, C(n,n) → 1, C(n,n-1) → n.
• Factorial sum factoring: Sums and differences of factorials with related arguments are factored out, e.g., n! - (n-1)! → (n-1)! * (n-1), (n+1)! + n! → n! * (n+2).

0.50.2 2026-02-12

Numerics

• Centralized overflow protection: Improved robustness of Rational and ExactNumericValue arithmetic by centralizing overflow checks and automatic promotion to BigInt.
• [#287](https://github.com/cortex-js/compute-engine/issues/287) Improved precision for large integer products: Multiplications and additions of large integers that would previously lose precision (exceeding Number.MAX_SAFE_INTEGER) are now automatically promoted to BigInt to maintain exact results.

Symbols

• #288 Allow reassigning a symbol from operator to value: ce.assign() no longer throws when assigning a plain value to a symbol that was previously declared as a function. Existing expressions using the symbol as a function head will produce a type error at evaluation time if the new value is not callable.

Evaluation

• Fixed scope leaks: Ensured that evaluation contexts are correctly popped even when an error or timeout occurs in BoxedFunction.evaluate(), findUnivariateRoots(), and rule-boxing operations.
• Improved numerical evaluation performance: Sum, Product, Divide, and statistical operators (Mean, Variance, etc.) now correctly propagate the numericApproximation option, significantly speeding up large numerical calculations by avoiding expensive exact arithmetic.

0.50.1 2026-02-11

Compilation

• CompilationResult.preamble for shader targets: compile() with interval-wgsl and interval-glsl targets now returns a preamble field containing the interval arithmetic library (struct definitions, helper functions). Previously, the compiled code referenced functions like ia_div and ia_sin that were not included in the output. Use preamble + code for a self-contained shader, or call compileShaderFunction() on the target directly.

0.50.0 2026-02-11

Breaking API Changes

This release includes several breaking changes to the public API.

The most significant is the restructuring of the Expression type hierarchy and the introduction of type-guarded role interfaces, which improves type safety and API ergonomics but requires updates to code that accessed role-specific properties directly on expression instances.

See MIGRATION_GUIDE_0.50.0.md for details.

Naming Alignment: Expression, MathJsonExpression, and ExpressionInput

• The compute-engine runtime type is now Expression (preferred name). BoxedExpression is retained as a deprecated alias for migration.
• The MathJSON type is now MathJsonExpression (the old MathJSON Expression name has been removed from the math-json entrypoint).
• SemiBoxedExpression is now ExpressionInput (with a deprecated alias for migration).

Role-Specific Properties Moved to Type-Guarded Interfaces

Properties that were previously on all Expression instances (returning undefined when not applicable) have been moved to role interfaces. They are now only accessible after narrowing with a type guard.

Removed from Expression	Access via
.symbol	isSymbol(expr) or isSymbol(expr, 'Pi') then expr.symbol
.string	isString(expr) then expr.string
.ops, .nops, .op1/.op2/.op3	isFunction(expr) or isFunction(expr, 'Add') then expr.ops etc.
.numericValue, .isNumberLiteral	isNumber(expr) then expr.numericValue
.tensor	isTensor(expr) then expr.tensor

// Before
if (expr.symbol !== null) console.log(expr.symbol);

// After
import { isSymbol, sym } from '@cortex-js/compute-engine';

if (isSymbol(expr)) console.log(expr.symbol);
// isSymbol() accepts an optional symbol name:
if (isSymbol(expr, 'Pi')) { /* expr is the Pi symbol */ }
// or use the convenience helper:
if (sym(expr) === 'Pi') { /* ... */ }

// isFunction() accepts an optional operator name:
if (isFunction(expr, 'Add')) {
  // expr is narrowed to a function AND has operator 'Add'
  console.log(expr.ops);
}

Properties that remain on Expression: .operator, .re/.im, .shape, all arithmetic methods (.add(), .mul(), etc.), and all numeric predicates (.isPositive, .isInteger, etc.).

Expression Creation: form Replaces canonical/structural

The canonical (boolean or array) and structural (boolean) options on ce.box(), ce.function(), and ce.parse() have been unified into a single form option.

ce.box(['Add', 1, 'x'], { form: 'canonical' }); // default
ce.box(['Add', 1, 'x'], { form: 'raw' });        // no canonicalization, no binding
ce.function('Add', [1, 'x'], { form: 'structural' }); // bound, not fully canonical
ce.box(['Add', 1, 'x'], { form: ['Number', 'Order'] }); // selective passes

New Free Functions

Top-level free functions are now available for common operations and use a shared ComputeEngine instance created on first call.

Function	Purpose
getDefaultEngine()	Return the shared default ComputeEngine instance.
parse(latex)	Parse a LaTeX string into an Expression.
simplify(exprOrLatex)	Simplify an expression or LaTeX input.
evaluate(exprOrLatex)	Evaluate an expression or LaTeX input symbolically.
N(exprOrLatex)	Numerically evaluate an expression or LaTeX input.
assign(id, value) / assign(record)	Assign one symbol value or many at once.
expand(exprOrLatex)	Expand distributively at the top level (Expression | null).
expandAll(exprOrLatex)	Expand distributively recursively (Expression | null).
solve(exprOrLatex, vars?)	Solve equations/systems (returns solve result variants).
factor(exprOrLatex)	Factor an expression.
compile(exprOrLatex, options?)	Compile to a target language with CompilationResult.

import {
  getDefaultEngine,
  parse,
  simplify,
  evaluate,
  N,
  assign,
  expand,
  expandAll,
  solve,
  factor,
  compile,
} from '@cortex-js/compute-engine';

assign('x', 3);

const expr = parse('x^2 - 5x + 6');
solve(expr, 'x');           // [2, 3]
factor('(2x)(4y)');         // 8xy
compile('x^2 + 1').run({ x: 3 }); // 10

Except for parse(), assign(), and getDefaultEngine(), these free functions accept either a LaTeX string or an existing Expression.

Free Function Notes

• compile() is now a top-level entry point returning CompilationResult. Custom compilation targets are managed with ce.registerCompilationTarget() and ce.unregisterCompilationTarget().
• expand() and expandAll() return null when an expression is not expandable.
• solve() is available as a top-level wrapper over equation/system solving.
• factor() is the top-level factoring entry point. Specialized helpers such as factorPolynomial() and factorQuadratic() remain expression-only APIs.

trigSimplify() Method Removed

Use simplify({ strategy: 'fu' }) instead, which is equivalent.

// Before
const result = expr.trigSimplify();

// After
const result = expr.simplify({ strategy: 'fu' });

Library System

The constructor now accepts a libraries option for controlling which libraries are loaded. Libraries declare their dependencies and are loaded in topological order.

// Load specific standard libraries
const ce = new ComputeEngine({
  libraries: ['core', 'arithmetic', 'trigonometry'],
});

// Add a custom library
const ce = new ComputeEngine({
  libraries: [
    ...ComputeEngine.getStandardLibrary(),
    { name: 'physics', requires: ['arithmetic'], definitions: { /* ... */ } },
  ],
});

User-Extensible Simplification Rules

ce.simplificationRules is now a public getter/setter. Users can push additional rules or replace the entire rule set.

ce.simplificationRules.push({
  match: ['Power', ['Sin', '_x'], 2],
  replace: ['Subtract', 1, ['Power', ['Cos', '_x'], 2]],
});

Canonicalization

• Exact numeric folding during canonicalization: canonicalAdd and canonicalMultiply now fold exact numeric operands at canonicalization time, making behavior consistent with canonicalDivide which already folded coefficients. This means expressions are reduced earlier in the pipeline without waiting for a .simplify() call.

  What gets folded (exact values):

  • Integers: Add(2, x, 5) → Add(x, 7)
  • Rationals: Add(1/3, x, 2/3) → Add(x, 1)
  • Radicals: Add(√2, x, √2) → Add(x, 2√2)
  • Mixed exact: Multiply(2, x, 5) → Multiply(10, x)
  • Full reduction: Add(2, 3) → 5, Multiply(2, 3) → 6
  • Identity elimination: Multiply(1/2, x, 2) → x
  • Complex promotion: Add(1, Complex(0, -1)) → Complex(1, -1)

  What is NOT folded (non-exact values):

  • Machine floats: Add(1.5, x, 0.5) remains Add(x, 0.5, 1.5)
  • Infinity/NaN: Multiply(0, ∞) correctly returns NaN
  • Single numeric: Multiply(5, Pi) is unchanged (nothing to fold)

  The folding uses the existing ExactNumericValue arithmetic, which automatically handles radical grouping (√2 + √2 = 2√2) and rational simplification (1/3 + 2/3 = 1).

• Exact numeric folding in canonicalPower: Integer powers of numeric literals are now folded during canonicalization when the exponent is an integer with |e| ≤ 64. For machine-number bases, the result must be a safe integer; for exact numeric values (rationals, radicals), NumericValue.pow() is used.

  • Power(2, 3) → 8
  • Power(3, 2) → 9
  • Power(1/2, 2) → 1/4
  • Power(-2, 3) → -8
  • Power(2, 100) remains unevaluated (exponent exceeds limit)

• Complex promotion handles non-adjacent operands: canonicalAdd now combines a real float with imaginary terms even when they are not adjacent in the operand list. Previously, only a real immediately followed by an imaginary was promoted to a complex number.

Type Inference

• Type handlers for 25 operators: Added explicit type handlers to operators that were missing them, enabling the type system to return precise types instead of the broad signature return type.
  • Arithmetic: Factorial, Factorial2, Sign return finite_integer; Ceil and Floor return finite_integer for finite inputs, integer otherwise.
  • Trigonometry: Arctan uses numericTypeHandler (returns finite_real for real inputs, finite_number for complex).
  • Complex: Real, Imaginary, Argument return finite_real.
  • Number theory: Totient, Sigma0, Sigma1, Eulerian, Stirling, NPartition return finite_integer; SigmaMinus1 returns finite_rational.
  • Combinatorics: Choose, Fibonacci, Binomial, Multinomial, Subfactorial, BellNumber return finite_integer.
  • Truncate, GCD, LCM type handlers: Truncate returns finite_integer for finite inputs (matching Ceil/Floor); GCD and LCM always return finite_integer.

Solving

• And operator support for systems of equations: solve() now accepts And(Equal(...), Equal(...)) in addition to List(Equal(...), Equal(...)) for representing systems of equations. Both forms route through the same linear, polynomial, and inequality solvers.

• Parametric solution type filtering: filterSolutionByTypes now uses === false instead of !== true for type predicate checks. This allows underdetermined (parametric) solutions to pass through when type predicates return undefined (unknown) rather than being incorrectly rejected.

• Or operator support in solve(): Solving Or(Equal(x,1), Equal(x,2)) returns the union of solutions from each branch, with deduplication. Works for both univariate (returns array of values) and multivariate (returns array of records) cases.

• Mixed equality + inequality systems: solve() now handles systems combining Equal and inequality operators (Less, LessEqual, Greater, GreaterEqual). Equalities are solved first, then solutions are filtered against the inequalities.

• Parametric solutions omit free variables: Underdetermined linear systems no longer include free variables (self-referential entries) in the result record. Only dependent variables with non-trivial expressions are returned.

Special Functions

• Numeric evaluation for Digamma, Trigamma, PolyGamma, Beta, Zeta, LambertW: These six functions now evaluate numerically when .N() is called, at both machine precision and arbitrary precision (bignum). Returns unevaluated without numeric approximation.

  • Digamma/Trigamma: recurrence + asymptotic with Bernoulli numbers
  • PolyGamma: generalized recurrence for arbitrary order n
  • Beta: via gamma, with log-gamma fallback for large arguments
  • Zeta: Cohen-Villegas-Zagier acceleration, functional equation for \operatorname{Re}(s)<0
  • LambertW: Halley's method with branch-point handling

• Arbitrary-precision (bignum) variants for special functions: When ce.precision > 15, Digamma, Trigamma, PolyGamma, Beta, Zeta, and LambertW now compute results to the requested precision using bignum arithmetic. The asymptotic shift threshold scales with precision to maintain accuracy (e.g., ce.precision = 50 produces 50-digit results for Digamma and Zeta).

• Numeric evaluation for Bessel functions (BesselJ, BesselY, BesselI, BesselK): Integer-order Bessel functions now evaluate numerically.

  • BesselJ: power series for small |x|, Miller's backward recurrence for intermediate values, Hankel asymptotic expansion for large |x|
  • BesselY: DLMF 10.8.3 series for Y_0/Y_1, forward recurrence for higher orders, shared Hankel asymptotic with BesselJ
  • BesselI: power series + asymptotic expansion
  • BesselK: series for K_0, Wronskian-derived K_1, forward recurrence for higher orders, asymptotic for large x

• Numeric evaluation for Airy functions (AiryAi, AiryBi): Power series using Maclaurin coefficients for |x| \leq 5, asymptotic expansions (exponential decay for Ai, exponential growth for Bi at positive x, oscillatory for negative x) for large arguments.

Linear Algebra

(Fix #285)

• \begin{vmatrix} now parses to Determinant: The vmatrix LaTeX environment now produces ["Determinant", ["Matrix", ...]] instead of ["Matrix", ..., "'||'"]. Serialization round-trips correctly back to \begin{vmatrix}...\end{vmatrix} when the argument is a Matrix expression, and uses \det\left(...\right) for symbol arguments.

• \begin{Vmatrix} now parses to Norm: The Vmatrix LaTeX environment now produces ["Norm", ["Matrix", ...]] instead of ["Matrix", ..., "'‖‖'"]. Serialization round-trips to \begin{Vmatrix}...\end{Vmatrix} when the argument is a Matrix, and uses \left\Vert...\right\Vert for symbol arguments.

• A^{-1} produces Inverse for matrix-typed symbols and matrix expressions: When a symbol is declared with type matrix, parsing A^{-1} now returns ["Inverse", "A"] instead of ["Power", "A", -1]. This also works for inline matrix expressions, e.g. \begin{pmatrix}...\end{pmatrix}^{-1}. Undeclared symbols still fall through to the default Power/Divide handling, and function symbols still produce InverseFunction (e.g., \sin^{-1} → Arcsin).

• Inverse serializes as ^{-1}: ["Inverse", "A"] now serializes to A^{-1} instead of \mathrm{Inverse}(A).

• Power(A, -1) canonicalizes to Inverse(A) for matrices: When A has a matrix type, ce.box(["Power", "A", -1]) now canonicalizes to ["Inverse", "A"] instead of ["Divide", 1, "A"].

• \det(A) and \tr(A) now parse correctly: Fixed Determinant and Trace LaTeX dictionary entries to use latexTrigger (\det, \tr) instead of symbolTrigger, which only matches plain identifiers. Both functions also accept plain text forms (det(A), tr(A)).

• \det A and \tr A work without parentheses: Determinant and Trace now accept implicit arguments, so \det A parses as ["Determinant", "A"] (like \cos x parses as ["Cos", "x"]). Implicit arguments bind at multiplication precedence, so \det 2A + 1 parses as det(2A) + 1.

• Determinant serialization uses \det A for simple arguments: Symbol arguments serialize as \det A instead of \det\left(A\right). Matrix arguments still serialize as \begin{vmatrix}...\end{vmatrix}.

• Added standard LaTeX operators \ker, \dim, \deg, \hom: These commands are now in the MathJSON LaTeX dictionary as function entries with implicit arguments, so forms like \ker V, \dim V, \deg p, and \hom(V, W) parse correctly and serialize back to the corresponding standard operator notation. The corresponding function symbols (Kernel, Dimension, Degree, Hom) are also registered in the linear algebra library.

• Implemented runtime evaluation for Kernel, Dimension, Degree, and Hom:

  • Kernel now computes a numeric null-space basis (for scalar/vector/matrix real inputs) and returns it as a list of basis vectors.
  • Dimension now evaluates finite dimensions for concrete tensors and collections, and computes dim(Hom(V, W)) = dim(V) * dim(W) when both dimensions are inferable.
  • Degree now evaluates polynomial degree for polynomial-form expressions while keeping ambiguous bare symbols (for example Degree(p)) unevaluated.
  • Hom now evaluates/simplifies its arguments while preserving the symbolic Hom(...) form.

LaTeX Parsing

• arguments: 'implicit' option for function dictionary entries: Function entries in the LaTeX dictionary can now set arguments: 'implicit' to accept bare arguments without parentheses (e.g., \det A), matching the behavior of trig functions. The default remains 'enclosure' (parentheses required). Applied to \det, \tr, \Re, \Im, \arg, \max, \min, \sup, \inf.

Simplification

• Infinity handling for 24+ functions: arctan(∞), arccot(±∞), tanh/coth/sech/csch(±∞), arsinh(-∞), arcosh(-∞), arccoth(±∞), arcsch(±∞), π^∞, ∞^n, (-∞)^{-n}, log_∞(x), log_{0.5}(∞), √∞, ∛∞ now all return correct limits.

• Root edge cases: Root(x, 0) → NaN, Root(0, n), Root(1, n), Root(+∞, n), and Sqrt(+∞) now handled correctly.

• Division edge cases: a/a → 1 now works for compound expressions (e.g., (π+1)/(π+1)); 2/0 → ComplexInfinity and 1/(1/0) → 0 propagate correctly.

• Logarithm edge cases: Fixed infinity detection in simplify-log.ts (was using sym() which fails on BoxedNumber infinity values); added log_∞(∞) → NaN, base-aware log_c(0), guards for log_1(x) and log_c(c^x) evaluation.

• Absolute value of odd functions: |arcsin(x)|, |sinh(x)|, |arsinh(x)|, |artanh(x)| now simplify to f(|x|).

• Even function with abs argument: cosh(|x+2|) → cosh(x+2).

• Trig period shifts: cot(π+x) → cot(x), csc(π+x) → -csc(x).

• Ln simplification in Add/Multiply operands: ln(x^3) − 3·ln(x) → 0 and ln(x^√2) → √2·ln(x) now work; cost function bypassed for log rules that are mathematically valid but structurally more expensive.

• Preserved function identity: Removed unconditional expansions of sinh/cosh → exp, arsinh/arcosh/artanh → ln, and arcsin → arctan2 that prevented abs/odd-function rules from firing.

Compilation

• WGSL (WebGPU Shading Language) Compilation Target: New built-in WGSL target for compiling mathematical expressions to WebGPU shaders.

  // Via the registry
  const result = compile(expr, { to: 'wgsl' });

  WGSL-specific differences from GLSL:

  • inverseSqrt (camelCase) instead of inversesqrt
  • % operator for mod instead of mod() function
  • vec2f/vec3f/vec4f constructors instead of vec2/vec3/vec4
  • array<f32, n>() instead of float[n]()
  • fn name(x: f32) -> f32 instead of float name(float x)
  • @vertex/@fragment/@compute entry points with struct-based I/O
  • @group/@binding uniform declarations and @workgroup_size for compute

• Interval WGSL Compilation Target: New interval-wgsl target for interval arithmetic in WebGPU shaders, mirroring the existing interval-glsl target. Since WGSL does not support function overloading, the library uses _v suffixes for internal vec2f-parameter implementations (e.g., ia_add_v), while the public API (ia_add, ia_sin, etc.) takes IntervalResult values.

Resolved Issues

• Sequence type inference now returns a proper tuple type: Multi-argument Sequence expressions previously returned 'any' as their inferred type, losing all type information. They now return a tuple<...> type with each element's individual type preserved (e.g., Sequence(1, "a") types as tuple<integer, string>), consistent with the Tuple operator.

• Subscript parsing now checks for collection type: The LaTeX subscript (_) parser now checks whether the LHS is a collection (symbol declared as indexed_collection, or a list literal) and produces At() directly at parse time, consistent with bracket indexing (x[i]). Multi-index subscripts on collections (A_{k,j}) are now correctly unpacked into separate At arguments instead of being wrapped in a Tuple.

• NumericValue(0).mul(Infinity) now returns NaN: All three NumericValue subclasses (MachineNumericValue, BigNumericValue, ExactNumericValue) had an early-return if (this.isZero) return this in mul(), which returned 0 without checking if the other operand was infinity. 0 × ±∞ is now correctly indeterminate (NaN), and ±∞ × 0 is handled symmetrically.

• Power simplification (a^n)^m -> a^{nm} now correctly guarded: The rule was applied unconditionally, which is mathematically incorrect when the base can be negative and exponents are non-integer. The classic counterexample: ((-1)^2)^{1/2} = 1, but (-1)^{2·1/2} = -1. The rule is now only applied when: (1) the base is non-negative, (2) the outer exponent is an integer, or (3) the inner exponent is an odd integer. This fix applies to canonicalization (canonicalPower), the pow() helper, and simplification (simplifyPower). As a result, (x^2)^{1/2} now correctly simplifies to |x| instead of x.

• Power distribution rules now guarded for non-integer exponents: Three additional power distribution rules in pow() were applied unconditionally, producing wrong results when the exponent is non-integer and operands are negative. (1) (a/b)^c -> a^c / b^c — e.g. ((-2)(-3))^{1/2} = sqrt(6) but distributing gives (-2)^{1/2} * (-3)^{1/2} = -sqrt(6). (2) (a*b)^c -> a^c * b^c — same class of bug. (3) (-x)^n used n % 2 === 0 to test parity, but for non-integer n (e.g. 0.5), 0.5 % 2 = 0.5 falls to the odd branch, giving (-x)^{0.5} -> -(x^{0.5}) which is wrong. All three rules, plus the corresponding canonicalPower() Divide rule, now require integer exponents (or non-negative operands) before distributing.

• Sqrt/Root exponent rearrangement now guarded: Two more rules in pow() unconditionally rearranged exponents. (1) (√a)^b -> √(a^b) rearranges (a^{1/2})^b to (a^b)^{1/2}, which is wrong for negative a (e.g. (√(-4))^3 = -8i but √((-4)^3) = 8i). Now only applied when a >= 0. The even-integer branches ((√a)^2 -> a, (√a)^{2k} -> a^k) remain unconditional since integer outer exponents are always safe. (2) Root(a,b)^c -> a^{c/b} combined exponents unconditionally. Now guarded with a >= 0 or c is integer. Audit of simplify-power.ts confirmed all rules there are already properly guarded.

• Relational operators now evaluate: Seven relational operators (TildeFullEqual, TildeEqual, Approx, ApproxEqual, ApproxNotEqual, Precedes, Succeeds) previously had canonical handlers but no evaluate handlers, so expressions like Approx(3.14, 3.14) returned unevaluated. The approximate-equality family (TildeFullEqual, TildeEqual, Approx, ApproxEqual) now checks whether |a - b| <= tolerance via ce.chop(), with support for multi-argument chains. Precedes and Succeeds evaluate as numeric < and > respectively. Negated variants (NotApprox, NotTildeFullEqual, etc.) work automatically through the Not operator.

• BoxedNumber.operator now returns specific numeric types: The operator property on BoxedNumber instances previously returned the generic 'Number' for all numeric values. It now returns specific types that match the internal type system: 'Integer' for integers, 'Rational' for non-integer rationals, 'Real' for floating-point numbers, 'Complex' for complex numbers with non-zero imaginary part, and 'NaN', 'PositiveInfinity', 'NegativeInfinity' for special values. This improves API consistency with the type property and enables more precise pattern matching and type discrimination in user code. Breaking change: Code that explicitly checks for .operator === 'Number' will need to be updated to check for specific numeric types or use the isNumber() type guard instead.

• Non-XIDC Unicode characters in symbol names now encoded correctly: When parsing LaTeX symbols containing non-identifier Unicode characters via \unicode{...}, \char, or ^^XX escapes (e.g., figure dash U+2012 in \operatorname{speed\unicode{"2012}of\unicode{"2012}sound}), the characters are now encoded as ____XXXXXX (4 underscores + 6 hex digits) in the symbol name. This encoding is valid per isValidSymbol() and round-trips correctly: the serializer decodes ____XXXXXX back to \unicode{"XXXX"} in LaTeX output. Previously, these characters passed through raw and caused symbol validation to fail.

• Assign to compound symbol names no longer misinterpreted as sequence definitions (fixes #286): ce.box(["Assign", "t_half", 10]) previously failed because the Assign evaluate handler split any symbol containing _ and treated it as a subscripted sequence definition. User-provided compound symbols like t_half or half_life are now assigned correctly. Sequence definitions via parsed LaTeX (e.g., L_0 := 1) continue to work as before.

0.35.6 2026-02-07

Resolved Issues

• Monte Carlo improper integrals: Fixed two bugs in monteCarloEstimate() that produced incorrect results (typically NaN or Infinity) for improper integrals. The change-of-variables estimator was inverted (f(x) / \mathrm{jacobian} instead of f(x) * \mathrm{jacobian}), and the finite-interval scale factor b - a was applied to transformed domains where it is infinite. Affects NIntegrate and compiled integrate for any integral with infinite bounds.

Compilation

• Truncate, Remainder, and Mod for JS/GLSL targets: Added Truncate (Math.trunc / trunc), Remainder, and Mod to the JavaScript and GLSL compilation targets, matching the Python target which already had them.

• Interval trunc and remainder: Added trunc() and remainder() to the interval arithmetic library. trunc has proper discontinuity detection (behaves like floor for positive, ceil for negative, continuous at zero). remainder(a, b) = a - b * round(a/b) composes existing interval operations with discontinuity detection inherited from round. Added corresponding mappings to both interval JavaScript and interval GLSL targets.

• Interval Lb, Log, and Root for GLSL: Added ia_log2, ia_log10, and Root to the interval GLSL target for consistency with the interval JavaScript target.

• Reverse cross-reference test: Added a test that verifies all core CE math functions have compilation support in every target. Currently all 5 targets have full coverage of the 47 compilable math functions.

0.35.5 2026-02-06

Resolved Issues

• Compilation Target Function Name Mismatches: Fixed several function keys in compilation targets that did not match their canonical library operator names, causing silent compilation failures and runtime errors ("Unexpected value"). Affected mappings: Ceiling → Ceil, Sgn → Sign, LogGamma → GammaLn, Arcsinh → Arsinh, Arccosh → Arcosh, Arctanh → Artanh, Re → Real, Im → Imaginary, Arg → Argument across all five compilation targets.

• Missing Library Operator Definitions: Added library definitions for Exp2, Fract, Log10, Log2, Remainder, and Truncate which were referenced by compilation targets but had no corresponding library entries. Exp2 canonicalizes to Power(2, x), Log10/Log2 canonicalize to Log with the appropriate base, and Fract, Remainder, Truncate have direct numeric evaluation.

• Derivative Rule for GammaLn: Fixed the derivative table entry that used the non-canonical name LogGamma instead of GammaLn, preventing the derivative d/dx GammaLn(x) = Digamma(x) from being computed.

0.35.4 2026-02-06

Interval Arithmetic

• Discontinuity Continuity Direction: Singular interval results now include an optional continuity field ('left' or 'right') indicating from which side the function is continuous at a jump discontinuity. Floor, Round, Fract, and Mod report 'right' (right-continuous), Ceil reports 'left' (left-continuous). Pole-type singularities (e.g., tan, 1/x) leave the field undefined. This is reflected in both the JavaScript and GLSL interval arithmetic targets (new IA_SINGULAR_RIGHT and IA_SINGULAR_LEFT status constants in GLSL).

0.35.3 2026-02-06

Compilation

• Expanded Function Support Across All Targets: Added comprehensive function mappings to all five compilation targets (JavaScript, GLSL, Interval GLSL, Interval JavaScript, Python): reciprocal trig (Cot, Csc, Sec), inverse reciprocal trig (Arccot, Arccsc, Arcsec), hyperbolic (Sinh, Cosh, Tanh), reciprocal hyperbolic (Coth, Csch, Sech), inverse hyperbolic (Arcosh, Arsinh, Artanh, Arcoth, Arcsch, Arsech), and elementary functions (Sgn, Lb, Log with base, Square, Root, Fract).

• Interval Discontinuity Detection: Floor, Ceil, Round, Sign, Fract, and Mod now correctly report singularities when an interval spans a discontinuity point, in both the JavaScript and GLSL interval arithmetic targets. Previously these functions returned normal interval bounds even across jump discontinuities, which could cause incorrect connecting lines in plotted curves.

• New Interval Functions: Added Round, Fract, and Mod to the interval arithmetic targets (both JS and GLSL) with proper discontinuity detection.

0.35.2 2026-02-05

Resolved Issues

• Decimal Number Representation: Numbers written with a decimal point (e.g., 6.02e23) are now correctly treated as approximate decimal values (BigNumericValue) rather than exact integers. Previously, 6.02e23 was incorrectly converted to the exact bigint 602000000000000000000000, which implied false precision and caused memory inefficiency for very large exponents. Numbers without a decimal point (e.g., 602e21) continue to be treated as exact integers when possible. This change aligns with the documented behavior of the parseNumbers: 'auto' option.

• Scientific Notation Serialization (#284): Fixed toLatex() with scientific and adaptiveScientific notation options to produce properly normalized output. Previously, numbers like 6.02e23 would serialize as 602\cdot10^{21} instead of the expected 6.02\cdot10^{23}. The output now depends only on the numeric value and formatting options, not on the internal representation.

• Numeric Sum Precision: Fixed precision loss when summing large integers with rational values (e.g., 12345678^3 + 1/3). The ExactNumericValue.sum() method now uses bignumRe instead of re to preserve full precision when handling large integer values from BigNumericValue.

• Broadcastable Functions with Union/Any Types (#235): Broadcastable (threadable) functions like Multiply and Add no longer reject arguments whose type is a union of numeric and collection types (e.g., number | list) or any. Previously, declaring a symbol as ce.declare('a', 'number | list') and using it in ce.box(['Multiply', 'a', 'b']) would produce an incompatible-type error.

• Division Canonicalization Over-Simplification (#227): Fixed A/A being incorrectly simplified to 1 during canonicalization for constant expressions that evaluate to infinity or zero, such as tan(π/2)/tan(π/2). This now correctly evaluates to NaN (since ∞/∞ is indeterminate) instead of 1. Expressions with free variables (e.g., x/x, sin(x)/sin(x)) continue to simplify to 1 per standard algebraic convention. Also fixed deferred constant divisions like 0/(1-1) and (1-1)/(1-1) to properly evaluate to NaN instead of remaining as unevaluated expressions.

0.35.1 2026-02-03

Resolved Issues

• Interval Arithmetic (JS/GLSL): Fixed interval evaluation of compound arguments (e.g. sin(2x), sin(x+x), sin(x^2), cos(2x)) by propagating interval results through trig, elementary, and comparison functions in interval-js, and by adding IntervalResult overloads to the GLSL interval library for interval-glsl.

0.35.0 2026-02-02

Parsing

• Large Integer Precision: Fixed precision loss when parsing integers exceeding Number.MAX_SAFE_INTEGER with parseNumbers: 'rational'. Large integers and rational numerators now use BigInt arithmetic to preserve exact values. Fixes #283.

Compilation

• Interval Arithmetic Targets: Added two new compilation targets for reliable singularity detection:
  • interval-js - Compiles to JavaScript using interval arithmetic
  • interval-glsl - Compiles to GLSL for GPU-based interval evaluation

0.34.0 2026-02-01

Parsing

• \mathopen and \mathclose: The LaTeX parser supports \mathopen and \mathclose delimiter prefixes for matchfix operators (explicit delimiter spacing control), e.g. \mathopen(a, b\mathclose) and \mathopen{(}a, b\mathclose{)}.

• Interval Notation Parsing: Added support for parsing mathematical interval notation from LaTeX, including half-open intervals. Addresses #254.

  // Half-open intervals (American notation)
  ce.parse('[3, 4)').json;   // → ["Interval", 3, ["Open", 4]]
  ce.parse('(3, 4]').json;   // → ["Interval", ["Open", 3], 4]
  
  // Open intervals (ISO/European notation)
  ce.parse(']3, 4[').json;   // → ["Interval", ["Open", 3], ["Open", 4]]
  
  // LaTeX bracket commands and sizing prefixes
  ce.parse('\\lbrack 3, 4\\rparen').json;  // → ["Interval", 3, ["Open", 4]]
  ce.parse('\\left[ 3, 4 \\right)').json;  // → ["Interval", 3, ["Open", 4]]
  ce.parse('\\bigl( 3, 4 \\bigr]').json;   // → ["Interval", ["Open", 3], 4]

  Contextual Parsing: Lists and tuples are automatically converted to intervals when used in set contexts (Element, Union, Intersection, etc.):

  ce.parse('x \\in [0, 1]').json;
  // → ["Element", "x", ["Interval", 0, 1]]
  
  ce.parse('[0, 1] \\cup [2, 3]').json;
  // → ["Union", ["Interval", 0, 1], ["Interval", 2, 3]]
  
  // Standalone notation remains backward compatible
  ce.parse('[0, 1]').json;  // → ["List", 0, 1]
  ce.parse('(0, 1)').json;  // → ["Tuple", 0, 1]

Compilation

• Custom Operator Compilation: The compile() method now supports overriding operators to use function calls instead of native operators. This enables compilation of vector/matrix operations and custom domain-specific languages. Addresses #240.

  // Override operators for vector operations
  const expr = ce.parse('v + w');
  const compiled = expr.compile({
    operators: {
      Add: ['add', 11],      // Convert + to add() function
      Multiply: ['mul', 12]  // Convert * to mul() function
    },
    functions: {
      add: (a, b) => a.map((v, i) => v + b[i]),
      mul: (a, b) => a.map((v, i) => v * b[i])
    }
  });
  
  const result = compiled({ v: [1, 2, 3], w: [4, 5, 6] });
  // → [5, 7, 9]

  Highlights:

  • Map operators via an object or a function
  • Function-name operators compile to calls; symbol operators compile to infix
  • Supports scalar/collection arguments and partial overrides

• Exported Compilation Interfaces: Advanced users can now create custom compilation targets by using the exported CompileTarget interface, BaseCompiler class, and JavaScriptTarget class.

  import { BaseCompiler, JavaScriptTarget } from '@cortex-js/compute-engine';
  
  // Create a custom compilation target
  const customTarget = {
    language: 'my-dsl',
    operators: (op) => ({ Add: ['ADD', 11], Multiply: ['MUL', 12] }[op]),
    functions: (id) => id.toUpperCase(),
    var: (id) => `VAR("${id}")`,
    string: (s) => `"${s}"`,
    number: (n) => n.toString(),
    ws: () => ' ',
    preamble: '',
    indent: 0,
  };
  
  const expr = ce.parse('x + y * 2');
  const code = BaseCompiler.compile(expr, customTarget);
  // → "ADD(VAR("x"), MUL(VAR("y"), 2))"

  Exported building blocks include CompileTarget, LanguageTarget, CompilationOptions, CompiledExecutable, BaseCompiler, JavaScriptTarget, and GLSLTarget (plus helper types like CompiledOperators and CompiledFunctions).

• Compilation Plugin Architecture: The Compute Engine now supports registering custom compilation targets, allowing you to compile mathematical expressions to any target language beyond the built-in JavaScript and GLSL targets.

  import { ComputeEngine, BaseCompiler } from '@cortex-js/compute-engine';
  
  const ce = new ComputeEngine();
  
  // Define a custom Python target
  class PythonTarget {
    // ... implementation (see documentation)
  }
  
  // Register the custom target
  ce.registerCompilationTarget('python', new PythonTarget());
  
  // Compile to Python
  const expr = ce.parse('\\sin(x) + \\cos(y)');
  const pythonCode = expr.compile({ to: 'python' });
  console.log(pythonCode.toString());
  // → math.sin(x) + math.cos(y)
  
  // Switch between targets
  const jsFunc = expr.compile({ to: 'javascript' });
  const glslCode = expr.compile({ to: 'glsl' });

  Notes:

  • Built-in targets: javascript (executable) and glsl (shader code)
  • Add targets via ce.registerCompilationTarget(name, target)
  • Switch targets with compile({ to: ... }) (or override once with target)

• Python/NumPy Compilation Target: Added a complete Python/NumPy compilation target for scientific computing workflows. The PythonTarget class compiles mathematical expressions to NumPy-compatible Python code.

  import { ComputeEngine, PythonTarget } from '@cortex-js/compute-engine';
  
  const ce = new ComputeEngine();
  const python = new PythonTarget({ includeImports: true });
  
  // Register the target
  ce.registerCompilationTarget('python', python);
  
  // Compile expressions to Python
  const expr = ce.parse('\\sin(x) + \\cos(y)');
  const code = expr.compile({ to: 'python' });
  console.log(code.toString());
  // → import numpy as np
  //
  //   np.sin(x) + np.cos(y)
  
  // Generate complete Python functions
  const func = python.compileFunction(
    ce.parse('\\sqrt{x^2 + y^2}'),
    'magnitude',
    ['x', 'y'],
    'Calculate vector magnitude'
  );
  // Generates:
  // import numpy as np
  //
  // def magnitude(x, y):
  //     """Calculate vector magnitude"""
  //     return np.sqrt(x ** 2 + y ** 2)

  Highlights:

  • NumPy-compatible output (including arrays)
  • Function mapping for common math + linear algebra
  • Helpers for full functions, lambdas, and vectorized code

  See the Python/NumPy Target Guide for complete documentation and examples.

• GLSL Compilation Target: New built-in GLSL (OpenGL Shading Language) target for compiling mathematical expressions to WebGL shaders.

  const expr = ce.parse('x^2 + y^2');
  const glslCode = expr.compile({ to: 'glsl' });
  console.log(glslCode.toString());
  // → pow(x, 2.0) + pow(y, 2.0)
  
  // Generate complete GLSL functions
  import { GLSLTarget } from '@cortex-js/compute-engine';
  const glsl = new GLSLTarget();
  
  const distExpr = ce.parse('\\sqrt{x^2 + y^2 + z^2}');
  const func = glsl.compileFunction(distExpr, 'distance3D', 'float', [
    ['x', 'float'],
    ['y', 'float'],
    ['z', 'float'],
  ]);
  console.log(func);
  // → float distance3D(float x, float y, float z) {
  //     return sqrt(pow(x, 2.0) + pow(y, 2.0) + pow(z, 2.0));
  //   }
  
  // Generate complete shaders
  const shader = glsl.compileShader({
    type: 'fragment',
    version: '300 es',
    outputs: [{ name: 'fragColor', type: 'vec4' }],
    body: [
      {
        variable: 'fragColor',
        expression: ce.box(['List', 1, 0, 0, 1]),
      },
    ],
  });

  Highlights:

  • Native vector/matrix operators and constructors
  • Float literal formatting (2.0)
  • Helpers for functions and complete shaders

Algebra

• Polynomial Factoring: The Factor function now supports comprehensive polynomial factoring including perfect square trinomials, difference of squares, and quadratic factoring with rational roots. Addresses #180 and #33.

  // Perfect square trinomials
  ce.parse('x^2 + 2x + 1').factor().latex;
  // → "(x+1)^2"
  
  ce.parse('4x^2 + 12x + 9').factor().latex;
  // → "(2x+3)^2"
  
  // Difference of squares
  ce.parse('x^2 - 4').factor().latex;
  // → "(x-2)(x+2)"
  
  // Quadratic with rational roots
  ce.box(['Factor', ['Add', ['Power', 'x', 2], ['Multiply', 5, 'x'], 6], 'x'])
    .evaluate().latex;
  // → "(x+2)(x+3)"

  Automatic Factoring in sqrt Simplification: Square roots now automatically factor their arguments before applying simplification rules, enabling expressions like √(x²+2x+1) to simplify to |x+1|.

  // Issue #180 - Now works!
  ce.parse('\\sqrt{x^2 + 2x + 1}').simplify().latex;
  // → "\\vert x+1\\vert"
  
  ce.parse('\\sqrt{4x^2 + 12x + 9}').simplify().latex;
  // → "\\vert 2x+3\\vert"
  
  ce.parse('\\sqrt{a^2 + 2ab + b^2}').simplify().latex;
  // → "\\vert a+b\\vert"

  Includes perfect square trinomials, difference of squares, and quadratics with rational roots. Helper functions are exported for advanced usage (factorPerfectSquare, factorDifferenceOfSquares, factorQuadratic, factorPolynomial).

  MathJSON API:

  ["Factor", expr]              // Auto-detect variable
  ["Factor", expr, variable]    // Explicit variable specification

  The enhanced factoring system works seamlessly with existing polynomial functions like Expand, Together, Cancel, PolynomialGCD, and others.

Simplification

• Absolute Value Power Simplification: Fixed simplification of |x^n| expressions with even and rational exponents. Previously, expressions like |x²| and |x^{2/3}| were not simplified. Now they correctly simplify based on the parity of the exponent's numerator. Addresses #181.

  ce.parse('|x^2|').simplify().latex;      // → "x^2" (even exponent)
  ce.parse('|x^3|').simplify().latex;      // → "|x|^3" (odd exponent)
  ce.parse('|x^{2/3}|').simplify().latex;  // → "x^{2/3}" (even numerator)
  ce.parse('|x^{3/2}|').simplify().latex;  // → "|x|^{3/2}" (odd numerator)

• Assumption-Based Simplification: Simplification rules use assumptions about symbol signs:

  ce.assume(ce.parse('x > 0'));
  ce.parse('\\sqrt{x^2}').simplify().latex;  // → "x" (was "|x|")
  ce.parse('|x|').simplify().latex;          // → "x" (was "|x|")
  
  ce.assume(ce.parse('y < 0'));
  ce.parse('\\sqrt{y^2}').simplify().latex;  // → "-y"
  ce.parse('|y|').simplify().latex;          // → "-y"

• Nested Root Simplification: Nested roots simplify to a single root:

  ce.box(['Sqrt', ['Sqrt', 'x']]).simplify()     // → root(4)(x)
  ce.box(['Root', ['Root', 'x', 3], 2]).simplify() // → root(6)(x)
  ce.box(['Sqrt', ['Root', 'x', 3]]).simplify()  // → root(6)(x)

  Applies to all combinations: sqrt(sqrt(x)), root(sqrt(x), n), sqrt(root(x, n)), and root(root(x, m), n).

• Extended Coefficient Factoring in Power Combination: The power combination rule now handles additional coefficient forms when combining same-base powers in products:

  • Multi-prime coefficients: 12·2ˣ·3ˣ → 2^(x+2)·3^(x+1) (since 12 = 2²·3). All primes in the factorization must have a matching base. Non-matching multi-prime coefficients like 6·2ˣ are left unchanged.
  • Negative coefficients: -4·2ˣ → -2^(x+2), -8·2ˣ → -2^(x+3). The absolute value is factored and the sign is preserved.
  • Rational-radical coefficients: √2·2ˣ → 2^(x+½), 2√2·2ˣ → 2^(x+3/2), (√2/2)·2ˣ → 2^(x-½). Decomposes (num/den)·√radical into prime contributions from all three components (radical primes get half-integer exponents, numerator primes get positive exponents, denominator primes get negative exponents).
  • Rational coefficients: 2ˣ/4 → 2^(x-2), 3ˣ/9 → 3^(x-2). Factors both numerator (positive exponents) and denominator (negative exponents).

• Improved Cost Function for Negated Powers: Negate(Power(...)) now costs 3 + cost(exponent), consistent with the cost of Multiply(-1, Power(...)). This makes the cost model more accurate when comparing negated power forms.

Assumptions & Types

• Improved ask() Queries: ce.ask() now matches patterns with wildcards correctly, can answer common "bound" queries such as ask(["Greater", "x", "_k"]) and ask(["Greater", "_x", "_k"]), normalizes inequality patterns for matching (e.g. ask(["Greater", "_x", 0])), and falls back to verify() for closed predicates when the fact is known but not stored as an explicit assumption.

• Tri-state verify(): Implemented ce.verify() as a truth query that returns true, false or undefined when a predicate cannot be determined from the current assumptions and declarations. And/Or/Not use 3-valued logic.

• Element/NotElement Type Membership: Element(x, T) and NotElement(x, T) now support type-style RHS (e.g. real, finite_real, number, any) in addition to set collections (e.g. RealNumbers, Integers).

• Value Resolution from Equality Assumptions: After ce.assume(['Equal', symbol, value]), the symbol now evaluates to the assumed value:

  ce.assume(ce.box(['Equal', 'one', 1]));
  ce.box('one').evaluate();               // → 1 (was: 'one')
  ce.box(['Equal', 'one', 1]).evaluate(); // → True (was: ['Equal', 'one', 1])
  ce.box(['Equal', 'one', 0]).evaluate(); // → False
  ce.box('one').type.matches('integer');  // → true

  This also fixes comparison evaluation: Equal(symbol, assumed_value) now correctly evaluates to True instead of staying symbolic.

• Inequality Evaluation Using Assumptions: Inequality comparisons can use transitive bounds extracted from assumptions.

  ce.assume(ce.box(['Greater', 'x', 4]));
  ce.box(['Greater', 'x', 0]).evaluate();  // → True (x > 4 > 0)
  ce.box(['Less', 'x', 0]).evaluate();     // → False
  ce.box('x').isGreater(0);                // → true
  ce.box('x').isPositive;                  // → true

• Type Inference from Assumptions: Inequalities infer real; equalities infer from the value.

  ce.assume(ce.box(['Greater', 'x', 4]));
  ce.box('x').type.toString();  // → 'real' (was: 'unknown')
  
  ce.assume(ce.box(['Equal', 'one', 1]));
  ce.box('one').type.toString();  // → 'integer' (was: 'unknown')

• Tautology and Contradiction Detection: ce.assume() returns 'tautology' for redundant assumptions and 'contradiction' for conflicts.

  ce.assume(ce.box(['Greater', 'x', 4]));
  
  // Redundant assumption (x > 4 implies x > 0)
  ce.assume(ce.box(['Greater', 'x', 0]));  // → 'tautology' (was: 'ok')
  
  // Conflicting assumption (x > 4 contradicts x < 0)
  ce.assume(ce.box(['Less', 'x', 0]));     // → 'contradiction'
  
  // Same assumption repeated
  ce.assume(ce.box(['Equal', 'one', 1]));
  ce.assume(ce.box(['Equal', 'one', 1]));  // → 'tautology'
  
  // Conflicting equality
  ce.assume(ce.box(['Less', 'one', 0]));   // → 'contradiction'

Solving

• Systems of Linear Equations: The solve() method now handles systems of linear equations parsed from LaTeX \begin{cases}...\end{cases} environments. Returns an object mapping variable names to their solutions.

  const e = ce.parse('\\begin{cases}x+y=70\\\\2x-4y=80\\end{cases}');
  const result = e.solve(['x', 'y']);
  console.log(result.x.json);  // 60
  console.log(result.y.json);  // 10
  
  // 3x3 systems work too
  const e2 = ce.parse('\\begin{cases}x+y+z=6\\\\2x+y-z=1\\\\x-y+2z=5\\end{cases}');
  const result2 = e2.solve(['x', 'y', 'z']);
  // → { x: 1, y: 2, z: 3 }

  Non-linear systems that don't match known patterns and inconsistent systems return null.

• Non-linear Polynomial Systems: The solve() method now handles certain non-linear polynomial systems with 2 equations and 2 variables:

  • Product + sum pattern: Systems like xy = p, x + y = s are solved by recognizing that x and y are roots of the quadratic t² - st + p = 0.

  • Substitution method: When one equation is linear in one variable, it substitutes into the other equation and solves the resulting univariate equation.

  Returns an array of solution objects (multiple solutions possible):

  // Product + sum pattern
  const e = ce.parse('\\begin{cases}xy=6\\\\x+y=5\\end{cases}');
  const result = e.solve(['x', 'y']);
  // → [{ x: 2, y: 3 }, { x: 3, y: 2 }]
  
  // Substitution method
  const e2 = ce.parse('\\begin{cases}x+y=5\\\\x^2+y=7\\end{cases}');
  const result2 = e2.solve(['x', 'y']);
  // → [{ x: 2, y: 3 }, { x: -1, y: 6 }]

  Only real solutions are returned; complex solutions are filtered out.

• Exact Rational Arithmetic in Linear Systems: The linear system solver now uses exact rational arithmetic throughout the Gaussian elimination process. Systems with fractional coefficients produce exact fractional results rather than floating-point approximations.

  const e = ce.parse('\\begin{cases}x+y=1\\\\x-y=1/2\\end{cases}');
  const result = e.solve(['x', 'y']);
  console.log(result.x.json);  // ["Rational", 3, 4]  (exact 3/4)
  console.log(result.y.json);  // ["Rational", 1, 4]  (exact 1/4)
  
  // Fractional coefficients
  const e2 = ce.parse('\\begin{cases}x/3+y/2=1\\\\x/4+y/5=1\\end{cases}');
  const result2 = e2.solve(['x', 'y']);
  // → { x: 36/7, y: -10/7 }

• Linear Inequality Systems: The solve() method now handles systems of linear inequalities in 2 variables, returning the vertices of the feasible region (convex polygon). Supports all inequality operators: <, <=, >, >=.

  // Triangle: x >= 0, y >= 0, x + y <= 10
  const e = ce.parse('\\begin{cases}x\\geq 0\\\\y\\geq 0\\\\x+y\\leq 10\\end{cases}');
  const result = e.solve(['x', 'y']);
  // → [{ x: 0, y: 0 }, { x: 10, y: 0 }, { x: 0, y: 10 }]
  
  // Square: 0 <= x <= 5, 0 <= y <= 5
  const square = ce.parse('\\begin{cases}x\\geq 0\\\\x\\leq 5\\\\y\\geq 0\\\\y\\leq 5\\end{cases}');
  square.solve(['x', 'y']);
  // → [{ x: 0, y: 0 }, { x: 5, y: 0 }, { x: 5, y: 5 }, { x: 0, y: 5 }]

  Vertices are returned in counterclockwise convex hull order. Returns null for infeasible systems or non-linear constraints.

• Under-determined Systems (Parametric Solutions): The solve() method now returns parametric solutions for under-determined linear systems (fewer equations than variables) instead of returning null. Free variables appear as themselves in the solution, with other variables expressed in terms of them.

  // Single equation with two variables
  const e = ce.parse('\\begin{cases}x+y=5\\end{cases}');
  const result = e.solve(['x', 'y']);
  // → { x: -y + 5, y: y }  (y is a free variable)
  
  // Two equations with three variables
  const e2 = ce.parse('\\begin{cases}x+y+z=6\\\\x-y=2\\end{cases}');
  const result2 = e2.solve(['x', 'y', 'z']);
  // → { x: -z/2 + 4, y: -z/2 + 2, z: z }  (z is a free variable)

  Inconsistent systems still return null.

• Extended Sqrt Equation Solving: The equation solver now handles sqrt equations of the form √(f(x)) = g(x) by squaring both sides and solving the resulting polynomial. Extraneous roots are automatically filtered.

  ce.parse('\\sqrt{x+1} = x').solve('x');      // → [1.618...] (golden ratio)
  ce.parse('\\sqrt{2x+3} = x - 1').solve('x'); // → [4.449...]
  ce.parse('\\sqrt{3x-2} = x').solve('x');     // → [1, 2]
  ce.parse('\\sqrt{x} = x').solve('x');        // → [0, 1]

• Two Sqrt Equation Solving: The equation solver now handles equations with two sqrt terms of the form √(f(x)) + √(g(x)) = e using double squaring. Both addition and subtraction forms are supported, and extraneous roots are automatically filtered.

  ce.parse('\\sqrt{x+1} + \\sqrt{x+4} = 3').solve('x');  // → [0]
  ce.parse('\\sqrt{x} + \\sqrt{x+7} = 7').solve('x');    // → [9]
  ce.parse('\\sqrt{x+5} - \\sqrt{x-3} = 2').solve('x');  // → [4]
  ce.parse('\\sqrt{2x+1} + \\sqrt{x-1} = 4').solve('x'); // → [46 - 8√29] ≈ 2.919

• Nested Sqrt Equation Solving: The equation solver now handles nested sqrt equations of the form √(x + √x) = a using substitution. These patterns have √x inside the argument of an outer sqrt. The solver uses u = √x substitution, solves the resulting quadratic, and filters negative u values.

  ce.parse('\\sqrt{x + 2\\sqrt{x}} = 3').solve('x');  // → [11 - 2√10] ≈ 4.675
  ce.parse('\\sqrt{x + \\sqrt{x}} = 2').solve('x');   // → [9/2 - √17/2] ≈ 2.438
  ce.parse('\\sqrt{x - \\sqrt{x}} = 1').solve('x');   // → [φ²] ≈ 2.618

• Quadratic Equations Without Constant Term: Added support for solving quadratic equations of the form ax² + bx = 0 (missing constant term). These are solved by factoring: x(ax + b) = 0 → x = 0 or x = -b/a.

  ce.parse('x^2 + 3x = 0').solve('x');  // → [0, -3]
  ce.parse('2x^2 - 4x = 0').solve('x'); // → [0, 2]

Subscripts & Indexing

• Subscript Evaluation Handler: Define custom evaluation functions for subscripted symbols like mathematical sequences using subscriptEvaluate:

  // Define a Fibonacci sequence
  ce.declare('F', {
    subscriptEvaluate: (subscript, { engine }) => {
      const n = subscript.re;
      if (!Number.isInteger(n) || n < 0) return undefined;
      // Calculate Fibonacci number...
      return engine.number(fibValue);
    },
  });
  
  ce.parse('F_{10}').evaluate();  // → 55
  ce.parse('F_5').evaluate();     // → 5
  ce.parse('F_n').evaluate();     // → stays symbolic (handler returns undefined)

  Both simple subscripts (F_5) and complex subscripts (F_{5}) are supported. When the handler returns undefined, the expression stays symbolic. Subscripted expressions with subscriptEvaluate have type number and can be used in arithmetic operations: ce.parse('F_{5} + F_{3}').evaluate() works correctly.

• Type-Aware Subscript Handling: Subscripts on symbols declared as collection types (list, tuple, matrix, etc.) now automatically convert to At() indexing operations:

  ce.declare('v', 'list<number>');
  ce.parse('v_n');      // → At(v, n)
  ce.parse('v_{n+1}');  // → At(v, n+1)
  ce.parse('v_{i,j}');  // → At(v, Tuple(i, j))

  This works for both simple subscripts (v_n) and complex subscripts (v_{n+1}). The type of the At() expression is correctly inferred from the collection's element type, allowing subscripted collection elements to be used in arithmetic.

• Complex Subscripts in Arithmetic (Issue #273): Subscript expressions like a_{n+1} can now be used in arithmetic operations without type errors:

  ce.parse('a_{n+1} + 1');     // → Add(Subscript(a, n+1), 1)
  ce.parse('2 * a_{n+1}');     // → Multiply(2, Subscript(a, n+1))
  ce.parse('a_{n+1}^2');       // → Power(Subscript(a, n+1), 2)

  Previously, complex subscripts would fail with "incompatible-type" errors when used in arithmetic contexts.

• Multi-Index At() Support: The At function now supports multiple indices for accessing nested collections (e.g., matrices):

  const matrix = ce.box(['List', ['List', 2, 3, 4], ['List', 6, 7, 9]]);
  ce.box(['At', matrix, 1, 2]).evaluate();  // → 3 (row 1, column 2)

  The signature was updated from single index to variadic: (value: indexed_collection, index: (number|string)+) -> unknown

• Text Subscripts: Added support for \text{} in subscripts, allowing descriptive subscript names:

  ce.parse('x_{\\text{max}}');  // → symbol "x_max"
  ce.parse('v_{\\text{initial}}');  // → symbol "v_initial"

Sequences

• Declarative Sequence Definitions: Define mathematical sequences using recurrence relations with the new declareSequence() method:

  // Fibonacci sequence
  ce.declareSequence('F', {
    base: { 0: 0, 1: 1 },
    recurrence: 'F_{n-1} + F_{n-2}',
  });
  ce.parse('F_{10}').evaluate();  // → 55
  ce.parse('F_{20}').evaluate();  // → 6765
  
  // Arithmetic sequence: a_n = a_{n-1} + 2, a_0 = 1
  ce.declareSequence('A', {
    base: { 0: 1 },
    recurrence: 'A_{n-1} + 2',
  });
  ce.parse('A_{5}').evaluate();  // → 11
  
  // Factorial via recurrence
  ce.declareSequence('H', {
    base: { 0: 1 },
    recurrence: 'n \\cdot H_{n-1}',
  });
  ce.parse('H_{5}').evaluate();  // → 120

  Features:

  • Base cases as index → value mapping
  • Recurrence relation as LaTeX string or BoxedExpression
  • Automatic memoization for efficient evaluation (configurable)
  • Custom index variable name (default: n)
  • Domain constraints (min/max valid indices)
  • Symbolic subscripts stay symbolic (e.g., F_k remains unevaluated)

  Alternatively, sequences can be defined using natural LaTeX assignment notation:

  // Arithmetic sequence via LaTeX
  ce.parse('L_0 := 1').evaluate();
  ce.parse('L_n := L_{n-1} + 2').evaluate();
  ce.parse('L_{5}').evaluate();  // → 11
  
  // Fibonacci via LaTeX
  ce.parse('F_0 := 0').evaluate();
  ce.parse('F_1 := 1').evaluate();
  ce.parse('F_n := F_{n-1} + F_{n-2}').evaluate();
  ce.parse('F_{10}').evaluate();  // → 55

  Base cases and recurrence can be defined in any order. The sequence is finalized when both are present.

• Sequence Status API: Query the status of sequence definitions with getSequenceStatus():

  ce.parse('F_0 := 0').evaluate();
  ce.getSequenceStatus('F');
  // → { status: 'pending', hasBase: true, hasRecurrence: false, baseIndices: [0] }
  
  ce.parse('F_n := F_{n-1} + F_{n-2}').evaluate();
  ce.getSequenceStatus('F');
  // → { status: 'complete', hasBase: true, hasRecurrence: true, baseIndices: [0] }
  
  ce.getSequenceStatus('x');
  // → { status: 'not-a-sequence', hasBase: false, hasRecurrence: false }

• Sequence Introspection API: Inspect and manage defined sequences:

  // Get sequence information
  ce.getSequence('F');
  // → { name: 'F', variable: 'n', baseIndices: [0, 1], memoize: true, cacheSize: 5 }
  
  // List all defined sequences
  ce.listSequences();  // → ['F', 'A', 'H']
  
  // Check if a symbol is a sequence
  ce.isSequence('F');  // → true
  ce.isSequence('x');  // → false
  
  // Manage memoization cache
  ce.getSequenceCache('F');  // → Map { 2 => 1, 3 => 2, ... }
  ce.clearSequenceCache('F');  // Clear cache for specific sequence
  ce.clearSequenceCache();     // Clear all sequence caches

• Generate Sequence Terms: Generate a list of sequence terms with getSequenceTerms():

  ce.declareSequence('F', {
    base: { 0: 0, 1: 1 },
    recurrence: 'F_{n-1} + F_{n-2}',
  });
  
  ce.getSequenceTerms('F', 0, 10);
  // → [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
  
  // With step parameter (every other term)
  ce.getSequenceTerms('F', 0, 10, 2);
  // → [0, 1, 3, 8, 21, 55]

• Sum and Product over Sequences: Sum and Product now work seamlessly with user-defined sequences:

  ce.declareSequence('F', {
    base: { 0: 0, 1: 1 },
    recurrence: 'F_{n-1} + F_{n-2}',
  });
  
  ce.parse('\\sum_{k=0}^{10} F_k').evaluate();  // → 143
  ce.parse('\\prod_{k=1}^{5} A_k').evaluate();  // Works with any defined sequence

• OEIS Integration: Look up sequences in the Online Encyclopedia of Integer Sequences (OEIS) and verify your sequences against known mathematical sequences:

  // Look up a sequence by its terms
  const results = await ce.lookupOEIS([0, 1, 1, 2, 3, 5, 8, 13]);
  // → [{ id: 'A000045', name: 'Fibonacci numbers', terms: [...], url: '...' }]
  
  // Check if your sequence matches a known OEIS sequence
  ce.declareSequence('F', {
    base: { 0: 0, 1: 1 },
    recurrence: 'F_{n-1} + F_{n-2}',
  });
  
  const result = await ce.checkSequenceOEIS('F', 10);
  // → { matches: [{ id: 'A000045', name: 'Fibonacci numbers', ... }], terms: [...] }

  Note: OEIS lookups require network access to oeis.org.

• Multi-Index Sequences: Define sequences with multiple indices like Pascal's triangle P_{n,k} or grid-based recurrences:

  // Pascal's Triangle: P_{n,k} = P_{n-1,k-1} + P_{n-1,k}
  ce.declareSequence('P', {
    variables: ['n', 'k'],
    base: { 'n,0': 1, 'n,n': 1 },  // Pattern-based base cases
    recurrence: 'P_{n-1,k-1} + P_{n-1,k}',
    domain: { n: { min: 0 }, k: { min: 0 } },
    constraints: 'k <= n',  // k must not exceed n
  });
  
  ce.parse('P_{5,2}').evaluate();  // → 10
  ce.parse('P_{10,5}').evaluate(); // → 252

  Features:

  • Multiple index variables with variables: ['n', 'k']
  • Pattern-based base cases: 'n,0' matches any (n, 0), 'n,n' matches diagonal
  • Per-variable domain constraints
  • Constraint expressions (e.g., 'k <= n')
  • Composite key memoization (e.g., '5,2')
  • Full introspection support with isMultiIndex flag

  Pattern matching for base cases:

  • Exact values: '0,0' matches only (0, 0)
  • Wildcards: 'n,0' matches any value for n with k=0
  • Equality: 'n,n' matches when both indices are equal
  • Priority: exact matches are checked before patterns

Special Functions

• Special Function Definitions: Added type signatures for special mathematical functions, enabling them to be used in expressions without type errors:

  • Zeta - Riemann zeta function \zeta(s)
  • Beta - Euler beta function B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b)
  • LambertW - Lambert W function (product logarithm)
  • BesselJ, BesselY, BesselI, BesselK - Bessel functions of first/second kind
  • AiryAi, AiryBi - Airy functions

  These functions now have proper signatures and can be composed with other expressions: ce.box(['Add', 1, ['LambertW', 'x']]) works correctly.

• Special Function LaTeX Parsing: Added LaTeX parsing support for special functions: \zeta(s), \Beta(a,b), \operatorname{W}(x), Bessel functions via \operatorname{J}, \operatorname{Y}, etc., and Airy functions via \operatorname{Ai}, \operatorname{Bi}.

Calculus

• LambertW Derivative: Added derivative rule for the Lambert W function: d/dx W(x) = W(x)/(x·(1+W(x)))

• Bessel Function Derivatives: Added derivative support for all four Bessel function types using order-dependent recurrence relations:

  ce.box(['D', ['BesselJ', 'n', 'x'], 'x']).evaluate();
  // → 1/2 * BesselJ(n-1, x) - 1/2 * BesselJ(n+1, x)
  
  ce.box(['D', ['BesselI', 'n', 'x'], 'x']).evaluate();
  // → 1/2 * BesselI(n-1, x) + 1/2 * BesselI(n+1, x)
  
  ce.box(['D', ['BesselK', 'n', 'x'], 'x']).evaluate();
  // → -1/2 * BesselK(n-1, x) - 1/2 * BesselK(n+1, x)

  Chain rule is automatically applied for composite arguments.

• Multi-Argument Function Derivatives: Added derivative support for:

  • Log(x, base) - Logarithm with custom base:

    ce.box(['D', ['Log', 'x', 2], 'x']).evaluate();  // → 1/(x·ln(2))
    ce.box(['D', ['Log', 'x', 'a'], 'x']).evaluate(); // → 1/(x·ln(a))

    Also handles cases where both x and base depend on the variable by applying the quotient rule to ln(x)/ln(base).

  • Discrete functions (Mod, GCD, LCM) - Return 0 as these are step functions with derivative 0 almost everywhere:

    ce.box(['D', ['Mod', 'x', 5], 'x']).evaluate();  // → 0
    ce.box(['D', ['GCD', 'x', 6], 'x']).evaluate();  // → 0

• Integration of 1/(x·ln(x)) Pattern: Added support for integrating expressions where the denominator is a product and one factor is the derivative of another:

  ce.parse('\\int \\frac{1}{x\\ln x} dx').evaluate();  // → ln(|ln(x)|)
  ce.parse('\\int \\frac{3}{x\\ln x} dx').evaluate();  // → 3·ln(|ln(x)|)

  This uses u-substitution: since 1/x = d/dx(ln(x)), the integral becomes ∫ h'(x)/h(x) dx = ln|h(x)|.

• Cyclic Integration for e^x with Trigonometric Functions: Added support for integrating products of exponentials and trigonometric functions that require the "solve for the integral" technique:

  ce.parse('\\int e^x \\sin x dx').evaluate();
  // → -1/2·cos(x)·e^x + 1/2·sin(x)·e^x
  
  ce.parse('\\int e^x \\cos x dx').evaluate();
  // → 1/2·sin(x)·e^x + 1/2·cos(x)·e^x
  
  // Also works with linear arguments:
  ce.parse('\\int e^x \\sin(2x) dx').evaluate();
  // → -2/5·cos(2x)·e^x + 1/5·sin(2x)·e^x
  
  ce.parse('\\int e^x \\cos(2x) dx').evaluate();
  // → 1/5·cos(2x)·e^x + 2/5·sin(2x)·e^x

  These patterns cannot be solved by standard integration by parts (which would lead to infinite recursion) and instead use direct formulas:

  • ∫ e^x·sin(ax+b) dx = (e^x/(a²+1))·(sin(ax+b) - a·cos(ax+b))
  • ∫ e^x·cos(ax+b) dx = (e^x/(a²+1))·(a·sin(ax+b) + cos(ax+b))

• Derivative Recursion Safety: Added recursion protection to differentiate() with a depth limit (MAX_DIFFERENTIATION_DEPTH), returning undefined when the limit is exceeded.

• Equation Equivalence in isEqual() (Issue #275): Two equations are now recognized as equivalent if they have the same solution set:

  ce.parse('2x+1=0').isEqual(ce.parse('x=-1/2'));   // → true
  ce.parse('3x+1=0').isEqual(ce.parse('6x+2=0'));   // → true

  Uses sampling to check whether (LHS₁-RHS₁)/(LHS₂-RHS₂) is a non-zero constant.

Logic

• Boolean Simplification Rules: Added absorption laws and improved boolean expression simplification:

  • Absorption: A ∧ (A ∨ B) → A and A ∨ (A ∧ B) → A
  • Idempotence: A ∧ A → A and A ∨ A → A
  • Complementation: A ∧ ¬A → False and A ∨ ¬A → True
  • Identity: A ∧ True → A and A ∨ False → A
  • Domination: A ∧ False → False and A ∨ True → True
  • Double negation: ¬¬A → A

  These rules are applied automatically during simplification:

  ce.box(['And', 'A', ['Or', 'A', 'B']]).simplify();  // → A
  ce.box(['Or', 'A', ['And', 'A', 'B']]).simplify();  // → A

• Prime Implicants and Minimal Normal Forms: Added Quine-McCluskey algorithm for finding prime implicants/implicates and computing minimal CNF/DNF:

  • PrimeImplicants(expr) - Find all prime implicants (minimal product terms)
  • PrimeImplicates(expr) - Find all prime implicates (minimal sum clauses)
  • MinimalDNF(expr) - Convert to minimal DNF using prime implicant cover
  • MinimalCNF(expr) - Convert to minimal CNF using prime implicate cover

  // Find prime implicants (terms that can't be further simplified)
  ce.box(['PrimeImplicants', ['Or', ['And', 'A', 'B'], ['And', 'A', ['Not', 'B']]]]).evaluate();
  // → [A] (AB and A¬B combine to just A)
  
  // Compute minimal DNF
  ce.box(['MinimalDNF', ['Or',
    ['And', 'A', 'B'],
    ['And', 'A', ['Not', 'B']],
    ['And', ['Not', 'A'], 'B']
  ]]).evaluate();
  // → A ∨ B (simplified from 3 terms to 2)

  Limited to 12 variables to prevent exponential blowup; larger expressions return unevaluated.

Linear Algebra

• Matrix Decompositions: Added four matrix decomposition functions for numerical linear algebra:

  • LUDecomposition(A) → [P, L, U] - LU factorization with partial pivoting
  • QRDecomposition(A) → [Q, R] - QR factorization using Householder reflections
  • CholeskyDecomposition(A) → L - Cholesky factorization for positive definite matrices
  • SVD(A) → [U, Σ, V] - Singular Value Decomposition

  ce.box(['LUDecomposition', [[4, 3], [6, 3]]]).evaluate();
  // → [P, L, U] where PA = LU
  
  ce.box(['QRDecomposition', [[1, 2], [3, 4]]]).evaluate();
  // → [Q, R] where A = QR, Q orthogonal, R upper triangular
  
  ce.box(['CholeskyDecomposition', [[4, 2], [2, 2]]]).evaluate();
  // → L where A = LL^T
  
  ce.box(['SVD', [[1, 2], [3, 4]]]).evaluate();
  // → [U, Σ, V] where A = UΣV^T

Fixed

• replace() Literal Matching in Object Rules: .replace({ match: 'a', replace: 2 }) no longer treats 'a' as a wildcard (string rules like "a*x -> 2*x" still auto-wildcard).

  const expr = ce.box(['Add', ['Multiply', 'a', 'x'], 'b']);
  expr.replace({match: 'a', replace: 2}, {recursive: true});
  // → 2x + b (was: 2 - incorrectly matched entire expression)

• forget() Clears Assumed Values: ce.forget() now clears values set by equality assumptions across all evaluation context frames.

  ce.assume(ce.box(['Equal', 'x', 5]));
  ce.box('x').evaluate();  // → 5
  ce.forget('x');
  ce.box('x').evaluate();  // → 'x' (was: 5)

• Scoped Assumptions Clean Up on popScope(): Assumptions made inside a scope no longer leak after popScope().

  ce.pushScope();
  ce.assume(ce.box(['Equal', 'y', 10]));
  ce.box('y').evaluate();  // → 10
  ce.popScope();
  ce.box('y').evaluate();  // → 'y' (was: 10)

• Extraneous Root Filtering for Sqrt Equations: Candidate solutions are now validated against the original expression (before clearing denominators / harmonization) to filter extraneous roots.

  Examples of equations that now correctly filter extraneous roots:

  • √x = x - 2 → returns [4] (filters out x=1)
  • √x + x - 2 = 0 → returns [1] (filters out x=4)
  • √x - x + 2 = 0 → returns [4] (filters out x=1)
  • x - 2√x - 3 = 0 → returns [9] (filters out x=1)
  • 2x + 3√x - 2 = 0 → returns [1/4] (filters out x=4)

• Simplification (#178):

  • Safer division canonicalization for denominators that may simplify to 0
  • Implicit multiplication powers: xx → x^2
  • Targeted exp/log rewriting for \exp(\log(x)±y)

0.33.0 2026-01-30

Resolved Issues

Arithmetic and Infinity

• Division by Zero: Improved handling of division by zero:

  • 0/0 returns NaN (indeterminate form)
  • a/0 where a ≠ 0 returns ComplexInfinity (~∞) as a "better NaN" that indicates an infinite result with unknown sign
  • This applies to all forms including 1/0, x/0, and rational literals

• Infinity Sign Propagation: Fixed infinity multiplication not propagating signs correctly. Now ∞ * (-2) = -∞ and -∞ * 2 = -∞ as expected.

• Infinity Division: Fixed ∞/∞ incorrectly returning 1. Now correctly returns NaN (indeterminate form). The a/a → 1 simplification rule now excludes infinity values.

Trigonometry

• Trigonometric Period Identities: Fixed incorrect sign handling for csc(π+x) and cot(π+x):

  • csc(π+x) now correctly simplifies to -csc(x) (was incorrectly csc(x))
  • cot(π+x) now correctly simplifies to cot(x) (was incorrectly -cot(x), cotangent has period π)

• Trigonometric Co-function Identities: Fixed co-function identities not applying to canonical form expressions. Now correctly simplifies:

  • sin(π/2 - x) → cos(x)
  • cos(π/2 - x) → sin(x)
  • tan(π/2 - x) → cot(x)
  • cot(π/2 - x) → tan(x)
  • sec(π/2 - x) → csc(x)
  • csc(π/2 - x) → sec(x)

• Double Angle with Coefficient: Fixed 2sin(x)cos(x) not simplifying to sin(2x). The product-to-sum identity now handles coefficients:

  • 2sin(x)cos(x) → sin(2x)
  • c·sin(x)cos(x) → c·sin(2x)/2 for any coefficient c

• Trigonometric Product Identities: Improved handling of trig products in simplification. The Multiply rule now correctly defers to trig-specific rules for patterns like sin(x)*cos(x) and tan(x)*cot(x), ensuring these are simplified to sin(2x)/2 and 1 respectively.

Logarithms and Exponentials

• Logarithm-Exponential Composition: Fixed log(exp(x)) incorrectly simplifying to x. Now correctly returns x/ln(10) ≈ 0.434x since log₁₀(eˣ) = x·log₁₀(e) = x/ln(10). The identity log(exp(x)) = x only holds for natural logarithm.

• Logarithm of e: Added simplification for log(e) → 1/ln(10) ≈ 0.434 and log_c(e) → 1/ln(c) for any base c.

• Logarithm Combination Base Preservation: Fixed log(x) + log(y) (base 10) incorrectly becoming ln(xy). Now correctly produces log(xy) preserving the original base.

• Logarithm Quotient Rule: Added expansion rule for logarithm of quotients. ln(x/y) now simplifies to ln(x) - ln(y) when x and y are known positive. Similarly for any base: log_c(x/y) → log_c(x) - log_c(y).

• Exponential-Logarithm Composition: Added simplification for exp(log(x)) where log has a different base than e. Now e^log(x) → x^{1/ln(10)} and more generally e^log_c(x) → x^{1/ln(c)} for any base c.

Powers and Exponents

• Zero Power with Symbolic Exponent: Fixed 0^π and similar expressions with positive symbolic exponents not simplifying. Now 0^x → 0 when x is known to be positive (including π, e, etc.).

• Exponent Evaluation in Products: Fixed (x³)² · (y²)² not simplifying to x⁶y⁴. Numeric subexpressions in exponents (like 2×3 in x^{2×3}) are now evaluated when the expression is part of a product.

• Negative Exponents on Fractions: Fixed (a/b)^{-n} not simplifying properly. Now (x³/y²)^{-2} correctly simplifies to y⁴/x⁶ during canonicalization by distributing the negative exponent.

• Negative Base with Fractional Exponent: Fixed (-ax)^{p/q} returning complex results when p and q are both odd. Now correctly factors out the negative sign: (-2x)^{3/5} → -(2x)^{3/5} = -2^{3/5}·x^{3/5}, giving real results. This affects products like (-2x)^{3/5}·x which now correctly simplify to -2^{3/5}·x^{8/5} instead of returning an imaginary value.

Radicals

• Radical Perfect Square Factoring: Fixed √(x²y) not simplifying to |x|√y. Adjusted cost function to penalize radicals containing perfect squares, enabling the simplification rule to apply.

• Generalized Root Extraction: Added comprehensive root simplification rules:

  • √[n]{x^m} → x^{m/n} for odd roots (always valid)
  • √[n]{x^m} → |x|^{m/n} for even roots with integer result
  • √{x^{odd}} → |x|^n · √x factoring (e.g., √{x⁵} → |x|²√x)
  • Handles all combinations: √[4]{x⁶} → |x|^{3/2}, √[3]{x⁶} → x²

• Symbolic Radicals Preservation: Fixed numeric radicals (√2, ∛5, 2^{3/5}) being evaluated to floating-point approximations during multiplication. Now x * √2 stays as √2 · x instead of 1.414... · x, and x * 2^{1/3} stays as x · ∛2 instead of 1.259... · x. This preserves exact irrational values and allows proper algebraic manipulation. Use .N() to get numeric approximations when needed.

LaTeX Parsing

• LaTeX \exp() Juxtaposition: Fixed adjacent \exp() calls not parsing as multiplication. Now \exp(x)\exp(2) correctly parses as e^x · e^2 instead of producing a parse error. The expression then simplifies to e^{x+2} as expected.

Features

Trigonometry

• Fu Algorithm for Trigonometric Simplification: Implemented the Fu algorithm based on Fu, Zhong, and Zeng's paper "Automated and readable simplification of trigonometric expressions" (2006). This provides systematic, high-quality trigonometric simplification through:

  • Transformation Rules (TR1-TR22): Comprehensive set of rewrite rules including reciprocal conversions (sec→1/cos), ratio forms (tan→sin/cos), Pythagorean substitutions (sin²+cos²=1), power reductions, product-to-sum, sum-to-product, angle expansion/contraction, and Morrie's law for cosine product chains.

  • Rule Lists (RL1, RL2): Organized application sequences for tan/cot expressions and sin/cos expressions respectively, with greedy selection of optimal results.

  • Cost Function: Minimizes trigonometric function count as primary metric, with leaf count as secondary, to find the most readable form.

  Usage:

  // Option 1: Use strategy option with simplify()
  const result = expr.simplify({ strategy: 'fu' });
  
  // Option 2: Dedicated trigSimplify() method
  const result = expr.trigSimplify();

  Examples:

  • sin(x)⁴ - cos(x)⁴ → -cos(2x)
  • tan(x)·cot(x) → 1
  • sin²(x) + cos²(x) → 1
  • 2sin(x)cos(x) → sin(2x)
  • cos(x)·cos(2x)·cos(4x) → sin(8x)/(8sin(x)) (Morrie's law)

  Enhanced Transformations:

  • TRmorrie with Rational Coefficients: Morrie's law now handles angles that are rational multiples of π, such as cos(π/9)·cos(2π/9)·cos(4π/9) → 1/8. The algorithm detects maximal geometric sequences and handles cases where the sine terms cancel to produce pure fractions.

  • TR12i Tangent Sum Identity: Recognizes the pattern tan(A) + tan(B) - k·tan(A)·tan(B) and simplifies to -tan(C) when A + B + C = π and k = tan(C). Works with standard angles (π/6, π/4, π/3, etc.) and handles sign variations.

  • TRpythagorean for Compound Expressions: Detects sin²(x) + cos²(x) pairs within larger Add expressions and simplifies them to 1, e.g., sin²(x) + cos²(x) + 2 → 3.

  • Early TR9 Sum-to-Product: Applies sum-to-product transformation before angle expansion to catch patterns like sin(x+h) + sin(x-h) → 2sin(x)cos(h) that would otherwise be expanded and lose their simplified form.

  • Dual Strategy Approach: The Fu strategy now tries both "Fu first" and "simplify first" approaches and picks the best result. This handles both Morrie-like patterns (which need Fu before evaluation) and period reduction patterns (which need simplification first for angle contraction).

• Trigonometric Periodicity Reduction: Trigonometric functions now simplify arguments containing integer multiples of π:

  • sin(5π + k) → -sin(k) (period 2π, with sign change for odd multiples)
  • cos(4π + k) → cos(k) (period 2π)
  • tan(3π + k) → tan(k) (period π)
  • Works for all six trig functions: sin, cos, tan, cot, sec, csc
  • Handles both positive and negative multiples of π

• Pythagorean Trigonometric Identities: Added simplification rules for all Pythagorean identities:

  • sin²(x) + cos²(x) → 1
  • 1 - sin²(x) → cos²(x) and 1 - cos²(x) → sin²(x)
  • sin²(x) - 1 → -cos²(x) and cos²(x) - 1 → -sin²(x)
  • tan²(x) + 1 → sec²(x) and sec²(x) - 1 → tan²(x)
  • 1 + cot²(x) → csc²(x) and csc²(x) - 1 → cot²(x)
  • a·sin²(x) + a·cos²(x) → a (with coefficient)

• Trigonometric Equation Solving: The solve() method now handles basic trigonometric equations:

  • sin(x) = a → x = arcsin(a) and x = π - arcsin(a) (two solutions)
  • cos(x) = a → x = arccos(a) and x = -arccos(a) (two solutions)
  • tan(x) = a → x = arctan(a) (one solution per period)
  • cot(x) = a → x = arccot(a)
  • Supports coefficient form: a·sin(x) + b = 0
  • Domain validation: returns no solutions when |a| > 1 for sin/cos
  • Automatic deduplication of equivalent solutions (e.g., cos(x) = 1 → single solution 0)

Calculus

• (#163) Additional Derivative Notations: Added support for parsing multiple derivative notations beyond Leibniz notation:

  • Newton's dot notation for time derivatives: \dot{x} → ["D", "x", "t"], \ddot{x} for second derivative, \dddot{x} and \ddddot{x} for higher orders. The time variable is configurable via the new timeDerivativeVariable parser option (default: "t").

  • Lagrange prime notation with arguments: f'(x) now parses to ["D", ["f", "x"], "x"], inferring the differentiation variable from the function argument. Works for f''(x), f'''(x), etc. for higher derivatives.

  • Euler's subscript notation: D_x f → ["D", "f", "x"] and D^2_x f or D_x^2 f for second derivatives.

  • Derivative serialization: D expressions now serialize to Leibniz notation (\frac{\mathrm{d}}{\mathrm{d}x}f) for consistent round-trip parsing.

• Derivative Rules for Special Functions: Added derivative formulas for:

  • d/dx Digamma(x) = Trigamma(x)
  • d/dx Erf(x), d/dx Erfc(x), d/dx Erfi(x)
  • d/dx FresnelS(x), d/dx FresnelC(x)
  • d/dx LogGamma(x) = Digamma(x)

Special Functions

• Special Function Definitions: Added type signatures for Digamma, Trigamma, and PolyGamma functions to the library:
  • Digamma(x) - The digamma function ψ(x), logarithmic derivative of Gamma
  • Trigamma(x) - The trigamma function ψ₁(x), derivative of digamma
  • PolyGamma(n, x) - The polygamma function ψₙ(x), nth derivative of digamma

Logarithms and Exponentials

• Logarithm Combination Rules: Added simplification rules that combine logarithms with the same base:

  • ln(x) + ln(y) → ln(xy) (addition combines via multiplication)
  • ln(x) - ln(y) → ln(x/y) (subtraction combines via division)
  • log_c(x) + log_c(y) → log_c(xy) (works with any base)
  • log_c(x) - log_c(y) → log_c(x/y)
  • Handles multiple terms: ln(a) + ln(b) - ln(c) → ln(ab/c)

• Exponential e Simplification: Added rules for combining powers of e:

  • eˣ · eʸ → e^(x+y) (same-base multiplication)
  • eˣ / eʸ → e^(x-y) (same-base division)
  • eˣ · e → e^(x+1) and eˣ / e → e^(x-1)
  • Preserves symbolic form instead of evaluating e^n numerically

Powers and Exponents

• Negative Base Power Simplification: Added rules to simplify powers with negated bases:

  • (-x)^n → x^n when n is even (e.g., (-x)^4 → x^4)
  • (-x)^n → -x^n when n is odd (e.g., (-x)^3 → -x^3)
  • (-x)^{n/m} → x^{n/m} when n is even and m is odd
  • (-x)^{n/m} → -x^{n/m} when both n and m are odd
  • (-1)^{p/q} → -1 when both p and q are odd (real odd root)

• Power Distribution: Added rule to distribute integer exponents over products:

  • (ab)^n → a^n · b^n when n is an integer
  • Example: (x³y²)² → x⁶y⁴
  • Example: (-2x)² → 4x²

• Same-Base Power Combination: Improved power combination for products with 3+ terms:

  • a³ · a · a² → a⁶ (combines all same-base terms)
  • Works with unknown symbols when sum of exponents is positive
  • Handles mixed products: b³c²dx⁷ya⁵gb²x⁵(3b) → 3dgyx¹²b⁶a⁵c²

Sum and Product

• (#133) Element-based Indexing Sets for Sum/Product: Added support for \in notation in summation and product subscripts:

  • Parsing: \sum_{n \in \{1,2,3\}} n now correctly parses to ["Sum", "n", ["Element", "n", ["Set", 1, 2, 3]]] instead of silently dropping the constraint.

  • Evaluation: Sums and products over finite sets, lists, and ranges are now evaluated correctly:

    • \sum_{n \in \{1,2,3\}} n → 6
    • \sum_{n \in \{1,2,3\}} n^2 → 14
    • \prod_{k \in \{1,2,3,4\}} k → 24

  • Serialization: Element-based indexing sets serialize back to LaTeX with proper \in notation: \sum_{n\in \{1, 2, 3\}}n

  • Range support: Works with Range expressions via ce.box(): ["Sum", "n", ["Element", "n", ["Range", 1, 5]]] → 15

  • Bracket notation as Range: Two-element integer lists in bracket notation [a,b] are now treated as Range(a,b) when used in Element context:

    • \sum_{n \in [1,5]} n → 15 (iterates 1, 2, 3, 4, 5)
    • Previously returned 6 (treated as List with just elements 1 and 5)

  • Interval support: Interval expressions work with Element-based indexing, including support for Open and Closed boundary markers:

    • ["Interval", 1, 5] → iterates integers 1, 2, 3, 4, 5 (closed bounds)
    • ["Interval", ["Open", 0], 5] → iterates 1, 2, 3, 4, 5 (excludes 0)
    • ["Interval", 1, ["Open", 6]] → iterates 1, 2, 3, 4, 5 (excludes 6)

  • Infinite series with Element notation: Known infinite integer sets are converted to their equivalent Limits form and iterated (capped at 1,000,000):

    • NonNegativeIntegers (ℕ₀) → iterates from 0, like \sum_{n=0}^{\infty}
    • PositiveIntegers (ℤ⁺) → iterates from 1, like \sum_{n=1}^{\infty}
    • Convergent series produce numeric approximations: \sum_{n \in \Z^+} \frac{1}{n^2} → ≈1.6449 (close to π²/6)

  • Non-enumerable domains stay symbolic: When the domain cannot be enumerated (unknown symbol, non-iterable infinite set, or symbolic bounds), the expression stays symbolic instead of returning NaN:

    • \sum_{n \in S} n with unknown S → stays as ["Sum", "n", ["Element", "n", "S"]]
    • \sum_{n \in \Z} n → stays symbolic (bidirectional, can't forward iterate)
    • \sum_{x \in \R} f(x) → stays symbolic (non-countable)
    • \sum_{n \in [1,a]} n with symbolic bound → stays symbolic
    • Previously these would all return NaN with no explanation

  • Multiple Element indexing sets: Comma-separated Element expressions now parse and evaluate correctly:

    • \sum_{n \in A, m \in B} (n+m) → ["Sum", ..., ["Element", "n", "A"], ["Element", "m", "B"]]
    • Nested sums like \sum_{i \in A}\sum_{j \in B} i \cdot j evaluate correctly
    • Mixed indexing sets (Element + Limits) work together

  • Condition/filter support in Element expressions: Conditions can be attached to Element expressions to filter values from the set:

    • \sum_{n \in S, n > 0} n → sums only positive values from S
    • \sum_{n \in S, n \ge 2} n → sums values ≥ 2 from S
    • \prod_{k \in S, k < 0} k → multiplies only negative values from S
    • Supported operators: >, >=, <, <=, !=
    • Conditions are attached as the 4th operand of Element: ["Element", "n", "S", ["Greater", "n", 0]]

Linear Algebra

• Matrix Multiplication: Added MatrixMultiply function supporting:

  • Matrix × Matrix: A (m×n) × B (n×p) → result (m×p)
  • Matrix × Vector: A (m×n) × v (n) → result (m)
  • Vector × Matrix: v (m) × B (m×n) → result (n)
  • Vector × Vector (dot product): v1 (n) · v2 (n) → scalar
  • Proper dimension validation with incompatible-dimensions errors
  • LaTeX serialization using \cdot notation

• Matrix Addition and Scalar Broadcasting: Add now supports element-wise operations on tensors (matrices and vectors):

  • Matrix + Matrix: Element-wise addition (shapes must match)
  • Scalar + Matrix: Broadcasts scalar to all elements
  • Vector + Vector: Element-wise addition
  • Scalar + Vector: Broadcasts scalar to all elements
  • Symbolic support: [[a,b],[c,d]] + [[1,2],[3,4]] evaluates correctly
  • Proper dimension validation with incompatible-dimensions errors

• Matrix Construction Functions: Added convenience functions for creating common matrices:

  • IdentityMatrix(n): Creates an n×n identity matrix
  • ZeroMatrix(m, n?): Creates an m×n matrix of zeros (square if n omitted)
  • OnesMatrix(m, n?): Creates an m×n matrix of ones (square if n omitted)

• Matrix and Vector Norms: Added Norm function for computing various norms:

  • Vector norms: L1 (sum of absolute values), L2 (Euclidean, default), L-infinity (max absolute value), and general Lp norms
  • Matrix norms: Frobenius (default, sqrt of sum of squared elements), L1 (max column sum), L-infinity (max row sum)
  • Scalar norms return the absolute value

• Eigenvalues and Eigenvectors: Added functions for eigenvalue decomposition:

  • Eigenvalues(matrix): Returns list of eigenvalues (2×2: symbolic via characteristic polynomial; 3×3: Cardano's formula; larger: numeric QR)
  • Eigenvectors(matrix): Returns list of corresponding eigenvectors using null space computation via Gaussian elimination
  • Eigen(matrix): Returns tuple of (eigenvalues, eigenvectors)

• Diagonal Function: Now fully implemented with bidirectional behavior:

  • Vector → Matrix: Creates a diagonal matrix from a vector (Diagonal([1,2,3]) → 3×3 diagonal matrix)
  • Matrix → Vector: Extracts the diagonal as a vector (Diagonal([[1,2],[3,4]]) → [1,4])

• Higher-Rank Tensor Operations: Extended Transpose, ConjugateTranspose, and Trace to work with rank > 2 tensors:

  • Transpose: Swaps last two axes by default (batch transpose), or specify explicit axes with ['Transpose', T, axis1, axis2]
  • ConjugateTranspose: Same axis behavior as Transpose, plus element-wise complex conjugation
  • Trace (batch trace): Returns a tensor of traces over the last two axes. For a [2,2,2] tensor, returns [trace of T[0], trace of T[1]]. Optional axis parameters: ['Trace', T, axis1, axis2]

• Reshape Cycling: Implements APL-style ravel cycling. When reshaping to a larger shape, elements cycle from the beginning: Reshape([1,2,3], (2,2)) → [[1,2],[3,1]]

• Scalar Handling: Most linear algebra functions now handle scalar inputs:

  • Flatten(42) → [42] (single-element list)
  • Transpose(42) → 42 (identity)
  • Determinant(42) → 42 (1×1 matrix determinant)
  • Trace(42) → 42 (1×1 matrix trace)
  • Inverse(42) → 1/42 (scalar reciprocal)
  • ConjugateTranspose(42) → 42 (conjugate of real is itself)
  • Reshape(42, (2,2)) → [[42,42],[42,42]] (scalar replication)

• Improved Error Messages: Operations requiring square matrices (Determinant, Trace, Inverse) now return expected-square-matrix error for vectors and tensors (rank > 2).

Performance

• Pattern Matching Optimization: Significantly improved performance of commutative pattern matching by adding early rejection guards:
  • Arity Guard: Patterns without sequence wildcards (__/___) now immediately reject expressions with mismatched operand counts instead of attempting factorial permutations
  • Anchor Fingerprint: Patterns with literal or symbolic anchors verify anchor presence before attempting permutation matching, eliminating impossible matches in O(n) time
  • Universal Anchoring: Extended the efficient anchor-based backtracking algorithm to all patterns with anchors, not just those with sequence wildcards
  • Hash Bucketing: For patterns with many anchors (4+) against large expressions (6+ operands), uses hash-based indexing to reduce anchor lookup from O(n×m) to O(n+m) average case
  • Example: Matching a + b + c + 1 against x + y + z now rejects immediately (arity mismatch: 4 vs 3) instead of trying 24 permutations

Resolved Issues

Arithmetic

• Indeterminate Form Handling: Fixed incorrect results for mathematical indeterminate forms:

  • 0 * ∞ now correctly returns NaN (previously returned ∞)
  • ∞ / ∞ now correctly returns NaN (previously returned 1)
  • ∞^0 now correctly returns NaN (was already correct)
  • All combinations (0 * (-∞), (-∞) / ∞, etc.) are handled correctly

• (#176) Power Combination Simplification: Fixed simplification failing to combine powers with the same base when one factor has an implicit exponent or when there are 3+ operands. Previously, expressions like 2 * 2^x, e * e^x * e^{-x}, and x^2 * x would not simplify. Now correctly simplifies to 2^(x+1), e, and x^3 respectively. The fix includes:

  • Extended power combination rules to support numeric literal bases
  • Added functional rule to handle n-ary Multiply expressions (3+ operands)
  • Adjusted simplification cost threshold from 1.2 to 1.3 to accept mathematically valid simplifications where exponents become slightly more complex (e.g., 2 * 2^x → 2^(x+1))

• Symbolic Factorial: Fixed (n-1)! incorrectly evaluating to NaN instead of staying symbolic. The factorial evaluate function was attempting numeric computation on symbolic arguments. Now correctly returns undefined (keeping the expression symbolic) when the argument is not a number literal.

Linear Algebra

• Matrix Operations Type Validation: Fixed matrix operations (Shape, Rank, Flatten, Transpose, Determinant, Inverse, Trace, etc.) returning incorrect results or failing with type errors. The root cause was a type mismatch: function signatures expected matrix type (a 2D list with dimensions), but BoxedTensor.type returned list<number> without dimensions. Now BoxedTensor, BoxedFunction, and BoxedSymbol correctly derive shape and rank from their type's dimensions. Additionally, linear algebra functions now properly evaluate their operands before checking if they are tensors.

Calculus

• Numerical Integration: Fixed \int_0^1 \sin(x) dx returning NaN when evaluated numerically with .N(). The integrand was already wrapped in a Function expression by the canonical form, but the numerical evaluation code was wrapping it again, creating a nested function that returned a function instead of a number. Now correctly checks if the integrand is already a Function before wrapping.

LaTeX Parsing and Serialization

• Subscript Function Calls: Fixed parsing of function calls with subscripted names like f_\text{a}(5). Previously, this was incorrectly parsed as a Tuple instead of a function call because Subscript expressions weren't being canonicalized before the function call check. Now correctly recognizes that f_a(5) is a function call when the subscript canonicalizes to a symbol.

• (#130) Prefix/Postfix Operator LaTeX Serialization: Fixed incorrect LaTeX output for prefix operators (like Negate) and postfix operators (like Factorial) when applied to expressions with lower precedence. Previously, Negate(Add(a, b)) incorrectly serialized as -a+b instead of -(a+b), causing round-trip failures where parsing the output produced a mathematically different expression. Similarly, Factorial(Add(a, b)) now correctly serializes as (a+b)! instead of a+b!. The fix ensures operands are wrapped in parentheses when their precedence is lower than the operator's precedence.

• (#156) Logical Operator Precedence: Fixed parsing of logical operators \vee (Or) and \wedge (And) with relational operators. Previously, expressions like 3=4\vee 7=8 were incorrectly parsed with the wrong precedence. Now correctly parses as ["Or", ["Equal", 3, 4], ["Equal", 7, 8]]. Logical operators have lower precedence (230-235) than comparison operators (245) and set relations (240), so compound propositions parse correctly without requiring parentheses.

• (#156) Logical Connective Arrows: Added support for additional arrow notation in logical expressions:

  • \rightarrow now parses as Implies (previously parsed as To for set/function mapping)
  • \leftrightarrow now parses as Equivalent (previously produced an "unexpected-command" error)
  • Long arrow variants now supported: \Longrightarrow, \longrightarrow → Implies; \Longleftrightarrow, \longleftrightarrow → Equivalent
  • The existing variants \Rightarrow, \Leftrightarrow, \implies, \iff continue to work
  • \to remains available for function/set mapping notation (e.g., f: A \to B)

Simplification

• Rules Cache Isolation: Fixed rules cache building failing with "Invalid rule" errors when user expressions had previously polluted the global scope. For example, parsing x(y+z) would add x as a symbol with function type to the global scope. Later, when the simplification rules cache was built, rule parsing would fail because wildcards like _x in rules would be type-checked against the polluted scope where x had incompatible type. The fix ensures rule parsing uses a clean scope that inherits only from the system scope (containing built-in definitions), not from user-polluted scopes.

• Simplification Rules: Added and fixed several simplification rules:

  • x + x now correctly simplifies to 2x (term combination)
  • e^x * e^{-x} now correctly simplifies to 1 (exponential inverse)
  • sin(∞) and cos(∞) now correctly evaluate to NaN
  • tanh(∞) now correctly evaluates to 1, tanh(-∞) to -1
  • log_b(x^n) now correctly simplifies to n * log_b(x) (log power rule)
  • Improved cost function to prefer n * ln(x) form over ln(x^n)
  • Trigonometric functions now reduce arguments by their period (e.g., cos(5π + k) simplifies using cos(π + k) = -cos(k))

• (#178) Non-Canonical Expression Simplification: Fixed .simplify() not working on expressions parsed with { canonical: false }. Previously, ce.parse('x+x', { canonical: false }).simplify() would return x+x instead of 2x. The bug was in the simplification loop detection: when canonicalizing before simplification, the non-canonical form was recorded in the "seen" set, and since isSame() considers non-canonical and canonical forms equivalent, the canonical form was incorrectly detected as already processed. Now the simplification correctly starts fresh when canonicalizing, allowing full simplification to proceed.

0.32.0 2026-01-28

Resolved Issues

Calculus

• (#230) Root Derivatives: Fixed the D operator not differentiating expressions containing the Root operator (n-th roots). Previously, D(Root(x, 3), x) (derivative of ∛x) would return an unevaluated derivative expression instead of computing the result. Now correctly returns 1/(3x^(2/3)), equivalent to the expected (1/3)·x^(-2/3). The fix adds a special case in the differentiate function to handle Root(base, n) by applying the power rule with exponent 1/n.

• Abs Derivative: Fixed d/dx |x| returning an error when evaluated with a variable that has an assigned value. The derivative formula now uses Sign(x) instead of a complex Which expression that couldn't be evaluated symbolically.

• Step Function Derivatives: Fixed D(floor(x), x), D(ceil(x), x), and D(round(x), x) causing infinite recursion. These step functions now correctly return 0 (the derivative is 0 almost everywhere). Also fixed a bug where derivative formulas that evaluate to 0 weren't recognized due to a falsy check.

• Inverse Trig Integrals: Fixed incorrect integration formulas for arcsin, arccos, and arctan. The previous formulas were completely wrong. Correct:

  • ∫ arcsin(x) dx = x·arcsin(x) + √(1-x²)
  • ∫ arccos(x) dx = x·arccos(x) - √(1-x²)
  • ∫ arctan(x) dx = x·arctan(x) - (1/2)·ln(1+x²)

• Erfc Derivative: Fixed incorrect derivative formula for erfc(x). Now correctly returns -2/√π · e^(-x²) (the negative of the erf derivative).

• LogGamma Derivative: Added derivative rule for LogGamma(x) which returns Digamma(x) (the digamma/psi function).

• Special Function Derivatives: Fixed derivative formulas for several special functions and removed incorrect ones:

  • Fixed d/dx erfi(x) = (2/√π)·e^(x²) (imaginary error function)
  • Fixed d/dx S(x) = sin(πx²/2) (Fresnel sine integral)
  • Fixed d/dx C(x) = cos(πx²/2) (Fresnel cosine integral)
  • Removed incorrect derivative formulas for Zeta, Digamma, PolyGamma, Beta, LambertW, Bessel functions, and Airy functions (these now return symbolic derivatives like Digamma'(x) instead of wrong numeric results)

• Symbolic Derivative Evaluation: Fixed derivatives of unknown functions returning 0 instead of symbolic derivatives. For example, D(Digamma(x), x) now correctly returns Digamma'(x) (as Apply(Derivative(Digamma, 1), x)) instead of incorrectly returning 0.

LaTeX Parsing and Serialization

• (#256) Subscript Symbol Parsing: Fixed parsing of single-letter symbols with subscripts. Previously, i_A was incorrectly parsed as ["Subscript", ["Complex", 0, 1], "A"] because i was recognized as the imaginary unit before the subscript was processed. Now i_A correctly parses as the symbol i_A. This applies to all single-letter symbols including constants like e and i. Complex subscripts containing operators (n+1), commas (n,m), or parentheses ((n+1)) still produce Subscript expressions.

• LaTeX Serialization: Fixed TypeScript error in power serialization where denom (a number | null) was incorrectly passed where an Expression was expected. Now correctly uses operand(exp, 2) to get the expression form.

• (#168) Absolute Value: Fixed parsing of nested absolute value expressions that start with a double bar (e.g. ||3-5|-4|), which previously produced an invalid structure instead of evaluating correctly.

• (#244) Serialization: Fixed LaTeX and ASCIIMath serialization ambiguity for negative bases and negated powers. Powers now render (-2)^2 (instead of -2^2) when the base is negative, and negated powers now render as -(2^2) rather than -2^2.

• (#243) LaTeX Parsing: Fixed logic operator precedence causing expressions like x = 1 \vee x = 2 to be parsed incorrectly as x = (1 ∨ x) = 2 instead of (x = 1) ∨ (x = 2). Comparison operators (=, <, >, etc.) now correctly bind tighter than logic operators (\land, \lor, \veebar, etc.).

• (#264) Serialization: Fixed LaTeX serialization of quantified expressions (ForAll, Exists, ExistsUnique, NotForAll, NotExists). Previously, only the quantifier symbol was output (e.g., \forall x instead of \forall x, x>y). The body of the quantified expression is now correctly serialized.

• (#257) LaTeX Parsing: Fixed \gcd command not parsing function arguments correctly. Previously \gcd\left(24,37\right) would parse as ["Tuple", "GCD", ["Tuple", 24, 37]] instead of the expected ["GCD", 24, 37]. The \operatorname{gcd} form was unaffected. Also added support for \lcm as a LaTeX command (in addition to the existing \operatorname{lcm}).

• (#223) Serialization: Fixed scientific/engineering LaTeX serialization dropping the leading coefficient for exact powers of ten. For example, 1000 now serializes to 1\cdot10^{3} (or 1\times10^{3} depending on exponentProduct) instead of 10^{3}.

• LaTeX Parsing: Fixed \cosh incorrectly mapping to Csch instead of Cosh.

• (#255) LaTeX Parsing: Fixed multi-letter subscripts like A_{CD} causing "incompatible-type" errors in arithmetic operations. Multi-letter subscripts without parentheses are now interpreted as compound symbol names (e.g., A_{CD} → A_CD, x_{ij} → x_ij, T_{max} → T_max). Use parentheses for expression subscripts: A_{(CD)} creates a Subscript expression where CD represents implicit multiplication. The Delimiter wrapper is now stripped from subscript expressions for cleaner output.

First-Order Logic

• (#263) Quantifier Scope: Fixed quantifier scope in First-Order Logic expressions. Previously, \forall x.P(x)\rightarrow Q(x) was parsed with the implication inside the quantifier scope: ["ForAll", "x", ["To", P(x), Q(x)]]. Now it correctly follows standard FOL conventions where the quantifier binds only the immediately following formula: ["To", ["ForAll", "x", P(x)], Q(x)]. This applies to all quantifiers (ForAll, Exists, ExistsUnique, NotForAll, NotExists) and all logical connectives (\rightarrow, \to, \implies, \land, \lor, \iff). Use explicit parentheses for wider scope: \forall x.(P(x)\rightarrow Q(x)). Also fixed quantifier type signatures to properly return boolean, enabling correct type checking when quantified expressions are used as arguments to logical operators.

Simplification

• Sign Simplification: Fixed Sign(x).simplify() returning 1 instead of -1 when x is negative. The simplification rule incorrectly returned ce.One for both positive and negative cases.

Type System

• Ceil Type Signature: Fixed Ceil function signature from (real) -> integer to (number) -> integer to match Floor. This resolves "incompatible-type" errors when computing derivatives of ceiling expressions or using Ceil in contexts expecting a general number type.

Polynomials

• Polynomial Degree Detection: Fixed polynomialDegree() returning 0 for expressions like e^x or e^(-x^2) when it should return -1 (not a polynomial). When the base of a power is constant but the exponent depends on the variable, this is not a polynomial. This bug caused infinite recursion in simplification when simplifying expressions containing exponentials, such as the derivative of erf(x) which is (2/√π)·e^(-x²).

Pattern Matching

• (#258) Pattern Matching: Fixed BoxedExpression.match() returning null when matching patterns against canonicalized expressions. Several cases are now handled:
  • Rational patterns now match expressions like ['Rational', 'x', 2] which are canonicalized to ['Multiply', ['Rational', 1, 2], 'x']
  • Power patterns now match ['Power', 'x', -1] which is canonicalized to ['Divide', 1, 'x'], returning {_base: x, _exp: -1}
  • Power patterns now match ['Root', 'x', 3] (cube root), returning {_base: x, _exp: ['Divide', 1, 3]}

Sum and Product

• (#252) Sum/Product: Fixed Sum and Product returning NaN when the body contains free variables (variables not bound by the index). For example, \sum_{n=1}^{10}(x) now correctly evaluates to 10x instead of NaN, and \prod_{n=1}^{5}(x) evaluates to x^5. Mixed expressions like \sum_{n=1}^{10}(n \cdot x) now return 55x. Also fixed toString() for Sum and Product expressions with non-trivial bodies (e.g., Multiply) which were incorrectly displayed as int().

Equation Solving

• (#242) Solve: Fixed solve() returning an empty array for equations with variables in fractions. For example, F = 3g/h solved for g now correctly returns Fh/3 instead of an empty array. The solver now clears denominators before applying solve rules, enabling it to handle expressions like a + bx/c = 0. Also added support for solving equations where the variable is in the denominator (e.g., a/x = b now returns x = a/b).

• (#220) Solve: Fixed solve() returning an empty array for equations involving square roots of the unknown, e.g. 2x = \sqrt{5x}. The solver now handles equations of the form ax + b√x + c = 0 using quadratic substitution. Also added support for solving logarithmic equations like a·ln(x) + b = 0 which returns x = e^(-b/a).

Improvements

First-Order Logic

• (#263) First-Order Logic: Added several improvements for working with First-Order Logic expressions:
  • Configurable quantifier scope: New quantifierScope parsing option controls how quantifier scope is determined. Use "tight" (default) for standard FOL conventions where quantifiers bind only the immediately following formula, or "loose" for scope extending to the end of the expression.

    ce.parse('\\forall x. P(x)', { quantifierScope: 'tight' })  // default
    ce.parse('\\forall x. P(x)', { quantifierScope: 'loose' })

  • Automatic predicate inference: Single uppercase letters followed by parentheses (e.g., P(x), Q(a,b)) are now automatically recognized as predicate/function applications without requiring explicit declaration. This enables natural FOL syntax like \forall x. P(x) \rightarrow Q(x) to work out of the box.
  • Quantifier evaluation over finite domains: Quantifiers (ForAll, Exists, ExistsUnique, NotForAll, NotExists) now evaluate to boolean values when the bound variable is constrained to a finite set. For example:

    ce.box(['ForAll', ['Element', 'x', ['Set', 1, 2, 3]], ['Greater', 'x', 0]]).evaluate()
    // Returns True (all values in {1,2,3} are > 0)
    ce.box(['Exists', ['Element', 'x', ['Set', 1, 2, 3]], ['Greater', 'x', 2]]).evaluate()
    // Returns True (3 > 2)
    ce.box(['ExistsUnique', ['Element', 'x', ['Set', 1, 2, 3]], ['Equal', 'x', 2]]).evaluate()
    // Returns True (only one element equals 2)

    Supports Set, List, Range, and integer Interval domains up to 1000 elements. Nested quantifiers are evaluated over the Cartesian product of their domains.
  • Symbolic simplification for quantifiers: Quantifiers now simplify automatically in special cases:
    • ∀x. True → True, ∀x. False → False
    • ∃x. True → True, ∃x. False → False
    • ∀x. P → P (when P doesn't contain x)
    • ∃x. P → P (when P doesn't contain x)
  • CNF/DNF conversion: New ToCNF and ToDNF functions convert boolean expressions to Conjunctive Normal Form and Disjunctive Normal Form respectively:

    ce.box(['ToCNF', ['Or', ['And', 'A', 'B'], 'C']]).evaluate()
    // Returns (A ∨ C) ∧ (B ∨ C)
    ce.box(['ToDNF', ['And', ['Or', 'A', 'B'], 'C']]).evaluate()
    // Returns (A ∧ C) ∨ (B ∧ C)

    Handles And, Or, Not, Implies, Equivalent, Xor, Nand, and Nor operators using De Morgan's laws and distribution.
  • Boolean operator evaluation: Added evaluation support for Xor, Nand, and Nor operators with True/False arguments:

    ce.box(['Xor', 'True', 'False']).evaluate()   // Returns True
    ce.box(['Nand', 'True', 'True']).evaluate()   // Returns False
    ce.box(['Nor', 'False', 'False']).evaluate()  // Returns True

  • N-ary boolean operators: Xor, Nand, and Nor now support any number of arguments:
    • Xor(a, b, c, ...) returns true when an odd number of arguments are true
    • Nand(a, b, c, ...) returns the negation of And(a, b, c, ...)
    • Nor(a, b, c, ...) returns the negation of Or(a, b, c, ...)
  • Satisfiability checking: New IsSatisfiable function checks if a boolean expression can be made true with some assignment of variables:

    ce.box(['IsSatisfiable', ['And', 'A', ['Not', 'A']]]).evaluate()  // False
    ce.box(['IsSatisfiable', ['Or', 'A', 'B']]).evaluate()            // True

  • Tautology checking: New IsTautology function checks if a boolean expression is true for all possible variable assignments:

    ce.box(['IsTautology', ['Or', 'A', ['Not', 'A']]]).evaluate()     // True
    ce.box(['IsTautology', ['And', 'A', 'B']]).evaluate()             // False

  • Truth table generation: New TruthTable function generates a complete truth table for a boolean expression:

    ce.box(['TruthTable', ['And', 'A', 'B']]).evaluate()
    // Returns [["A","B","Result"],["False","False","False"],...]

  • Explicit Predicate function: Added a new Predicate function to explicitly represent predicate applications in First-Order Logic. Inside quantifier scopes (\forall, \exists, etc.), single uppercase letters followed by parentheses are now parsed as ["Predicate", "P", "x"] instead of ["P", "x"]. This distinguishes predicates from regular function applications and avoids naming conflicts with library functions.

    ce.parse('\\forall x. P(x)').json
    // Returns ["ForAll", "x", ["Predicate", "P", "x"]]

    Outside quantifier scopes, P(x) is still parsed as ["P", "x"] to maintain backward compatibility with function definitions like Q(x) := ....
  • D(f, x) no longer maps to derivative: The LaTeX notation D(f, x) is not standard mathematical notation for derivatives and previously caused confusion with the D derivative function in MathJSON. Now D(f, x) in LaTeX parses as ["Predicate", "D", "f", "x"] instead of the derivative. Use Leibniz notation (\frac{d}{dx}f) for derivatives in LaTeX, or construct the derivative directly in MathJSON: ["D", expr, "x"].
  • N(x) no longer maps to numeric evaluation: Similarly, N(x) in LaTeX is CAS-specific notation, not standard math notation. Now N(x) parses as ["Predicate", "N", "x"] instead of the numeric evaluation function. This allows N to be used as a variable (e.g., "for all N in Naturals"). Use the .N() method for numeric evaluation, or construct it directly in MathJSON: ["N", expr].

Polynomials

• Polynomial Simplification: The simplify() function now automatically cancels common polynomial factors in univariate rational expressions. For example, (x² - 1)/(x - 1) simplifies to x + 1, (x³ - x)/(x² - 1) simplifies to x, and (x + 1)/(x² + 3x + 2) simplifies to 1/(x + 2). Previously, this required explicitly calling the Cancel function with a variable argument.

Sum and Product

• Sum/Product Simplification: Added simplification rules for Sum and Product expressions with symbolic bounds:
  • Constant body: \sum_{n=1}^{b}(x) simplifies to b * x
  • Triangular numbers (general bounds): \sum_{n=a}^{b}(n) simplifies to (b(b+1) - a(a-1))/2
  • Sum of squares: \sum_{n=1}^{b}(n^2) simplifies to b(b+1)(2b+1)/6
  • Sum of cubes: \sum_{n=1}^{b}(n^3) simplifies to [b(b+1)/2]^2
  • Geometric series: \sum_{n=0}^{b}(r^n) simplifies to (1-r^(b+1))/(1-r)
  • Alternating unit series: \sum_{n=0}^{b}((-1)^n) simplifies to (1+(-1)^b)/2
  • Alternating linear series: \sum_{n=0}^{b}((-1)^n * n) simplifies to (-1)^b * floor((b+1)/2)
  • Arithmetic progression: \sum_{n=0}^{b}(a + d*n) simplifies to (b+1)(a + db/2)
  • Sum of binomial coefficients: \sum_{k=0}^{n}C(n,k) simplifies to 2^n
  • Alternating binomial sum: \sum_{k=0}^{n}((-1)^k * C(n,k)) simplifies to 0
  • Weighted binomial sum: \sum_{k=0}^{n}(k * C(n,k)) simplifies to n * 2^(n-1)
  • Partial fractions (telescoping): \sum_{k=1}^{n}(1/(k(k+1))) simplifies to n/(n+1)
  • Partial fractions (telescoping): \sum_{k=2}^{n}(1/(k(k-1))) simplifies to (n-1)/n
  • Weighted squared binomial sum: \sum_{k=0}^{n}(k^2 * C(n,k)) simplifies to n(n+1) * 2^(n-2)
  • Weighted cubed binomial sum: \sum_{k=0}^{n}(k^3 * C(n,k)) simplifies to n²(n+3) * 2^(n-3)
  • Alternating weighted binomial sum: \sum_{k=0}^{n}((-1)^k * k * C(n,k)) simplifies to 0 (n ≥ 2)
  • Sum of binomial squares: \sum_{k=0}^{n}(C(n,k)^2) simplifies to C(2n, n)
  • Sum of consecutive products: \sum_{k=1}^{n}(k(k+1)) simplifies to n(n+1)(n+2)/3
  • Arithmetic progression (general bounds): \sum_{n=m}^{b}(a + d*n) simplifies to (b-m+1)(a + d(m+b)/2)
  • Product of constant: \prod_{n=1}^{b}(x) simplifies to x^b
  • Factorial: \prod_{n=1}^{b}(n) simplifies to b!
  • Shifted factorial: \prod_{n=1}^{b}(n+c) simplifies to (b+c)!/c!
  • Odd double factorial: \prod_{n=1}^{b}(2n-1) simplifies to (2b-1)!!
  • Even double factorial: \prod_{n=1}^{b}(2n) simplifies to 2^b * b!
  • Rising factorial (Pochhammer): \prod_{k=0}^{n-1}(x+k) simplifies to (x)_n
  • Falling factorial: \prod_{k=0}^{n-1}(x-k) simplifies to x!/(x-n)!
  • Telescoping product: \prod_{k=1}^{n}((k+1)/k) simplifies to n+1
  • Wallis-like product: \prod_{k=2}^{n}(1 - 1/k^2) simplifies to (n+1)/(2n)
  • Factor out constants: \sum_{n=1}^{b}(c \cdot f(n)) simplifies to c \cdot \sum_{n=1}^{b}(f(n)), and similarly for products where the constant is raised to the power of the iteration count
  • Nested sums/products: inner sums/products are simplified first, enabling cascading simplification
  • Edge cases: empty ranges (upper < lower) return identity elements (0 for Sum, 1 for Product), and single-iteration ranges substitute the bound value

0.31.0 2026-01-27

Breaking Changes

• The [Length] function has been renamed to [Count].
• The xsize property of collections has been renamed to count.
• The xcontains() method of collections has been renamed to contains().
• Handling of dictionaries (["Dictionary"] expressions and \{dict:...\} shorthand) has been improved.
• Inverse hyperbolic functions have been renamed to follow the ISO 80000-2 standard: Arcsinh → Arsinh, Arccosh → Arcosh, Arctanh → Artanh, Arccoth → Arcoth, Arcsech → Arsech, Arccsch → Arcsch. The "ar" prefix (for "area") is mathematically correct since these functions relate to areas on a hyperbola, not arc lengths. Both LaTeX spellings (\arsinh and \arcsinh) are accepted as input (Postel's law).

Resolved Issues

LaTeX Parsing

• Metadata Preservation: Fixed verbatimLatex not being preserved when parsing with preserveLatex: true. The original LaTeX source is now correctly stored on parsed expressions (when using non-canonical mode). Also fixed metadata (latex, wikidata) being lost when boxing MathJSON objects that contain these attributes.

• String Parsing: Fixed parsing of \text{...} with preserveLatex: true which was incorrectly returning an "invalid-symbol" error instead of a string expression.

Calculus

• Derivatives: d/dx e^x now correctly simplifies to e^x instead of ln(e) * e^x. The hasSymbolicTranscendental() function now recognizes that transcendentals which simplify to exact rational values (like ln(e) = 1) should not be preserved symbolically.

• Derivatives: d/dx log(x) now returns 1 / (x * ln(10)) symbolically instead of evaluating to 0.434... / x. Fixed by using substitution instead of function application when applying derivative formulas, which preserves symbolic transcendental constants.

Arithmetic

• Rationals: Fixed reducedRational() to properly normalize negative denominators before the early return check. Previously 1/-2 would not canonicalize to -1/2.

• Arithmetic: Fixed .mul() to preserve logarithms symbolically. Previously multiplying expressions containing Ln or Log would evaluate the logarithm to its numeric value.

Serialization

• Serialization: Fixed case inconsistency in toString() output for trigonometric functions. Some functions like Cot were being serialized with capital letters while others like csc were lowercase. All trig functions now consistently serialize in lowercase (e.g., cot(x) instead of Cot(x)).

• Serialization: Improved display of inverse trig derivatives and similar expressions:

  • Negative exponents like x^(-1/2) now display as 1/sqrt(x) in both LaTeX and ASCII-math output
  • When a sum starts with a negative term and contains a positive constant, the constant is moved to the front (e.g., -x^2 + 1 displays as 1 - x^2) while preserving polynomial ordering (e.g., x^2 - x + 3 stays unchanged)
  • d/dx arcsin(x) now displays as 1/sqrt(1-x^2) instead of (-x^2+1)^(-1/2)

• Scientific Notation: Fixed normalization of scientific notation for fractional values (e.g., numbers less than 1).

Sum and Product

• Compilation: Fixed compilation of Sum and Product expressions.

• Sum/Product: Fixed sum and prod library functions to correctly handle substitution of index variables.

New Features and Improvements

Serialization

• Number Serialization: Added adaptiveScientific notation mode. When serializing numbers to LaTeX, this mode uses scientific notation but avoids exponents within a configurable range (controlled by avoidExponentsInRange). This provides a balance between readability and precision for numbers across different orders of magnitude.

Type System

• Refactored the type parser to use a modular architecture. This allows for better extensibility and maintainability of the type system.

Pattern Matching

• Pattern Matching: The validatePattern() function is now exported from the public API. Use it to check patterns for invalid combinations like consecutive sequence wildcards before using them.

Polynomials

• Polynomial Arithmetic: Added new library functions for polynomial operations:
  • PolynomialDegree(expr, var) - Get the degree of a polynomial
  • CoefficientList(expr, var) - Get the list of coefficients
  • PolynomialQuotient(dividend, divisor, var) - Polynomial division quotient
  • PolynomialRemainder(dividend, divisor, var) - Polynomial division remainder
  • PolynomialGCD(a, b, var) - Greatest common divisor of polynomials
  • Cancel(expr, var) - Cancel common factors in rational expressions

Calculus

• Integration: Significantly expanded symbolic integration capabilities:
  • Polynomial division: Integrals like ∫ x²/(x²+1) dx now correctly divide first, yielding x - arctan(x)
  • Repeated linear roots: ∫ 1/(x-1)² dx = -1/(x-1) and higher powers
  • Derivative pattern recognition: ∫ f'(x)/f(x) dx = ln|f(x)| is now recognized automatically
  • Completing the square: Irreducible quadratics like ∫ 1/(x²+2x+2) dx now yield arctan(x+1)
  • Reduction formulas: ∫ 1/(x²+1)² dx now works using reduction formulas
  • Mixed partial fractions: ∫ 1/((x-1)(x²+1)) dx now decomposes correctly
  • Factor cancellation: ∫ (x+1)/(x²+3x+2) dx simplifies before integrating
  • Inverse hyperbolic: Added ∫ 1/√(x²+1) dx = arcsinh(x) and ∫ 1/√(x²-1) dx = arccosh(x)
  • Arcsec pattern: Added ∫ 1/(x·√(x²-1)) dx = arcsec(x)
  • Trigonometric substitution: Added support for ∫√(a²-x²) dx, ∫√(x²+a²) dx, and ∫√(x²-a²) dx using trig/hyperbolic substitution