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Upper incomplete gamma for s < -1 at small x; exact square roots of negative integers over integers - #447
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Upper incomplete gamma for s < -1 at small x; exact square roots of negative integers over integers#447enumeratio wants to merge 1 commit into
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…ith DLMF 8.8.2, and the compiled JavaScript Gamma(s, z) that dropped z; build exact pure-imaginary values for negative integer radicals so Sqrt(-13)/3 is not NaN. Fix cortex-js#446
This was referenced Oct 16, 2026
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Every engine we ship or evaluate examples on now sets compute-engine's `ce.roundingTies = "to-even"` (option since 0.149, cortex-js/compute-engine#417), so `Round(5/2)` is 2 and `Round(-2.5)` is -2, as in Wolfram, SymPy, mpmath, Sage and Julia. CE's own default is untouched. ## Where it is set One function, `setEngineDefaults(ce)` in `@enumeratio/engine`, called from: - `declareEvaluation` (the base library): covers every worker, session/kernel engine, and every planned engine (census `fullEngine`, reference `declaredEngine` and the isolated evaluator's `configure`) - the frontend engine (lazy import, so the page chunk stays engine-free), the CLI `Session`, the site prerender engine, the forms collector - `createEngine` / `createLatexEngine` in `@enumeratio/engine/testing` (not `bareEngine`) Bare engines that snapshot CE's own bindings (census/manifest scripts, ce-patches landed checks, oracle comparison engines) are left as they are. The CLI and web packages gain a workspace dependency on `@enumeratio/engine`. ## Changed rows (Round) Each is a tie; the new value is Wolfram's. Three ids that said the opposite are renamed (no alias): `rounds-half-away-from-zero-not-banker-s-rounding` to `rounds-half-to-even`, `and-by-the-same-convention-2-5-rounds-to-3-not-2` to `minus-2-5-rounds-to-minus-2`, `an-exact-half-rounds-away-from-zero-as-well` to `an-exact-half-rounds-to-even`; the sidecars moved with them. Rows below are listed by their new ids. | row | old | new | | --- | --- | --- | | `round-neg-1-over-2-exact-tie` | -1 | 0 | | `rounds-half-to-even` | 3 | 2 | | `minus-2-5-rounds-to-minus-2` | -3 | -2 | | `round-5-over-2-exact-rational-tie` | 3 | 2 | | `an-exact-half-rounds-to-even` | 1 | 0 | Captions and `index.md` prose now say ties go to even. The `rounding-mode` classification is dropped for Wolfram, Sage, SymPy, mpmath and Julia, whose rows now agree (rescanned: wolfram, sage, sympy, mpmath, julia). Rust's `round` goes away from zero, so those rows are now classified `rounding-mode` for Rust only. No other `known` was touched. The other Round rows (7/2, 3.5, the list row, 19.995 at step 1/100, ...) are unchanged. No other examples.tsv row depends on tie rounding. Also updated: the Ceil and Floor notes, and the site's Wolfram source-output row for `Round(2.5)`, which claimed a divergence. ## Two-argument Gamma compile `Gamma(s, x)` with a symbolic `x` compiled (JS target) to `_SYS.gamma(2, _.x + 1)`, which drops x (cortex-js/compute-engine#447). `declareAnalytic` now lets `Gamma` compile with one operand only, so the two-operand call falls back to the interpreter; a test checks compiled `Gamma` and `GammaRegularized` either decline or match evaluate. `GammaRegularized` compiled correctly and is unchanged. Lift the guard (the `1` in the `Gamma` row of the loop in `declare.ts`) with the CE bump that carries #447. Nothing else in the repo lowers two-argument Gamma (ce-patches WGSL, frontend `complex-eval`, the combinatorics compile scripts and the bench catalogue use only the one-argument form).
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Landed on main as 38e5d18 with the dual-review fixes: the shift loop is capped at order 4096 and returns Infinity on overflow; two-operand Gamma is a complex-continuation head so the compiled lane agrees with .N() for a real z < 0; 1 − eᵗ is recognized by the new exponential-divisor pole family. Thank you. |
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…al-divisor poles Lands PR #447 by enumeratio with review fixes, plus one maintainer fix found while reviewing PR #445. - Gamma(s, z) for a non-integer s < −1 and z < 1: start from Γ(s+n, z) with −1 < s+n < 0 and apply DLMF 8.8.2 n times (the Legendre fraction did not converge there; Gamma(-1.5, 0.001) was off by 1.2e-5). The shift is capped at order 4096 and an overflow returns Infinity. Measured against mpmath: relative error below 4e-14 over 132 points. - Compiled Gamma(s, z) keeps its second operand (it compiled as Γ(s)), and is a complex-continuation head: a real z < 0 takes the complex kernel unless s is a positive integer, as .N() does. The interval target declines the two-operand form. - Sqrt(-13)/3 is (√13/3)i, not NaN: a negative integer radical builds the exact pure-imaginary value in _numericValue. Sqrt(-13)/Sqrt(-3) is √39/3, not −√39/3. - Interior-pole detection knows an exponential divisor k·bᵘ + c with u linear (eˣ − 1, 1 − eˣ, 2ᵗ − 8): ∫₋₁¹ eˣ/(eˣ − 1) dx is Indeterminate, where it evaluated to 1. The free-symbol scan still lets an assumption decide 1/(eᵗ − a). New ROADMAP entry: an integer coefficient times √(−c) stays a product, and √(−c) with c ≥ 10⁶ keeps its square factors (correct, unreduced).
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Fix #446.
Gamma(s, x)for a non-integers < -1andx < 1. The Legendre continued fraction does not converge in its cap there, and the error grew asxshrank:Gamma(-1.5, 0.001)was21020.691761…where mpmath has21020.937167…(1.2e-5), andGamma(-1.5, 1e-6)was off by 4.2e-5. We now start fromΓ(s+n, x)with-1 < s+n < 0and applyΓ(a-1, x) = (Γ(a, x) - x^{a-1}e^{-x})/(a-1)(DLMF 8.8.2)ntimes; atx < 1the subtracted term dominates, so nothing cancels. Within 0.05 of a pole ofΓ(s+n)the baseΓ(a) - γ(a, x)cancels, so the real kernel defers to the complex kernel there. Measured against mpmath, the relative error is below 3e-15 on s in {-0.5, -1.5, -2.5, -3.5} by x in {1e-6, 1e-3, 1e-2, 0.1, 1}, and below 5e-14 ons = -(k+f)with k up to 12, 0.001 <= f <= 0.999 and 1e-10 <= x < 1. There is no bignum kernel for this head, soce.precisiondoes not change the result.The compiled JavaScript lane had a separate bug:
Gamma(x, y)compiled to_SYS.gamma(x)and droppedy(Gamma(-1.5, 0.001)ran as2.3633). It now calls the upper incomplete gamma, with the complex kernel for a complex operand. The interval target declines the two-operand form, as the GPU targets do.Sqrt(-13)/3and its relatives. A negative integer radical went to the float lane, which tookMath.sqrt(-13).Sqrt(-13)/3,Sqrt(-2)/3,Sqrt(-13)/2,Sqrt(-13)/3.0andSqrt(-13)*Rational(1,3)wereNaN, and so wasArcosh(Sqrt(-13)/3)._numericValuenow builds the exact pure-imaginary value√c·ifor a radical-c, so canonicalization,evaluate()and.N()agree:Sqrt(-13)/3is(√13/3)iandArcosh(Sqrt(-13)/3)is1.0171572086841316 + 1.5707963267948966i. While there,Sqrt(-13)/Sqrt(-3)was-√39/3; it is√39/3.