fix: avoid cancellation for large negative m in math/base/special/ellipk

[Abhist17]

Description

What is the purpose of this pull request?

This pull request:

• forms 1 - x as 1/(1-m) in math/base/special/ellipk (JS and C), where x = m/(m-1) is the transformed parameter for m < 0, instead of subtracting x from one.

The subtraction cancels as m grows more negative, and below -2^53 x rounds to exactly one:

m	develop	this PR	mpmath
-1e10	1.2899219784372459e-4	1.2899219825792643e-4	1.2899219825792638e-4
-1e15	5.899571246472065e-7	5.899444819017533e-7	5.899444819017532e-7
-3.7e15	3.190675447053322e-7	3.174519615424412e-7	3.174519615424411e-7
-1e16	Infinity	1.9806975105072258e-7	1.9806975105072255e-7
-1e300	Infinity	3.4677405831022676e-148	3.4677405831022676e-148

This also explains the FIXME: investigate the reason for this large discrepancy on the large-negative test (tolerance 204,773,163 ULPs). The C++ runner computes those fixtures with Boost through the same transformation (k = sqrt(m/(1+m))), so the expected values themselves were off by 1e-8 to 6e-8 relative to mpmath. The fixed implementation is off by 1e-16.

• replaces test/fixtures/cpp/large_negative.json with a Julia runner that evaluates K(m) = π / (2 agm(1, sqrt(1-m))) in 2048-bit BigFloat. That identity holds for every m < 1 without transforming m. It writes large_negative.json (same range as before) and a new huge_negative.json (-1e10 to -1e300). The large-negative tolerance drops from 204,773,163 to 5 ULPs, the new set passes at 4, and the FIXME is gone.
• keeps ellipk( -Infinity ) returning NaN, as documented.

Related Issues

Does this pull request have any related issues?

None. math/base/special/ellipe calls ellipk( x ) with the same rounded x for m < 0, and returns 1 below -2^53. I'll send that as a follow-up, since it depends on this.

Questions

Any questions for reviewers of this pull request?

No.

Other

Any other information relevant to this pull request? This may include screenshots, references, and/or implementation notes.

test.js and test.native.js: 17007 pass.

Checklist

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• Read, understood, and followed the contributing guidelines.

AI Assistance

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• Code generation (e.g., when writing an implementation or fixing a bug)
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[stdlib-bot]

Coverage Report

Package	Statements	Branches	Functions	Lines
math/base/special/ellipk	$\\color{green}872/872$ $\\color{green}+100.00\\%$	$\\color{green}84/84$ $\\color{green}+100.00\\%$	$\\color{green}14/14$ $\\color{green}+100.00\\%$	$\\color{green}872/872$ $\\color{green}+100.00\\%$

The above coverage report was generated for the changes in this PR.