fix: add the missing sqrt(-ln(x)) >= 18 branch to math/base/special/erfcinv

[Abhist17]

Description

What is the purpose of this pull request?

This pull request:

• adds the missing rational approximation for 18 <= sqrt(-ln(x)) < 44 to math/base/special/erfcinv (JS and C), which fixes its accuracy for x below about 1e-140.

The implementation follows Boost's erf_inv_imp, which for small x uses a series of rational approximations in p = sqrt(-ln(x)), one each for p < 3, p < 6, p < 18, p < 44 and p >= 44. The JSDoc here already lists all five error bounds, but the code stops at the third. Every p >= 6 goes through the [6, 18) approximation, and past p = 18 (x ~ 1e-141) it is extrapolating. For doubles, p reaches 27.28... at the smallest subnormal, so the [18, 44) branch is the only missing one, and p >= 44 can't occur.

Error against 2048-bit erfcinv (Julia, SpecialFunctions) on x = 10^-k for k = 100..323, plus subnormals:

	max error	points over 2 ULP
develop	1,340 ULP (at 5e-324)	131 of 228
this PR	1 ULP	0

Changes:

• scripts/evalrational.js gains Boost's P/Q coefficients for the [18, 44) range (padded with zeros like the other tables). It generates lib/rational_p6q6.js and the rational_p6q6 function in src/main.c, inside new BEGIN/END markers.
• lib/main.js and src/main.c gain Y6 = 0.99714565277099609375 and the q < 18 / q >= 18 split.
• Tests: there's a new x_tiny.json fixture (1,001 points, x log-spaced over [1e-323, 1e-15] plus 5e-324), generated by a gen_tiny function in runner.jl that evaluates erfcinv in 2048-bit BigFloat, with a 2-ULP tolerance. On develop, 416 of its assertions fail. With this change, all 7,013 assertions pass in both test.js and test.native.js.

Related Issues

Does this pull request have any related issues?

This pull request has the following related issues:

• fix: evaluate stats/base/dists/normal/quantile via erfcinv to keep the lower tail #15615 (normal/quantile via erfcinv), whose lower-tail fixture stops at 1e-250 because of 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.

• Running the whole runner.jl with a current JSON.jl fails on the existing x_1.9999..8_2.json block, because erfcinv( 2.0 ) is -Inf and newer JSON.jl refuses to write it. So I generated only the new fixture, with the same gen_tiny code, and the existing fixture files are unchanged.
• The generator also rewrites rational_p1q1.js to rational_p5q5.js, but only their copyright year changed, so I left them out of the diff.

Checklist

Please ensure the following tasks are completed before submitting this pull request.

• Read, understood, and followed the contributing guidelines.

AI Assistance

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• Yes
• No

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• Code generation (e.g., when writing an implementation or fixing a bug)
• Test/benchmark generation
• Documentation (including examples)
• Research and understanding

Disclosure

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This PR was written primarily by Claude Code.

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Coverage Report

Package	Statements	Branches	Functions	Lines
math/base/special/erfcinv	$\\color{red}707/731$ $\\color{green}+96.72\\%$	$\\color{red}53/66$ $\\color{green}+80.30\\%$	$\\color{green}8/8$ $\\color{green}+100.00\\%$	$\\color{red}707/731$ $\\color{green}+96.72\\%$

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