fix: avoid overflow and underflow for scales far from 1 in stats/base/dists/rayleigh/pdf

[Abhist17]

Description

What is the purpose of this pull request?

This pull request:

• avoids overflow and underflow for scale parameters far from 1 in stats/base/dists/rayleigh/pdf (JS, factory and C).

For sigma outside about [1.5e-154, 1.3e154], sigma^2 (or x^2) overflows or underflows, so the PDF comes out NaN or 0 where it is finite:

var pdf = require( '@stdlib/stats/base/dists/rayleigh/pdf' );

pdf( 1.0e200, 1.0e200 );    // develop: NaN   this PR: 6.0653065971263344e-201
pdf( 1.0e-200, 1.0e-200 );  // develop: NaN   this PR: 6.065306597126334e+199
pdf( 1.0e150, 1.0e200 );    // develop: 0     this PR: 1e-250

This is the follow-up I mentioned in #15923, using the same approach. The existing expression is kept wherever it is safe — sigma^2 normal with 2*sigma^2 finite (the 1e154 case you found on #15923), x^2 finite, x/sigma^2 normal — and on all 3,000 points of the existing fixtures the result is bit-for-bit identical to develop. Otherwise it uses x/sigma: (r/sigma) * exp(-r^2/2), falling back to exp( ln(r) - ln(sigma) - r^2/2 ) when x/sigma^2 overflows, i.e. for a tiny sigma, where the exponential decides the result.

Tests, and why the existing fixtures weren't regenerated this time. On #15923 you asked for the old fixtures to be regenerated in extended precision. I tried the same here. Against a 2048-bit reference, develop — and this PR, being bit-identical there — is up to 691 ULP off on the existing fixtures, at x/sigma ≈ 34. That's exp(-r^2/2) amplifying the rounding of x^2/(2*sigma^2) by its own magnitude (~590 at that point), which any double-precision evaluation inherits. The old Float64 Julia fixtures share that rounding, so they pass at 2 ULP. Regenerating them would mean loosening those tolerances to ~700 ULP for a part of the function this PR doesn't change.

So gen and the three existing fixtures are untouched. A separate gen_extended (2048-bit BigFloat, textbook formula) writes two new fixtures:

• tiny_scale.json: sigma in [1e-200, 1e-155], where sigma^2 underflows.
• huge_scale.json: sigma in [1e155, 1e200], where it overflows.

Both use x/sigma in [1e-3, 1], so they test the new branch rather than that conditioning. On them this PR is within 2 ULP, the existing tolerance; develop fails all 2,000 points in both test.pdf.js and test.factory.js. test.pdf.js / test.factory.js / test.native.js pass 5,014 each. If you'd still prefer the old fixtures regenerated, I'll do that and set those three blocks' tolerances from the measured errors instead.

Related Issues

Does this pull request have any related issues?

None. Follow-up to #15923 (rayleigh/logpdf), as mentioned there.

Questions

Any questions for reviewers of this pull request?

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Coverage Report

Package	Statements	Branches	Functions	Lines
stats/base/dists/rayleigh/pdf	$\\color{red}334/338$ $\\color{green}+98.82\\%$	$\\color{red}39/41$ $\\color{green}+95.12\\%$	$\\color{green}4/4$ $\\color{green}+100.00\\%$	$\\color{red}334/338$ $\\color{green}+98.82\\%$

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