fix: avoid cancellation for small p in stats/base/dists/geometric/logcdf

[Abhist17]

Description

What is the purpose of this pull request?

This pull request:

• fixes the precision of stats/base/dists/geometric/logcdf for small p (JS, factory and C).

The function computed log1p( -pow( 1.0-p, x+1.0 ) ). Forming 1 - p rounds away the low-order bits of p, and for p < 2^-53 it rounds to 1, so the result is -Infinity:

var logcdf = require( '@stdlib/stats/base/dists/geometric/logcdf' );

logcdf( 0.0, 1.0e-20 );
// develop: -Infinity
// this PR: -46.051701859880914 (= ln(1e-20))

logcdf( 9.0, 1.0e-10 );
// develop: -20.723265754206043
// this PR: -20.72326583739641

For p < 0.5 it now uses log1mexp( (x+1) * log1p(-p) ). For p >= 0.5, 1 - p is exact, so the old form stays. There it is more accurate than going through log1p, because for p near 1 the error in (x+1) * log1p(-p) gets multiplied by its own magnitude.

Fixtures and tests:

• runner.jl now evaluates the textbook formula ln(1 - (1-p)^(floor(x)+1)) in 2048-bit BigFloat instead of Distributions.logcdf. Distributions computes in double precision with the same loss. Distributions is dropped from REQUIRE. small_p.json and large_p.json are regenerated with the same input ranges.

• New tiny_p.json: p log-spaced over [1e-300, 0.1], x in [0, 100].

• Max error against the new fixtures, JS and native:

  fixture	develop	this PR
  large_p	2 ULP	2 ULP
  small_p	56 ULP	9 ULP
  tiny_p	-Infinity for p < 2^-53	3 ULP

  The tolerances are set to those values (were 42 and 15). On develop, 1,006 of the 3,012 assertions in test.logcdf.js fail. With this change they all pass.

Related Issues

Does this pull request have any related issues?

None. Companion to #15765 (geometric/cdf).

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.

No.

Checklist

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

• Read, understood, and followed the contributing guidelines.

AI Assistance

When authoring the changes proposed in this PR, did you use any kind of AI assistance?

• Yes
• No

If you answered "yes" above, how did you use AI assistance?

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

Disclosure

If you answered "yes" to using AI assistance, please provide a short disclosure indicating how you used AI assistance. This helps reviewers determine how much scrutiny to apply when reviewing your contribution. Example disclosures: "This PR was written primarily by Claude Code." or "I consulted ChatGPT to understand the codebase, but the proposed changes were fully authored manually by myself.".

This PR was written primarily by Claude Code. I found the bug by reading the stats/base/dists functions for 1 - p cancellation.

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[stdlib-bot]

Coverage Report

Package	Statements	Branches	Functions	Lines
stats/base/dists/geometric/logcdf	$\\color{green}307/307$ $\\color{green}+100.00\\%$	$\\color{green}31/31$ $\\color{green}+100.00\\%$	$\\color{green}4/4$ $\\color{green}+100.00\\%$	$\\color{green}307/307$ $\\color{green}+100.00\\%$

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

[0PrashantYadav0]

Suggested change

	p = exp10.( range( -300.0, stop = -1.0, length = 1000 ) );
	p = exp10.( range( -323.0, stop = -1.0, length = 1000 ) );

Non-blocking: this also covers subnormal p, and the tests still pass. Unlike #15797 (comment), no extra branch is needed here, since x+1 >= 1 keeps (x+1)*log1p(-p) exact.

[Abhist17]

@0PrashantYadav0 thanks. I tried the -323 range: the subnormal end is fine (e.g. logcdf( 10, 1e-320 ) matches ln(11p)), but with the new grid one point near the top of the range, x = 99.23, p = 0.0476, comes out 5 ULP off against the 3-ULP tolerance. That's the existing formula at ordinary p, not this change, and I didn't want to re-roll x until a draw passed. So I've left the range at -300 for now. Happy to switch it and set that block to 5 ULP if you'd rather have the subnormal coverage.