[BUG]: math/base/special/gammainc is off by up to 0.03 for x > 1000 when a is close to x #16533

Description

Encountered an error when evaluating gammainc with x > 1000 and a near x. The large-x asymptotic series (tgamma_i_large_x_series.js) starts its falling factorial at a instead of a - 1, so results are off by up to a few percent. This propagates to stats/base/dists/poisson/cdf, chisquare/cdf, gamma/cdf and others for large parameters.

In lib/node_modules/@stdlib/math/base/special/gammainc/lib/tgamma_i_large_x_series.js, var a = a1; should be var a = a1 - 1.0;, matching Boost's incomplete_tgamma_large_x_series (a_poch(a - 1)). With that one-line change the four values below match SciPy. The C implementation in src/main.c (ac = a; at line 703, in the same series) has the same off-by-one.

Found with a differential-testing tool; the report was drafted with AI assistance and reviewed and edited by me.

Related Issues

None.

Questions

No.

Demo

https://trackoor.github.io/stdlib-gammainc-large-x-repro/ (source: https://github.com/trackoor/stdlib-gammainc-large-x-repro). It shows the first three values from the jsDelivr ESM build; the chisqCDF value is only wrong in the npm build.

Reproduction

• npm i @stdlib/math-base-special-gammainc@0.3.1 @stdlib/stats-base-dists-poisson-cdf@0.3.1 @stdlib/stats-base-dists-chisquare-cdf@0.3.1
• save as t.js and run node t.js:

var gammainc = require( '@stdlib/math-base-special-gammainc' );
var poissonCDF = require( '@stdlib/stats-base-dists-poisson-cdf' );
var chisqCDF = require( '@stdlib/stats-base-dists-chisquare-cdf' );

console.log( gammainc( 1367.3436948118986, 1409.203708740708 ) );
console.log( gammainc( 1001, 1000.5, true, true ) );
console.log( poissonCDF( 999, 1001 ) );
console.log( chisqCDF( 2500, 2400 ) );

Expected Results

0.13190499189857086
0.489491914600245
0.48318854707021375
0.9241270884571916

(SciPy 1.10.1: gammainc(1409.203708740708, 1367.3436948118986), gammaincc(1000.5, 1001), poisson.cdf(999, 1001), chi2.cdf(2500, 2400))

Actual Results

0.09952119051642438
0.501850998882592
0.49530155533410974
0.9231500432608719

Version

@stdlib/math-base-special-gammainc 0.3.1; same code on develop 734972c

Environments

Node.js

Browser Version

Chrome 152 for the demo page; the report itself uses Node.js.

Node.js / npm Version

v26.13.0

Platform

Linux

Checklist

• Read and understood the Code of Conduct.
• Searched for existing issues and pull requests.

Credit: https://ucsbplse.github.io/