feat: add C implementation of math/base/special/kernel-betainc

[nirmaljb]

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Progresses #649 .

Description

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This pull request:

• Progresses a part of : [RFC]: Add C implementations to base special math functions (tracking issue) #649

Related Issues

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This pull request has the following related issues:

• [RFC]: Add C implementations to base special math functions (tracking issue) #649

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I used codex to find references, but the proposed changes were fully authored manually by myself.

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[stdlib-bot]

Coverage Report

Package	Statements	Branches	Functions	Lines
math/base/special/kernel-betainc	$\\color{red}1699/1788$ $\\color{green}+0.00\\%$	$\\color{red}239/256$ $\\color{green}+0.00\\%$	$\\color{green}16/16$ $\\color{green}+0.00\\%$	$\\color{red}1699/1788$ $\\color{green}+0.00\\%$

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

[nirmaljb]

/stdlib update-copyright-years

[nirmaljb]

Thank you for the suggestions! @gunjjoshi

[nirmaljb]

Fixed the issues
Thank you for pointing out! @gunjjoshi

[nirmaljb]

Note: The following three Rocco Romeo cases are not added out because they require long double (80-bit extended) precision to match the Wolfram reference values. Boost passes these by internally promoting double to long double (its default promote_double policy). JavaScript and C double-only computation loses too much precision in the intermediate x^a * y^b / Beta(a,b) power term for these extreme x values.

1. beta( 2, 4, ldexp(1+1/1024, -351) ): rel error ~9.3e-10 (~4.2M eps)
2. beta( 2, 4, ldexp(1+1/2048, -351) ): rel error ~2.4e-7 (~1B eps)
3. ibeta( 3, 5, ldexp(1+15/16, -268) ): rel error ~6.5e-3 (~29B eps)

[nirmaljb]

@Planeshifter and @Neerajpathak07 I've addressed the requested changes. I have also updated the fixtures as well as the tests, covering everything from the boost implementation.
There is are few abnormally huge ulp value for asymptotically large a,b values, which need investigation.

[0PrashantYadav0]

@Planeshifter @kgryte I have a PR which depend on this, so please review and merge if it looks good to you.

[Planeshifter]

Let's land this one! Finally! Thanks everyone for pushing this PR forward and @nirmaljb especially for opening it!

[stdlib-bot]

PR Commit Message

feat: add C implementation of `math/base/special/kernel-betainc`

PR-URL: https://github.com/stdlib-js/stdlib/pull/10279
Ref: https://github.com/stdlib-js/stdlib/issues/649

Co-authored-by: Gunj Joshi <gunjjoshi8372@gmail.com>
Co-authored-by: Neeraj Pathak <neerajrpathak710@gmail.com>
Co-authored-by: Philipp Burckhardt <pburckhardt@outlook.com>
Co-authored-by: stdlib-bot <noreply@stdlib.io>
Reviewed-by: Athan Reines <kgryte@gmail.com>
Reviewed-by: Karan Anand <anandkarancompsci@gmail.com>
Reviewed-by: Gunj Joshi <gunjjoshi8372@gmail.com>
Reviewed-by: Neeraj Pathak <neerajrpathak710@gmail.com>
Reviewed-by: Philipp Burckhardt <pburckhardt@outlook.com>
Signed-off-by: Neeraj Pathak <neerajrpathak710@gmail.com>
Signed-off-by: Nirmal Jyoti Biswas <59522492+nirmaljb@users.noreply.github.com>

Please review the above commit message and make any necessary adjustments.

⚠️ Action Required: This PR references tracking issues. Please update the PR description to replace any "Closes", "Fixes", or "Resolves" keywords with "Ref" when referencing these issues.

[Neerajpathak07]

Indeed this was a very crucial package and a huge thanks to @nirmaljb for speed-running this effort!!!