refactor: update stats/base/dists/triangular/mgf implementation and test tolerances

[anandkaranubc]

Resolves #4704

Description

What is the purpose of this pull request?

This pull request:

• updates the implementation of the triangular MGF in main.c, main.js, and factory.js to align with the latest Julia standards. It also updates test tolerances in test.native.js for consistency.
• update fixtures for triangular MGF based on Julia implementation (using the same Julia version 1.5)

Related Issues

Does this pull request have any related issues?

This pull request:

• resolves #4704

Questions

Any questions for reviewers of this pull request?

• There is a helper function called phi2 which has been added to the C and JS files. I have added it as a private helper function in the same files. Should it be moved to a separate module for JS and a separate C file?

Other

Any other information relevant to this pull request? This may include screenshots, references, and/or implementation notes.

• The PR reduces the tolerance thresholds for test.native.js from a factor of 1.0e7 to 4.0. The required tolerance should be 1.0, but only 2 test cases fail with a 1.0 factor tolerance. I tried investigating the root cause by changing the order of terms and avoiding operator fusion, but no changes were observed. I will need some advice on how to fix this.

Checklist

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

• Read, understood, and followed the contributing guidelines.

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@stdlib-js/reviewers

[stdlib-bot]

Coverage Report

Package	Statements	Branches	Functions	Lines
stats/base/dists/triangular/mgf	$\color{green}380/380$ $\color{green}+100.00\%$	$\color{green}38/38$ $\color{green}+100.00\%$	$\color{green}5/5$ $\color{green}+100.00\%$	$\color{green}380/380$ $\color{green}+100.00\%$

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

[anandkaranubc]

Hii @kgryte,

Over to you for review. A follow up question in addition to the ones above:

• To ensure 100% coverage, should I update runner.jl to include examples from all 4 cases, or should I just manually add additional test cases in a new tape function?

[kgryte]

@anandkaranubc Would you mind updating and rerunning the runner?

[anandkaranubc]

Can I get some feedback on this too please, @kgryte? Thanks!

[anandkaranubc]

Should the README.md also be changed with the updated formula for mgf calculation?

$$M_X(t) := \mathbb{E}\!\left[e^{tX}\right] = \begin{cases} \exp(ct) \frac{(c-a)\phi_2((a-c)t) + (b-c)\phi_2((b-c)t)}{b-a} & \text{if } a < c < b \\\ \exp(ct) \phi_2((a-c)t) & \text{if } a = c < b \\\ \exp(ct) \phi_2((b-c)t) & \text{if } a < c = b \\\ \exp(ct) & \text{if } a = c = b \end{cases}$$ $$\phi_2(x) := \begin{cases} 1, & \text{if } x = 0 \\\ \frac{2 \left( \exp(x) - 1 - x \right)}{x^2} & \text{if } x \neq 0 \end{cases}$$

I notice that this would also mean changing the svg link provided for the equation.

A side note: The Julia implementation does this case by case breakdown to avoid unnecessary computation.

[kgryte]

Should the README.md also be changed with the updated formula for mgf calculation?

No. The equations in READMEs should not necessarily reflect implementation logic. That is an implementation detail. Unless the package explicitly concerns a particular algorithm, the README contains equations as you'd find in a textbook or Wikipedia. The MGF currently in the README follows the latter.

[anandkaranubc]

Once #5638 gets merged, the code coverage will be 100%.

Edit: That PR has been merged and the code coverage is 100% now!

[anandkaranubc]

I can confirm that if CFLAGS="-ffp-contract=off" is used, the tolerance can be set to 1.0 here.

[anandkaranubc]

/stdlib merge

[Planeshifter]

Let's get this one in! 🚀

[stdlib-bot]

PR Commit Message

refactor: update `stats/base/dists/triangular/mgf` implementation and test tolerances

PR-URL: https://github.com/stdlib-js/stdlib/pull/4768
Ref: https://github.com/stdlib-js/stdlib/issues/4704

Co-authored-by: Philipp Burckhardt <[email protected]>
Co-authored-by: stdlib-bot <[email protected]>
Reviewed-by: Philipp Burckhardt <[email protected]>
Reviewed-by: Athan Reines <[email protected]>
Signed-off-by: Philipp Burckhardt <[email protected]>

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