Repository: stdlib-js/stdlib
Pull request #4768: refactor: update `stats/base/dists/triangular/mgf` implementation and test tolerances
Author: login: anandkaranubc
Labels: name: Statistics
description: Issue or pull request related to statistical functionality.
name: Ready To Merge
description: A pull request which is ready to be merged.
Created: 2025-01-15T18:20:27Z | Merged: 2025-03-06T02:18:02Z

Resolves [#4704](https://github.com/stdlib-js/stdlib/issues/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](https://github.com/JuliaStats/Distributions.jl/blob/957f0c0f79307edefb9afa75c682ef18b35dd982/src/univariate/continuous/triangular.jl#L139-L171) 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](https://github.com/stdlib-js/stdlib/issues/4704)

## Questions

> Any questions for reviewers of this pull request?

- There is a helper function called [phi2](https://github.com/JuliaStats/Distributions.jl/blob/957f0c0f79307edefb9afa75c682ef18b35dd982/src/univariate/continuous/triangular.jl#L139-L151) 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.

-   [x] Read, understood, and followed the [contributing guidelines][contributing].

* * *

@stdlib-js/reviewers

[contributing]: https://github.com/stdlib-js/stdlib/blob/develop/CONTRIBUTING.md



Diff:
diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/benchmark/benchmark.js b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/benchmark/benchmark.js
index e14334a122ef..90d2cc2b940e 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/benchmark/benchmark.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/benchmark/benchmark.js
@@ -22,7 +22,7 @@
 
 var bench = require( '@stdlib/bench' );
 var Float64Array = require( '@stdlib/array/float64' );
-var randu = require( '@stdlib/random/base/randu' );
+var uniform = require( '@stdlib/random/base/uniform' );
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
 var EPS = require( '@stdlib/constants/float64/eps' );
 var pkg = require( './../package.json' ).name;
@@ -47,10 +47,10 @@ bench( pkg, function benchmark( b ) {
 	mode = new Float64Array( len );
 
 	for ( i = 0; i < len; i++ ) {
-		t[ i ] = randu() * 5.0;
-		min[ i ] = randu() * 10.0;
-		max[ i ] = min[ i ] + ( randu() * 40.0 ) + EPS;
-		mode[ i ] = min[ i ] + ( ( max[ i ] - min[ i ] ) * randu() );
+		t[ i ] = uniform( 0.0, 5.0 );
+		min[ i ] = uniform( 0.0, 10.0 );
+		max[ i ] = uniform( min[ i ] + EPS, min[ i ] + 40.0 );
+		mode[ i ] = uniform( min[ i ], max[ i ]);
 	}
 
 	b.tic();
@@ -73,6 +73,7 @@ bench( pkg+':factory', function benchmark( b ) {
 	var mode;
 	var min;
 	var max;
+	var len;
 	var t;
 	var y;
 	var i;
@@ -81,11 +82,15 @@ bench( pkg+':factory', function benchmark( b ) {
 	max = 1.5;
 	mode = 0.5;
 	mymgf = mgf.factory( min, max, mode );
+	len = 100;
+	t = new Float64Array( len );
+	for ( i = 0; i < len; i++ ) {
+		t[ i ] = uniform( 0.0, 5.0 );
+	}
 
 	b.tic();
 	for ( i = 0; i < b.iterations; i++ ) {
-		t = randu() * 5.0;
-		y = mymgf( t );
+		y = mymgf( t[ i % len ]);
 		if ( isnan( y ) ) {
 			b.fail( 'should not return NaN' );
 		}
diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/benchmark/benchmark.native.js b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/benchmark/benchmark.native.js
index a00907214e1d..f1d1571d782e 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/benchmark/benchmark.native.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/benchmark/benchmark.native.js
@@ -23,7 +23,7 @@
 var resolve = require( 'path' ).resolve;
 var bench = require( '@stdlib/bench' );
 var Float64Array = require( '@stdlib/array/float64' );
-var randu = require( '@stdlib/random/base/randu' );
+var uniform = require( '@stdlib/random/base/uniform' );
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
 var tryRequire = require( '@stdlib/utils/try-require' );
 var EPS = require( '@stdlib/constants/float64/eps' );
@@ -56,10 +56,10 @@ bench( pkg+'::native', opts, function benchmark( b ) {
 	mode = new Float64Array( len );
 
 	for ( i = 0; i < len; i++ ) {
-		t[ i ] = randu() * 5.0;
-		min[ i ] = randu() * 10.0;
-		max[ i ] = min[ i ] + ( randu() * 40.0 ) + EPS;
-		mode[ i ] = min[ i ] + ( ( max[ i ] - min[ i ] ) * randu() );
+		t[ i ] = uniform( 0.0, 5.0 );
+		min[ i ] = uniform( 0.0, 10.0 );
+		max[ i ] = uniform( min[ i ] + EPS, min[ i ] + 40.0 );
+		mode[ i ] = uniform( min[ i ], max[ i ]);
 	}
 
 	b.tic();
diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/factory.js b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/factory.js
index 4c1e941e5325..9de246f4dbad 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/factory.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/factory.js
@@ -23,7 +23,7 @@
 var constantFunction = require( '@stdlib/utils/constant-function' );
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
 var exp = require( '@stdlib/math/base/special/exp' );
-var pow = require( '@stdlib/math/base/special/pow' );
+var phi2 = require( './phi2.js' );
 
 
 // MAIN //
@@ -45,10 +45,6 @@ var pow = require( '@stdlib/math/base/special/pow' );
 * // returns ~10.205
 */
 function factory( a, b, c ) {
-	var bmc;
-	var bma;
-	var cma;
-
 	if (
 		isnan( a ) ||
 		isnan( b ) ||
@@ -58,9 +54,6 @@ function factory( a, b, c ) {
 	) {
 		return constantFunction( NaN );
 	}
-	bmc = b - c;
-	bma = b - a;
-	cma = c - a;
 	return mgf;
 
 	/**
@@ -75,19 +68,19 @@ function factory( a, b, c ) {
 	* // returns <number>
 	*/
 	function mgf( t ) {
-		var ret;
-
 		if ( isnan( t ) ) {
 			return NaN;
 		}
-		if ( t === 0.0 ) {
-			return 1.0;
+		if ( a < c ) {
+			if ( c < b ) {
+				return exp( c*t ) * ( ( (c-a)*phi2( (a-c)*t ) ) + ( (b-c)*phi2( (b-c)*t ) ) ) / ( b-a ); // eslint-disable-line max-len
+			}
+			return exp( c*t ) * phi2( ( a-c ) * t );
+		}
+		if ( c < b ) {
+			return exp( c*t ) * phi2( ( b-c ) * t );
 		}
-		ret = ( bmc * exp( a*t ) ) - ( bma * exp( c*t ) );
-		ret += cma * exp( b*t );
-		ret *= 2.0;
-		ret /= bma * cma * bmc * pow( t, 2.0 );
-		return ret;
+		return exp( c*t );
 	}
 }
 
diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/main.js b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/main.js
index c9fe40e86f76..34feaa1ffec3 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/main.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/main.js
@@ -22,7 +22,7 @@
 
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
 var exp = require( '@stdlib/math/base/special/exp' );
-var pow = require( '@stdlib/math/base/special/pow' );
+var phi2 = require( './phi2.js' );
 
 
 // MAIN //
@@ -73,11 +73,6 @@ var pow = require( '@stdlib/math/base/special/pow' );
 * // returns NaN
 */
 function mgf( t, a, b, c ) {
-	var bmc;
-	var bma;
-	var cma;
-	var ret;
-
 	if (
 		isnan( t ) ||
 		isnan( a ) ||
@@ -88,17 +83,16 @@ function mgf( t, a, b, c ) {
 	) {
 		return NaN;
 	}
-	if ( t === 0.0 ) {
-		return 1.0;
+	if ( a < c ) {
+		if ( c < b ) {
+			return exp( c*t ) * ( ( (c-a)*phi2( (a-c)*t ) ) + ( (b-c)*phi2( (b-c)*t ) ) ) / ( b-a ); // eslint-disable-line max-len
+		}
+		return exp( c*t ) * phi2( ( a-c ) * t );
+	}
+	if ( c < b ) {
+		return exp( c*t ) * phi2( ( b-c ) * t );
 	}
-	bmc = b - c;
-	bma = b - a;
-	cma = c - a;
-	ret = ( bmc * exp( a*t ) ) - ( bma * exp( c*t ) );
-	ret += cma * exp( b*t );
-	ret *= 2.0;
-	ret /= bma * cma * bmc * pow( t, 2.0 );
-	return ret;
+	return exp( c*t );
 }
 
 
diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/phi2.js b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/phi2.js
new file mode 100644
index 000000000000..45ef5569502b
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/phi2.js
@@ -0,0 +1,45 @@
+/**
+* @license Apache-2.0
+*
+* Copyright (c) 2025 The Stdlib Authors.
+*
+* Licensed under the Apache License, Version 2.0 (the "License");
+* you may not use this file except in compliance with the License.
+* You may obtain a copy of the License at
+*
+*    http://www.apache.org/licenses/LICENSE-2.0
+*
+* Unless required by applicable law or agreed to in writing, software
+* distributed under the License is distributed on an "AS IS" BASIS,
+* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+* See the License for the specific language governing permissions and
+* limitations under the License.
+*/
+
+'use strict';
+
+// MODULES //
+
+var expm1 = require( '@stdlib/math/base/special/expm1' );
+
+
+// MAIN //
+
+/**
+* Helper function for repeated computation in the MGF formula.
+*
+* @private
+* @param {number} x - input value
+* @returns {number} evaluated result
+*/
+function phi2( x ) {
+	if ( x === 0.0 ) {
+		return 1.0;
+	}
+	return ( 2.0 * ( expm1( x ) - x ) ) / ( x*x );
+}
+
+
+// EXPORTS //
+
+module.exports = phi2;
diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/manifest.json b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/manifest.json
index a54b0ef8e252..7e6448af4a8c 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/manifest.json
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/manifest.json
@@ -41,7 +41,7 @@
         "@stdlib/math/base/napi/quaternary",
         "@stdlib/math/base/assert/is-nan",
         "@stdlib/math/base/special/exp",
-        "@stdlib/math/base/special/pow"
+        "@stdlib/math/base/special/expm1"
       ]
     },
     {
diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/src/main.c b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/src/main.c
index b9deb936b74e..e489baa7bb29 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/src/main.c
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/src/main.c
@@ -18,8 +18,21 @@
 
 #include "stdlib/stats/base/dists/triangular/mgf.h"
 #include "stdlib/math/base/assert/is_nan.h"
+#include "stdlib/math/base/special/expm1.h"
 #include "stdlib/math/base/special/exp.h"
-#include "stdlib/math/base/special/pow.h"
+
+/**
+* Helper function for repeated computation in the MGF formula.
+*
+* @param x   input value
+* @return    evaluated result
+*/
+static double phi2( const double x ) {
+	if ( x == 0.0 ) {
+		return 1.0;
+	}
+	return ( 2.0 * ( stdlib_base_expm1( x ) - x ) ) / ( x*x );
+}
 
 /**
 * Evaluates the moment-generating function (MGF) for a triangular distribution with lower limit `a`, upper limit `b`, and mode `c` at a value `t`.
@@ -35,29 +48,19 @@
 * // returns ~1.021
 */
 double stdlib_base_dists_triangular_mgf( const double t, const double a, const double b, const double c ) {
-	double bmc;
-	double bma;
-	double cma;
-	double ret;
-	if (
-		stdlib_base_is_nan( t ) ||
-		stdlib_base_is_nan( a ) ||
-		stdlib_base_is_nan( b ) ||
-		stdlib_base_is_nan( c ) ||
-		a > c ||
-		c > b
-	) {
-		return 0.0/0.0; // NaN
+	if ( stdlib_base_is_nan( t ) || stdlib_base_is_nan( a ) || stdlib_base_is_nan( b ) || stdlib_base_is_nan( c ) || a > c || c > b ) {
+		return 0.0 / 0.0; // NaN
 	}
-	if ( t == 0.0 ) {
-		return 1.0;
+	if ( a < c ) {
+		if ( c < b ) {
+			double term1 = ( c-a ) * phi2( ( a-c ) * t );
+			double term2 = ( b-c ) * phi2( ( b-c ) * t );
+			return stdlib_base_exp( c*t ) * ( term1+term2 ) / ( b-a );
+		}
+		return stdlib_base_exp( c*t ) * phi2( ( a-c ) * t );
+	}
+	if ( c < b ) {
+		return stdlib_base_exp( c*t ) * phi2( ( b-c ) * t );
 	}
-	bmc = b - c;
-	bma = b - a;
-	cma = c - a;
-	ret = ( bmc * stdlib_base_exp( a*t ) ) - ( bma * stdlib_base_exp( c*t ) );
-	ret += cma * stdlib_base_exp( b*t );
-	ret *= 2.0;
-	ret /= bma * cma * bmc * stdlib_base_pow( t, 2.0 );
-	return ret;
+	return stdlib_base_exp( c*t );
 }
diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/a_eq_c_eq_b.json b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/a_eq_c_eq_b.json
new file mode 100644
index 000000000000..0bf9b9a1bf18
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/a_eq_c_eq_b.json
@@ -0,0 +1 @@
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diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/a_eq_c_less_b.json b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/a_eq_c_less_b.json
new file mode 100644
index 000000000000..f48c59c53ad0
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/a_eq_c_less_b.json
@@ -0,0 +1 @@
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diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/a_less_c_eq_b.json b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/a_less_c_eq_b.json
new file mode 100644
index 000000000000..d2a71726e3d7
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/a_less_c_eq_b.json
@@ -0,0 +1 @@
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diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/a_less_c_less_b.json b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/a_less_c_less_b.json
new file mode 100644
index 000000000000..12e34e02b954
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/a_less_c_less_b.json
@@ -0,0 +1 @@
+{"expected":[3.9463644191618437e8,1.4497351920109137e10,4.1074712179428875e8,3.6169077767718832e22,16580.27833636114,6.225142544426928e8,21481.274735691564,2.330322411864421e16,1.2102317270202719e14,2355.6231780437593,9.028354650269449e19,1.0806483128648321e10,2.2946816299056303e20,2.851860602573042e9,2.598088435686875e7,1.054299694130179e19,218291.89831133792,5.953560227026181e12,8.646036172230397e7,3.000018628731164e6,3.64333506804337e10,1.7530223415454742e10,1.9049072221851784e14,12187.261523526584,2.957014666106759e20,5.651540620943412e6,7.887090509293138e24,75.22395257378083,8.177628849188512,450635.41051562544,1.2975642303348,2.0014008271998537e6,204.60788075045923,9909.38779135916,7738.561227476148,44938.02733438308,5.7918629230642805e9,11.763361128190128,2.27146048705203,7.662440054052186e15,1.819666687952312e20,1.2421547820463576e17,4.715525822513456e18,2.2300170338077884e11,2.5525248652807184e16,1.120503374814716e37,2.8094270053923562e7,9.760219802322487,30171.30617596861,9.442452616651326e21,1.068525697274912e20,1.3275571204026045e18,1.5170065981084805e7,2.692336946502507e8,141.12888229563634,1.073856509392885e7,1.3661825934074216e26,168662.6182718251,1300.556763031143,3.728794118249073e19,204.90689807162374,307378.69789910706,2390.0299148985973,1.7377321056099625e9,3.232284094225539e8,3.606405437872262e13,1.0614016763386235e6,7.515350012560409e29,1.9417298113442273e18,5.915450262451302e8,5.540901412234641e19,2.708722270652364e7,2.1576088963492584e9,179201.60192756957,2.299885068245786,1.3276144919382122e6,2.793574804450276e6,843.2200958096947,1.4590409249898888,9.539020635226266e8,1.4229659228002265e8,283.7671371593937,17659.873554351074,10.891144624094078,7.315275648846904e18,1.5983570641401812e14,5715.419616089615,7.17928126115883,8.141695517229158e15,1.5992674514038542e23,1.886831922920137e12,4443.964057559408,5.367578070505459e15,7.567128082603241e11,155.46541575834547,6.127046238421183e18,1.821233275752761e16,2.6329408414297784,8.820559930385853e9,3.3308393478874136,4.3040978342705155e6,2.6778442656192263e6,2.721405763398425e8,4.328084660149496,1.8277380815453557e12,954.3611170984707,3.29663919589352,1.8892148379661857e27,9520.415685702592,520.5198358121285,1.616681493949403e20,691.4596467766564,1.0182785621891079e9,2.183135662758753,1.6640193592394044,164.123016381577,1.0991688765208704e10,38607.65188790388,5.6650239076832354e19,2054.536437818905,764115.0251809483,1.0573478947728266e14,100442.72928864245,3.4154746954152666e7,11.001826142158748,125409.46261999992,3812.1138989095357,55116.6010749231,5.729416973060232e13,6.535407470426384e14,58.54859660796268,7.701207040837639,1151.8903424638577,2.336579842053172e18,7499.649980525344,523108.730824743,74.35190870643433,2.0071897732266327e6,7.296848618031836,209174.97248176896,457.9894102282729,4.879005266607343e6,3.258961105446164,33.47359570870041,4.098459710458623e7,7.4511141336162435e6,3.2751101983132953,2.6733767000860214e11,4.779766670135318e10,24195.897620905977,26.587104957463154,3.1939468626340553e7,1.0091782399951173e6,1.5490185241403294e9,4.0459210535934577e6,9.687903993561086e7,69.67687150741995,4.15333184972256e10,7.137447219270627e9,9.047099702778763e10,3.8573827925721606e11,1.3710993551429298,1.918518458427518e16,1.9528924879652088,61049.798728172944,630939.7685073084,6.138033568816853e9,1.8355514610145554e8,4.784032992431914e11,11.504519214559416,2.7261084330668983e10,1.7408735060596455e7,1.0925238415273638e9,5.51057507452623e8,3.391507458355366e8,1.6250121701636608e15,1.5158785416906141e24,3.549730492638622e6,6.7042382359014375e19,2.753107912745559,6.135367785482389e15,8.975155309858536e22,1.81700700013844e21,9.591095989813544e10,22.953804372779057,2.873080439388278e13,6.265044301763033e15,3.017779771970134e7,2.6782300451815497,7.510153652959615e31,1318.037510068706,9.554263098445922,10.7805604308716,616.7726903827605,7.060725348709316e8,9293.441408780247,3.6547010606604557e9,447143.10397922754,61233.128417803026,35354.79106961009,3.2499651694836025e14,4.099785998091115e14,103128.19541169087,30944.972825997094,510905.84775651095,2.053800214192073,363.7664974005881,7.134362378882178e13,2.6760660145625982e8,3.553518062686279e11,2.5142073940127906e6,5.589367079246823e6,139.32367314466046,6.119664495090367e13,9.921618341646904e10,3.5325246794679388e16,2.746411387560658e21,2.3021430596741523e22,206.71560439151298,1.1001470429832495e6,13051.422539415758,20.76819808676909,2.990939431089521e6,10.619613047892035,3.682079108622935e8,9.568706330729235e14,7035.924791138457,42542.50238867519,2.036450501346158e9,1.0097671600445502e17,5.413543743386649e21,7.235658023606416e9,2.6004949880774264e27,3.7187409114592957e11,19346.34637918883,2.691683726780438e7,57393.90537551124,1.5178458028012318e10,4003.009956636051,4.1445248244925684e8,2.744417966639474e9,582.2874277761526,3.5938071508026637,2222.500721714869,1.7506103566879433e27,253.46847959772614,7.121224177901112e6,2.579397367409532e20,2.444872598664338e21,29.822705023947133,2.2622039131061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diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/large_range.json b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/large_range.json
index 61e38e8356f6..12e34e02b954 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/large_range.json
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/large_range.json
@@ -1 +1 @@
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diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/medium_range.json b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/medium_range.json
index 16697c29330a..67b3140c2c01 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/medium_range.json
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/medium_range.json
@@ -1 +1 @@
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diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/runner.jl b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/runner.jl
index 9fac32a69018..1afc54ddff0e 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/runner.jl
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/runner.jl
@@ -93,3 +93,15 @@ b = ( rand( 1000 ) .* 80.0 ) .+ a;
 c = a .+ ( b .- a ) .* rand();
 x = rand( 1000 );
 gen( x, a, b, c, "large_range.json" );
+
+# Case: a < c < b
+gen( x, a, b, c, "a_less_c_less_b.json" );
+
+# Case: a < c = b
+gen( x, a, b, b, "a_less_c_eq_b.json" );
+
+# Case: a = c < b
+gen( x, a, b, a, "a_eq_c_less_b.json" );
+
+# Case: a = c = b
+gen( x, a, a, a, "a_eq_c_eq_b.json" );
diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/small_range.json b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/small_range.json
index c3cee33a4b06..dfe41c3b26b8 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/small_range.json
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/fixtures/julia/small_range.json
@@ -1 +1 @@
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diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.factory.js b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.factory.js
index e09ee2891521..06e5699cc2e2 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.factory.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.factory.js
@@ -34,6 +34,10 @@ var factory = require( './../lib/factory.js' );
 var smallRange = require( './fixtures/julia/small_range.json' );
 var mediumRange = require( './fixtures/julia/medium_range.json' );
 var largeRange = require( './fixtures/julia/large_range.json' );
+var aLessCLessB = require( './fixtures/julia/a_less_c_less_b.json' );
+var aLessCEqB = require( './fixtures/julia/a_less_c_eq_b.json' );
+var aEqCLessB = require( './fixtures/julia/a_eq_c_less_b.json' );
+var aEqCEqB = require( './fixtures/julia/a_eq_c_eq_b.json' );
 
 
 // TESTS //
@@ -239,3 +243,127 @@ tape( 'the created function evaluates the MGF for `x` given a large range `b - a
 	}
 	t.end();
 });
+
+tape( 'the created function evaluates the MGF for `x` given the case: a < c < b', function test( t ) {
+	var expected;
+	var delta;
+	var mgf;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aLessCLessB.expected;
+	x = aLessCLessB.x;
+	a = aLessCLessB.a;
+	b = aLessCLessB.b;
+	c = aLessCLessB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		mgf = factory( a[i], b[i], c[i] );
+		y = mgf( x[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 1.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
+
+tape( 'the created function evaluates the MGF for `x` given the case: a < c = b', function test( t ) {
+	var expected;
+	var delta;
+	var mgf;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aLessCEqB.expected;
+	x = aLessCEqB.x;
+	a = aLessCEqB.a;
+	b = aLessCEqB.b;
+	c = aLessCEqB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		mgf = factory( a[i], b[i], c[i] );
+		y = mgf( x[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 1.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
+
+tape( 'the created function evaluates the MGF for `x` given the case: a = c < b', function test( t ) {
+	var expected;
+	var delta;
+	var mgf;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aEqCLessB.expected;
+	x = aEqCLessB.x;
+	a = aEqCLessB.a;
+	b = aEqCLessB.b;
+	c = aEqCLessB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		mgf = factory( a[i], b[i], c[i] );
+		y = mgf( x[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 1.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
+
+tape( 'the created function evaluates the MGF for `x` given the case: a = c = b', function test( t ) {
+	var expected;
+	var delta;
+	var mgf;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aEqCEqB.expected;
+	x = aEqCEqB.x;
+	a = aEqCEqB.a;
+	b = aEqCEqB.b;
+	c = aEqCEqB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		mgf = factory( a[i], b[i], c[i] );
+		y = mgf( x[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 1.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.mgf.js b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.mgf.js
index cb18e08e3d04..1f326c75a7dc 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.mgf.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.mgf.js
@@ -32,6 +32,10 @@ var mgf = require( './../lib' );
 var smallRange = require( './fixtures/julia/small_range.json' );
 var mediumRange = require( './fixtures/julia/medium_range.json' );
 var largeRange = require( './fixtures/julia/large_range.json' );
+var aLessCLessB = require( './fixtures/julia/a_less_c_less_b.json' );
+var aLessCEqB = require( './fixtures/julia/a_less_c_eq_b.json' );
+var aEqCLessB = require( './fixtures/julia/a_eq_c_less_b.json' );
+var aEqCEqB = require( './fixtures/julia/a_eq_c_eq_b.json' );
 
 
 // TESTS //
@@ -176,3 +180,119 @@ tape( 'the function evaluates the MGF for `x` given a large range `b - a`', func
 	}
 	t.end();
 });
+
+tape( 'the function evaluates the MGF for `x` given the case: a < c < b', function test( t ) {
+	var expected;
+	var delta;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aLessCLessB.expected;
+	x = aLessCLessB.x;
+	a = aLessCLessB.a;
+	b = aLessCLessB.b;
+	c = aLessCLessB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[i], a[i], b[i], c[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 1.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
+
+tape( 'the function evaluates the MGF for `x` given the case: a < c = b', function test( t ) {
+	var expected;
+	var delta;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aLessCEqB.expected;
+	x = aLessCEqB.x;
+	a = aLessCEqB.a;
+	b = aLessCEqB.b;
+	c = aLessCEqB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[i], a[i], b[i], c[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 1.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
+
+tape( 'the function evaluates the MGF for `x` given the case: a = c < b', function test( t ) {
+	var expected;
+	var delta;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aEqCLessB.expected;
+	x = aEqCLessB.x;
+	a = aEqCLessB.a;
+	b = aEqCLessB.b;
+	c = aEqCLessB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[i], a[i], b[i], c[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 1.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
+
+tape( 'the function evaluates the MGF for `x` given the case: a = c = b', function test( t ) {
+	var expected;
+	var delta;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aEqCEqB.expected;
+	x = aEqCEqB.x;
+	a = aEqCEqB.a;
+	b = aEqCEqB.b;
+	c = aEqCEqB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[i], a[i], b[i], c[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 1.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
diff --git a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.native.js b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.native.js
index 47e6e6ebcbc7..11bfd214f22d 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.native.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.native.js
@@ -33,6 +33,10 @@ var tryRequire = require( '@stdlib/utils/try-require' );
 var smallRange = require( './fixtures/julia/small_range.json' );
 var mediumRange = require( './fixtures/julia/medium_range.json' );
 var largeRange = require( './fixtures/julia/large_range.json' );
+var aLessCLessB = require( './fixtures/julia/a_less_c_less_b.json' );
+var aLessCEqB = require( './fixtures/julia/a_less_c_eq_b.json' );
+var aEqCLessB = require( './fixtures/julia/a_eq_c_less_b.json' );
+var aEqCEqB = require( './fixtures/julia/a_eq_c_eq_b.json' );
 
 
 // VARIABLES //
@@ -122,8 +126,8 @@ tape( 'the function evaluates the MGF for `x` given a small range `b - a`', opts
 		} else {
 			delta = abs( y - expected[ i ] );
 
-			// FIXME: the tolerance for the C implementation is widely different from the JavaScript implementation, and it is not clear why.
-			tol = 1.0e7 * EPS * abs( expected[ i ] );
+			// NOTE: the tolerance here is larger than for the JavaScript implementation due to compiler optimizations which may be performed resulting in result divergence. For discussion, see https://github.com/stdlib-js/stdlib/pull/2298#discussion_r1624765205
+			tol = 2.0 * EPS * abs( expected[ i ] );
 			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
 		}
 	}
@@ -152,9 +156,7 @@ tape( 'the function evaluates the MGF for `x` given a medium range `b - a`', opt
 			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
 		} else {
 			delta = abs( y - expected[ i ] );
-
-			// FIXME: the tolerance for the C implementation is widely different from the JavaScript implementation, and it is not clear why.
-			tol = 1.0e7 * EPS * abs( expected[ i ] );
+			tol = 1.0 * EPS * abs( expected[ i ] );
 			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
 		}
 	}
@@ -183,9 +185,123 @@ tape( 'the function evaluates the MGF for `x` given a large range `b - a`', opts
 			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
 		} else {
 			delta = abs( y - expected[ i ] );
+			tol = 1.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
+
+tape( 'the function evaluates the MGF for `x` given the case: a < c < b', opts, function test( t ) {
+	var expected;
+	var delta;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aLessCLessB.expected;
+	x = aLessCLessB.x;
+	a = aLessCLessB.a;
+	b = aLessCLessB.b;
+	c = aLessCLessB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[i], a[i], b[i], c[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 1.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
+
+tape( 'the function evaluates the MGF for `x` given the case: a < c = b', opts, function test( t ) {
+	var expected;
+	var delta;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aLessCEqB.expected;
+	x = aLessCEqB.x;
+	a = aLessCEqB.a;
+	b = aLessCEqB.b;
+	c = aLessCEqB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[i], a[i], b[i], c[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 1.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
+
+tape( 'the function evaluates the MGF for `x` given the case: a = c < b', opts, function test( t ) {
+	var expected;
+	var delta;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aEqCLessB.expected;
+	x = aEqCLessB.x;
+	a = aEqCLessB.a;
+	b = aEqCLessB.b;
+	c = aEqCLessB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[i], a[i], b[i], c[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 1.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
+
+tape( 'the function evaluates the MGF for `x` given the case: a = c = b', opts, function test( t ) {
+	var expected;
+	var delta;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
 
-			// FIXME: the tolerance for the C implementation is widely different from the JavaScript implementation, and it is not clear why.
-			tol = 1.0e7 * EPS * abs( expected[ i ] );
+	expected = aEqCEqB.expected;
+	x = aEqCEqB.x;
+	a = aEqCEqB.a;
+	b = aEqCEqB.b;
+	c = aEqCEqB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[i], a[i], b[i], c[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 1.0 * EPS * abs( expected[ i ] );
 			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
 		}
 	}


Reviews:
user: login: kgryte
state: COMMENTED
author_association: MEMBER
submitted_at: 2025-01-16T02:20:11Z
user: login: kgryte
state: COMMENTED
author_association: MEMBER
submitted_at: 2025-01-18T11:15:48Z
user: login: kgryte
state: COMMENTED
author_association: MEMBER
submitted_at: 2025-01-18T11:18:08Z
user: login: kgryte
state: COMMENTED
author_association: MEMBER
submitted_at: 2025-01-18T11:20:12Z
user: login: kgryte
state: COMMENTED
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submitted_at: 2025-01-18T11:20:47Z
user: login: kgryte
state: COMMENTED
author_association: MEMBER
submitted_at: 2025-01-18T11:20:57Z
user: login: kgryte
state: COMMENTED
author_association: MEMBER
submitted_at: 2025-01-18T11:21:41Z
user: login: kgryte
state: COMMENTED
author_association: MEMBER
submitted_at: 2025-01-18T11:22:11Z
user: login: kgryte
state: COMMENTED
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submitted_at: 2025-01-18T11:22:33Z
user: login: kgryte
state: COMMENTED
author_association: MEMBER
submitted_at: 2025-01-18T11:25:37Z
user: login: kgryte
state: COMMENTED
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submitted_at: 2025-01-18T11:26:52Z
user: login: kgryte
state: COMMENTED
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submitted_at: 2025-01-18T22:22:28Z
user: login: anandkaranubc
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-01-21T15:47:39Z
user: login: anandkaranubc
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-01-22T02:14:38Z
user: login: anandkaranubc
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-01-22T02:16:17Z
user: login: kgryte
state: COMMENTED
author_association: MEMBER
submitted_at: 2025-01-22T03:45:33Z
user: login: anandkaranubc
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-01-22T03:47:55Z
user: login: anandkaranubc
state: COMMENTED
author_association: CONTRIBUTOR
submitted_at: 2025-03-01T19:24:26Z
user: login: Planeshifter
state: COMMENTED
author_association: MEMBER
submitted_at: 2025-03-06T02:06:33Z
user: login: Planeshifter
body: Let's get this one in! 🚀 
state: APPROVED
author_association: MEMBER
submitted_at: 2025-03-06T02:07:51Z

Review comments:
diff_hunk: @@ -44,23 +61,16 @@ var pow = require( '@stdlib/math/base/special/pow' );
 * y = mgf( 2.0 );
 * // returns ~10.205
 */
-function factory( a, b, c ) {
-	var bmc;
-	var bma;
-	var cma;
-
+function factory(a, b, c) {
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/factory.js
user: login: kgryte
body: ```suggestion
function factory( a, b, c ) {
```

stdlib conventions are to use spaces within brackets. Applies here and elsewhere throughout this PR.
created_at: 2025-01-16T02:20:11Z
author_association: MEMBER
diff_hunk: @@ -23,7 +23,24 @@
 var constantFunction = require( '@stdlib/utils/constant-function' );
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
 var exp = require( '@stdlib/math/base/special/exp' );
-var pow = require( '@stdlib/math/base/special/pow' );
+var expm1 = require( '@stdlib/math/base/special/expm1' );
+
+
+// FUNCTIONS //
+
+/**
+* Helper function for repeated computation in the MGF formula.
+*
+* @private
+* @param {number} x - input value
+* @returns {number} evaluated result
+*/
+function phi2( x ) {
+	if ( x === 0 ) {
+		return 1;
+	}
+	return ( 2 * ( expm1( x ) - x ) ) / ( x * x );
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/factory.js
user: login: kgryte
body: ```suggestion
	if ( x === 0.0 ) {
		return 1.0;
	}
	return ( 2.0 * ( expm1( x ) - x ) ) / ( x * x );
```

`phi2` generally expects and returns doubles. As such, similar to C, we add decimals to indicate floating-point values.
created_at: 2025-01-18T11:15:48Z
author_association: MEMBER
diff_hunk: @@ -75,19 +85,24 @@ function factory( a, b, c ) {
 	* // returns <number>
 	*/
 	function mgf( t ) {
-		var ret;
-
 		if ( isnan( t ) ) {
 			return NaN;
 		}
-		if ( t === 0.0 ) {
-			return 1.0;
+		if ( a < c ) {
+			if ( c < b ) {
+				return (
+					exp( c * t ) * (
+						( ( c - a ) * phi2( ( a - c ) * t ) ) +
+						( ( b - c ) * phi2( ( b - c ) * t ) )
+					) / ( b - a )
+				);
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/factory.js
user: login: kgryte
body: ```suggestion
				return exp( c*t ) * ( ( (c-a)*phi2( (a-c)*t ) ) + ( (b-c)*phi2( (b-c)*t ) ) ) / ( b-a ); // eslint-disable-line max-len
```
created_at: 2025-01-18T11:18:08Z
author_association: MEMBER
diff_hunk: @@ -75,19 +85,24 @@ function factory( a, b, c ) {
 	* // returns <number>
 	*/
 	function mgf( t ) {
-		var ret;
-
 		if ( isnan( t ) ) {
 			return NaN;
 		}
-		if ( t === 0.0 ) {
-			return 1.0;
+		if ( a < c ) {
+			if ( c < b ) {
+				return (
+					exp( c * t ) * (
+						( ( c - a ) * phi2( ( a - c ) * t ) ) +
+						( ( b - c ) * phi2( ( b - c ) * t ) )
+					) / ( b - a )
+				);
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/factory.js
user: login: kgryte
body: This is a minor quibble. In short, my preference is to use negative space to visually group related symbols and to draw the eye to the most salient information. This is philosophy underlying our use of negative space throughout the project.
created_at: 2025-01-18T11:20:12Z
author_association: MEMBER
diff_hunk: @@ -22,7 +22,24 @@
 
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
 var exp = require( '@stdlib/math/base/special/exp' );
-var pow = require( '@stdlib/math/base/special/pow' );
+var expm1 = require( '@stdlib/math/base/special/expm1' );
+
+
+// FUNCTIONS //
+
+/**
+* Helper function for repeated computation in the MGF formula.
+*
+* @private
+* @param {number} x - input value
+* @returns {number} evaluated result
+*/
+function phi2( x ) {
+	if ( x === 0 ) {
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/main.js
user: login: kgryte
body: Same comments as other file.
created_at: 2025-01-18T11:20:47Z
author_association: MEMBER
diff_hunk: @@ -88,17 +100,21 @@ function mgf( t, a, b, c ) {
 	) {
 		return NaN;
 	}
-	if ( t === 0.0 ) {
-		return 1.0;
+	if ( a < c ) {
+		if ( c < b ) {
+			return (
+				exp( c * t ) * (
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/main.js
user: login: kgryte
body: Same comments.
created_at: 2025-01-18T11:20:56Z
author_association: MEMBER
diff_hunk: @@ -23,7 +23,24 @@
 var constantFunction = require( '@stdlib/utils/constant-function' );
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
 var exp = require( '@stdlib/math/base/special/exp' );
-var pow = require( '@stdlib/math/base/special/pow' );
+var expm1 = require( '@stdlib/math/base/special/expm1' );
+
+
+// FUNCTIONS //
+
+/**
+* Helper function for repeated computation in the MGF formula.
+*
+* @private
+* @param {number} x - input value
+* @returns {number} evaluated result
+*/
+function phi2( x ) {
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/factory.js
user: login: kgryte
body: For the JS implementation, I suggest moving `phi2` to a separate file and then importing in both this file and in `main.js`. This will reduce code duplication.
created_at: 2025-01-18T11:21:41Z
author_association: MEMBER
diff_hunk: @@ -18,8 +18,22 @@
 
 #include "stdlib/stats/base/dists/triangular/mgf.h"
 #include "stdlib/math/base/assert/is_nan.h"
+#include "stdlib/math/base/special/expm1.h"
 #include "stdlib/math/base/special/exp.h"
-#include "stdlib/math/base/special/pow.h"
+
+/**
+* Helper function for repeated computation in the MGF formula.
+*
+* @private
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/src/main.c
user: login: kgryte
body: ```suggestion
```

`static` already conveys "private".
created_at: 2025-01-18T11:22:11Z
author_association: MEMBER
diff_hunk: @@ -18,8 +18,22 @@
 
 #include "stdlib/stats/base/dists/triangular/mgf.h"
 #include "stdlib/math/base/assert/is_nan.h"
+#include "stdlib/math/base/special/expm1.h"
 #include "stdlib/math/base/special/exp.h"
-#include "stdlib/math/base/special/pow.h"
+
+/**
+* Helper function for repeated computation in the MGF formula.
+*
+* @private
+* @param x   input value
+* @return    evaluated result
+*/
+static double phi2( const double x ) {
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/src/main.c
user: login: kgryte
body: This, however, is fine and does **not** need to be split out into a separate file.
created_at: 2025-01-18T11:22:33Z
author_association: MEMBER
diff_hunk: @@ -224,3 +228,127 @@ tape( 'the created function evaluates the MGF for `x` given a large range `b - a
 	}
 	t.end();
 });
+
+tape( 'the function evaluates the MGF for `x` given the case: a < c < b', function test( t ) {
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.factory.js
user: login: kgryte
body: ```suggestion
tape( 'the created function evaluates the MGF for `x` given the case: a < c < b', function test( t ) {
```

The change applies here and below and ensures consistency with the descriptions above.
created_at: 2025-01-18T11:25:37Z
author_association: MEMBER
diff_hunk: @@ -165,9 +165,123 @@ tape( 'the function evaluates the MGF for `x` given a large range `b - a`', opts
 			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
 		} else {
 			delta = abs( y - expected[ i ] );
+			tol = 4.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
 
-			// FIXME: the tolerance for the C implementation is widely different from the JavaScript implementation, and it is not clear why.
-			tol = 1.0e7 * EPS * abs( expected[ i ] );
+tape( 'the function evaluates the MGF for `x` given the case: a < c < b', opts, function test( t ) {
+	var expected;
+	var delta;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aLessCLessB.expected;
+	x = aLessCLessB.x;
+	a = aLessCLessB.a;
+	b = aLessCLessB.b;
+	c = aLessCLessB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[i], a[i], b[i], c[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 4.0 * EPS * abs( expected[ i ] );
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.native.js
user: login: kgryte
body: While muuuuch closer to JS, it isn't a perfect match. Any idea why?
created_at: 2025-01-18T11:26:52Z
author_association: MEMBER
diff_hunk: @@ -165,9 +165,123 @@ tape( 'the function evaluates the MGF for `x` given a large range `b - a`', opts
 			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
 		} else {
 			delta = abs( y - expected[ i ] );
+			tol = 4.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
 
-			// FIXME: the tolerance for the C implementation is widely different from the JavaScript implementation, and it is not clear why.
-			tol = 1.0e7 * EPS * abs( expected[ i ] );
+tape( 'the function evaluates the MGF for `x` given the case: a < c < b', opts, function test( t ) {
+	var expected;
+	var delta;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aLessCLessB.expected;
+	x = aLessCLessB.x;
+	a = aLessCLessB.a;
+	b = aLessCLessB.b;
+	c = aLessCLessB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[i], a[i], b[i], c[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 4.0 * EPS * abs( expected[ i ] );
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.native.js
user: login: kgryte
body: After re-reading your OP, have you tried recompiling but disabling certain compiler optimizations?
created_at: 2025-01-18T22:22:27Z
author_association: MEMBER
diff_hunk: @@ -165,9 +165,123 @@ tape( 'the function evaluates the MGF for `x` given a large range `b - a`', opts
 			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
 		} else {
 			delta = abs( y - expected[ i ] );
+			tol = 4.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
 
-			// FIXME: the tolerance for the C implementation is widely different from the JavaScript implementation, and it is not clear why.
-			tol = 1.0e7 * EPS * abs( expected[ i ] );
+tape( 'the function evaluates the MGF for `x` given the case: a < c < b', opts, function test( t ) {
+	var expected;
+	var delta;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aLessCLessB.expected;
+	x = aLessCLessB.x;
+	a = aLessCLessB.a;
+	b = aLessCLessB.b;
+	c = aLessCLessB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[i], a[i], b[i], c[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 4.0 * EPS * abs( expected[ i ] );
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.native.js
user: login: anandkaranubc
body: Update: I added `CFLAGS` to the `Makefile` to disable compiler optimizations (`-O0 -g`). This did impact the test results, with only one test failing after adjusting all tolerances from a factor of `4.0` to `1.0`.

I'll bring this up in today's OH to clarify if this is the best approach or if there's an alternative solution. I've also added a question to the OH issue. See you soon!
created_at: 2025-01-21T15:47:39Z
author_association: CONTRIBUTOR
diff_hunk: @@ -165,9 +165,123 @@ tape( 'the function evaluates the MGF for `x` given a large range `b - a`', opts
 			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
 		} else {
 			delta = abs( y - expected[ i ] );
+			tol = 4.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
 
-			// FIXME: the tolerance for the C implementation is widely different from the JavaScript implementation, and it is not clear why.
-			tol = 1.0e7 * EPS * abs( expected[ i ] );
+tape( 'the function evaluates the MGF for `x` given the case: a < c < b', opts, function test( t ) {
+	var expected;
+	var delta;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aLessCLessB.expected;
+	x = aLessCLessB.x;
+	a = aLessCLessB.a;
+	b = aLessCLessB.b;
+	c = aLessCLessB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[i], a[i], b[i], c[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 4.0 * EPS * abs( expected[ i ] );
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.native.js
user: login: anandkaranubc
body: Hi again @kgryte,  

Here's an update on the issue: I realized that I was considering higher tolerances for all cases, but in most cases, a factor of `1.0` works just fine.  

There is, however, one test that fails in this test category:  
https://github.com/stdlib-js/stdlib/blob/0937887902aed132ffe165cc713f053bb28fe369/lib/node_modules/%40stdlib/stats/base/dists/triangular/mgf/test/test.native.js#L88  

I reviewed the discussion in https://github.com/stdlib-js/stdlib/pull/2298, but it seems that using `CFLAGS="-ffp-contract=off"` or `CFLAGS="-ffp-contract=on"` doesn't make a difference in this case.  

I tracked down the first mismatch in the results between the JS and C implementations to the calculation of `expm1`:  
https://github.com/stdlib-js/stdlib/blob/0937887902aed132ffe165cc713f053bb28fe369/lib/node_modules/%40stdlib/stats/base/dists/triangular/mgf/src/main.c#L34  

The C implementation gives:  
`stdlib_base_expm1((a - c) * t)` & `stdlib_base_expm1((b - c) * t)`  
- term1: `-0.75243526099293723`  
- term2: `0.51382856876570271`  

The JS implementation gives:  
`expm1((a - c) * t)` & `expm1((b - c) * t)`  
- term1: `-0.7524352609929372`  
- term2: `0.5138285687657028`  

This doesn't seem to be the result of simple rounding errors like those seen in the other issue. As you mentioned, this is likely due to different implementations or approximations used for these functions. Any thoughts? :)
created_at: 2025-01-22T02:14:38Z
author_association: CONTRIBUTOR
diff_hunk: @@ -165,9 +165,123 @@ tape( 'the function evaluates the MGF for `x` given a large range `b - a`', opts
 			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
 		} else {
 			delta = abs( y - expected[ i ] );
+			tol = 4.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
 
-			// FIXME: the tolerance for the C implementation is widely different from the JavaScript implementation, and it is not clear why.
-			tol = 1.0e7 * EPS * abs( expected[ i ] );
+tape( 'the function evaluates the MGF for `x` given the case: a < c < b', opts, function test( t ) {
+	var expected;
+	var delta;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aLessCLessB.expected;
+	x = aLessCLessB.x;
+	a = aLessCLessB.a;
+	b = aLessCLessB.b;
+	c = aLessCLessB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[i], a[i], b[i], c[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 4.0 * EPS * abs( expected[ i ] );
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.native.js
user: login: anandkaranubc
body: Also, for the one test that fails, a factor of `2.0` instead of `1.0` works. Added a comment in the recent commit.
created_at: 2025-01-22T02:16:17Z
author_association: CONTRIBUTOR
diff_hunk: @@ -165,9 +165,123 @@ tape( 'the function evaluates the MGF for `x` given a large range `b - a`', opts
 			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
 		} else {
 			delta = abs( y - expected[ i ] );
+			tol = 4.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
 
-			// FIXME: the tolerance for the C implementation is widely different from the JavaScript implementation, and it is not clear why.
-			tol = 1.0e7 * EPS * abs( expected[ i ] );
+tape( 'the function evaluates the MGF for `x` given the case: a < c < b', opts, function test( t ) {
+	var expected;
+	var delta;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aLessCLessB.expected;
+	x = aLessCLessB.x;
+	a = aLessCLessB.a;
+	b = aLessCLessB.b;
+	c = aLessCLessB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[i], a[i], b[i], c[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 4.0 * EPS * abs( expected[ i ] );
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.native.js
user: login: kgryte
body: Adjusting to a tolerance of `2.0` should be fine. In the test comment, you can link to this discussion.
created_at: 2025-01-22T03:45:33Z
author_association: MEMBER
diff_hunk: @@ -165,9 +165,123 @@ tape( 'the function evaluates the MGF for `x` given a large range `b - a`', opts
 			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
 		} else {
 			delta = abs( y - expected[ i ] );
+			tol = 4.0 * EPS * abs( expected[ i ] );
+			t.ok( delta <= tol, 'within tolerance. x: '+x[ i ]+'. a: '+a[i]+'. b: '+b[i]+'. c: '+c[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
+		}
+	}
+	t.end();
+});
 
-			// FIXME: the tolerance for the C implementation is widely different from the JavaScript implementation, and it is not clear why.
-			tol = 1.0e7 * EPS * abs( expected[ i ] );
+tape( 'the function evaluates the MGF for `x` given the case: a < c < b', opts, function test( t ) {
+	var expected;
+	var delta;
+	var tol;
+	var x;
+	var a;
+	var b;
+	var c;
+	var y;
+	var i;
+
+	expected = aLessCLessB.expected;
+	x = aLessCLessB.x;
+	a = aLessCLessB.a;
+	b = aLessCLessB.b;
+	c = aLessCLessB.c;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[i], a[i], b[i], c[i] );
+		if ( y === expected[i] ) {
+			t.equal( y, expected[i], 'x: '+x[i]+', a: '+a[i]+', b: '+b[i]+', c: '+c[i]+', y: '+y+', expected: '+expected[i] );
+		} else {
+			delta = abs( y - expected[ i ] );
+			tol = 4.0 * EPS * abs( expected[ i ] );
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.native.js
user: login: anandkaranubc
body: Already done. Hope this looks good:

https://github.com/stdlib-js/stdlib/blob/2d4f12d50db7baa89db55d9f1fb9c20564efb9d4/lib/node_modules/%40stdlib/stats/base/dists/triangular/mgf/test/test.native.js#L111
created_at: 2025-01-22T03:47:55Z
author_association: CONTRIBUTOR
diff_hunk: @@ -104,8 +108,8 @@ tape( 'the function evaluates the MGF for `x` given a small range `b - a`', opts
 		} else {
 			delta = abs( y - expected[ i ] );
 
-			// FIXME: the tolerance for the C implementation is widely different from the JavaScript implementation, and it is not clear why.
-			tol = 1.0e7 * EPS * abs( expected[ i ] );
+			// NOTE: the tolerance here is larger than for the JavaScript implementation due to compiler optimizations which may be performed resulting in result divergence. For discussion, see https://github.com/stdlib-js/stdlib/pull/2298#discussion_r1624765205
+			tol = 2.0 * EPS * abs( expected[ i ] );
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/test/test.native.js
user: login: anandkaranubc
body: I can confirm that if `CFLAGS="-ffp-contract=off"` is used, the tolerance can be set to `1.0` here.
created_at: 2025-03-01T19:24:26Z
author_association: CONTRIBUTOR
diff_hunk: @@ -0,0 +1,42 @@
+/**
+* @license Apache-2.0
+*
+* Copyright (c) 2025 The Stdlib Authors.
+*
+* Licensed under the Apache License, Version 2.0 (the "License");
+* you may not use this file except in compliance with the License.
+* You may obtain a copy of the License at
+*
+*    http://www.apache.org/licenses/LICENSE-2.0
+*
+* Unless required by applicable law or agreed to in writing, software
+* distributed under the License is distributed on an "AS IS" BASIS,
+* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+* See the License for the specific language governing permissions and
+* limitations under the License.
+*/
+
+'use strict';
+
+// MODULES //
+
+var expm1 = require( '@stdlib/math/base/special/expm1' );
+
+
+// MAIN //
+
+/**
+* Helper function for repeated computation in the MGF formula.
+*
+* @private
+* @param {number} x - input value
+* @returns {number} evaluated result
+*/
+function phi2( x ) {
+	if ( x === 0.0 ) {
+		return 1.0;
+	}
+	return ( 2.0 * ( expm1( x ) - x ) ) / ( x*x );
+}
+
+module.exports = phi2;
path: lib/node_modules/@stdlib/stats/base/dists/triangular/mgf/lib/phi2.js
user: login: Planeshifter
body: ```suggestion
}


// EXPORTS //

module.exports = phi2;
```
created_at: 2025-03-06T02:06:32Z
author_association: MEMBER

PR comments:
user: login: stdlib-bot
created_at: 2025-01-15T18:22:29Z
body: ## Coverage Report

| Package | Statements | Branches | Functions | Lines |
| --------- | ------------ | ---------- | ----------- | ----- |
| <a href= https://coverage.stdlib.io/pr-4768/stats/base/dists/triangular/mgf/index.html >stats/base/dists/triangular/mgf</a> | $\color{green}380/380$ <br> $\color{green}+100.00\\%$ | $\color{green}38/38$ <br> $\color{green}+100.00\\%$ | $\color{green}5/5$ <br> $\color{green}+100.00\\%$ | $\color{green}380/380$ <br> $\color{green}+100.00\\%$ |

The above coverage report was generated for the [changes in this PR](https://github.com/stdlib-js/stdlib/compare/6a802bc9ee090d54bc05b665bb10e4365ba7d917...0e26b69d361fb74072ab057d79568ba240f08361).
author_association: CONTRIBUTOR
user: login: anandkaranubc
created_at: 2025-01-15T21:26:48Z
body: 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](https://github.com/JuliaStats/Distributions.jl/blob/957f0c0f79307edefb9afa75c682ef18b35dd982/src/univariate/continuous/triangular.jl#L156-L169), or should I just manually add additional test cases in a new tape function?
author_association: CONTRIBUTOR
user: login: kgryte
created_at: 2025-01-16T02:19:12Z
body: @anandkaranubc Would you mind updating and rerunning the runner?
author_association: MEMBER
user: login: anandkaranubc
created_at: 2025-01-18T07:50:15Z
body: Can I get some feedback on this too please, @kgryte? Thanks!
author_association: CONTRIBUTOR
user: login: anandkaranubc
created_at: 2025-01-22T14:54:21Z
body: Should the `README.md` also be changed with the updated formula for mgf calculation?

```math
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}
```

```math
\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.
author_association: CONTRIBUTOR
user: login: kgryte
created_at: 2025-01-22T18:56:25Z
body: > 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.
author_association: MEMBER
user: login: anandkaranubc
created_at: 2025-03-01T19:22:26Z
body: Once https://github.com/stdlib-js/stdlib/pull/5638 gets merged, the code coverage will be 100%.

Edit: That PR has been merged and the code coverage is 100% now!

author_association: CONTRIBUTOR
user: login: anandkaranubc
created_at: 2025-03-01T23:15:30Z
body: /stdlib merge
author_association: CONTRIBUTOR
user: login: stdlib-bot
created_at: 2025-03-06T02:09:54Z
body: ### PR Commit Message

```text
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 <pburckhardt@outlook.com>
Co-authored-by: stdlib-bot <noreply@stdlib.io>
Reviewed-by: Philipp Burckhardt <pburckhardt@outlook.com>
Reviewed-by: Athan Reines <kgryte@gmail.com>
Signed-off-by: Philipp Burckhardt <pburckhardt@outlook.com>
```

*Please review the above commit message and make any necessary adjustments.*
author_association: CONTRIBUTOR