From 23c6125747c0d706fc52cf155e1278aaee7e9484 Mon Sep 17 00:00:00 2001
From: 0PrashantYadav0 <devprashantkyadav@gmail.com>
Date: Sat, 17 Oct 2026 21:37:31 +0530
Subject: [PATCH 1/2] fix: correct MGF in
 `stats/base/dists/negative-binomial/mgf`

---
type: pre_commit_static_analysis_report
description: Results of running static analysis checks when committing changes.
report:
  - task: lint_filenames
    status: passed
  - task: lint_editorconfig
    status: passed
  - task: lint_markdown_pkg_readmes
    status: passed
  - task: lint_markdown_docs
    status: na
  - task: lint_markdown
    status: na
  - task: lint_package_json
    status: na
  - task: lint_repl_help
    status: passed
  - task: lint_javascript_src
    status: passed
  - task: lint_javascript_cli
    status: na
  - task: lint_javascript_examples
    status: na
  - task: lint_javascript_tests
    status: passed
  - task: lint_javascript_benchmarks
    status: passed
  - task: lint_python
    status: na
  - task: lint_r
    status: na
  - task: lint_c_src
    status: missing_dependencies
  - task: lint_c_examples
    status: na
  - task: lint_c_benchmarks
    status: na
  - task: lint_c_tests_fixtures
    status: na
  - task: lint_shell
    status: na
  - task: lint_typescript_declarations
    status: passed
  - task: lint_typescript_tests
    status: passed
  - task: lint_license_headers
    status: passed
---
---
 .../dists/negative-binomial/ctor/README.md    |  2 +-
 .../negative-binomial/ctor/docs/repl.txt      |  2 +-
 .../dists/negative-binomial/ctor/lib/main.js  |  2 +-
 .../dists/negative-binomial/mgf/README.md     | 30 ++++----
 .../mgf/benchmark/benchmark.js                |  2 +-
 .../mgf/benchmark/benchmark.native.js         |  2 +-
 .../dists/negative-binomial/mgf/docs/repl.txt | 14 ++--
 .../mgf/docs/types/index.d.ts                 | 26 +++----
 .../negative-binomial/mgf/lib/factory.js      | 10 +--
 .../dists/negative-binomial/mgf/lib/index.js  | 14 ++--
 .../dists/negative-binomial/mgf/lib/main.js   | 18 ++---
 .../dists/negative-binomial/mgf/lib/native.js | 10 +--
 .../dists/negative-binomial/mgf/manifest.json | 12 +--
 .../dists/negative-binomial/mgf/src/main.c    | 10 +--
 .../mgf/test/fixtures/julia/REQUIRE           |  3 +
 .../mgf/test/fixtures/julia/data.json         |  1 +
 .../mgf/test/fixtures/julia/runner.jl         | 77 +++++++++++++++++++
 .../mgf/test/test.factory.js                  | 34 ++++----
 .../negative-binomial/mgf/test/test.mgf.js    | 32 ++++----
 .../negative-binomial/mgf/test/test.native.js | 31 ++++----
 20 files changed, 213 insertions(+), 119 deletions(-)
 create mode 100644 lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/fixtures/julia/REQUIRE
 create mode 100644 lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/fixtures/julia/data.json
 create mode 100644 lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/fixtures/julia/runner.jl

diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/ctor/README.md b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/ctor/README.md
index 39cceeb992d3..c6688e73db48 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/ctor/README.md
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/ctor/README.md
@@ -204,7 +204,7 @@ Evaluates the [moment-generating function][mgf] (MGF).
 var nbinomial = new NegativeBinomial( 4.0, 0.2 );
 
 var y = nbinomial.mgf( 0.1 );
-// returns ~1.66
+// returns ~8.878
 ```
 
 #### NegativeBinomial.prototype.pmf( x )
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/ctor/docs/repl.txt b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/ctor/docs/repl.txt
index 77b059c22331..3180847f7bad 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/ctor/docs/repl.txt
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/ctor/docs/repl.txt
@@ -81,7 +81,7 @@
     > nbinomial.logpmf( 3.0 )
     ~-2.837
     > nbinomial.mgf( 0.2 )
-    ~36.675
+    ~7.405
     > nbinomial.pmf( 3.0 )
     ~0.059
     > nbinomial.quantile( 0.8 )
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/ctor/lib/main.js b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/ctor/lib/main.js
index 021ebde1818c..d06fd49b3c36 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/ctor/lib/main.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/ctor/lib/main.js
@@ -329,7 +329,7 @@ setReadOnly( NegativeBinomial.prototype, 'logpmf', negativeBinomialLogPMF );
 * var nbinomial = new NegativeBinomial( 4.0, 0.2 );
 *
 * var v = nbinomial.mgf( 0.1 );
-* // returns ~1.66
+* // returns ~8.878
 */
 setReadOnly( NegativeBinomial.prototype, 'mgf', negativeBinomialMGF );
 
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/README.md b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/README.md
index 13ccf548f2bd..37b2f9979fe5 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/README.md
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/README.md
@@ -28,20 +28,20 @@ limitations under the License.
 
 The [moment-generating function][mgf] for a [negative binomial][negative-binomial-distribution] random variable is
 
-<!-- <equation class="equation" label="eq:negative_binomial_mgf_function" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \biggl(\frac{\left( 1- p \right) e^t }{1 - p e^t}\biggr)^{\!r} \text{ for }t<-\log p" alt="Moment-generating function (MGF) for a negative binomial distribution."> -->
+<!-- <equation class="equation" label="eq:negative_binomial_mgf_function" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \biggl(\frac{p}{1 - \left( 1-p \right) e^t}\biggr)^{\!r} \text{ for }t<-\log(1-p)" alt="Moment-generating function (MGF) for a negative binomial distribution."> -->
 
 ```math
-M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \biggl(\frac{\left( 1- p \right) e^t }{1 - p e^t}\biggr)^{\!r} \text{ for }t<-\log p
+M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \biggl(\frac{p}{1 - \left( 1-p \right) e^t}\biggr)^{\!r} \text{ for }t<-\log(1-p)
 ```
 
-<!-- <div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \biggl(\frac{\left( 1- p \right) e^t }{1 - p e^t}\biggr)^{\!r} \text{ for }t&lt;-\log p" data-equation="eq:negative_binomial_mgf_function">
+<!-- <div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \biggl(\frac{p}{1 - \left( 1-p \right) e^t}\biggr)^{\!r} \text{ for }t&lt;-\log(1-p)" data-equation="eq:negative_binomial_mgf_function">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/docs/img/equation_negative_binomial_mgf_function.svg" alt="Moment-generating function (MGF) for a negative binomial distribution.">
     <br>
 </div> -->
 
 <!-- </equation> -->
 
-where `r > 0` is the number of failures until the experiment is stopped and `0 <= p <= 1` is the success probability.
+where `r > 0` is the number of successes until the experiment is stopped and `0 <= p <= 1` is the success probability.
 
 </section>
 
@@ -63,26 +63,26 @@ Evaluates the [moment-generating function][mgf] for a [negative binomial][negati
 
 ```javascript
 var y = mgf( 0.05, 20.0, 0.8 );
-// returns ~267.839
+// returns ~1.294
 
-y = mgf( 0.1, 20.0, 0.1 );
-// returns ~9.347
+y = mgf( 0.1, 20.0, 0.9 );
+// returns ~1.265
 ```
 
 While `r` can be interpreted as the number of successes until the experiment is stopped, the [negative binomial][negative-binomial-distribution] distribution is also defined for non-integers `r`. In this case, `r` denotes shape parameter of the [gamma mixing distribution][negative-binomial-mixture-representation].
 
 ```javascript
 var y = mgf( 0.1, 15.5, 0.5 );
-// returns ~26.375
+// returns ~5.598
 
-y = mgf( 0.5, 7.4, 0.4 );
-// returns ~2675.677
+y = mgf( 0.5, 7.4, 0.8 );
+// returns ~3.704
 ```
 
-If `t >= -ln( p )`, the function returns `NaN`.
+If `t >= -ln( 1-p )`, the function returns `NaN`.
 
 ```javascript
-var y = mgf( 0.7, 15.5, 0.5 ); // -ln( p ) = ~0.693
+var y = mgf( 0.7, 15.5, 0.5 ); // -ln( 1-p ) = ~0.693
 // returns NaN
 ```
 
@@ -126,10 +126,10 @@ Returns a function for evaluating the [moment-generating function][mgf] of a [ne
 ```javascript
 var myMGF = mgf.factory( 4.3, 0.4 );
 var y = myMGF( 0.2 );
-// returns ~4.696
+// returns ~5.672
 
 y = myMGF( 0.4 );
-// returns ~30.83
+// returns ~315.814
 ```
 
 </section>
@@ -203,7 +203,7 @@ Evaluates the [moment-generating function][mgf] for a [negative binomial][negati
 
 ```c
 double out = stdlib_base_dists_negative_binomial_mgf( 0.05, 20.0, 0.8 );
-// returns ~267.839
+// returns ~1.294
 ```
 
 The function accepts the following arguments:
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/benchmark/benchmark.js b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/benchmark/benchmark.js
index 0a1a5c490832..af152a63d364 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/benchmark/benchmark.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/benchmark/benchmark.js
@@ -82,7 +82,7 @@ bench( format( '%s::factory', pkg ), function benchmark( b ) {
 	len = 100;
 	t = new Float64Array( len );
 	for ( i = 0; i < len; i++ ) {
-		t[ i ] = uniform( 0.0, -ln( p ) );
+		t[ i ] = uniform( 0.0, -ln( 1.0 - p ) );
 	}
 
 	b.tic();
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/benchmark/benchmark.native.js b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/benchmark/benchmark.native.js
index 2762517f62e5..021557b7e170 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/benchmark/benchmark.native.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/benchmark/benchmark.native.js
@@ -57,7 +57,7 @@ bench( format( '%s::native', pkg ), opts, function benchmark( b ) {
 	p = new Float64Array( len );
 	for ( i = 0; i < len; i++ ) {
 		p[ i ] = uniform( EPS, 1.0 );
-		t[ i ] = uniform( 0.0, -ln( p[ i ] ) );
+		t[ i ] = uniform( 0.0, -ln( 1.0 - p[ i ] ) );
 		r[ i ] = discreteUniform( 1, 100 );
 	}
 
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/docs/repl.txt b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/docs/repl.txt
index 38ee6a8e4c88..3d3c47f339bf 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/docs/repl.txt
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/docs/repl.txt
@@ -31,11 +31,11 @@
     Examples
     --------
     > var y = {{alias}}( 0.05, 20.0, 0.8 )
-    ~267.839
-    > y = {{alias}}( 0.1, 20.0, 0.1 )
-    ~9.347
-    > y = {{alias}}( 0.5, 10.0, 0.4 )
-    ~42822.023
+    ~1.294
+    > y = {{alias}}( 0.1, 20.0, 0.9 )
+    ~1.265
+    > y = {{alias}}( 0.5, 10.0, 0.8 )
+    ~5.868
 
     > y = {{alias}}( 0.1, 0.0, 0.5 )
     NaN
@@ -77,9 +77,9 @@
     --------
     > var myMGF = {{alias}}.factory( 4.3, 0.4 );
     > var y = myMGF( 0.2 )
-    ~4.696
+    ~5.672
     > y = myMGF( 0.4 )
-    ~30.83
+    ~315.814
 
     See Also
     --------
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/docs/types/index.d.ts b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/docs/types/index.d.ts
index 6d394632e0a9..a96edb18f593 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/docs/types/index.d.ts
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/docs/types/index.d.ts
@@ -45,15 +45,15 @@ interface MGF {
 	*
 	* @example
 	* var y = mgf( 0.05, 20.0, 0.8 );
-	* // returns ~267.839
+	* // returns ~1.294
 	*
 	* @example
-	* var y = mgf( 0.1, 20.0, 0.1 );
-	* // returns ~9.347
+	* var y = mgf( 0.1, 20.0, 0.9 );
+	* // returns ~1.265
 	*
 	* @example
-	* var y = mgf( 0.5, 10.0, 0.4 );
-	* // returns ~42822.023
+	* var y = mgf( 0.5, 10.0, 0.8 );
+	* // returns ~5.868
 	*
 	* @example
 	* var y = mgf( 0.1, 0.0, 0.5 );
@@ -95,7 +95,7 @@ interface MGF {
 	* @example
 	* var myMGF = mgf.factory( 4.3, 0.4 );
 	* var y = myMGF( 0.2 );
-	* // returns ~4.696
+	* // returns ~5.672
 	*/
 	factory( r: number, p: number ): Unary;
 }
@@ -110,20 +110,20 @@ interface MGF {
 *
 * @example
 * var y = mgf( 0.05, 20.0, 0.8 );
-* // returns ~267.839
+* // returns ~1.294
 *
-* y = mgf( 0.1, 20.0, 0.1 );
-* // returns ~9.347
+* y = mgf( 0.1, 20.0, 0.9 );
+* // returns ~1.265
 *
-* y = mgf( 0.5, 10.0, 0.4 );
-* // returns ~42822.023
+* y = mgf( 0.5, 10.0, 0.8 );
+* // returns ~5.868
 *
 * var myMGF = mgf.factory( 4.3, 0.4 );
 * y = myMGF( 0.2 );
-* // returns ~4.696
+* // returns ~5.672
 *
 * y = myMGF( 0.4 );
-* // returns ~30.83
+* // returns ~315.814
 */
 declare var mgf: MGF;
 
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/factory.js b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/factory.js
index cb9e5c465b9d..523c39217fa1 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/factory.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/factory.js
@@ -22,9 +22,9 @@
 
 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 expm1 = require( '@stdlib/math/base/special/expm1' );
 var pow = require( '@stdlib/math/base/special/pow' );
-var ln = require( '@stdlib/math/base/special/ln' );
+var log1p = require( '@stdlib/math/base/special/log1p' );
 
 
 // MAIN //
@@ -39,7 +39,7 @@ var ln = require( '@stdlib/math/base/special/ln' );
 * @example
 * var mgf = factory( 4.3, 0.4 );
 * var y = mgf( 0.2 );
-* // returns ~4.696
+* // returns ~5.672
 */
 function factory( r, p ) {
 	if (
@@ -65,10 +65,10 @@ function factory( r, p ) {
 	* // returns <number>
 	*/
 	function mgf( t ) {
-		if ( isnan( t ) || t >= -ln( p ) ) {
+		if ( isnan( t ) || t >= -log1p( -p ) ) {
 			return NaN;
 		}
-		return pow( ( (1.0 - p) * exp( t ) ) / ( 1.0 - (p * exp( t )) ), r );
+		return pow( p / ( p - ((1.0 - p) * expm1( t )) ), r );
 	}
 }
 
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/index.js b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/index.js
index b269b9a13f39..e44d39430afe 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/index.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/index.js
@@ -27,20 +27,20 @@
 * var mgf = require( '@stdlib/stats/base/dists/negative-binomial/mgf' );
 *
 * var y = mgf( 0.05, 20.0, 0.8 );
-* // returns ~267.839
+* // returns ~1.294
 *
-* y = mgf( 0.1, 20.0, 0.1 );
-* // returns ~9.347
+* y = mgf( 0.1, 20.0, 0.9 );
+* // returns ~1.265
 *
-* y = mgf( 0.5, 10.0, 0.4 );
-* // returns ~42822.023
+* y = mgf( 0.5, 10.0, 0.8 );
+* // returns ~5.868
 *
 * var myMGF = mgf.factory( 4.3, 0.4 );
 * y = myMGF( 0.2 );
-* // returns ~4.696
+* // returns ~5.672
 *
 * y = myMGF( 0.4 );
-* // returns ~30.83
+* // returns ~315.814
 */
 
 // MODULES //
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/main.js b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/main.js
index d4e2ea5c5206..a36bb5e723bb 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/main.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/main.js
@@ -21,9 +21,9 @@
 // MODULES //
 
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
-var exp = require( '@stdlib/math/base/special/exp' );
+var expm1 = require( '@stdlib/math/base/special/expm1' );
 var pow = require( '@stdlib/math/base/special/pow' );
-var ln = require( '@stdlib/math/base/special/ln' );
+var log1p = require( '@stdlib/math/base/special/log1p' );
 
 
 // MAIN //
@@ -38,15 +38,15 @@ var ln = require( '@stdlib/math/base/special/ln' );
 *
 * @example
 * var y = mgf( 0.05, 20.0, 0.8 );
-* // returns ~267.839
+* // returns ~1.294
 *
 * @example
-* var y = mgf( 0.1, 20.0, 0.1 );
-* // returns ~9.347
+* var y = mgf( 0.1, 20.0, 0.9 );
+* // returns ~1.265
 *
 * @example
-* var y = mgf( 0.5, 10.0, 0.4 );
-* // returns ~42822.023
+* var y = mgf( 0.5, 10.0, 0.8 );
+* // returns ~5.868
 *
 * @example
 * var y = mgf( 0.1, 0.0, 0.5 );
@@ -84,11 +84,11 @@ function mgf( t, r, p ) {
 		r <= 0.0 ||
 		p < 0.0 ||
 		p > 1.0 ||
-		t >= -ln( p )
+		t >= -log1p( -p )
 	) {
 		return NaN;
 	}
-	return pow( ( (1.0 - p) * exp( t ) ) / ( 1.0 - (p * exp( t )) ), r );
+	return pow( p / ( p - ((1.0 - p) * expm1( t )) ), r );
 }
 
 
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/native.js b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/native.js
index 0e4bef593ebf..08a069f46f09 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/native.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/native.js
@@ -36,15 +36,15 @@ var addon = require( './../src/addon.node' );
 *
 * @example
 * var y = mgf( 0.05, 20.0, 0.8 );
-* // returns ~267.839
+* // returns ~1.294
 *
 * @example
-* var y = mgf( 0.1, 20.0, 0.1 );
-* // returns ~9.347
+* var y = mgf( 0.1, 20.0, 0.9 );
+* // returns ~1.265
 *
 * @example
-* var y = mgf( 0.5, 10.0, 0.4 );
-* // returns ~42822.023
+* var y = mgf( 0.5, 10.0, 0.8 );
+* // returns ~5.868
 *
 * @example
 * var y = mgf( 0.1, 0.0, 0.5 );
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/manifest.json b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/manifest.json
index c82926d08d89..5d9343c532b6 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/manifest.json
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/manifest.json
@@ -40,9 +40,9 @@
       "dependencies": [
         "@stdlib/math/base/napi/ternary",
         "@stdlib/math/base/assert/is-nan",
-        "@stdlib/math/base/special/exp",
+        "@stdlib/math/base/special/expm1",
         "@stdlib/math/base/special/pow",
-        "@stdlib/math/base/special/ln"
+        "@stdlib/math/base/special/log1p"
       ]
     },
     {
@@ -58,9 +58,9 @@
       "libpath": [],
       "dependencies": [
         "@stdlib/math/base/assert/is-nan",
-        "@stdlib/math/base/special/exp",
+        "@stdlib/math/base/special/expm1",
         "@stdlib/math/base/special/pow",
-        "@stdlib/math/base/special/ln"
+        "@stdlib/math/base/special/log1p"
       ]
     },
     {
@@ -76,9 +76,9 @@
       "libpath": [],
       "dependencies": [
         "@stdlib/math/base/assert/is-nan",
-        "@stdlib/math/base/special/exp",
+        "@stdlib/math/base/special/expm1",
         "@stdlib/math/base/special/pow",
-        "@stdlib/math/base/special/ln"
+        "@stdlib/math/base/special/log1p"
       ]
     }
   ]
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/src/main.c b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/src/main.c
index 694daeae5ec2..9f02a4bb9630 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/src/main.c
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/src/main.c
@@ -18,8 +18,8 @@
 
 #include "stdlib/stats/base/dists/negative-binomial/mgf.h"
 #include "stdlib/math/base/assert/is_nan.h"
-#include "stdlib/math/base/special/exp.h"
-#include "stdlib/math/base/special/ln.h"
+#include "stdlib/math/base/special/expm1.h"
+#include "stdlib/math/base/special/log1p.h"
 #include "stdlib/math/base/special/pow.h"
 
 /**
@@ -32,7 +32,7 @@
 *
 * @example
 * double y = stdlib_base_dists_negative_binomial_mgf( 0.05, 20.0, 0.8 );
-* // returns ~267.839
+* // returns ~1.294
 */
 double stdlib_base_dists_negative_binomial_mgf( const double t, const double r, const double p ) {
 	if (
@@ -42,9 +42,9 @@ double stdlib_base_dists_negative_binomial_mgf( const double t, const double r,
 		r <= 0.0 ||
 		p < 0.0 ||
 		p > 1.0 ||
-		t >= -stdlib_base_ln( p )
+		t >= -stdlib_base_log1p( -p )
 	) {
 		return 0.0 / 0.0; // NaN
 	}
-	return stdlib_base_pow( ( ( 1.0 - p ) * stdlib_base_exp( t ) ) / ( 1.0 - ( p * stdlib_base_exp( t ) ) ), r );
+	return stdlib_base_pow( p / ( p - ( ( 1.0 - p ) * stdlib_base_expm1( t ) ) ), r );
 }
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/fixtures/julia/REQUIRE b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/fixtures/julia/REQUIRE
new file mode 100644
index 000000000000..67010df15256
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/fixtures/julia/REQUIRE
@@ -0,0 +1,3 @@
+Distributions 0.25.131
+julia 1.13
+JSON 1.11
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/fixtures/julia/data.json b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/fixtures/julia/data.json
new file mode 100644
index 000000000000..f99f12de271e
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/fixtures/julia/data.json
@@ -0,0 +1 @@
+{"expected":[0.06890934575151222,0.4941781295922668,1.3394958557710335,23.18397998816791,0.02815127079737662,0.6377452044732435,0.1994435451594667,0.08242985085845353,8593.050182097037,0.5977310596257505,0.047435674266251254,0.07799322574180961,2.164230910919319,0.671960268093534,1.4632378544151379,0.513040457470329,0.19934759230543114,0.7094991946800381,0.3437455987255384,0.39202981062907216,0.5955934598841374,0.10379048609479638,1.2260000166728937,8.493876064162695,155.19281402767746,0.009669463763447312,0.6312322344800922,1.4953552477019647,1.2016186528681267,5.896527788705739,26.370143305977837,0.4944117135503037,1.7138843853915295,0.882195686081171,0.014426838921468855,0.2240589180118839,0.9325623930146314,0.056663812425535955,0.9531071245998249,0.13270001742168358,0.025192828575167536,700.0579321058585,1.1187140493445304,5.968794911664068,0.09599068832489559,41.27926330575226,0.04713882419423173,0.522048398461613,1.9642249628591746,0.03485297089020545,2514.2048514364806,33.145715614015664,1.813007363376206e-5,2.3940311641282226,9646.062393630815,0.36534514286258984,0.7700328268920564,2.109286783555107,6.109139374063561,0.06769308570652909,2.1575714099279533,3.312523669619963,0.8618679737850661,0.45095971031013304,0.012871198977444558,0.013037916481886914,0.2468267872679239,72.9940959133736,1.814219099153404,0.8314751397098924,0.004440613422671804,0.6581043042322426,0.00022615116625839492,0.0028410107643892005,0.023395818841026993,0.3898176301061028,0.3020906745626764,0.14165650068035762,0.0943486227015525,0.3822255014397336,0.4611654392758874,0.0200683774174844,0.08136621484067783,0.13682212224973406,0.3534460279639,0.0627096037570452,0.6516791816639018,0.6109430344348472,11.356341930462076,1.4370780056386079,0.005961554553054288,7.482410902146766,0.38376831528450284,4.72192150403054,5.909099223777797,0.22186601075743187,2.2669809710256156e-5,0.02129019435151617,0.3465431875250685,0.15600459602050404,0.38854429811956476,0.835029973263071,103.14801548960857,0.18287136338524354,6.369509509690719,0.1142326997385906,0.12628183448524125,0.020514942087673144,0.9150743181010346,8.299153979651965,0.011171343511078931,0.8841168671039246,0.4750067306283507,14.364345846391656,0.6705249045184359,0.5682226158416114,1.387530527611275,0.003417124509606575,0.932699104937332,0.03728356368492467,23.44749979279943,0.5204825508892417,2.8061995134117974,1.2438897649092544,0.6341039667119958,1.1884453527432308,0.08322741874780688,7.885976138038255,0.0024352377991217096,0.863521381740556,4.387871591814542,0.9758760630444892,0.8465497045936348,0.08544303563733652,29.807453466320553,0.4471206425621383,0.6724146625969892,0.1345053637076971,0.03616757748628392,11059.71802147351,1.1379530284418353,1.867283677712406,2.0338159152929776,1.6754493221128792,0.0013582497516695918,0.22226638424174866,0.013327224469361857,0.15163575871128857,0.4856245690237708,1.1087742060179908,0.018003793587560143,6018.364803023151,0.9512172175542476,192.58220148013302,614.1021817357722,3389.067979973575,0.645972491934482,0.008827678163834818,2.111765140329444,0.40611654648214407,1.0833777728584055,10.321353642279766,12.47152605358008,0.09566936575654285,102.04115558275899,105.79988443387073,158.9722737851258,5.219258909221754,18.73088377133263,0.27203310475904363,5.77522163894722,0.8502588460159756,0.22255404512225793,2.0031338810663417,0.08545559719481309,2.545011963554441,271.1809767944452,1.9813028957058954,0.00038103710712917167,0.6784594826944097,1.230987664378313,0.6638644393800703,8.798266070313097,0.09550114182053021,0.31369649200275296,6.693548096976819,3509.503089440216,5.244553160549046,42.1005834814392,0.00016717419552957712,1.1925674994816424,0.9138402989079608,1.1185257937072632,18.46303170735112,0.0059352080628391184,1.415359228660431,0.008132594358151264,0.5233533779617635,1.1787263735878557,0.00044556538417746196,0.9638463459212073,0.0017993220869667835,15.581161563485727,18.9531491109302,0.036586963470411585,1.826756912817175,0.0006819391373000953,2.5138472890371664,1.4879144465911092,0.045490076912439587,34.205885386290845,1.691902607765415,15.087044538179377,0.11099469334770765,3.8800014761898884,0.0007186316871694612,0.028482554099345418,0.6017099643835627,0.15195434884893147,0.2128180128979773,0.4155005488993551,0.018581408276697017,0.04055476987603633,0.0008499414010618114,0.904411132209085,228.84541435587863,1.1586793729200053,10.943365861682397,0.02293798749733916,0.0005076354005945967,1.0202278098593724,0.08602077371889402,1.0709142657924786,45.48811431666724,0.6913300835003605,1.4889539745916018,2.0036170126631165,0.00372200147103615,0.3037270366269534,0.6006777482398109,0.7881613382658348,1.1223714261705269,0.1844820171444708,1.507816198045954,1.7265491742041077,0.4144517406765377,0.002569094632671344,0.026174986025088186,441.54521367520397,1.45980001791579,0.07817394815673655,1.2356932767215663,42.82074640566988,0.10012761889882953,5.607642253260505,0.010822322943944128,10.169115987167283,0.2715947340006616,0.00503865541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diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/fixtures/julia/runner.jl b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/fixtures/julia/runner.jl
new file mode 100644
index 000000000000..2d4a861afe26
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/fixtures/julia/runner.jl
@@ -0,0 +1,77 @@
+#!/usr/bin/env julia
+#
+# @license Apache-2.0
+#
+# Copyright (c) 2026 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.
+
+import Distributions: mgf, NegativeBinomial
+import JSON
+
+"""
+	gen( t, r, p, name )
+
+Generate fixture data and write to file.
+
+# Arguments
+
+* `t`: input value
+* `r`: number of successes until experiment is stopped
+* `p`: success probability
+* `name::AbstractString`: output filename
+
+# Examples
+
+``` julia
+julia> p = rand( 1000 );
+julia> t = ( rand( 1000 ) .- 1.0 ) .* -log1p.( -p );
+julia> r = ( rand( 1000 ) .* 20 ) .+ 0.5;
+julia> gen( t, r, p, "data.json" );
+```
+"""
+function gen( t, r, p, name )
+	z = Array{Float64}( undef, length(t) );
+	for i in eachindex(t)
+		z[ i ] = mgf( NegativeBinomial( r[i], p[i] ), t[i] );
+	end
+
+	# Store data to be written to file as a collection:
+	data = Dict([
+		("t", t),
+		("r", r),
+		("p", p),
+		("expected", z)
+	]);
+
+	# Based on the script directory, create an output filepath:
+	filepath = joinpath( dir, name );
+
+	# Write the data to the output filepath as JSON:
+	outfile = open( filepath, "w" );
+	write( outfile, JSON.json(data) );
+	write( outfile, "\n" );
+	close( outfile );
+end
+
+# Get the filename:
+file = @__FILE__;
+
+# Extract the directory in which this file resides:
+dir = dirname( file );
+
+# Generate fixtures (`t` must be less than `-ln(1-p)`):
+p = rand( 1000 );
+t = ( ( rand( 1000 ) .* 1.5 ) .- 1.0 ) .* -log1p.( -p );
+r = ( rand( 1000 ) .* 20 ) .+ 0.5;
+gen( t, r, p, "data.json" );
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/test.factory.js b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/test.factory.js
index 463733b98b4b..18625c35ccdc 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/test.factory.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/test.factory.js
@@ -21,15 +21,18 @@
 // MODULES //
 
 var tape = require( 'tape' );
-var isNumber = require( '@stdlib/assert/is-number' );
+var isAlmostSameValue = require( '@stdlib/assert/is-almost-same-value' );
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
-var randu = require( '@stdlib/random/base/randu' );
-var ln = require( '@stdlib/math/base/special/ln' );
 var PINF = require( '@stdlib/constants/float64/pinf' );
 var NINF = require( '@stdlib/constants/float64/ninf' );
 var factory = require( './../lib/factory.js' );
 
 
+// FIXTURES //
+
+var data = require( './fixtures/julia/data.json' );
+
+
 // TESTS //
 
 tape( 'main export is a function', function test( t ) {
@@ -121,36 +124,37 @@ tape( 'if provided a success probability `p` outside `[0,1]`, the created functi
 	t.end();
 });
 
-tape( 'the created function returns `NaN` if provided `t >= -ln( p )`', function test( t ) {
+tape( 'the created function returns `NaN` if provided `t >= -ln( 1-p )`', function test( t ) {
 	var mgf;
 	var y;
 
-	mgf = factory( 10, 0.5 ); // -ln( p ) = ~0.693
+	mgf = factory( 10, 0.5 ); // -ln( 1-p ) = ~0.693
 	y = mgf( 0.7 );
 	t.strictEqual( isnan( y ), true, 'returns expected value' );
 
-	mgf = factory( 10, 0.2 ); // -ln( p ) = ~1.609
+	mgf = factory( 10, 0.2 ); // -ln( 1-p ) = ~0.223
 	y = mgf( 1.7 );
 	t.strictEqual( isnan( y ), true, 'returns expected value' );
 	t.end();
 });
 
 tape( 'the created function evaluates the mgf', function test( t ) {
+	var expected;
 	var mgf;
 	var r;
 	var p;
-	var i;
 	var x;
 	var y;
+	var i;
 
-	// TODO: Add test fixtures (function is not available in R and Julia)
-	r = 10;
-	p = 0.5;
-	mgf = factory( r, p );
-	for ( i = 0; i < 100; i++ ) {
-		x = ln( p ) - randu();
-		y = mgf( x );
-		t.strictEqual( !isnan( y ) && isNumber( y ), true, 'returns expected value' );
+	expected = data.expected;
+	x = data.t;
+	r = data.r;
+	p = data.p;
+	for ( i = 0; i < x.length; i++ ) {
+		mgf = factory( r[ i ], p[ i ] );
+		y = mgf( x[ i ] );
+		t.strictEqual( isAlmostSameValue( y, expected[ i ], 40 ), true, 'returns expected value' );
 	}
 	t.end();
 });
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/test.mgf.js b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/test.mgf.js
index d70bf3beadd3..d6b22a6cfe61 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/test.mgf.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/test.mgf.js
@@ -21,15 +21,18 @@
 // MODULES //
 
 var tape = require( 'tape' );
-var isNumber = require( '@stdlib/assert/is-number' );
+var isAlmostSameValue = require( '@stdlib/assert/is-almost-same-value' );
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
-var randu = require( '@stdlib/random/base/randu' );
-var ln = require( '@stdlib/math/base/special/ln' );
 var PINF = require( '@stdlib/constants/float64/pinf' );
 var NINF = require( '@stdlib/constants/float64/ninf' );
 var mgf = require( './../lib' );
 
 
+// FIXTURES //
+
+var data = require( './fixtures/julia/data.json' );
+
+
 // TESTS //
 
 tape( 'main export is a function', function test( t ) {
@@ -84,28 +87,29 @@ tape( 'if provided a success probability `p` outside of `[0,1]`, the function re
 	t.end();
 });
 
-tape( 'if provided `t >= -ln( p )`, the function returns `NaN`', function test( t ) {
-	var y = mgf( 0.7, 10.0, 0.5 ); // -ln( p ) = ~0.693
+tape( 'if provided `t >= -ln( 1-p )`, the function returns `NaN`', function test( t ) {
+	var y = mgf( 0.7, 10.0, 0.5 ); // -ln( 1-p ) = ~0.693
 	t.strictEqual( isnan( y ), true, 'returns expected value' );
-	y = mgf( 1.7, 10.0, 0.2 ); // -ln( p ) = ~1.609
+	y = mgf( 1.7, 10.0, 0.2 ); // -ln( 1-p ) = ~0.223
 	t.strictEqual( isnan( y ), true, 'returns expected value' );
 	t.end();
 });
 
 tape( 'the function evaluates the mgf', function test( t ) {
-	var i;
+	var expected;
 	var r;
 	var p;
 	var x;
 	var y;
+	var i;
 
-	// TODO: Add test fixtures (function is not available in R and Julia)
-	for ( i = 0; i < 100; i++ ) {
-		r = randu() * 20.0;
-		p = randu();
-		x = ln( p ) - randu();
-		y = mgf( x, r, p );
-		t.strictEqual( !isnan( y ) && isNumber( y ), true, 'returns expected value' );
+	expected = data.expected;
+	x = data.t;
+	r = data.r;
+	p = data.p;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[ i ], r[ i ], p[ i ] );
+		t.strictEqual( isAlmostSameValue( y, expected[ i ], 40 ), true, 'returns expected value' );
 	}
 	t.end();
 });
diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/test.native.js b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/test.native.js
index 42d3858c4873..a85f128d0d1a 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/test.native.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/test/test.native.js
@@ -23,14 +23,17 @@
 var resolve = require( 'path' ).resolve;
 var tape = require( 'tape' );
 var tryRequire = require( '@stdlib/utils/try-require' );
-var isNumber = require( '@stdlib/assert/is-number' );
+var isAlmostSameValue = require( '@stdlib/assert/is-almost-same-value' );
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
-var randu = require( '@stdlib/random/base/randu' );
-var ln = require( '@stdlib/math/base/special/ln' );
 var PINF = require( '@stdlib/constants/float64/pinf' );
 var NINF = require( '@stdlib/constants/float64/ninf' );
 
 
+// FIXTURES //
+
+var data = require( './fixtures/julia/data.json' );
+
+
 // VARIABLES //
 
 var mgf = tryRequire( resolve( __dirname, './../lib/native.js' ) );
@@ -93,27 +96,29 @@ tape( 'if provided a success probability `p` outside of `[0,1]`, the function re
 	t.end();
 });
 
-tape( 'if provided `t >= -ln( p )`, the function returns `NaN`', opts, function test( t ) {
-	var y = mgf( 0.7, 10.0, 0.5 ); // -ln( p ) = ~0.693
+tape( 'if provided `t >= -ln( 1-p )`, the function returns `NaN`', opts, function test( t ) {
+	var y = mgf( 0.7, 10.0, 0.5 ); // -ln( 1-p ) = ~0.693
 	t.strictEqual( isnan( y ), true, 'returns expected value' );
-	y = mgf( 1.7, 10.0, 0.2 ); // -ln( p ) = ~1.609
+	y = mgf( 1.7, 10.0, 0.2 ); // -ln( 1-p ) = ~0.223
 	t.strictEqual( isnan( y ), true, 'returns expected value' );
 	t.end();
 });
 
 tape( 'the function evaluates the mgf', opts, function test( t ) {
-	var i;
+	var expected;
 	var r;
 	var p;
 	var x;
 	var y;
+	var i;
 
-	for ( i = 0; i < 100; i++ ) {
-		r = randu() * 20.0;
-		p = randu();
-		x = ln( p ) - randu();
-		y = mgf( x, r, p );
-		t.strictEqual( !isnan( y ) && isNumber( y ), true, 'returns expected value' );
+	expected = data.expected;
+	x = data.t;
+	r = data.r;
+	p = data.p;
+	for ( i = 0; i < x.length; i++ ) {
+		y = mgf( x[ i ], r[ i ], p[ i ] );
+		t.strictEqual( isAlmostSameValue( y, expected[ i ], 50 ), true, 'returns expected value' );
 	}
 	t.end();
 });

From 5ce9dfdbfe132f297886b67d769d552b8b44a60c Mon Sep 17 00:00:00 2001
From: 0PrashantYadav0 <devprashantkyadav@gmail.com>
Date: Sat, 17 Oct 2026 22:23:37 +0530
Subject: [PATCH 2/2] bench: fix input range for `t`

---
type: pre_commit_static_analysis_report
description: Results of running static analysis checks when committing changes.
report:
  - task: lint_filenames
    status: passed
  - task: lint_editorconfig
    status: passed
  - task: lint_markdown_pkg_readmes
    status: na
  - task: lint_markdown_docs
    status: na
  - task: lint_markdown
    status: na
  - task: lint_package_json
    status: na
  - task: lint_repl_help
    status: na
  - task: lint_javascript_src
    status: na
  - task: lint_javascript_cli
    status: na
  - task: lint_javascript_examples
    status: na
  - task: lint_javascript_tests
    status: na
  - task: lint_javascript_benchmarks
    status: passed
  - task: lint_python
    status: na
  - task: lint_r
    status: na
  - task: lint_c_src
    status: na
  - task: lint_c_examples
    status: na
  - task: lint_c_benchmarks
    status: na
  - task: lint_c_tests_fixtures
    status: na
  - task: lint_shell
    status: na
  - task: lint_typescript_declarations
    status: passed
  - task: lint_typescript_tests
    status: na
  - task: lint_license_headers
    status: passed
---
---
 .../base/dists/negative-binomial/mgf/benchmark/benchmark.js     | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/benchmark/benchmark.js b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/benchmark/benchmark.js
index af152a63d364..cefdbf1536a5 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/benchmark/benchmark.js
+++ b/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/benchmark/benchmark.js
@@ -48,7 +48,7 @@ bench( pkg, function benchmark( b ) {
 	p = new Float64Array( len );
 	for ( i = 0; i < len; i++ ) {
 		p[ i ] = uniform( EPS, 1.0 );
-		t[ i ] = uniform( 0.0, -ln( p[ i ] ) );
+		t[ i ] = uniform( 0.0, -ln( 1.0 - p[ i ] ) );
 		r[ i ] = discreteUniform( 1, 100 );
 	}