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feat: add Wald distribution PDF
PR-URL: stdlib-js#9324 Closes: stdlib-js#209 Reviewed-by: Philipp Burckhardt <pburckhardt@outlook.com>
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‎lib/node_modules/@stdlib/stats/base/dists/wald/pdf/README.md‎

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<!--
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@license Apache-2.0
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Copyright (c) 2026 The Stdlib Authors.
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Licensed under the Apache License, Version 2.0 (the "License");
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you may not use this file except in compliance with the License.
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You may obtain a copy of the License at
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http://www.apache.org/licenses/LICENSE-2.0
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Unless required by applicable law or agreed to in writing, software
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distributed under the License is distributed on an "AS IS" BASIS,
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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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See the License for the specific language governing permissions and
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limitations under the License.
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-->
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# Probability Density Function
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> [Wald][wald-distribution] distribution probability density function (PDF).
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<section class="intro">
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The [probability density function][pdf] (PDF) for a [Wald][wald-distribution] random variable is
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<!-- <equation class="equation" label="eq:wald_pdf" align="center" raw="f(x;\mu,\lambda)=\sqrt{\frac{\lambda}{2\pi x^3}}\, e^{-\frac{\lambda(x-\mu)^2}{2\mu^2 x}}" alt="Probability density function (PDF) for a Wald distribution."> -->
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```math
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f(x;\mu,\lambda)=\sqrt{\frac{\lambda}{2\pi x^3}}\, e^{-\frac{\lambda(x-\mu)^2}{2\mu^2 x}}
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```
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<!-- <div class="equation" align="center" data-raw-text="f(x;\mu,\lambda)=\sqrt{\frac{\lambda}{2\pi x^3}}\, e^{-\frac{\lambda(x-\mu)^2}{2\mu^2 x}}" data-equation="eq:wald_pdf">
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<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/wald/pdf/docs/img/equation_wald_pdf.svg" alt="Probability density function (PDF) for a Wald distribution.">
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<br>
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</div> -->
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<!-- </equation> -->
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where `µ > 0` is the mean and `λ > 0` is the shape parameter.
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</section>
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<!-- /.intro -->
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<section class="usage">
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## Usage
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```javascript
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var pdf = require( '@stdlib/stats/base/dists/wald/pdf' );
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```
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#### pdf( x, mu, lambda )
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Evaluates the [probability density function][pdf] (PDF) for a [Wald][wald-distribution] distribution with parameters `mu` (mean) and `lambda` (shape parameter).
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```javascript
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var y = pdf( 2.0, 1.0, 1.0 );
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// returns ~0.110
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y = pdf( 0.5, 2.0, 3.0 );
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// returns ~0.362
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```
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If provided `NaN` as any argument, the function returns `NaN`.
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```javascript
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var y = pdf( NaN, 1.0, 1.0 );
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// returns NaN
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y = pdf( 1.0, NaN, 1.0 );
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// returns NaN
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y = pdf( 1.0, 1.0, NaN );
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// returns NaN
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```
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If provided `mu <= 0`, the function returns `NaN`.
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```javascript
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var y = pdf( 2.0, 0.0, 1.0 );
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// returns NaN
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y = pdf( 2.0, -1.0, 1.0 );
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// returns NaN
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```
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If provided `lambda < 0`, the function returns `NaN`.
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```javascript
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var y = pdf( 2.0, 1.0, -1.0 );
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// returns NaN
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```
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If provided `lambda = 0`, the function evaluates the [PDF][pdf] of a [degenerate distribution][degenerate-distribution] centered at `mu`.
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```javascript
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var y = pdf( 2.0, 1.0, 0.0 );
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// returns 0.0
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y = pdf( 1.0, 1.0, 0.0 );
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// returns Infinity
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```
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If provided `x <= 0`, the function returns `0.0`.
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```javascript
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var y = pdf( 0.0, 1.0, 1.0 );
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// returns 0.0
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y = pdf( -1.0, 1.0, 1.0 );
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// returns 0.0
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```
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#### pdf.factory( mu, lambda )
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Partially applies `mu` and `lambda` to create a reusable `function` for evaluating the PDF.
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```javascript
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var mypdf = pdf.factory( 1.0, 1.0 );
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var y = mypdf( 2.0 );
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// returns ~0.110
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y = mypdf( 0.5 );
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// returns ~0.879
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```
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</section>
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<!-- /.usage -->
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<section class="examples">
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## Examples
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<!-- eslint no-undef: "error" -->
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```javascript
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var uniform = require( '@stdlib/random/array/uniform' );
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var logEachMap = require( '@stdlib/console/log-each-map' );
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var EPS = require( '@stdlib/constants/float64/eps' );
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var pdf = require( '@stdlib/stats/base/dists/wald/pdf' );
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var opts = {
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'dtype': 'float64'
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};
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var x = uniform( 10, EPS, 10.0, opts );
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var mu = uniform( 10, EPS, 10.0, opts );
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var lambda = uniform( 10, EPS, 20.0, opts );
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logEachMap( 'x: %0.4f, µ: %0.4f, λ: %0.4f, f(x;µ,λ): %0.4f', x, mu, lambda, pdf );
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```
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</section>
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<!-- /.examples -->
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<!-- C interface documentation. -->
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* * *
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<section class="c">
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## C APIs
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<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->
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<section class="intro">
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</section>
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<!-- /.intro -->
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<!-- C usage documentation. -->
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<section class="usage">
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### Usage
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```c
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#include "stdlib/stats/base/dists/wald/pdf.h"
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```
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#### stdlib_base_dists_wald_pdf( x, mu, lambda )
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Evaluates the [probability density function][pdf] (PDF) for a [Wald][wald-distribution] distribution with parameters `mu` (mean) and `lambda` (shape parameter).
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```c
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double y = stdlib_base_dists_wald_pdf( 2.0, 1.0, 1.0 );
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// returns ~0.110
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```
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The function accepts the following arguments:
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- **x**: `[in] double` input value.
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- **mu**: `[in] double` mean.
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- **lambda**: `[in] double` shape parameter.
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```c
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double stdlib_base_dists_wald_pdf( const double x, const double mu, const double lambda );
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```
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</section>
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<!-- /.usage -->
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<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
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<section class="notes">
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</section>
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<!-- /.notes -->
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<!-- C API usage examples. -->
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<section class="examples">
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### Examples
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```c
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#include "stdlib/stats/base/dists/wald/pdf.h"
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#include "stdlib/constants/float64/eps.h"
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#include <stdlib.h>
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#include <stdio.h>
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static double random_uniform( const double min, const double max ) {
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double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
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return min + ( v*(max-min) );
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}
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int main( void ) {
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double lambda;
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double mu;
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double x;
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double y;
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int i;
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for ( i = 0; i < 10; i++ ) {
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x = random_uniform( 0.0, 10.0 );
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mu = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 10.0 );
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lambda = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 20.0 );
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y = stdlib_base_dists_wald_pdf( x, mu, lambda );
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printf( "x: %lf, µ: %lf, λ: %lf, f(x;µ,λ): %lf\n", x, mu, lambda, y );
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}
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}
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```
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</section>
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<!-- /.examples -->
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</section>
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<!-- /.c -->
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<!-- Section to include cited references. If references are included, add a horizontal rule *before* the section. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
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<section class="references">
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</section>
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<!-- /.references -->
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<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->
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<section class="related">
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</section>
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<!-- /.related -->
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<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
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<section class="links">
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[pdf]: https://en.wikipedia.org/wiki/Probability_density_function
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[wald-distribution]: https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution
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[degenerate-distribution]: https://en.wikipedia.org/wiki/Degenerate_distribution
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</section>
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<!-- /.links -->

‎lib/node_modules/@stdlib/stats/base/dists/wald/pdf/benchmark/benchmark.js‎

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/**
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* @license Apache-2.0
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*
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* Copyright (c) 2026 The Stdlib Authors.
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*
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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'use strict';
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// MODULES //
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var bench = require( '@stdlib/bench' );
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var runiform = require( '@stdlib/random/array/uniform' );
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var uniform = require( '@stdlib/random/base/uniform' );
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var isnan = require( '@stdlib/math/base/assert/is-nan' );
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var EPS = require( '@stdlib/constants/float64/eps' );
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var format = require( '@stdlib/string/format' );
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var pkg = require( './../package.json' ).name;
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var pdf = require( './../lib' );
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// MAIN //
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bench( pkg, function benchmark( b ) {
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var lambda;
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var len;
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var mu;
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var x;
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var y;
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var i;
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len = 100;
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x = runiform( len, EPS, 100.0 );
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mu = runiform( len, EPS, 50.0 );
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lambda = runiform( len, EPS, 20.0 );
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b.tic();
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for ( i = 0; i < b.iterations; i++ ) {
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y = pdf( x[ i%len ], mu[ i%len ], lambda[ i%len ] );
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if ( isnan( y ) ) {
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b.fail( 'should not return NaN' );
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}
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}
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b.toc();
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if ( isnan( y ) ) {
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b.fail( 'should not return NaN' );
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}
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b.pass( 'benchmark finished' );
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b.end();
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});
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bench( format( '%s:factory', pkg ), function benchmark( b ) {
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var lambda;
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var mypdf;
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var mu;
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var x;
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var y;
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var i;
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mu = 1.0;
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lambda = 1.5;
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mypdf = pdf.factory( mu, lambda );
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b.tic();
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for ( i = 0; i < b.iterations; i++ ) {
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x = uniform( EPS, 10.0 );
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y = mypdf( x );
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if ( isnan( y ) ) {
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b.fail( 'should not return NaN' );
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}
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}
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b.toc();
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if ( isnan( y ) ) {
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b.fail( 'should not return NaN' );
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}
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b.pass( 'benchmark finished' );
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b.end();
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});

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