npm package: @stdlib/stats-base-dists-cauchy-quantile
Description: Cauchy distribution quantile function.
Author: The Stdlib Authors
Latest version: 0.3.1

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# Quantile Function

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> [Cauchy][cauchy-distribution] distribution [quantile function][quantile-function].

<section class="intro">

The [quantile function][quantile-function] for a [Cauchy][cauchy-distribution] random variable is

<!-- <equation class="equation" label="eq:cauchy_cauchy_quantile_function" align="center" raw="Q(p; x_0,\gamma) = x_0 + \gamma\,\tan\left[\pi\left(p-\tfrac{1}{2}\right)\right]" alt="Quantile function for a Cauchy distribution."> -->

<div class="equation" align="center" data-raw-text="Q(p; x_0,\gamma) = x_0 + \gamma\,\tan\left[\pi\left(p-\tfrac{1}{2}\right)\right]" data-equation="eq:cauchy_cauchy_quantile_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/cauchy/quantile/docs/img/equation_cauchy_cauchy_quantile_function.svg" alt="Quantile function for a Cauchy distribution.">
    <br>
</div>

<!-- </equation> -->

for `0 <= p <= 1`, where `x0` is the location parameter and `gamma > 0` is the scale parameter.

</section>

<!-- /.intro -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-cauchy-quantile
```

</section>

<section class="usage">

## Usage

```javascript
var quantile = require( '@stdlib/stats-base-dists-cauchy-quantile' );
```

#### quantile( p, x0, gamma )

Evaluates the [quantile function][quantile-function] for a [Cauchy][cauchy-distribution] distribution with parameters `x0` (location parameter) and `gamma > 0` (scale parameter).

```javascript
var y = quantile( 0.5, 0.0, 1.0 );
// returns 0.0

y = quantile( 0.2, 4.0, 2.0 );
// returns ~1.247

y = quantile( 0.9, 4.0, 2.0 );
// returns ~10.155
```

If provided a probability `p` outside the interval `[0,1]`, the function returns `NaN`.

```javascript
var y = quantile( 1.9, 0.0, 1.0 );
// returns NaN

y = quantile( -0.1, 0.0, 1.0 );
// returns NaN
```

If provided `NaN` as any argument, the function returns `NaN`.

```javascript
var y = quantile( NaN, 0.0, 1.0 );
// returns NaN

y = quantile( 0.0, NaN, 1.0 );
// returns NaN

y = quantile( 0.0, 0.0, NaN );
// returns NaN
```

If provided `gamma <= 0`, the function returns `NaN`.

```javascript
var y = quantile( 0.4, 0.0, -1.0 );
// returns NaN

y = quantile( 0.4, 0.0, 0.0 );
// returns NaN
```

#### quantile.factory( x0, gamma )

Returns a function for evaluating the [quantile function][quantile-function] of a [Cauchy][cauchy-distribution] distribution with location parameter `x0` and scale parameter `gamma > 0`.

```javascript
var myquantile = quantile.factory( 10.0, 2.0 );

var y = myquantile( 0.2 );
// returns ~7.247

y = myquantile( 0.8 );
// returns ~12.753
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var EPS = require( '@stdlib/constants-float64-eps' );
var quantile = require( '@stdlib/stats-base-dists-cauchy-quantile' );

var opts = {
    'dtype': 'float64'
};
var gamma = uniform( 10, EPS, 20.0, opts );
var x0 = uniform( 10, -5.0, 5.0, opts );
var p = uniform( 10, 0.0, 1.0, opts );

logEachMap( 'p: %0.4f, x0: %0.4f γ: %0.4f, Q(p;x0,γ): %0.4f', p, x0, gamma, quantile );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/stats/base/dists/cauchy/quantile.h"
```

#### stdlib_base_dists_cauchy_quantile( p, x0, gamma )

Evaluates the [quantile function][quantile-function] of a [Cauchy][cauchy-distribution] distribution with location parameter `x0` and scale parameter `gamma`.

```c
double out = stdlib_base_dists_cauchy_quantile( 0.3, 2.0, 2.0 );
// returns ~0.547
```

The function accepts the following arguments:

-   **p**: `[in] double` probability.
-   **x0**: `[in] double` location parameter.
-   **gamma**: `[in] double` scale parameter.

```c
double stdlib_base_dists_cauchy_quantile( const double p, const double x0, const double gamma );
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
#include "stdlib/stats/base/dists/cauchy/quantile.h"
#include "stdlib/constants/float64/eps.h"
#include <stdlib.h>
#include <stdio.h>

static double random_uniform( const double min, const double max ) {
    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
    return min + ( v*(max-min) );
}

int main( void ) {
    double gamma;
    double x0;
    double y;
    double p;
    int i;

    for ( i = 0; i < 25; i++ ) {
        p = random_uniform( 0.0, 1.0 );
        x0 = random_uniform( -5.0, 5.0 );
        gamma = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 20.0 );
        y = stdlib_base_dists_cauchy_quantile( p, x0, gamma );
        printf( "p: %lf, x0: %lf, γ: %lf, Q(p;x0,γ): %lf\n", p, x0, gamma, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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<section class="references">

</section>

<!-- /.references -->

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<section class="related">

</section>

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* * *

## Notice

This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].

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## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/stats-base-dists-cauchy-quantile/main/LICENSE

[quantile-function]: https://en.wikipedia.org/wiki/Quantile_function

[cauchy-distribution]: https://en.wikipedia.org/wiki/Cauchy_distribution

</section>

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