npm package: @stdlib/math-base-special-erfcx
Description: Scaled complementary error function.
Author: The Stdlib Authors
Latest version: 0.3.1

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# erfcx

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

> Scaled [complementary error function][complementary-error-function].

<section class="intro">

The scaled [complementary error function][complementary-error-function] is defined as

<!-- <equation class="equation" label="eq:scaled_complementary_error_function" align="center" raw="\operatorname{erfcx}(x) = e^{x^2} \operatorname{erfc}(x)" alt="Scaled complementary error function."> -->

<div class="equation" align="center" data-raw-text="\operatorname{erfcx}(x) = e^{x^2} \operatorname{erfc}(x)" data-equation="eq:scaled_complementary_error_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@d97f372ed0bdcad1d7d373fad56a663ec94abe9e/lib/node_modules/@stdlib/math/base/special/erfcx/docs/img/equation_scaled_complementary_error_function.svg" alt="Scaled complementary error function.">
    <br>
</div> -->

<!-- </equation> -->

For large values, the scaled [complementary error function][complementary-error-function] is approximately equal to

<!-- <equation class="equation" label="eq:scaled_complementary_error_function_approximation_large_x" align="center" raw="\left( \frac{1}{\sqrt{\pi}} \right) \frac{1}{x}" alt="Approximation for large x"> -->

<!-- <div class="equation" align="center" data-raw-text="\left( \frac{1}{\sqrt{\pi}} \right) \frac{1}{x}" data-equation="eq:scaled_complementary_error_function_approximation_large_x">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@ab78b6493866fa172404de2b5d087f050d7c8807/lib/node_modules/@stdlib/math/base/special/erfcx/docs/img/equation_scaled_complementary_error_function_approximation_large_x.svg" alt="Approximation for large x">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="installation">

## Installation

```bash
npm install @stdlib/math-base-special-erfcx
```

</section>

<section class="usage">

## Usage

```javascript
var erfcx = require( '@stdlib/math-base-special-erfcx' );
```

#### erfcx( x )

Evaluates the scaled [complementary error function][complementary-error-function].

```javascript
var y = erfcx( 0.0 );
// returns 1.0

y = erfcx( 1.0 );
// returns ~0.4276

y = erfcx( -1.0 );
// returns ~5.01

y = erfcx( 50.0 );
// returns ~0.011

y = erfcx( -50.0 );
// returns +Infinity
```

If provided `NaN`, the function returns `NaN`.

```javascript
var y = erfcx( NaN );
// returns NaN
```

</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 erfcx = require( '@stdlib/math-base-special-erfcx' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 100, -30.0, 30.0, opts );

logEachMap( 'x: %0.4f, erfcx(x): %0.4f', x, erfcx );
```

</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/math/base/special/erfcx.h"
```

#### stdlib_base_erfcx( x )

Evaluates the [scaled complementary error function][complementary-error-function].

```c
double y  = stdlib_base_erfcx( 0.0 );
// returns 1.0

y  = stdlib_base_erfcx( 1.0 );
// returns ~0.4276
```

The function accepts the following arguments:

-   **x**: `[in] double` input value.

```c
double stdlib_base_erfcx( const double x );
```

</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/math/base/special/erfcx.h"
#include <stdio.h>

int main( void ) {
    const double x[] = { 0.0, 0.22, 0.44, 0.67, 0.89, 1.11, 1.33, 1.56, 1.78, 2.0 };

    double v;
    int i;
    for ( i = 0; i < 10; i++ ) {
        v = stdlib_base_erfcx( x[ i ] );
        printf( "x: %lf, erfcx(x): %lf\n", x[ i ], v );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

* * *

## See Also

-   <span class="package-name">[`@stdlib/math-base/special/erfc`][@stdlib/math/base/special/erfc]</span><span class="delimiter">: </span><span class="description">complementary error function.</span>
-   <span class="package-name">[`@stdlib/math-base/special/erfcinv`][@stdlib/math/base/special/erfcinv]</span><span class="delimiter">: </span><span class="description">inverse complementary error function.</span>
-   <span class="package-name">[`@stdlib/math-base/special/erf`][@stdlib/math/base/special/erf]</span><span class="delimiter">: </span><span class="description">error function.</span>
-   <span class="package-name">[`@stdlib/math-base/special/erfinv`][@stdlib/math/base/special/erfinv]</span><span class="delimiter">: </span><span class="description">inverse error function.</span>

</section>

<!-- /.related -->

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<section class="main-repo" >

* * *

## 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].

#### Community

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---

## Copyright

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

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[complementary-error-function]: https://en.wikipedia.org/wiki/Error_function

<!-- <related-links> -->

[@stdlib/math/base/special/erfc]: https://www.npmjs.com/package/@stdlib/math-base-special-erfc

[@stdlib/math/base/special/erfcinv]: https://www.npmjs.com/package/@stdlib/math-base-special-erfcinv

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