npm package: @stdlib/math-base-special-erfc
Description: Complementary error function.
Author: The Stdlib Authors
Latest version: 0.2.5

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# erfc

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

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

<section class="intro">

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

<!-- <equation class="equation" label="eq:complementary_error_function" align="center" raw="\operatorname{erfc}(x) = 1 - \operatorname{erf}(x) = \frac{2}{\sqrt\pi} \int_x^{\infty} e^{-t^2}\, dt" alt="Complementary error function."> -->

<div class="equation" align="center" data-raw-text="\operatorname{erfc}(x) = 1 - \operatorname{erf}(x) = \frac{2}{\sqrt\pi} \int_x^{\infty} e^{-t^2}\, dt" data-equation="eq:complementary_error_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/erfc/docs/img/equation_complementary_error_function.svg" alt="Complementary error function.">
    <br>
</div> -->

<!-- </equation> -->

The [complementary error function][complementary-error-function] can also be expressed using Craig's formula

<!-- <equation class="equation" label="eq:craigs_formula" align="center" raw="\operatorname{erfc}(x) = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \exp \left( - \frac{x^2}{\sin^2 \theta} \right) d\theta" alt="Craig's formula of the complementary error function."> -->

<!-- <div class="equation" align="center" data-raw-text="\operatorname{erfc}(x) = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \exp \left( - \frac{x^2}{\sin^2 \theta} \right) d\theta" data-equation="eq:craigs_formula">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/math/base/special/erfc/docs/img/equation_craigs_formula.svg" alt="Craig's formula of the complementary error function.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### erfc( x )

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

```javascript
var y = erfc( 2.0 );
// returns ~0.0047

y = erfc( -1.0 );
// returns ~1.8427

y = erfc( Infinity );
// returns 0.0

y = erfc( -Infinity );
// returns 2.0
```

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

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

var opts = {
    'dtype': 'float64'
};
var x = uniform( 100, -10.0, 10.0, opts );

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

</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/erfc.h"
```

#### stdlib_base_erfc( x )

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

```c
double out = stdlib_base_erfc( 2.0 );
// returns ~0.0047

out = stdlib_base_erfc( -1.0 );
// returns ~1.8427
```

The function accepts the following arguments:

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

```c
double stdlib_base_erfc( 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/erfc.h"
#include <stdlib.h>
#include <stdio.h>

int main( void ) {
    const double x[] = { -10.0, -7.78, -5.56, -3.33, -1.11, 1.11, 3.33, 5.56, 7.78, 10.0 };

    double v;
    int i;
    for ( i = 0; i < 10; i++ ) {
        v = stdlib_base_erfc( x[ i ] );
        printf( "x: %lf, erfc(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/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>
-   <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/erfcx`][@stdlib/math/base/special/erfcx]</span><span class="delimiter">: </span><span class="description">scaled complementary 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> -->

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