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# erfcx

> 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="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 -->

<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="links">

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

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

[@stdlib/math/base/special/erfc]: https://github.com/stdlib-js/math/tree/main/base/special/erfc

[@stdlib/math/base/special/erfcinv]: https://github.com/stdlib-js/math/tree/main/base/special/erfcinv

[@stdlib/math/base/special/erf]: https://github.com/stdlib-js/math/tree/main/base/special/erf

[@stdlib/math/base/special/erfinv]: https://github.com/stdlib-js/math/tree/main/base/special/erfinv

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

</section>

<!-- /.links -->
