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# erfinv

> [Inverse error function][inverse-error-function].

<section class="intro">

The [inverse error function][inverse-error-function] is defined in terms of the [Maclaurin series][maclaurin-series]

<!-- <equation class="equation" label="eq:inverse_error_function" align="center" raw="\operatorname{erf}^{-1}(z)=\sum_{k=0}^\infty\frac{c_k}{2k+1}\left (\frac{\sqrt{\pi}}{2}z\right )^{2k+1}" alt="Inverse error function."> -->

<div class="equation" align="center" data-raw-text="\operatorname{erf}^{-1}(z)=\sum_{k=0}^\infty\frac{c_k}{2k+1}\left (\frac{\sqrt{\pi}}{2}z\right )^{2k+1}" data-equation="eq:inverse_error_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/math/base/special/erfinv/docs/img/equation_inverse_error_function.svg" alt="Inverse error function.">
    <br>
</div> -->

<!-- </equation> -->

where `c_0 = 1` and

<!-- <equation class="equation" label="eq:inverse_error_function_series_coefficients" align="center" raw="c_k=\sum_{m=0}^{k-1}\frac{c_m c_{k-1-m}}{(m+1)(2m+1)} = \left\{1,1,\frac{7}{6},\frac{127}{90},\frac{4369}{2520},\frac{34807}{16200},\ldots\right\}" alt="Series coefficients."> -->

<!-- <div class="equation" align="center" data-raw-text="c_k=\sum_{m=0}^{k-1}\frac{c_m c_{k-1-m}}{(m+1)(2m+1)} = \left\{1,1,\frac{7}{6},\frac{127}{90},\frac{4369}{2520},\frac{34807}{16200},\ldots\right\}" data-equation="eq:inverse_error_function_series_coefficients">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/math/base/special/erfinv/docs/img/equation_inverse_error_function_series_coefficients.svg" alt="Series coefficients.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### erfinv( x )

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

```javascript
var y = erfinv( 0.5 );
// returns ~0.4769

y = erfinv( 0.8 );
// returns ~0.9062

y = erfinv( -1.0 );
// returns -Infinity

y = erfinv( 1.0 );
// returns Infinity
```

The domain of `x` is restricted to `[-1,1]`. If `|x| > 1`, the function returns `NaN`.

```javascript
var y = erfinv( -3.14 );
// returns NaN
```

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

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

The [inverse error function][inverse-error-function] is an [odd function][odd-function]; i.e., `erfinv(-x) = -erfinv(x)`. Thus, in accordance with the [IEEE 754][ieee754] standard, if provided `-0`, the function returns `-0`.

```javascript
var y = erfinv( -0.0 );
// returns -0.0
```

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

var opts = {
    'dtype': 'float64'
};
var x = uniform( 100, -1.0, 1.0, opts );

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

</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/erfinv.h"
```

#### stdlib_base_erfinv( x )

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

```c
double out = stdlib_base_erfinv( 0.5 );
// returns ~0.4769

out = stdlib_base_erfinv( 0.8 );
// returns ~0.9062
```

The function accepts the following arguments:

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

```c
double stdlib_base_erfinv( 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/erfinv.h"
#include <stdlib.h>
#include <stdio.h>

int main( void ) {
    const double x[] = { -1.0, -0.78, -0.56, -0.33, -0.11, 0.11, 0.33, 0.56, 0.78, 1.0 };
    double v;
    int i;
    for ( i = 0; i < 10; i++ ) {
        v = stdlib_base_erfinv( x[ i ] );
        printf( "x: %lf, erfinv(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/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>

</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">

[inverse-error-function]: https://en.wikipedia.org/wiki/Error_function#Inverse_functions

[maclaurin-series]: http://mathworld.wolfram.com/MaclaurinSeries.html

[odd-function]: https://en.wikipedia.org/wiki/Even_and_odd_functions

[ieee754]: https://en.wikipedia.org/wiki/IEEE_754-1985

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

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

[@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

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

</section>

<!-- /.links -->
