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# factoriallnf

> Natural logarithm of the [factorial][factorial-function] of a single-precision floating-point number.

<section class="intro">

The natural logarithm of the factorial function may be expressed

<!-- <equation class="equation" label="eq:factorialln_function" align="center" raw="f(n)=\ln (n!)" alt="Equation of the natural logarithm of the factorial."> -->

<!-- </equation> -->

The [factorial function][factorial-function] may be defined as the product

<!-- <equation class="equation" label="eq:factorial_function" align="center" raw="n! = \prod_{k=1}^n k" alt="Factorial function definition"> -->

<!-- </equation> -->

or according to the recurrence relation

<!-- <equation class="equation" label="eq:factorial_recurrence_relation" align="center" raw="n! = \begin{cases}1 & \textrm{if } n = 0,\\(n-1)! \times n & \textrm{if } n > 1\end{cases}" alt="Factorial function recurrence relation"> -->

<!-- </equation> -->

Following the convention for an [empty product][empty-product], in all definitions,

<!-- <equation class="equation" label="eq:zero_factorial" align="center" raw="0! = 1" alt="Zero factorial"> -->

<!-- </equation> -->

The [Gamma][gamma-function] function extends the [factorial function][factorial-function] for non-integer values.

<!-- <equation class="equation" label="eq:factorial_function_and_gamma" align="center" raw="n! = \Gamma(n+1)" alt="Factorial function extension via the Gamma function"> -->

<!-- </equation> -->

The [factorial][factorial-function] of a **negative** integer is not defined.

Evaluating the natural logarithm of [factorial function][factorial-function] is useful as the [factorial function][factorial-function] can overflow for large `n`. Thus, `factoriallnf( n )` is generally preferred to `lnf( n! )`.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### factoriallnf( x )

Evaluates the natural logarithm of the [factorial][factorial-function] of a single-precision floating-point number. For input values other than negative integers, the function returns `ln( x! ) = ln( Γ(x+1) )`, where `Γ` is the [Gamma][gamma-function] function. For negative integers, the function returns `NaN`.

```javascript
var v = factoriallnf( 3.0 );
// returns ~1.792

v = factoriallnf( 2.4 );
// returns ~1.092

v = factoriallnf( -1.0 );
// returns NaN

v = factoriallnf( -1.5 );
// returns ~1.266
```

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

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

var opts = {
    'dtype': 'float32'
};
var x = uniform( 100, -10.0, 50.0, opts );

logEachMap( 'factoriallnf(%0.4f) = %0.4f', x, factoriallnf );
```

</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/factoriallnf.h"
```

#### stdlib_base_factoriallnf( x )

Evaluates the natural logarithm of the [factorial][factorial-function] of a single-precision floating-point number. For input values other than negative integers, the function returns `ln( x! ) = ln( Γ(x+1) )`, where `Γ` is the [Gamma][gamma-function] function. For negative integers, the function returns `NaN`.

```c
float out = stdlib_base_factoriallnf( 3.0f );
// returns ~1.792f

out = stdlib_base_factoriallnf( -1.5f );
// returns ~1.266f
```

The function accepts the following arguments:

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

```c
float stdlib_base_factoriallnf( const float 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/factoriallnf.h"
#include <stdio.h>

int main( void ) {
    const float x[] = { 2.0f, 3.0f, 5.0f, 8.0f };

    float y;
    int i;
    for ( i = 0; i < 4; i++ ) {
        y = stdlib_base_factoriallnf( x[ i ] );
        printf( "factoriallnf(%f) = %f\n", x[ i ], y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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

<section class="related">

</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">

[gamma-function]: https://en.wikipedia.org/wiki/Gamma_Function

[factorial-function]: https://en.wikipedia.org/wiki/Factorial

[empty-product]: https://en.wikipedia.org/wiki/Empty_product

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

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

</section>

<!-- /.links -->
