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# expm1rel

> Compute the relative error exponential.

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The relative error exponential is defined as

<!-- <equation class="equation" label="eq:relative_error_exponential" align="center" raw="f(x) = \frac{e^x - 1}{x}" alt="Equation for the relative error exponential."> -->

<div class="equation" align="center" data-raw-text="f(x) = \frac{e^x - 1}{x}" data-equation="eq:relative_error_exponential">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@237a80443a932a41f0195d0b9fc7699aa8c8e417/lib/node_modules/@stdlib/math/base/special/expm1rel/docs/img/equation_relative_error_exponential.svg" alt="Equation for the relative error exponential.">
    <br>
</div>

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</section>

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<section class="usage">

## Usage

```javascript
var expm1rel = require( '@stdlib/math/base/special/expm1rel' );
```

#### expm1rel( x )

Computes the relative error exponential.

```javascript
var v = expm1rel( 0.0 );
// returns 1.0

v = expm1rel( 1.0 );
// returns ~1.718

v = expm1rel( -1.0 );
// returns ~0.632

v = expm1rel( NaN );
// returns NaN
```

</section>

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## Notes

-   When `x` is near zero, `exp(x)-1` can suffer catastrophic cancellation (i.e., a significant loss in precision). `expm1rel` avoids such a loss in precision.

</section>

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<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 expm1rel = require( '@stdlib/math/base/special/expm1rel' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 100, -50.0, 50.0, opts );

logEachMap( '(e^%0.3f - 1)/%0.3f = %0.3f', x, x, expm1rel );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

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</section>

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### Usage

```c
#include "stdlib/math/base/special/expm1rel.h"
```

#### stdlib_base_expm1rel( x )

Computes the relative error exponential.

```c
double out = stdlib_base_expm1rel( 0.0 );
// returns 1.0

out = stdlib_base_expm1rel( 1.0 );
// returns ~1.718
```

The function accepts the following arguments:

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

```c
double stdlib_base_expm1rel( const double x );
```

</section>

<!-- /.usage -->

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</section>

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<section class="examples">

### Examples

```c
#include "stdlib/math/base/special/expm1rel.h"
#include <stdlib.h>
#include <stdio.h>

int main( void ) {
    double x;
    double v;
    int i;

    for ( i = 0; i < 100; i++ ) {
        x = ( ( (double)rand() / (double)RAND_MAX ) * 100.0 ) - 50.0;
        v = stdlib_base_expm1rel( x );
        printf( "(e^%lf - 1)/%lf = %lf\n", x, x, v );
    }
}
```

</section>

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</section>

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</section>

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* * *

## See Also

-   <span class="package-name">[`@stdlib/math/base/special/exp`][@stdlib/math/base/special/exp]</span><span class="delimiter">: </span><span class="description">natural exponential function.</span>
-   <span class="package-name">[`@stdlib/math/base/special/expm1`][@stdlib/math/base/special/expm1]</span><span class="delimiter">: </span><span class="description">compute exp(x) - 1.</span>

</section>

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

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

[@stdlib/math/base/special/expm1]: https://github.com/stdlib-js/math/tree/main/base/special/expm1

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</section>

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