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# Exponential Function

> [Exponential function][exponential-function].

<section class="intro">

The [exponential function][exponential-function] is defined as

<!-- <equation class="equation" label="eq:exponential_function" align="center" raw="y = b^x" alt="Exponential function"> -->

<div class="equation" align="center" data-raw-text="y = b^x" data-equation="eq:exponential_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@c1bdd27898df04d752ddb2dca37ca049e4c94d9b/lib/node_modules/@stdlib/math/base/special/fast/pow-int/docs/img/equation_exponential_function.svg" alt="Exponential function">
    <br>
</div>

<!-- </equation> -->

where `b` is the **base** and `x` is the **exponent**.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### pow( base, exponent )

Evaluates the [exponential function][exponential-function] given a signed 32-bit integer `exponent`.

```javascript
var v = pow( 2.0, 3 );
// returns 8.0

v = pow( 100.0, 0 );
// returns 1.0

v = pow( 3.14, 1 );
// returns 3.14

v = pow( -3.14, 1 );
// returns -3.14

v = pow( 2.0, -2 );
// returns 0.25

v = pow( NaN, 3 );
// returns NaN
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   This implementation is **not** recommended for high-precision applications due to error accumulation. As a trivial example, consider

    ```javascript
    var v = pow( 10.0, 308 );
    // returns 1.0000000000000006e+308
    ```

    where the expected result is `1.0e+308`.

-   If provided a negative `exponent`, the implementation first computes the reciprocal of the `base` and then evaluates the exponential function. This can introduce significant error. For example,

    ```javascript
    var v = pow( -459, -98 );
    // returns 1.3878956588399783e-261
    ```

    where the expected result is `1.3878956588399598e-261`. From the bit sequences,

    ```text
    0000100111000101110110100000000111001011001011010001000101010110
    0000100111000101110110100000000111001011001011010001000100000100
    ```

    one observes that the returned value differs in the last `7` bits of the significand.

</section>

<!-- /.notes -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

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

var x;
var y;
var v;

x = 10.0;
for ( y = 0; y < 309; y++ ) {
    v = pow( x, y );
    console.log( '%d^%d = %d', x, y, v );
}
```

</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/fast/pow_int.h"
```

#### stdlib_base_fast_pow( x, y )

Evaluates the [exponential function][exponential-function] given a signed 32-bit integer `exponent`.

```c
double out = stdlib_base_fast_pow( 2.0, 3 );
// returns 8.0

out = stdlib_base_fast_pow( 3.14, 0 );
// returns 1.0
```

The function accepts the following arguments:

-   **x**: `[in] double` base.
-   **y**: `[in] int32_t` exponent.

```c
double stdlib_base_fast_pow( const double x, const int32_t y );
```

</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/fast/pow_int.h"
#include <stdlib.h>
#include <stdio.h>
#include <stdint.h>

int main( void ) {
    const double x[] = { 3.14, 2.0, 2.0, 0.0 };
    const int32_t y[] = { 0, 3, -2, 0 };

    double z;
    int i;
    for ( i = 0; i < 4; i++ ) {
        z = stdlib_base_fast_pow( x[ i ], y[ i ] );
        printf( "pow( %lf, %d ) = %lf\n", x[ i ], y[ i ], z );
    }
}
```

</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/pow`][@stdlib/math/base/special/pow]</span><span class="delimiter">: </span><span class="description">exponential 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">

[exponential-function]: https://en.wikipedia.org/wiki/Exponential_function

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

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

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

</section>

<!-- /.links -->
