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# Binet's Formula

> Evaluate [Binet's formula][fibonacci-number] extended to real numbers.

<section class="intro">

[Binet's formula][fibonacci-number] refers to the closed-form solution for computing the nth [Fibonacci number][fibonacci-number] and may be expressed

<!-- <equation class="equation" label="eq:binets_formula" align="center" raw="F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}" alt="Binet's formula"> -->

<div class="equation" align="center" data-raw-text="F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}" data-equation="eq:binets_formula">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/binet/docs/img/equation_binets_formula.svg" alt="Binet's formula">
    <br>
</div> -->

<!-- </equation> -->

where `φ` is the [golden ratio][golden-ratio] and `ψ` is `1 - φ`. To extend [Fibonacci numbers][fibonacci-number] to real numbers, we may express [Binet's formula][fibonacci-number] as

<!-- <equation class="equation" label="eq:binets_formula_real_numbers" align="center" raw="F_x = \frac{\varphi^x - \varphi^{-x} \cdot \cos(\pi x)}{\sqrt{5}}" alt="Binet's formula extended to real numbers."> -->

<!-- <div class="equation" align="center" data-raw-text="F_x = \frac{\varphi^x - \varphi^{-x} \cdot \cos(\pi x)}{\sqrt{5}}" data-equation="eq:binets_formula_real_numbers">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/binet/docs/img/equation_binets_formula_real_numbers.svg" alt="Binet's formula extended to real numbers.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### binet( x )

Evaluates [Binet's formula][fibonacci-number] extended to real numbers.

```javascript
var v = binet( 0.0 );
// returns 0.0

v = binet( 1.0 );
// returns 1.0

v = binet( 2.0 );
// returns 1.0

v = binet( 3.0 );
// returns 2.0

v = binet( -1.0 );
// returns 1.0

v = binet( 3.14 );
// returns ~2.12
```

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

```javascript
var v = binet( NaN );
// returns NaN
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   The function returns only **approximate** [Fibonacci numbers][fibonacci-number] for nonnegative integers.
-   The function does **not** return complex numbers, guaranteeing real-valued return values.

</section>

<!-- /.notes -->

<section class="examples">

## Examples

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

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

var v;
var i;

for ( i = 0; i < 79; i++ ) {
    v = binet( i );
    console.log( 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/binet.h"
```

#### stdlib_base_binet( x )

Evaluates [Binet's formula][fibonacci-number] extended to real numbers.

```c
double out = stdlib_base_binet( 0.0 );
// returns 0.0

out = stdlib_base_binet( 1.0 );
// returns 1.0
```

The function accepts the following arguments:

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

```c
double stdlib_base_binet( 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/binet.h"
#include <stdio.h>

int main( void ) {
    const double x[] = { 0.0, 1.0, 2.0, 3.0, 4.0 };

    double y;
    int i;
    for ( i = 0; i < 5; i++ ) {
        y = stdlib_base_binet( x[ i ] );
        printf( "binet(%lf) = %lf\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">

* * *

## See Also

-   <span class="package-name">[`@stdlib/math/base/special/fibonacci`][@stdlib/math/base/special/fibonacci]</span><span class="delimiter">: </span><span class="description">compute the nth Fibonacci number.</span>
-   <span class="package-name">[`@stdlib/math/base/special/negafibonacci`][@stdlib/math/base/special/negafibonacci]</span><span class="delimiter">: </span><span class="description">compute the nth negaFibonacci number.</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">

[fibonacci-number]: https://en.wikipedia.org/wiki/Fibonacci_number

[golden-ratio]: https://en.wikipedia.org/wiki/Golden_ratio

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

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

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

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

</section>

<!-- /.links -->
