npm package: @stdlib/math-base-special-binet
Description: Evaluate Binet's formula extended to real numbers.
Author: The Stdlib Authors
Latest version: 0.3.1

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# Binet's Formula

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> 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="installation">

## Installation

```bash
npm install @stdlib/math-base-special-binet
```

</section>

<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="main-repo" >

* * *

## Notice

This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].

#### Community

[![Chat][chat-image]][chat-url]

---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

</section>

<!-- /.stdlib -->

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[umd]: https://github.com/umdjs/umd
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[deno-url]: https://github.com/stdlib-js/math-base-special-binet/tree/deno
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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/math-base-special-binet/main/LICENSE

[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://www.npmjs.com/package/@stdlib/math-base-special-fibonacci

[@stdlib/math/base/special/negafibonacci]: https://www.npmjs.com/package/@stdlib/math-base-special-negafibonacci

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

</section>

<!-- /.links -->