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# evalrational

> Evaluate a [rational function][rational-function] using double-precision floating-point arithmetic.

<section class="intro">

A [rational function][rational-function] `f(x)` is defined as

<!-- <equation class="equation" label="eq:rational_function" align="center" raw="f(x) = \frac{P(x)}{Q(x)}" alt="Rational function definition."> -->

<div class="equation" align="center" data-raw-text="f(x) = \frac{P(x)}{Q(x)}" data-equation="eq:rational_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/math/base/tools/evalrational/docs/img/equation_rational_function.svg" alt="Rational function definition.">
    <br>
</div> -->

<!-- </equation> -->

where both `P(x)` and `Q(x)` are polynomials in `x`. A [polynomial][polynomial] in `x` can be expressed

<!-- <equation class="equation" label="eq:polynomial" align="center" raw="c_nx^n + c_{n-1}x^{n-1} + \ldots + c_1x^1 + c_0 = \sum_{i=0}^{n} c_ix^i" alt="Polynomial expression."> -->

<!-- <div class="equation" align="center" data-raw-text="c_nx^n + c_{n-1}x^{n-1} + \ldots + c_1x^1 + c_0 = \sum_{i=0}^{n} c_ix^i" data-equation="eq:polynomial">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/math/base/tools/evalrational/docs/img/equation_polynomial.svg" alt="Polynomial expression.">
    <br>
</div>

<!-- </equation> -->

where `c_n, c_{n-1}, ..., c_0` are constants.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var evalrational = require( '@stdlib/math/base/tools/evalrational' );
```

#### evalrational( P, Q, x )

Evaluates a [rational function][rational-function] at a value `x` using double-precision floating-point arithmetic.

```javascript
var P = [ -6.0, -5.0 ];
var Q = [ 3.0, 0.5 ];

var v = evalrational( P, Q, 6.0 ); // => ( -6*6^0 - 5*6^1 ) / ( 3*6^0 + 0.5*6^1 ) = (-6-30)/(3+3)
// returns -6.0
```

For polynomials of different degree, the coefficient array for the lower degree [polynomial][polynomial] should be padded with zeros.

```javascript
// 2x^3 + 4x^2 - 5x^1 - 6x^0 => degree 4
var P = [ -6.0, -5.0, 4.0, 2.0 ];

// 0.5x^1 + 3x^0 => degree 2
var Q = [ 3.0, 0.5, 0.0, 0.0 ]; // zero-padded

var v = evalrational( P, Q, 6.0 ); // => ( -6*6^0 - 5*6^1 + 4*6^2 + 2*6^3 ) / ( 3*6^0 + 0.5*6^1 + 0*6^2 + 0*6^3 ) = (-6-30+144+432)/(3+3)
// returns 90.0
```

Coefficients should be ordered in **ascending** degree, thus matching summation notation.

#### evalrational.factory( P, Q )

Uses code generation to in-line coefficients and return a function for evaluating a [rational function][rational-function] using double-precision floating-point arithmetic.

```javascript
var P = [ 20.0, 8.0, 3.0 ];
var Q = [ 10.0, 9.0, 1.0 ];

var rational = evalrational.factory( P, Q );

var v = rational( 10.0 ); // => (20*10^0 + 8*10^1 + 3*10^2) / (10*10^0 + 9*10^1 + 1*10^2) = (20+80+300)/(10+90+100)
// returns 2.0

v = rational( 2.0 ); // => (20*2^0 + 8*2^1 + 3*2^2) / (10*2^0 + 9*2^1 + 1*2^2) = (20+16+12)/(10+18+4)
// returns 1.5
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   The coefficients `P` and `Q` are expected to be arrays of the **same** length.
-   For hot code paths in which coefficients are invariant, a compiled function will be more performant than `evalrational()`.
-   While code generation can boost performance, its use may be problematic in browser contexts enforcing a strict [content security policy][mdn-csp] (CSP). If running in or targeting an environment with a CSP, avoid using code generation.

</section>

<!-- /.notes -->

<section class="examples">
## Examples

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

```javascript
var discreteUniform = require( '@stdlib/random/array/discrete-uniform' );
var uniform = require( '@stdlib/random/base/uniform' );
var evalrational = require( '@stdlib/math/base/tools/evalrational' );

// Create two arrays of random coefficients...
var P = discreteUniform( 10, -100, 100 );
var Q = discreteUniform( 10, -100, 100 );

// Evaluate the rational function at random values...
var v;
var i;
for ( i = 0; i < 100; i++ ) {
    v = uniform( 0.0, 100.0 );
    console.log( 'f(%d) = %d', v, evalrational( P, Q, v ) );
}

// Generate an `evalrational` function...
var rational = evalrational.factory( P, Q );
for ( i = 0; i < 100; i++ ) {
    v = uniform( -50.0, 50.0 );
    console.log( 'f(%d) = %d', v, rational( v ) );
}
```

</section>

<!-- /.examples -->

<!-- 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/tools/evalpoly`][@stdlib/math/base/tools/evalpoly]</span><span class="delimiter">: </span><span class="description">evaluate a polynomial using double-precision floating-point arithmetic.</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">

[polynomial]: https://en.wikipedia.org/wiki/Polynomial

[rational-function]: https://en.wikipedia.org/wiki/Rational_function

[mdn-csp]: https://developer.mozilla.org/en-US/docs/Web/HTTP/CSP

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

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

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

</section>

<!-- /.links -->
