# evalrational
> Evaluate a [rational function][rational-function].
A [rational function][rational-function] `f(x)` is defined as
where both `P(x)` and `Q(x)` are polynomials in `x`. A [polynomial][polynomial] in `x` can be expressed
where `c_n, c_{n-1}, ..., c_0` are constants.
## Usage
```javascript
var evalrational = require( '@stdlib/math/base/tools/evalrational' );
```
#### evalrational( P, Q, x )
Evaluates a [rational function][rational-function] at a value `x`. The coefficients `P` and `Q` are expected to be arrays of the **same** length.
```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].
```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
```
## Notes
- 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.
## Examples
```javascript
var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var Float64Array = require( '@stdlib/array/float64' );
var evalrational = require( '@stdlib/math/base/tools/evalrational' );
var rational;
var sign;
var len;
var P;
var Q;
var v;
var i;
// Create two arrays of random coefficients...
len = 10;
P = new Float64Array( len );
Q = new Float64Array( len );
for ( i = 0; i < len; i++ ) {
if ( randu() < 0.5 ) {
sign = -1.0;
} else {
sign = 1.0;
}
P[ i ] = sign * round( randu()*100 );
Q[ i ] = sign * round( randu()*100 );
}
// Evaluate the rational function at random values...
for ( i = 0; i < 100; i++ ) {
v = randu() * 100.0;
console.log( 'f(%d) = %d', v, evalrational( P, Q, v ) );
}
// Generate an `evalrational` function...
rational = evalrational.factory( P, Q );
for ( i = 0; i < 100; i++ ) {
v = (randu()*100.0) - 50.0;
console.log( 'f(%d) = %d', v, rational( v ) );
}
```
[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
