# Epsilon Difference

Compute the relative difference of two real numbers in units of double-precision floating-point epsilon.

## Usage

``````var epsdiff = require( '@stdlib/math/base/utils/float64-epsilon-difference' );
``````

#### epsdiff( x, y[, scale] )

Computes the relative difference of two real numbers in units of double-precision floating-point epsilon.

``````var d = epsdiff( 12.15, 12.149999999999999 ); // => ~0.658ε
// returns ~0.658
``````

The following `scale` functions are supported:

• max-abs: maximum absolute value of `x` and `y` (default).
• max: maximum value of `x` and `y`.
• min-abs: minimum absolute value of `x` and `y`.
• min: minimum value of `x` and `y`.
• mean-abs: arithmetic mean of the absolute values of `x` and `y`.
• mean: arithmetic mean of `x` and `y`.
• x: `x` (noncommutative).
• y: `y` (noncommutative).

By default, the function scales the absolute difference by dividing the absolute difference by the maximum absolute value of `x` and `y`. To scale by a different function, specify a scale function name.

``````var d = epsdiff( 2.4341309458983933, 2.4341309458633909, 'mean-abs' ); // => ~64761.5ε => ~1.438e-11
// returns ~64761.5
``````

To use a custom scale function, provide a `function` which accepts two numeric arguments `x` and `y`.

``````function scale( x, y ) {
// Return the minimum value:
return ( x > y ) ? y : x;
}

var d = epsdiff( 1.0000000000000002, 1.0000000000000100, scale ); // => ~44ε
// returns ~44
``````

## Notes

• If computing the relative difference in units of epsilon will result in overflow, the function returns the maximum double-precision floating-point number.

``````var d = epsdiff( 1.0e304, 1.0, 'min' ); // => ~1.798e308ε => 1.0e304/ε overflows
// returns ~1.798e308
``````
• If the absolute difference of `x` and `y` is `0`, the relative difference is always `0`.

``````var d = epsdiff( 0.0, 0.0 );
// returns 0.0

d = epsdiff( 3.14, 3.14 );
// returns 0.0
``````
• If `x = y = +infinity` or `x = y = -infinity`, the function returns `NaN`.

``````var PINF = require( '@stdlib/constants/float64/pinf' );
var NINF = require( '@stdlib/constants/float64/ninf' );

var d = epsdiff( PINF, PINF );
// returns NaN

d = epsdiff( NINF, NINF );
// returns NaN
``````
• If `x = -y = +infinity` or `-x = y = +infinity`, the relative difference is `+infinity`.

``````var PINF = require( '@stdlib/constants/float64/pinf' );
var NINF = require( '@stdlib/constants/float64/ninf' );

var d = epsdiff( PINF, NINF );
// returns Infinity

d = epsdiff( NINF, PINF );
// returns Infinity
``````
• If a `scale` function returns `0`, the function returns `NaN`.

``````var d = epsdiff( -1.0, 1.0, 'mean' ); // => |2/0|
// returns NaN
``````

## Examples

``````var randu = require( '@stdlib/random/base/randu' );
var EPS = require( '@stdlib/constants/float64/eps' );
var epsdiff = require( '@stdlib/math/base/utils/float64-epsilon-difference' );

var sign;
var x;
var y;
var d;
var i;

for ( i = 0; i < 100; i++ ) {
x = randu();
sign = ( randu() > 0.5 ) ? 1.0 : -1.0;
y = x + ( sign*EPS*i );
d = epsdiff( x, y );
console.log( 'x = %d. y = %d. d = %dε.', x, y, d );
}
``````