# Epsilon Difference
> Compute the [relative difference][@stdlib/math/base/utils/relative-difference] of two real numbers in units of [double-precision floating-point epsilon][@stdlib/constants/float64/eps].
## Usage
```javascript
var epsdiff = require( '@stdlib/math/base/utils/float64-epsilon-difference' );
```
#### epsdiff( x, y\[, scale] )
Computes the [relative difference][@stdlib/math/base/utils/relative-difference] of two real numbers in units of [double-precision floating-point epsilon][@stdlib/constants/float64/eps].
```javascript
var d = epsdiff( 12.15, 12.149999999999999 ); // => ~0.658ε
// returns ~0.658
```
The following [`scale`][@stdlib/math/base/utils/relative-difference] functions are supported:
- **max-abs**: maximum [absolute value][@stdlib/math/base/special/abs] of `x` and `y` (_default_).
- **max**: maximum value of `x` and `y`.
- **min-abs**: minimum [absolute value][@stdlib/math/base/special/abs] of `x` and `y`.
- **min**: minimum value of `x` and `y`.
- **mean-abs**: arithmetic mean of the [absolute values][@stdlib/math/base/special/abs] 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][@stdlib/math/base/utils/absolute-difference] by dividing the [absolute difference][@stdlib/math/base/utils/absolute-difference] by the maximum [absolute value][@stdlib/math/base/special/abs] of `x` and `y`. To scale by a different function, specify a scale function name.
```javascript
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`.
```javascript
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][@stdlib/math/base/utils/relative-difference] in units of [epsilon][@stdlib/constants/float64/eps] will result in overflow, the function returns the [maximum double-precision floating-point number][@stdlib/constants/float64/max].
```javascript
var d = epsdiff( 1.0e304, 1.0, 'min' ); // => ~1.798e308ε => 1.0e304/ε overflows
// returns ~1.798e308
```
- If the [absolute difference][@stdlib/math/base/utils/absolute-difference] of `x` and `y` is `0`, the [relative difference][@stdlib/math/base/utils/relative-difference] is **always** `0`.
```javascript
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`.
```javascript
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][@stdlib/math/base/utils/relative-difference] is `+infinity`.
```javascript
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`.
```javascript
var d = epsdiff( -1.0, 1.0, 'mean' ); // => |2/0|
// returns NaN
```
## Examples
```javascript
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 );
}
```
[@stdlib/constants/float64/eps]: https://www.npmjs.com/package/@stdlib/constants-float64-eps
[@stdlib/constants/float64/max]: https://www.npmjs.com/package/@stdlib/constants-float64-max
[@stdlib/math/base/special/abs]: https://www.npmjs.com/package/@stdlib/math/tree/main/base/special/abs
[@stdlib/math/base/utils/absolute-difference]: https://www.npmjs.com/package/@stdlib/math/tree/main/base/utils/absolute-difference
[@stdlib/math/base/utils/relative-difference]: https://www.npmjs.com/package/@stdlib/math/tree/main/base/utils/relative-difference