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# 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].

<section class="usage">

## 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
```

</section>

<!-- /.usage -->

<section class="notes">

## 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
    ```

</section>

<!-- /.notes -->

<section class="examples">

## Examples

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

```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 );
}
```

</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/utils/absolute-difference`][@stdlib/math/base/utils/absolute-difference]</span><span class="delimiter">: </span><span class="description">compute the absolute difference of two real numbers.</span>
-   <span class="package-name">[`@stdlib/math/base/utils/relative-difference`][@stdlib/math/base/utils/relative-difference]</span><span class="delimiter">: </span><span class="description">compute the relative difference of two real numbers.</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">

[@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://github.com/stdlib-js/math/tree/main/base/special/abs

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

[@stdlib/math/base/utils/absolute-difference]: https://github.com/stdlib-js/math/tree/main/base/utils/absolute-difference

[@stdlib/math/base/utils/relative-difference]: https://github.com/stdlib-js/math/tree/main/base/utils/relative-difference

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

</section>

<!-- /.links -->
