Relative Difference

Compute the relative difference of two real numbers.

The relative difference of two real numbers is defined as

where |x-y| is the absolute difference and f(x,y) is a scale function. Common scale functions include

The choice of scale function depends on application context.

Installation

npm install @stdlib/math-base-utils-relative-difference

Usage

var reldiff = require( '@stdlib/math-base-utils-relative-difference' );

reldiff( x, y[, scale] )

Computes the relative difference of two real numbers.

var d = reldiff( 2.0, 5.0 ); // => 3/5
// returns 0.6

d = reldiff( -1.0, 3.14 ); // => 4.14/3.14
// returns ~1.318

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 = reldiff( -2.0, 5.0 ); // => |-7/5|
// returns 1.4

d = reldiff( -2.0, 5.0, 'max-abs' ); // => |-7/5|
// returns 1.4

d = reldiff( -2.0, 5.0, 'max' ); // => |-7/5|
// returns 1.4

d = reldiff( -2.0, 5.0, 'min-abs' ); // => |-7/2|
// returns 3.5

d = reldiff( -2.0, 5.0, 'min' ); // => |-7/-2|
// returns 3.5

d = reldiff( -2.0, 5.0, 'mean-abs' ); // => |-7/3.5|
// returns 2.0

d = reldiff( -2.0, 5.0, 'mean' ); // => |-7/1.5|
// returns ~4.67

d = reldiff( -2.0, 5.0, 'x' ); // => |-7/-2|
// returns 3.5

d = reldiff( 5.0, -2.0, 'x' ); // => |7/5|
// returns 1.4

d = reldiff( -2.0, 5.0, 'y' ); // => |-7/5|
// returns 1.4

d = reldiff( 5.0, -2.0, 'y' ); // => |7/-2|
// returns 3.5

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

var abs = require( '@stdlib/math-base-special-abs' );
var EPS = require( '@stdlib/constants-float64-eps' );

function scale( x, y ) {
    var s;
    x = abs( x );
    y = abs( y );

    // Maximum absolute value:
    s = (x < y ) ? y : x;

    // Scale in units of epsilon:
    return s * EPS;
}

var d = reldiff( 12.15, 12.149999999999999, scale );
// returns ~0.658

Notes

• If the absolute difference of x and y is 0, the relative difference is always 0.

  var d = reldiff( 0.0, 0.0 );
  // returns 0.0
  
  d = reldiff( 3.14, 3.14 );
  // returns 0.0

• If |x| = |y| = infinity, the function returns NaN.

  var d = reldiff( Infinity, Infinity );
  // returns NaN
  
  d = reldiff( -Infinity, -Infinity );
  // returns NaN

• If |x| = |-y| = infinity, the relative difference is +infinity.

  var d = reldiff( Infinity, -Infinity );
  // returns Infinity
  
  d = reldiff( -Infinity, Infinity );
  // returns Infinity

• If a scale function returns 0, the function returns NaN.

  var d = reldiff( 0.0, 2.0, 'mean' ); // => |2/1|
  // returns 2.0
  
  d = reldiff( -1.0, 1.0, 'mean' ); // => |2/0|
  // returns NaN

Examples

var randu = require( '@stdlib/random-base-randu' );
var reldiff = require( '@stdlib/math-base-utils-relative-difference' );

var scales = [ 'max-abs', 'max', 'min-abs', 'min', 'mean-abs', 'mean', 'x', 'y' ];
var x;
var y;
var d;
var i;
var j;

for ( i = 0; i < 100; i++ ) {
    x = ( randu()*1.0e4 ) - 5.0e3;
    y = ( randu()*1.0e4 ) - 5.0e3;
    for ( j = 0; j < scales.length; j++ ) {
        d = reldiff( x, y, scales[j] );
        console.log( 'x = %d. y = %d. d = %d. scale: %s.', x, y, d, scales[j] );
    }
}

See Also

• @stdlib/math-base/utils/absolute-difference: compute the absolute difference of two real numbers.
• @stdlib/math-base/utils/float64-epsilon-difference: compute the relative difference of two real numbers in units of double-precision floating-point epsilon.

Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

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License

See LICENSE.

Copyright

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