Floating point differences

floatdiff measures directed differences between floating point numbers by counting the spaces between them.

This distance was proposed by an anonymous reviewer to "On the definition of ulp(x)" (JM Muller 2005).

Various implementations included.

python

from floatdiff import floatdiff

floatdiff(1., 1. + 2.**-52) # 1.
floatdiff((1. + 5**0.5)/2, 1.6180339887) # -224707.
floatdiff(-0., 0.) # 1.

numpy

from floatdiff_numpy import floatdiff
from numpy import float32

floatdiff(1., [1. + 2.**-52, 1. + 2.**-50]) # array([1., 4.])
floatdiff(float32((1 + 5**0.5)/2), float32(1.6180339887)) # 0.
floatdiff(-0., float32(0.)) # TypeError

c

#include <stdint.h>
#include "floatdiff.c"

floatdiff(1., 1. + pow(2, -52)); /* 1. */
floatdiff((1. + sqrt(5))/2, 1.6180339887); /* -224707. */
floatdifff(-0., 0.) /* 1. */

go

floatdiff.Floatdiff(1, 1 + math.Pow(2, -52)) // 1.
floatdiff.Floatdiff((1 + math.Sqrt(5))/2, 1.6180339887) // -224707.
zero := float32(0)
floatdiff.Floatdiff32(-zero, 0) // 1.

Details

Each float or double gets an integer valuation rank(x) satisfying

rank(0.) == 0 # True

and

rank(nextafter(x)) == rank(x) + 1 # True .

Floats almost have this naturally when reinterpreted as integers, but are reversed for negative numbers.

We just reverse negative numbers' order.

Then

floatdiff(x, y) == float(rank(y) - rank(x)) # True

uses floating point for coverage of small and large distances.

A bits-precision equivalent conversion is given by bits.