Recipe18.13.Converting Numbers to Rationals via Farey Fractions

Recipe 18.13. Converting Numbers to Rationals via Farey Fractions

Credit: Scott David Daniels

Problem

You have a number v (of almost any type) and need to find a rational number (in reduced form) that is as close to v as possible but with a denominator no larger than a prescribed value.

Solution

Farey fractions, whose crucial properties were studied by Augustin Louis Cauchy, are an excellent way to find rational approximations of floating-point values:

def farey(v, lim):     """ No error checking on args.  lim = maximum denominator.         Results are (numerator, denominator); (1, 0) is "infinity".     """     if v < 0:         n, d = farey(-v, lim)         return -n, d     z = lim - lim    # Get a "0 of the right type" for denominator     lower, upper = (z, z+1), (z+1, z)     while True:         mediant = (lower[0] + upper[0]), (lower[1] + upper[1])         if v * mediant[1] > mediant[0]:             if lim < mediant[1]:                 return upper             lower = mediant         elif v * mediant[1] == mediant[0]:             if lim >= mediant[1]:                 return mediant             if lower[1] < upper[1]:                 return lower             return upper         else:             if lim < mediant[1]:                 return lower             upper = mediant

For example:

import math print farey(math.pi, 100) # emits: (22, 7)

Discussion

The rationals resulting from the algorithm shown in this recipe are in reduced form (meaning that numerator and denominator are mutually prime), but the proof, which was given by Cauchy, is rather subtle (see http://www.cut-the-knot.com/blue/Farey.html).

You can use farey to compute odds from a probability, such as:

probability = 0.333 n, d = farey(probability, 100) print "Odds are %d : %d" % (n, d-n) # emits: Odds are 1 : 2

This recipe's algorithm is ideally suited for reimplementation in a lower-level language (e.g., C, or assembly, or, maybe best, Pyrex) if you use it heavily. Since the code uses only multiplication and addition, it can play optimally to hardware strengths.

If you are using this recipe in an environment where you call it with a lot of values near 0.0, 1.0, or 0.5 (or other simple fractions), you may find that the algorithm's convergence is too slow. You can improve convergence in a continued fraction style, by appending to the first if in the farey function:

if v < 0: ... elif v < 0.5:     n, d = farey((v-v+1)/v, lim)     # lim is wrong; decide what you want     return d, n elif v > 1:     intpart = floor(v)     n, d = farey(v-intpart)     return n+intpart*d, d ...

James Farey was an English geologist and surveyor who wrote a letter to the Journal of Science in 1816. In that letter he observed that, while reading a privately published list of the decimal equivalents of fractions, he had noticed an interesting fact. Consider the set of all the fractions with values between 0 and 1, reduced to the lowest terms, with denominators not exceeding some integer N. Arrange the set in order of magnitude to get a sequence. For example, for N equal to 5, the Farey sequence is:

0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1

For any three consecutive fractions in this sequence (e.g., A/B, C/D, E/F), the middle fraction (C/D), called the mediant, is equal to the ratio (A + E)/(B + F). I enjoy envisioning Mr. Farey sitting up late on a rainy English night, reading tables of decimal expansions of fractions by an oil lamp. Calculation has come a long way since his day, and I'm pleased to be able to benefit from his work.

Recipe18.13.Converting Numbers to Rationals via Farey Fractions