Implementation limitations of float.as_integer_ratio()

Question

Recently, a correspondent mentioned float.as_integer_ratio(), new in Python 2.6, noting that typical floating point implementations are essentially rational approximations o

Answer 1

You get better approximations using

fractions.Fraction.from_float(math.pi).limit_denominator()

Fractions are included since maybe version 3.0. However, math.pi doesn't have enough accuracy to return a 30 digit approximation.

Answer 2

May I recommend gmpy's implementation of the Stern-Brocot tree:

>>> import gmpy
>>> import math
>>> gmpy.mpq(math.pi)
mpq(245850922,78256779)
>>> x=_
>>> float(x)
3.1415926535897931
>>> 

again, the result is "correct within the precision of 64-bit floats" (53-bit "so-called" mantissas;-), but:

>>> 245850922 + 78256779
324107701
>>> 884279719003555 + 281474976710656
1165754695714211L
>>> 428224593349304L + 136308121570117
564532714919421L

...gmpy's precision is obtained so much cheaper (in terms of sum of numerator and denominator values) than Arima's, much less Python 2.6's!-)

Answer 3

The algorithm used by as_integer_ratio only considers powers of 2 in the denominator. Here is a (probably) better algorithm.