Inaccurate Logarithm in Python

Question

I work daily with Python 2.4 at my company. I used the versatile logarithm function \'log\' from the standard math library, and when I entered log(2**31, 2) it returned 31.0

Answer 1

Always assume that floating point operations will have some error in them and check for equality taking that error into account (either a percentage value like 0.00001% or a fixed value like 0.00000000001). This inaccuracy is a given since not all decimal numbers can be represented in binary with a fixed number of bits precision.

Your particular case is not one of them if Python uses IEEE754 since 31 should be easily representable with even single precision. It's possible however that it loses precision in one of the many steps it takes to calculate log₂2³¹, simply because it doesn't have code to detect special cases like a direct power of two.

Answer 2

If you wish to calculate the highest power of 'k' in a number 'n'. Then the code below might be helpful:

import math
answer = math.ceil(math.log(n,k))
while k**answer>n:
    answer-=1

NOTE: You shouldn't use 'if' instead of 'while' because that will give wrong results in some cases like n=2**51-1 and k=2. In this example with 'if' the answer is 51 whereas with 'while' the answer is 50, which is correct.

Answer 3

IEEE double floating point numbers have 52 bits of precision. Since 10^15 < 2^52 < 10^16, a double has between 15 and 16 significant figures. The result 31.000000000000004 is correct to 16 figures, so it is as good as you can expect.