Inaccurate Logarithm in Python
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 31.000000000000004, which struck me as a bit odd.
I did the same thing with other powers of 2, and it worked perfectly. I ran 'log10(2**31) / log10(2)' and I got a round 31.0
I tried running the same original function in Python 3.0.1, assuming that it was fixed in a more advanced version.
Why does this happen? Is it possible that there are some inaccuracies in mathematical functions in Python?
This is to be expected with computer arithmetic. It is following particular rules, such as IEEE 754, that probably don't match the math you learned in school.
If this actually matters, use Python's decimal type.
Example:
from decimal import Decimal, Context
ctx = Context(prec=20)
two = Decimal(2)
ctx.divide(ctx.power(two, Decimal(31)).ln(ctx), two.ln(ctx))
You should read "What Every Computer Scientist Should Know About Floating-Point Arithmetic".
http://docs.sun.com/source/806-3568/ncg_goldberg.html
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.