pythonnumpyfloating-pointprecisionsingle-precision

double precision and single precision floating point numbers?


I was wondering why double precision and single precision numbers are sometimes equal and sometimes not. For example when I have the following they are not equal:

import numpy as np

x=np.float64(1./3.)
y=np.float32(1./3.)

but the following are equal:

x=np.float64(3.)
y=np.float32(3.)

I understand why the first set of x and y is not equal but I am not quite sure as to why the second set is equal.


Solution

  • This answer assumes single is IEEE 754 32 bit binary floating point, and double is the corresponding 64 bit type.

    Any value that can be represented exactly in a single can also be represented exactly as a double. That is the case for 3.0. The closest single and the closest double both have value exactly 3, and are equal.

    If a number cannot be represented exactly in a single, the double is likely to be a closer approximation and different from the single. That is the case for 1.0/3.0. The closest single is 0.3333333432674407958984375. The closest double is 0.333333333333333314829616256247390992939472198486328125.

    Both single and double are binary floating point. A number cannot be expressed exactly unless it is equal a fraction of the form A/(2**B), where A is an integer, B is a natural number, and "**" represents exponent. Numbers such as 0.1 and 0.2 that are terminating decimal fractions but not terminating binary fractions behave like 1/3.0. For example, the closest single to 0.1 is 0.100000001490116119384765625, the closest double is 0.1000000000000000055511151231257827021181583404541015625