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.
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