Today, I was looking through some C++ code (written by somebody else) and found this section:
double someValue = ...
if (someValue < std::numeric_limits<double>::epsilon() &&
someValue > -std::numeric_limits<double>::epsilon()) {
someValue = 0.0;
}
I'm trying to figure out whether this even makes sense.
The documentation for epsilon()
says:
The function returns the difference between 1 and the smallest value greater than 1 that is representable [by a double].
Does this apply to 0 as well, i.e. epsilon()
is the smallest value greater than 0? Or are there numbers between 0
and 0 + epsilon
that can be represented by a double
?
If not, then isn't the comparison equivalent to someValue == 0.0
?
Assuming 64-bit IEEE double, there is a 52-bit mantissa and 11-bit exponent. Let's break it to bits:
1.0000 00000000 00000000 00000000 00000000 00000000 00000000 × 2^0 = 1
The smallest representable number greater than 1:
1.0000 00000000 00000000 00000000 00000000 00000000 00000001 × 2^0 = 1 + 2^-52
Therefore:
epsilon = (1 + 2^-52) - 1 = 2^-52
Are there any numbers between 0 and epsilon? Plenty... E.g. the minimal positive representable (normal) number is:
1.0000 00000000 00000000 00000000 00000000 00000000 00000000 × 2^-1022 = 2^-1022
In fact there are (1022 - 52 + 1)×2^52 = 4372995238176751616
numbers between 0 and epsilon, which is 47% of all the positive representable numbers...