I know that IEEE754 double floats (64-bit floating number) is known to provide 52 bits of precision (or 53 bits including implicit 1). But I do not know the exact meaning of the precision.
Suppose we want to approximate a rational number $v$ using 64-bit floats, denoting the approximate version by $v'$. Clearly there exists an error $ e = |v-v'|$. Does the precision mean that $e$ is bounded by some number, like $ e < 2^{-53}$?
How many discrete points can be expressive in $[-1/2, 1/2)$ with IEEE 754 double floats?