Question

How many rep­re­sentable floats are there be­tween 0.0 and 0.5? And how many representable floats are there between 0.5 and 1.0? I’m more interested in the math behind it, and I need the answer for floats and doubles.

Answer 1

For IEEE754 floats, this is fairly straight forward. Fire up the Online Float Calculator and read on.

All pure powers of 2 are represented by a mantissa 0, which is actually 1.0 due to the implied leading 1. The exponent is corrected by a bias, so 1 and 0.5 are respectively 1.0 × 2⁰ and 1.0 × 2⁻¹, or in binary:

      S    Ex + 127    Mantissa - 1                Hex

1:    0    01111111    00000000000000000000000     0x3F800000
      +     0 + 127    1.0

0.5:  0    01111110    00000000000000000000000     0x3F000000
      +    -1 + 127    1.0

Since the floating point numbers represented in this form are ordered in the same order as their binary representation, we only need to take the difference of the integral value of the binary representation and conclude that there are 0x800000 = 2²³, i.e. 8,388,608 single-precision floating point values in the interval [0.5, 1.0).

Similarly, the answer is 2⁵² for double and 2⁶³ for long double.