Question

My assumptions for IEEE 754-2008:

binary16 – 2^16 distinct values,
binary32 – 2^32 distinct values,
…
binary128 – 2^128 distinct values.

Is this correct?

Answer 1

This is a trick question.

The floating-point formats define some special values. Whether you count these as distinct depends on your point of view. The following is for double-precision (binary64):

1. There are two representations of 0: with the sign bit 0 or 1 and both the exponent and mantissa all zero. The values are distinguishable by the fact that 1/+0 = infinity and 1/-0 = -infinity. But they compare equal.
2. There are 2 infinities, where the first 12 bits are 0x7ff or 0xfff and the mantissa is all zero. These are not finite real numbers, but they are values.
3. There is a whole range of Not-A-Number (NaN) values, having sign+exponent bits 0x7ff (signaling NaN) or 0xfff (“quiet” NaN) and a nonzero mantissa. Again, these are not real numbers but they are distinguishable values.

So, to summarize:

1. The total number of distinguishable values (real numbers or otherwise) is 2^64.
2. The number of distinct real numbers, excluding infinities and counting zero only once, is 2*(2^11-1)*2^52-1 = 18,437,736,874,454,810,623.

For binary16, the number of distinct real numbers is 2*(2^5-1)*2^10-1 = 63,487. For binary32, it’s 2*(2^8-1)*2^23-1=4,278,190,079. For binary128, it’s 2*(2^15-1)*2^112-1 or about 3.4*10^38.