What types of numbers are representable in binary floating-point?

Question

I\'ve read a lot about floats, but it\'s all unnecessarily involved. I think I\'ve got it pretty much understood, but there\'s just one thing I\'d like to know for

Answer 1

A finite number can be represented in the common IEEE 754 double-precision format if and only if it equals M•2^e for some integers M and e such that -2⁵³ < M < 2⁵³ and -1074 ≤ e ≤ 971.

For single precision, -2²⁴ < M < 2²⁴ and -149 ≤ e ≤ 104.

For double-precision, these are consequences of the facts that the double-precision format uses 52 bits to store a significand (which normally has 53 bits due to an implicit 1) and uses 11 bits to store an exponent. 11 bits encodes numbers from 0 to 2047, but 0 and 2047 are excluded for special purposes, and the encoded number is biased by 1023, so it represents unbiased exponents from -1022 to 1023. However, these unbiased exponents are for significands in the interval [1, 2), and those significands have fractions. To express the significand as an integer, I adjusted the exponent range by 52. Single-precision is similar, with 23 bits to store a 24-bit significand, 8 bits for the exponent, and a bias of 127.

Expressing the representable numbers using an integer times a power of two rather than the more common fractional significand simplifies some number theory and other reasoning about floating-point properties. I used it in this answer because it allows the set of representable values to be expressed concisely.