Largest odd integer that can be represented as a float

Question

I was wondering what largest odd integer that can be represented exactly as a float? And why there is a difference between the largest even integer represented as a float in this case.

I believe it would have to do with the base 2 exponents 2^n-1, however I am not familiar enough with data representation in C to see the distinction.

Answer 1

For IEEE-754 basic 32-bit binary floating-point, the largest representable odd integer is 2²⁴−1.

For IEEE-754 basic 64-bit binary floating-point, the largest representable odd integer is 2⁵³−1.

This is due to the fact that the formats have 24-bit and 53-bit significands. (The significand is the fraction part of a floating-point number.)

The values represented by the bits in the significand are scaled according to the exponent of the floating-point number. In order to represent an odd number, the floating-point number must have a bit in the significand that represents 2⁰. With a 24-bit significand, if the lowest bit represents 2⁰, then the highest bit represents 2²³. The largest value is obtained when all the bits are on, which makes the value 2⁰ + 2¹ + 2² + … 2²³, which equals 2²⁴−1.

More generally, the largest representable odd integer is normally scalbnf(1, FLT_MANT_DIG) - 1. This can also be computed as (2 - FLT_EPSILON) / FLT_EPSILON. (This assumes a normal case in which FLT_RADIX is even and FLT_MANT_DIG <= FLT_MAX_EXP. Note that if FLT_MANT_DIG == FLT_MAX_EXP, the latter expression, with FLT_EPSILON, should be used, because the former overflows.)

The abnormal cases, just for completeness:

• If FLT_RADIX is odd and FLT_MANT_DIG <= FLT_MAX_EXP, the largest representable odd integer is FLT_MAX if FLT_MANT_DIG is odd and FLT_MAX - scalbnf(FLT_EPSILON, FLT_MAX_EXP+1) otherwise.
• If FLT_RADIX is even and FLT_MANT_DIG > FLT_MAX_EXP, then: If FLT_MAX_EXP > 0, the largest representable odd integer is floorf(FLT_MAX). Otherwise, no odd integers are representable.
• If FLT_RADIX is odd and FLT_MANT_DIG > FLT_MAX_EXP, then: If FLT_MAX_EXP > 0, the largest representable odd integer is floorf(FLT_MAX) if FLT_MANT_DIG - FLT_MAX_EXP is odd or floorf(FLT_MAX)-1 otherwise. Otherwise, no odd integers are representable.