Smallest integer after only integers can be expressed IEEE 754

Question

What is the smallest integer after which only integers can be represented by IEEE 754 (single/double)?

My guess is that it occurs when the exponent is 254 (11111110b single). With that, no matter what one puts in the mantissa the number will be an integer. I just want to confirm that and what would be the equivalence in powers of 2.

Answer 1

Presuming that “after” refers to numbers with greater magnitude and not later in time, 2²³ for 32-bit binary floating-point and 2⁵² for 64-bit binary floating-point. (These are float and double in the most common programming language implementations with those type names.)

This is because the significands of the 32-bit and 64-bit formats have 24 and 53 bits, respectively. So, in the 32-bit format, if the high bit is scaled to 2²² by the exponent, the low bit would be 2^–1 (from 22 to –1, inclusive, is 24 positions). Since the significand contains a bit scaled to a non-integer value, the complete represented value could be a non-integer. If the high bit is scaled to 2²³ or greater, then the low bit has value at least 2⁰.

So, if an integer is less than 2²³, it is at most 2²³–1, which means its high significand bit is scaled to 2²², so the low bit is 2^–1. Indeed, 2²³–½ is exactly representable as a float.

The same reasoning applies to double.