floating point - Smallest integer after only integers can be expressed IEEE 754 -

what smallest integer after integers can represented ieee 754 (single/double)?

my guess occurs when exponent 254 (11111110b single). that, no matter 1 puts in mantissa number integer. want confirm , equivalence in powers of 2.

presuming “after” refers numbers greater magnitude , not later in time, 2²³ 32-bit binary floating-point , 2⁵² 64-bit binary floating-point. (these float , double in common programming language implementations type names.)

this because significands of 32-bit , 64-bit formats have 24 , 53 bits, respectively. so, in 32-bit format, if high bit scaled 2²² exponent, low bit 2^–1 (from 22 –1, inclusive, 24 positions). since significand contains bit scaled non-integer value, complete represented value non-integer. if high bit scaled 2²³ or greater, low bit has value @ least 2⁰.

so, if integer less 2²³, @ 2²³–1, means high significand bit scaled 2²², low bit 2^–1. indeed, 2²³–½ representable float.

the same reasoning applies double.