Which is the first integer that an IEEE 754 float is incapable of representing exactly?

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For clarity, if I’m using a language that implements IEE 754 floats and I declare:

float f0 = 0.f;
float f1 = 1.f;

…and then print them back out, I’ll get 0.0000 and 1.0000 – exactly.

But IEEE 754 isn’t capable of representing all the numbers along the real line. Close to zero, the ‘gaps’ are small; as you get further away, the gaps get larger.

So, my question is: for an IEEE 754 float, which is the first (closest to zero) integer which cannot be exactly represented? I’m only really concerned with 32-bit floats for now, although I’ll be interested to hear the answer for 64-bit if someone gives it!

I thought this would be as simple as calculating 2bits_of_mantissa and adding 1, where bits_of_mantissa is how many bits the standard exposes. I did this for 32-bit floats on my machine (MSVC++, Win64), and it seemed fine, though.

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