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typesfloating-pointieee-754

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

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.

Solution

  • 2mantissa bits + 1 + 1

    The +1 in the exponent (mantissa bits + 1) is because, if the mantissa contains abcdef... the number it represents is actually 1.abcdef... × 2^e, providing an extra implicit bit of precision.

    Therefore, the first integer that cannot be accurately represented and will be rounded is:

    • For 32-bit floats, 16,777,217 (224 + 1).
    • For 64-bit floats, 9,007,199,254,740,993 (253 + 1).

    Here's an example in CPython 3.10, which uses 64-bit floats:

    >>> 9007199254740993.0
9007199254740992.0