Crazy Little Thing Called Float

What's wrong with this C++ code?

int main() {
    float f = 16777217.f;
    while (f < 16777218.f) {
        f += 1.f;
    }
    printf("%f\n", f);
}

It will run forever.

It is pretty common knowledge that the best way to get broke is to use floats to store money. Clearly, not all rational numbers can be represented as floating-point numbers:

printf("%1.10f\n", 0.7f); // 0.6999999881

A little less known fact is that floats distribution is not uniform. The farther away from 0.0 the larger gaps without any representation.

Lets look at the numbers from the first code sample:

printf("%20.16f\n", 16777217.f); // 16777216.0000000000000000
printf("%20.16f\n", 16777218.f); // 16777218.0000000000000000

Now we can see that between 16,777,216 and 16,777,218 there is a pretty large gap. If you kept tracking your dollars as floats around 16M there would be gaps as big as 2$.

How fast will those gaps grow? At around 140,000,000 we can find gaps of size 16. At around 1,000,000, 000 of size 64.

To see how floating-point numbers are stored internally, you can play with this nice tool here.