How does this float square root approximation work?

Question

I found a rather strange but working square root approximation for floats; I really don\'t get it. Can someone explain me why this code works?

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Answer 1

Let y = sqrt(x),

it follows from the properties of logarithms that log(y) = 0.5 * log(x) (1)

Interpreting a normal float as an integer gives INT(x) = Ix = L * (log(x) + B - σ) (2)

where L = 2^N, N the number of bits of the significand, B is the exponent bias, and σ is a free factor to tune the approximation.

Combining (1) and (2) gives: Iy = 0.5 * (Ix + (L * (B - σ)))

Which is written in the code as (*(int*)&x >> 1) + 0x1fbb4000;

Find the σ so that the constant equals 0x1fbb4000 and determine whether it's optimal.