Writing your own square root function

Question

How do you write your own function for finding the most accurate square root of an integer?

After googling it, I found this (archived from its original link), but first, I didn't get it completely, and second, it is approximate too.

Assume square root as nearest integer (to the actual root) or a float.

Answer 1

The following computes floor(sqrt(N)) for N > 0:

x = 2^ceil(numbits(N)/2)
loop:
    y = floor((x + floor(N/x))/2)
    if y >= x
        return x
    x = y

This is a version of Newton's method given in Crandall & Pomerance, "Prime Numbers: A Computational Perspective". The reason you should use this version is that people who know what they're doing have proven that it converges exactly to the floor of the square root, and it's simple so the probability of making an implementation error is small. It's also fast (although it's possible to construct an even faster algorithm -- but doing that correctly is much more complex). A properly implemented binary search can be faster for very small N, but there you may as well use a lookup table.

To round to the nearest integer, just compute t = floor(sqrt(4N)) using the algorithm above. If the least significant bit of t is set, then choose x = (t+1)/2; otherwise choose t/2. Note that this rounds up on a tie; you could also round down (or round to even) by looking at whether the remainder is nonzero (i.e. whether t^2 == 4N).

Note that you don't need to use floating-point arithmetic. In fact, you shouldn't. This algorithm should be implemented entirely using integers (in particular, the floor() functions just indicate that regular integer division should be used).