This is a algorithm to find the square root of a number using Newton Raphson Method.
The Newton Raphson method is a way to find a good aproximation for the root of a real-valued function. The idea is to use the continously and differentiable of a function to approximate it by a straight line tangent to it.
The method needs to have a near point of the solution, lets say it is a x = x0. Then, a approximation would be given by
x1 = x0 - f(x0)/f'(x0).
The method consists in iterating the approximation above as many times as necessary to get the desire accuracy. Then, for a xn:
xn+1 = xn - f(xn)/f'(xn).
In that point of view, as we are dealing with solidity, it is a good point to try doing the fewer iterations to avoid high gas consumption. Besides that, it is also good to have a good initial point to avoid even more the number of iterations.