G'day all.
Quoting Tamas K Papp
2. Newton's method is not guaranteed to converge.
For computing square roots, it is. The square root function is exceedingly well-behaved. But you can make things better by choosing a smart initial estimate. The initial estimate in this version is so good that you only need a fixed number of iterations, so there are no loops here. mysqrt :: Double -> Double mysqrt x | x <= 0 = if x == 0 then 0 else error "mysqrt domain error" | r == 1 = sqrt2 * sqrtx | otherwise = sqrtx where (q,r) = exponent x `divMod` 2 sqrtx = scaleFloat q (sqrtSignificand (significand x)) -- Feel free to add more digits if this is insufficient sqrt2 = 1.41421356237309504880168872420969807856967 -- Compute the square root of a number in the range [0.5,1) sqrtSignificand :: Double -> Double sqrtSignificand x = iter . iter . iter $ d0 where -- d0 is a third-order Chebyshev approximation to sqrt x x' = x * 4.0 - 3.0 d2 = -6.177921038278929e-3 d1 = 2 * x' * d2 + 0.14589875298243973 d0 = x' * d1 - d2 + 0.5 * 1.7196949654923188 iter sx = 0.5 * (sx + x / sx) Cheers, Andrew Bromage