Square root using Newton's Method

sqrt_newton.py

#!/usr/bin/python

def sqrt_newton(x):
  if abs(x) > 1e16: raise ValueError("Too large to be calculated accurately.")
  f = lambda a: a ** 2 - x
  f1 = lambda a: 2.0 * a
  x1 = x / 2.0
  dx = 1e-16 * 10.0 ** (len(repr(x)) * 1.0)
  while abs(f(x1)) > dx:
    x1 = x1 - (f(x1) / f1(x1))
  return x1

if __name__ == "__main__":
  print [(sqrt_newton(10 ** x * 1.0), sqrt_newton(10 ** x * 5.0) ** 2.0) for x in range(1, 16)]