# SqrtDemo.py
# Charles Van Loan (cfv3)
# January 1, 2015

"""
This module compares math.sqrt with a sqrt function that is
based on (at most) twenty averaging steps.
"""

import math

def sqrt(x):
   """
   The square root of x.
   
   Performs at most 20 steps of Newton's method. Unless x
   is very large or very small, a relative accuracy of 10e-12 (or better)
   is achieved.
   
   Precondition: x is a positive float or int
   """
   L=float(x)
   W=1.0
   relErr = 10e-15
   its=0
   itMax = 200
   while abs(L-W)/L > relErr and its<=itMax:
      L = (L+W)/2
      W = x/L
      its+=1
   return L

# Test Script
if __name__=='__main__':
    print '\n\n     x               sqrt(x)        relError'
    print '------------------------------------------------------'
    k = -17
    while k<15:
       k+=2
       x = 10.0**k
       MySqrt = sqrt(x)
       TrueSqrt = math.sqrt(x)
       relErr = abs(MySqrt - TrueSqrt)/TrueSqrt
       print '%8.1e   %20.12e    %8.3e' % (x,sqrt(x),relErr)