Newton-Raphson O(1)

x = 23493


class SquareRootResult(Exception):
    def __init__(self, number, sqrt):
        self.number = number
        self.sqrt = sqrt


def Iterate(x, guess):
    guessedX = guess * guess

    difference = x - guessedX
    if difference > -0.0001 and difference < 0.0001:
        raise SquareRootResult(number=x, sqrt=guess)

    newGuess = 0.5 * (guess + (x / guess))
    return newGuess


def GetInitialGuess(x):
    # Unimportant code. From Wikipedia:
    # "one starts with an initial guess which is reasonably close to the true root"
    import math
    return math.sqrt(x)


numIterations = 0

try:
    guess = GetInitialGuess(x)
    while True:
        numIterations += 1
        guess = Iterate(x, guess)
except SquareRootResult as e:
    print "Square root of", e.number, "is", e.sqrt, "in", numIterations, "iterations."
except:
    print "This number has no square root after",  numIterations, "iterations."