How to perform square root without using math module?

Importing the math module only happens once, and you probably won't get much faster than the math module. There is also an older Stackoverflow question regarding Which is faster in Python: x**.5 or math.sqrt(x)?. It is not clear which method is faster.

Maybe take a look at NumPy and SciPy, not necessarily for the sqrt but if you're doing some heavy calculations they could be handy.

I'd think the math library would likely be as fast as anything you could write yourself. But if you want to write your own, here's one algorithm. I don't know Python, so I'll just write some pseudo-code.

function sqrt(x)
  lastGuess=x/2
  loop
    guess=(lastGuess+x/lastGuess)/2
    if abs(guess-lastGuess)<.000001 // or whatever threshold you want
      exit loop
    lastGuess=guess
  return guess

and the pseudocode translated to Python:

def sqrt(x):
    last_guess= x/2.0
    while True:
        guess= (last_guess + x/last_guess)/2
        if abs(guess - last_guess) < .000001: # example threshold
            return guess
        last_guess= guess