# Speed of calculating powers (in python)

### Question

I'm curious as to why it's so much faster to multiply than to take powers in python (though from what I've read this may well be true in many other languages too). For example it's much faster to do

``````x*x
``````

than

``````x**2
``````

I suppose the ** operator is more general and can also deal with fractional powers. But if that's why it's so much slower, why doesn't it perform a check for an int exponent and then just do the multiplication?

Edit: Here's some example code I tried...

``````def pow1(r, n):
for i in range(r):
p = i**n

def pow2(r, n):
for i in range(r):
p = 1
for j in range(n):
p *= i
``````

Now, pow2 is just a quick example and is clearly not optimised!
But even so I find that using n = 2 and r = 1,000,000, then pow1 takes ~ 2500ms and pow2 takes ~ 1700ms.
I admit that for large values of n, then pow1 does get much quicker than pow2. But that's not too surprising.

1
30
6/19/2009 8:20:02 PM

Basically naive multiplication is O(n) with a very low constant factor. Taking the power is O(log n) with a higher constant factor (There are special cases that need to be tested... fractional exponents, negative exponents, etc) . Edit: just to be clear, that's O(n) where n is the exponent.

Of course the naive approach will be faster for small n, you're only really implementing a small subset of exponential math so your constant factor is negligible.

24
6/19/2009 8:33:46 PM

Adding a check is an expense, too. Do you always want that check there? A compiled language could make the check for a constant exponent to see if it's a relatively small integer because there's no run-time cost, just a compile-time cost. An interpreted language might not make that check.

It's up to the particular implementation unless that kind of detail is specified by the language.

Python doesn't know what distribution of exponents you're going to feed it. If it's going to be 99% non-integer values, do you want the code to check for an integer every time, making runtime even slower?