Why is 2 * x * x faster than 2 * ( x * x ) in Python 3.x, for integers?

Question

The following Python 3.x integer multiplication takes on average between 1.66s and 1.77s:

import time
start_time = time.time()
num = 0
for x in range(0, 10000000         


        

Answer 1

First of all, note that we don't see the same thing in Python 2.x:

>>> timeit("for i in range(1000): 2*i*i")
51.00784397125244
>>> timeit("for i in range(1000): 2*(i*i)")
50.48330092430115

So this leads us to believe that this is due to how integers changed in Python 3: specifically, Python 3 uses long (arbitrarily large integers) everywhere.

For small enough integers (including the ones we're considering here), CPython actually just uses the O(MN) grade-school digit by digit multiplication algorithm (for larger integers it switches to the Karatsuba algorithm). You can see this yourself in the source.

The number of digits in x*x is roughly twice that of 2*x or x (since log(x²) = 2 log(x)). Note that a "digit" in this context is not a base-10 digit, but a 30-bit value (which are treated as single digits in CPython's implementation). Hence, 2 is a single-digit value, and x and 2*x are single-digit values for all iterations of the loop, but x*x is two-digit for x >= 2**15. Hence, for x >= 2**15, 2*x*x only requires single-by-single-digit multiplications whereas 2*(x*x) requires a single-by-single and a single-by-double-digit multiplication (since x*x has 2 30-bit digits).

Here's a direct way to see this (Python 3):

>>> timeit("a*b", "a,b = 2, 123456**2", number=100000000)
5.796971936999967
>>> timeit("a*b", "a,b = 2*123456, 123456", number=100000000)
4.3559221399999615

Again, compare this to Python 2, which doesn't use arbitrary-length integers everywhere:

>>> timeit("a*b", "a,b = 2, 123456**2", number=100000000)
3.0912468433380127
>>> timeit("a*b", "a,b = 2*123456, 123456", number=100000000)
3.1120400428771973

(One interesting note: If you look at the source, you'll see that the algorithm actually has a special case for squaring numbers (which we're doing here), but even still this is not enough to overcome the fact that 2*(x*x) just requires processing more digits.)