[issue8692] Use divide-and-conquer for faster factorials

[Alexander Belopolsky]

Alexander Belopolsky <belopolsky at users.sourceforge.net> added the comment:

I agree, recursive version of partial_product is much simpler to follow.  While allocation of all numbers can be avoided in iterative version, doing so would further complicate code with little benefit.

I still believe, however that an iterative version can benefit from redefining partial_product to be product(2*i+1 for i in range(start, stop)).

Divide and conquer algorithm for that is simply

def partial_product(start, stop):
    length = stop - start
    .. handle length = 1 and 2 ..
    middle = start + (length >> 1)
    return partial_product(start, middle) * partial_product(middle, stop)

I would also reconsider the decision of using iterative outer loop.  Recursive outer loop matching redefined partial_product() can be written as

def loop(n):
    p = r = 1
    if n > 2:
        p, r = loop(n >> 1)
        p *= partial_product((n >> 2) + (n >> 1 & 1),
                             (n >> 1) + (n & 1))
        r *= p
    return p, r

which I believe is not harder to follow than the iterative alternative and does not require bit_length calculation.

I am replacing my python implementation, factorial.py, with the one that uses algorithms outlined above.

If there is any interest, I can convert it to C.

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Added file: http://bugs.python.org/file17343/factorial.py

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