Question

I found FastFactorialFunctions describing a number of algorithms for computing the factorial. Unfortunately, the explanations are terse and I don’t feel like sifting through line after line of source code to understand the basic principles behind the algorithms.

Can anybody point me to more detailed descriptions of these (or other fast) algorithms for computing large exact factorials fast?

Factorials with prime factorization (Python)
describes the method of prime factorization, the technique common to all of the best-performing factorial algorithms. It also contains some nice example code in Python. The author links to a description of binary splitting and references an article in the Journal of Algorithms ("On the Complexity of Calculating Factorials") that looks promising, if I could only get my hands on it.

Answer 1

Check out this paper (PDF link) by Richard Fateman. The code samples are in Lisp, in but in any event, much of the secret boils down to minimizing the number of bignum (arbitrary precision integer) calculations you have to do.

Naturally, if you don’t need/have bignums, it’s trivial; either a lookup table or a simple loop will be fine.

EDIT: If you can use an approximate answer, you can either compute the logarithm of the factorial directly by summing log(k) for k = 2 ... n, or by using the venerable Stirling approximation. You want to work with the logarithm wherever possible to avoid overflow; in particular, a naive application of Stirling’s approximation will overflow in a lot of places where it doesn’t have to.