I know it's not technically a fast factorial algorithm, but I'm kinda surprised that there is no mention on the site of the AKS primality test (https://en.wikipedia.org/wiki/AKS_primality_test). It's operation is sort of like an FFT for canceling out factorials mod N.
I hoped this would help me solve some more Project Euler [1] problems. Unfortunately, the algorithms given are not explained in detail, so the learning experience is somewhat mediocre. Then again, I have ChatGPT to elucidate them for me.
This article [2] has some interesting details on the swinging factorial function n≀, but I can't seem to find the essay that it references: "Swing, divide and conquer the factorial", 2008.
My approach to the first 100 Euler eventually involved a bitmap of possible primes (only odd numbers, there's only one even prime) which was initially seeded by a 'magic number' set representing the lowest primes and would grow up to a target number of cached small / really fast primes.
Then the algorithms I understand less kick in. Most of those involve some form of modo math to wrap the number space based on one or more origami like folds and use well known and test algorithms which were previously exhaustively proven to cover the entire 32 bit number-space when utilized in a composite fashion.
I've always wondered if Stirling(n) can be used to arrive quickly in the vicinity of n!, and then use a search of some kind to get to the exact target.
If it runs fast there, it will run fast
everywhere, as integers are made
of Lisp conses on purpose,
so you can see Lisp built
from itself as if they were Peano
axioms:
That one is an approximation rather than returning all millions of exact big integer digits though (the approximation is more useful for real life statistics etc..., but doesn't look like what this article is targeting)
This article [2] has some interesting details on the swinging factorial function n≀, but I can't seem to find the essay that it references: "Swing, divide and conquer the factorial", 2008.
[1] https://projecteuler.net/
[2] https://oeis.org/A000142/a000142.pdf
http://www.luschny.de/math/factorial/SwingIntro.pdf
Then the algorithms I understand less kick in. Most of those involve some form of modo math to wrap the number space based on one or more origami like folds and use well known and test algorithms which were previously exhaustively proven to cover the entire 32 bit number-space when utilized in a composite fashion.
If it runs fast there, it will run fast everywhere, as integers are made of Lisp conses on purpose, so you can see Lisp built from itself as if they were Peano axioms:
https://www.t3x.org/zsp/
Test: Test with (trace fi) before running (fac 0) and (fac 10) Here you can see how (fi) works on every iteration.Integers are not actual integers, but lists.
'#3628800 it's '(3 6 2 8 8 0 0).
Open "nmath.l" to see how are digits implemented. Base.t it's interesting too, as it explains you some functions.