Fast Factorial Functions

N !

Fastfactorialfunctions

There are five algorithms which everyone who wants to compute the factorial n! = 1.2.3...n exactly should know.

The algorithm SplitRecursive, because it is simple and the fastest algorithm which does not use prime factorization.
The algorithm PrimeSwing, because it is the (asymptotical) fastest algorithm known to compute n!. The algorithm is based on the notion of the 'Swing Numbers' and computes n! via the prime factorization of these numbers.
The ingenious algorithm of Moessner which uses only additions! Though of no practical importance (because it is slow), it has the fascination of an unexpected solution.
The Poor Man's algorithm which uses no Big-Integer library and can be easily implemented in any computer language and is even fast up to 10000!.
The ParallelPrimeSwing algorithm, which is the PrimeSwing algorithm with improved performance using methods of concurrent programming and thus taking advantage of multiple core processors.
And here is an algorithm which nobody needs, for the Simple-Minded only:
long factorial(long n) { return n <= 1 ? 1 : n * factorial(n-1);} Just don't use it if n > 12.

	Link	Content
	Algorithms	A very short description of 21 algorithms for computing the factorial function n!.
X	FFF-MiniLib	The factorial function, the binomial function, the double factorial, the swing numbers and an efficient prime number sieve implemented in Scala and GO.
	Browse Code	Various algorithms implemented in Java, C# and C++.
	Benchmarks	New.Various benchmark results and a small sensation: With MPIR 2.3 you can calculate 100.000.000! in less than a minute provided you use a fast parallel algorithm.
	Conclusions	Which algorithm should we choose?
	Download	Download a test application and benchmark yourself.
X	Approximations	A unique collection! Approximation formulas.
	Gamma quot	Bounds for Gamma(x+1)/Gamma(x+1/2)
	Gamma shift	Why is Gamma(n)=(n-1)! and not Gamma(n)=n! ?
X	Hadamard	Hadamard's Gamma function and a new factorial function [MathJax version] requires math fonts.
	History	Not even Wikipedia knows this! The early history of the factorial function.
	Notation	On the notation n!
	Binary Split	For coders only. Go to the page of the day.
‼	Double Factorial	The fast double factorial function.
	Prime Factorial	Primfakultaet ('The Primorial', in German.)
	Bibliography	Bibliography on Inequalities for the Gamma function.
X	Bernoulli & Euler	Exotic Applications: Inclusions for the Bernoulli and Euler numbers.
	Binomial	Fast Binomial Function (Binomial Coefficients).
	Variations	A combinatorial generalization of the factorial.
X	Stieltjes' CF	On Stieltjes' Continued Fraction for the Gamma Function.
	al-Haytham / Lagrange	The ignorance of some western mathematicians. A deterministic factorial primality test.
	Factorial Digits	Number of decimal digits of 10ⁿ!
	Calculator	Calculate n! for n up to 9.999.999.999 .
	RPN-Factorial	The retro-factorial page!
	Permutations	Awesome! Permutation trees, the combinatorics of n!.
	Perm. trees	Download a pdf-poster with 120 permutation trees!
	Gamma LogGamma	Plots of the factorial (gamma) function.
	External links	Some bookmarks.

Fast-Factorial-Functions: The Homepage of Factorial Algorithms. (C) Peter Luschny, 2000-2011. All information and all source code in this directory is free under the Creative Commons Attribution-ShareAlike 3.0 Unported License (the same license which Wikipedia uses). This page is listed on the famous "Dictionary of Algorithms and Data Structures" at the National Institute of Standards and Technology's web site (NIST). Apr. 2003 / Apr. 2011 : 250,000 visitors! Thank you!

Rational Trees	Variants of Variations	Eulerian Polynomials	vonStaudt Primes	Irregular Primes	Generalized Binomial
Bernoulli Confusion	Partitions Stirling	Zeta Polynomials	Perfect Rulers	Swinging Primes	Clausen Numbers
Hadamard Gamma	History Factorial	Permutation Trees	Swiss Knife	Swinging Factorial	Worpitzky Triangle