
An example of a PrimeSwing computation:
As this example shows an efficient computation of the factorial function reduces to an efficient computation of the swinging factorial n≀. Some information about these numbers can be found here and here. The prime factorization of the swing numbers is crucial for the implementation of the PrimeSwing algorithm.
A concise description of this algorithm is given in this writeup (pdf) and in the SageMath link below (Algo 5).
Link  Content  
Algorithms  A very short description of 21 algorithms for computing the factorial function n!.  
X  Julia factorial  *NEW* The factorial function based on the swinging factorial which in turn is computed via prime factorization implemented in Julia. 
Mini Library  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++. 

SageMath  Implementations in SageMath.  
LISP  Implementations in Lisp.  
Benchmarks  Benchmark 2013: With MPIR 2.6 you can calculate 100.000.000! in less than a minute provided you use one of the fast algorithms described here.  
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)=(n1)! and not Gamma(n)=n! ?  
X 
Hadamard 
Hadamard's Gamma function and a new factorial function [MathJax version] 
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.  
Sage / Python  Implementation of the swing algorithm.  
‼  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. 
alHaytham / Lagrange 
The ignorance of some western mathematicians. A deterministic factorial primality test. 

Factorial Digits  Number of decimal digits of 10^{n}!  
Calculator  Calculate n! for n up to 9.999.999.999 .  
RPNFactorial  The retrofactorial page!  
Permutations  Awesome! Permutation trees, the combinatorics of n!.  
Perm. trees  Download a pdfposter with 120 permutation trees!  
Gamma LogGamma 
Plots of the factorial (gamma) function.  
External links  Some bookmarks. 
FastFactorialFunctions: The Homepage of Factorial Algorithms. (C) Peter Luschny, 20002017. All information and all source code in this directory is free under the Creative Commons AttributionShareAlike 3.0 Unported License. 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. 2017 : 800,000 visitors! Thank you!