Fast Factorial Functions

N !

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There are five algorithms which everyone who wants to compute the factorial n! = 1.2.3...n 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.
• If you do not attach great importance to high performance then get a BigInteger library and use:

  BigInt recfact(long start, long n) {
   long i;
   if (n <= 16) { 
   BigInt r = new BigInt(start);
   for (i = start + 1; i < start + n; i++) r *= i;
   return r;
   }
   i = n / 2;
   return recfact(start, i) * recfact(start + i, n - i);
  }
  BigInt factorial(long n) { return recfact(1, n); }
  

• 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!
  

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)=(n-1)! 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.

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.