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	How you would calculate factorials efficiently - libzahl - big integer library
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	---
	commit d751d9d8edc2d1ea099674efecb04a5033428511
	parent 5bfbc6a6984e2036bf39b15708647ac5fd003dc2
	Author: Mattias Andrée <[email protected]>
	Date: Fri, 13 May 2016 10:32:10 +0200

	How you would calculate factorials efficiently

	Signed-off-by: Mattias Andrée <[email protected]>

	Diffstat:
	M doc/not-implemented.tex \| 40 +++++++++++++++++++++++++++++…

	1 file changed, 40 insertions(+), 0 deletions(-)
	---
	diff --git a/doc/not-implemented.tex b/doc/not-implemented.tex
	@@ -174,6 +174,46 @@ important function for many combinatorial
	applications, therefore it may be implemented
	in the future if the demand is high enough.

	+An efficient, yet not optimal, implementation
	+of factorials that about halves the number of
	+required multiplications compared to the naïve
	+method can be derived from the observation
	+
	+\vspace{1em}
	+\( \displaystyle{
	+ n! = n!! ~ \lfloor n / 2 \rfloor! ~ 2^{\lfloor n / 2 \rfloor}
	+}\), $n$ odd.
	+\vspace{1em}
	+
	+\noindent
	+The resulting algorithm can be expressed
	+
	+\begin{alltt}
	+ void
	+ fact(z_t r, uint64_t n)
	+ \{
	+ z_t p, f, two;
	+ uint64_t *ns, s = 1, i = 1;
	+ zinit(p), zinit(f), zinit(two);
	+ zseti(r, 1), zseti(p, 1), zseti(f, n), zseti(two, 2);
	+ ns = alloca(zbits(f) * sizeof(*ns));
	+ while (n > 1) \{
	+ if (n & 1) \{
	+ ns[i++] = n;
	+ s += n >>= 1;
	+ \} else \{
	+ zmul(r, r, (zsetu(f, n), f));
	+ n -= 1;
	+ \}
	+ \}
	+ for (zseti(f, 1); i-- > 0; zmul(r, r, p);)
	+ for (n = ns[i]; zcmpu(f, n); zadd(f, f, two))
	+ zmul(p, p, f);
	+ zlsh(r, r, s);
	+ zfree(two), zfree(f), zfree(p);
	+ \}
	+\end{alltt}
	+

	\subsection{Subfactorial}
	\label{sec:Subfactorial}