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<a name="Factorial-Algorithm"></a>
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Next: <a href="Binomial-Coefficients-Algorithm.html#Binomial-Coefficients-Algorithm" accesskey="n" rel="next">Binomial Coefficients Algorithm</a>, Previous: <a href="Prime-Testing-Algorithm.html#Prime-Testing-Algorithm" accesskey="p" rel="prev">Prime Testing Algorithm</a>, Up: <a href="Other-Algorithms.html#Other-Algorithms" accesskey="u" rel="up">Other Algorithms</a> &nbsp; [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Factorial"></a>
<h4 class="subsection">15.7.2 Factorial</h4>
<a name="index-Factorial-algorithm"></a>
<p>Factorials are calculated by a combination of two algorithms. An idea is
shared among them: to compute the odd part of the factorial; a final step
takes account of the power of <em>2</em> term, by shifting.
</p>
<p>For small <em>n</em>, the odd factor of <em>n!</em> is computed with the simple
observation that it is equal to the product of all positive odd numbers
smaller than <em>n</em> times the odd factor of <em>[n/2]!</em>,
where <em>[x]</em> is the integer part of <em>x</em>, and so on
recursively. The procedure can be best illustrated with an example,
</p>
<blockquote>
<p><em>23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^{19}</em>
</p></blockquote>
<p>Current code collects all the factors in a single list, with a loop and no
recursion, and compute the product, with no special care for repeated chunks.
</p>
<p>When <em>n</em> is larger, computation pass trough prime sieving. An helper
function is used, as suggested by Peter Luschny:
</p>
<div class="example">
<pre class="example"> n
-----
n! | | L(p,n)
msf(n) = -------------- = | | p
[n/2]!^2.2^k p=3
</pre></div>
<p>Where <em>p</em> ranges on odd prime numbers. The exponent <em>k</em> is chosen to
obtain an odd integer number: <em>k</em> is the number of 1 bits in the binary
representation of <em>[n/2]</em>. The function L<em>(p,n)</em>
can be defined as zero when <em>p</em> is composite, and, for any prime
<em>p</em>, it is computed with:
</p>
<div class="example">
<pre class="example"> ---
\ n
L(p,n) = / [---] mod 2 &lt;= log (n) .
--- p^i p
i&gt;0
</pre></div>
<p>With this helper function, we are able to compute the odd part of <em>n!</em>
using the recursion implied by <em>n!=[n/2]!^2*msf(n)*2^k</em>. The recursion stops using the
small-<em>n</em> algorithm on some <em>[n/2^i]</em>.
</p>
<p>Both the above algorithms use binary splitting to compute the product of many
small factors. At first as many products as possible are accumulated in a
single register, generating a list of factors that fit in a machine word. This
list is then split into halves, and the product is computed recursively.
</p>
<p>Such splitting is more efficient than repeated Nx1 multiplies since it
forms big multiplies, allowing Karatsuba and higher algorithms to be used.
And even below the Karatsuba threshold a big block of work can be more
efficient for the basecase algorithm.
</p>
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Next: <a href="Binomial-Coefficients-Algorithm.html#Binomial-Coefficients-Algorithm" accesskey="n" rel="next">Binomial Coefficients Algorithm</a>, Previous: <a href="Prime-Testing-Algorithm.html#Prime-Testing-Algorithm" accesskey="p" rel="prev">Prime Testing Algorithm</a>, Up: <a href="Other-Algorithms.html#Other-Algorithms" accesskey="u" rel="up">Other Algorithms</a> &nbsp; [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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