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<div class="essay">
<h2>Magic Division -- Or Division by Integer Multiplication</h2>
<p>I noticed something strange in the assembly generated when compiling a program
using <code>gcc</code> an optimization turned off. Here it is in <a href="#figure1">Figure 1</a>
(though using optimization level 3 just to simplify, but it still shows in level 0).
In this generated assembly, we see instead of using the <code>div</code> operator,
something strange is going on with the number 1717986919 and shifting. My first
attempts at searching for this phenomenon online were not helpful. The first result
for the number was on Amazon as a book identifier. This number also shows up on
stack overflow, however it is no more helpful. No one on the post is talking about
why this number is in the generated code (or perhaps they all know this arcane
secret and don’t wish share). Eventually, I found out what was going on. Even
with no optimization, the compiler prefers to generate multiplication operator
over the division operator.
</p>
<p>We will consider unsigned integer division first. Let
<span class="math inline"><em>d</em></span>
be some constant divisor, and <span class="math inline"><em>n</em></span>
our numerator (for now let’s make <span class="math inline"><em>n</em></span>
a multiple of <span class="math inline"><em>d</em></span>). We will assume
integers are 32 bits. In order to computer
<span class="math inline"><em>n</em>/<em>d</em></span>,
first we need to know the values <span class="math inline"><em>l</em></span>
and <span class="math inline"><em>i</em></span> where
<span class="math inline"><em>d</em> = <em>l</em>2<sup><em>i</em></sup></span>.
So to begin out division, we can shift <span class="math inline"><em>n</em></span>
by <span class="math inline"><em>i</em></span> bits. Now we must divide this result
be <span class="math inline"><em>l</em></span>. Since
<span class="math inline"><em>l</em></span> will be odd, there exists some
inverse of <span class="math inline"><em>l</em></span>, called
<span class="math inline"><em>j</em></span> where <br />
<span class="math display"><em>l</em><em>j</em> = 1(mod 2<sup>32</sup>)</span><br />
So for any number <span class="math inline"><em>x</em></span>,
<span class="math inline"><em>x</em>/<em>l</em> = (<em>x</em><em>j</em>)/(<em>l</em><em>j</em>)=<em>x</em><em>j</em></span>.
So now all that is left after the shift is to multiply by
<span class="math inline"><em>j</em></span>, and we will
have computed <span class="math inline"><em>n</em>/<em>d</em></span>.</p>
<p>Dealing with remainder can go pretty in depth, and I recommend
reading the chapter in Hacker’s Delight about this topic
<a class="citation" href="#hacker">[1]</a>. Another write up can be found
in the MSDN blog archive <a class="citation" href="#msdn">[2]</a>.
</p>
<div class="figure">
<a name="figure1">
<h4>Figure 1</h4>
</a>
<span>Source:</span>
<div class="sourceCode" frame="single" numbers="left" language="c">
<pre><code class="sourceCode">
int div20(int n){
return a/20;
}
</code></pre>
</div>
<span>Compiled excerpt:</span>
<div class="sourceCode" frame="single" numbers="left">
<pre><code class="sourceCode">
.cfi_startproc
movl %edi, %eax
movl $1717986919, %edx
sarl $31, %edi
imull %edx
sarl $3, %edx
movl %edx, %eax
subl %edi, %eax
ret
.cfi_endproc
</code></pre>
</div>
</div>
<h3>References</h3>
<p><a name="hacker">1</a> Hacker’s Delight Chapter 10.
(<em><a href="http://www.hackersdelight.org/divcMore.pdf">
http://www.hackersdelight.org/divcMore.pdf
</a>
</em>)
</p>
<p><a name="msdn">2</a> MSDN Blog Archive
(<em><a href="https://blogs.msdn.microsoft.com/devdev/2005/12/12/integer-division-by-constants/">
https://blogs.msdn.microsoft.com/devdev/2005/12/12/integer-division-by-constants/
</a>
</em>)
</p>
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