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diff --git a/src/html/essay/magic-division.html b/src/html/essay/magic-division.html new file mode 100644 index 0000000..f40048d --- /dev/null +++ b/src/html/essay/magic-division.html @@ -0,0 +1,95 @@ +<!doctype html> +<html lang="en"> + <head> + <title>Magic Division</title> + <meta charset="UTF-8"> + <meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no"> + + </head> + <body> + <link rel="stylesheet" type="text/css" href="/css/styles.css"> + <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> + </div> + </body> +</html>
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