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diff --git a/src/html/essay/magic-division.html b/src/html/essay/magic-division.html index f40048d..10849b3 100644 --- a/src/html/essay/magic-division.html +++ b/src/html/essay/magic-division.html @@ -1,69 +1,74 @@ <!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> + +<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 + <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 + 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 + <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 + <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 + <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"> + <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"> + </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 @@ -74,22 +79,24 @@ movl %edx, %eax subl %edi, %eax ret .cfi_endproc - </code></pre></div> + </code></pre> + </div> </div> <h3>References</h3> - <p><a name="hacker">1</a> Hacker’s Delight Chapter 10. + <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 + http://www.hackersdelight.org/divcMore.pdf </a> </em>) </p> - <p><a name="msdn">2</a> MSDN Blog Archive + <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/ + https://blogs.msdn.microsoft.com/devdev/2005/12/12/integer-division-by-constants/ </a> </em>) </p> </div> - </body> +</body> + </html>
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