<!DOCTYPE html> <html lang="en"> <head> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> <meta name="generator" content="AsciiDoc 8.6.8"> <title>NumericLiteral</title> <link rel="stylesheet" href="./asciidoc.css" type="text/css"> <link rel="stylesheet" href="./pygments.css" type="text/css"> <script type="text/javascript" src="./asciidoc.js"></script> <script type="text/javascript"> /*<![CDATA[*/ asciidoc.install(); /*]]>*/ </script> <link rel="stylesheet" href="./mlton.css" type="text/css"/> </head> <body class="article"> <div id="banner"> <div id="banner-home"> <a href="./Home">MLton 20130715</a> </div> </div> <div id="header"> <h1>NumericLiteral</h1> </div> <div id="content"> <div id="preamble"> <div class="sectionbody"> <div class="paragraph"><p>Numeric literals in <a href="StandardML">Standard ML</a> can be written in either decimal or hexadecimal notation. Sometimes it can be convenient to write numbers down in other bases. Fortunately, using <a href="Fold">Fold</a>, it is possible to define a concise syntax for numeric literals that allows one to write numeric constants in any base and of various types (<span class="monospaced">int</span>, <span class="monospaced">IntInf.int</span>, <span class="monospaced">word</span>, and more).</p></div> <div class="paragraph"><p>We will define constants <span class="monospaced">I</span>, <span class="monospaced">II</span>, <span class="monospaced">W</span>, and <span class="monospaced">`</span> so that, for example,</p></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="n">I</span><span class="w"> </span><span class="mi">10</span><span class="w"> </span><span class="n">`</span><span class="mi">1</span><span class="n">`</span><span class="mi">2</span><span class="n">`</span><span class="mi">3</span><span class="w"> </span><span class="n">$</span><span class="w"></span> </pre></div></div></div> <div class="paragraph"><p>denotes <span class="monospaced">123:int</span> in base 10, while</p></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="n">II</span><span class="w"> </span><span class="mi">8</span><span class="w"> </span><span class="n">`</span><span class="mi">2</span><span class="n">`</span><span class="mi">3</span><span class="w"> </span><span class="n">$</span><span class="w"></span> </pre></div></div></div> <div class="paragraph"><p>denotes <span class="monospaced">19:IntInf.int</span> in base 8, and</p></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="n">W</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="n">`</span><span class="mi">1</span><span class="n">`</span><span class="mi">1</span><span class="n">`</span><span class="mi">0</span><span class="n">`</span><span class="mi">1</span><span class="w"> </span><span class="n">$</span><span class="w"></span> </pre></div></div></div> <div class="paragraph"><p>denotes <span class="monospaced">0w13: word</span>.</p></div> <div class="paragraph"><p>Here is the code.</p></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">Num</span><span class="w"> </span><span class="p">=</span><span class="w"></span> <span class="w"> </span><span class="k">struct</span><span class="w"></span> <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">make</span><span class="w"> </span><span class="p">(</span><span class="k">op</span><span class="w"> </span><span class="n">*</span><span class="p">,</span><span class="w"> </span><span class="k">op</span><span class="w"> </span><span class="n">+</span><span class="p">,</span><span class="w"> </span><span class="n">i2x</span><span class="p">)</span><span class="w"> </span><span class="n">iBase</span><span class="w"> </span><span class="p">=</span><span class="w"></span> <span class="w"> </span><span class="k">let</span><span class="w"></span> <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">xBase</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i2x</span><span class="w"> </span><span class="n">iBase</span><span class="w"></span> <span class="w"> </span><span class="k">in</span><span class="w"></span> <span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"></span> <span class="w"> </span><span class="p">((</span><span class="n">i2x</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"></span> <span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"></span> <span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="n"><=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">andalso</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="n"><</span><span class="w"> </span><span class="n">iBase</span><span class="w"> </span><span class="k">then</span><span class="w"></span> <span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">xBase</span><span class="w"> </span><span class="n">+</span><span class="w"> </span><span class="n">i2x</span><span class="w"> </span><span class="n">i</span><span class="w"></span> <span class="w"> </span><span class="k">else</span><span class="w"></span> <span class="w"> </span><span class="k">raise</span><span class="w"> </span><span class="n">Fail</span><span class="w"> </span><span class="p">(</span><span class="n">concat</span><span class="w"></span> <span class="w"> </span><span class="p">[</span><span class="s">"Num: "</span><span class="p">,</span><span class="w"> </span><span class="n">Int</span><span class="p">.</span><span class="n">toString</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"></span> <span class="w"> </span><span class="s">" is not a valid\</span> <span class="s"> \ digit in base "</span><span class="p">,</span><span class="w"></span> <span class="w"> </span><span class="n">Int</span><span class="p">.</span><span class="n">toString</span><span class="w"> </span><span class="n">iBase</span><span class="p">])),</span><span class="w"></span> <span class="w"> </span><span class="n">fst</span><span class="p">)</span><span class="w"></span> <span class="w"> </span><span class="k">end</span><span class="w"></span> <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">?</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">make</span><span class="w"> </span><span class="p">(</span><span class="k">op</span><span class="w"> </span><span class="n">*</span><span class="p">,</span><span class="w"> </span><span class="k">op</span><span class="w"> </span><span class="n">+</span><span class="p">,</span><span class="w"> </span><span class="n">id</span><span class="p">)</span><span class="w"> </span><span class="n">?</span><span class="w"></span> <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">II</span><span class="w"> </span><span class="n">?</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">make</span><span class="w"> </span><span class="p">(</span><span class="k">op</span><span class="w"> </span><span class="n">*</span><span class="p">,</span><span class="w"> </span><span class="k">op</span><span class="w"> </span><span class="n">+</span><span class="p">,</span><span class="w"> </span><span class="n">IntInf</span><span class="p">.</span><span class="n">fromInt</span><span class="p">)</span><span class="w"> </span><span class="n">?</span><span class="w"></span> <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">W</span><span class="w"> </span><span class="n">?</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">make</span><span class="w"> </span><span class="p">(</span><span class="k">op</span><span class="w"> </span><span class="n">*</span><span class="p">,</span><span class="w"> </span><span class="k">op</span><span class="w"> </span><span class="n">+</span><span class="p">,</span><span class="w"> </span><span class="n">Word</span><span class="p">.</span><span class="n">fromInt</span><span class="p">)</span><span class="w"> </span><span class="n">?</span><span class="w"></span> <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">`</span><span class="w"> </span><span class="n">?</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step1</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">step</span><span class="p">))</span><span class="w"> </span><span class="p">=></span><span class="w"></span> <span class="w"> </span><span class="p">(</span><span class="n">step</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">x</span><span class="p">),</span><span class="w"> </span><span class="n">step</span><span class="p">))</span><span class="w"> </span><span class="n">?</span><span class="w"></span> <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">10</span><span class="w"></span> <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">11</span><span class="w"></span> <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">12</span><span class="w"></span> <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">13</span><span class="w"></span> <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">e</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">14</span><span class="w"></span> <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">15</span><span class="w"></span> <span class="w"> </span><span class="k">end</span><span class="w"></span> </pre></div></div></div> <div class="paragraph"><p>where</p></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="k">fun</span><span class="w"> </span><span class="n">fst</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="p">_)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">x</span><span class="w"></span> </pre></div></div></div> <div class="paragraph"><p>The idea is for the fold to start with zero and to construct the result one digit at a time, with each stepper multiplying the previous result by the base and adding the next digit. The code is abstracted in two different ways for extra generality. First, the <span class="monospaced">make</span> function abstracts over the various primitive operations (addition, multiplication, etc) that are needed to construct a number. This allows the same code to be shared for constants <span class="monospaced">I</span>, <span class="monospaced">II</span>, <span class="monospaced">W</span> used to write down the various numeric types. It also allows users to add new constants for additional numeric types, by supplying the necessary arguments to make.</p></div> <div class="paragraph"><p>Second, the step function, <span class="monospaced">`</span>, is abstracted over the actual construction operation, which is created by make, and passed along the fold. This allows the same constant, <span class="monospaced">`</span>, to be used for all numeric types. The alternative approach, having a different step function for each numeric type, would be more painful to use.</p></div> <div class="paragraph"><p>On the surface, it appears that the code checks the digits dynamically to ensure they are valid for the base. However, MLton will simplify everything away at compile time, leaving just the final numeric constant.</p></div> </div> </div> </div> <div id="footnotes"><hr></div> <div id="footer"> <div id="footer-text"> </div> <div id="footer-badges"> </div> </div> </body> </html>