doc/guide/localhost/NumericLiteral

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  26 <h1>NumericLiteral</h1>
  27 </div>
  28 <div id="content">
  29 <div id="preamble">
  30 <div class="sectionbody">
  31 <div class="paragraph"><p>Numeric literals in <a href="StandardML">Standard ML</a> can be written in either
  32 decimal or hexadecimal notation.  Sometimes it can be convenient to
  33 write numbers down in other bases.  Fortunately, using <a href="Fold">Fold</a>, it is
  34 possible to define a concise syntax for numeric literals that allows
  35 one to write numeric constants in any base and of various types
  36 (<span class="monospaced">int</span>, <span class="monospaced">IntInf.int</span>, <span class="monospaced">word</span>, and more).</p></div>
  37 <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
  38 that, for example,</p></div>
  39 <div class="listingblock">
  40 <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">$</span><span class="w"></span>
  41 </pre></div></div></div>
  42 <div class="paragraph"><p>denotes <span class="monospaced">123:int</span> in base 10, while</p></div>
  43 <div class="listingblock">
  44 <div class="content"><div class="highlight"><pre><span class="n">II</span><span class="w"> </span><span class="mi">8</span><span class="w"> </span><span class="n">`</span><span class="mi">2</span><span class="n">`</span><span class="mi">3</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
  45 </pre></div></div></div>
  46 <div class="paragraph"><p>denotes <span class="monospaced">19:IntInf.int</span> in base 8, and</p></div>
  47 <div class="listingblock">
  48 <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>
  49 </pre></div></div></div>
  50 <div class="paragraph"><p>denotes <span class="monospaced">0w13: word</span>.</p></div>
  51 <div class="paragraph"><p>Here is the code.</p></div>
  52 <div class="listingblock">
  53 <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>
  54 <span class="w">   </span><span class="k">struct</span><span class="w"></span>
  55 <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>
  56 <span class="w">          </span><span class="k">let</span><span class="w"></span>
  57 <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>
  58 <span class="w">          </span><span class="k">in</span><span class="w"></span>
  59 <span class="w">             </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"></span>
  60 <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>
  61 <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">=&gt;</span><span class="w"></span>
  62 <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">&lt;=</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">&lt;</span><span class="w"> </span><span class="n">iBase</span><span class="w"> </span><span class="k">then</span><span class="w"></span>
  63 <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>
  64 <span class="w">                     </span><span class="k">else</span><span class="w"></span>
  65 <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>
  66 <span class="w">                                       </span><span class="p">[</span><span class="s">&quot;Num: &quot;</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>
  67 <span class="w">                                        </span><span class="s">&quot; is not a valid\</span>
  68 <span class="s">                                        \ digit in base &quot;</span><span class="p">,</span><span class="w"></span>
  69 <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>
  70 <span class="w">                 </span><span class="n">fst</span><span class="p">)</span><span class="w"></span>
  71 <span class="w">          </span><span class="k">end</span><span class="w"></span>
  72
  73 <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>
  74 <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>
  75 <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>
  76
  77 <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">=&gt;</span><span class="w"></span>
  78 <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>
  79
  80 <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>
  81 <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>
  82 <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>
  83 <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>
  84 <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>
  85 <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>
  86 <span class="w">   </span><span class="k">end</span><span class="w"></span>
  87 </pre></div></div></div>
  88 <div class="paragraph"><p>where</p></div>
  89 <div class="listingblock">
  90 <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>
  91 </pre></div></div></div>
  92 <div class="paragraph"><p>The idea is for the fold to start with zero and to construct the
  93 result one digit at a time, with each stepper multiplying the previous
  94 result by the base and adding the next digit.  The code is abstracted
  95 in two different ways for extra generality.  First, the <span class="monospaced">make</span>
  96 function abstracts over the various primitive operations (addition,
  97 multiplication, etc) that are needed to construct a number.  This
  98 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
  99 write down the various numeric types.  It also allows users to add new
 100 constants for additional numeric types, by supplying the necessary
 101 arguments to make.</p></div>
 102 <div class="paragraph"><p>Second, the step function, <span class="monospaced">&grave;</span>, is abstracted over the actual
 103 construction operation, which is created by make, and passed along the
 104 fold.  This allows the same constant, <span class="monospaced">&grave;</span>, to be used for all
 105 numeric types.  The alternative approach, having a different step
 106 function for each numeric type, would be more painful to use.</p></div>
 107 <div class="paragraph"><p>On the surface, it appears that the code checks the digits dynamically
 108 to ensure they are valid for the base.  However, MLton will simplify
 109 everything away at compile time, leaving just the final numeric
 110 constant.</p></div>
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