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  36   <section id="replacing-functions-in-the-dictionary">
  37 <h1>Replacing Functions in the Dictionary<a class="headerlink" href="#replacing-functions-in-the-dictionary" title="Permalink to this headline">¶</a></h1>
  38 <p>For now, there is no way to define new functions from within the Joy
  39 language. All functions (and the interpreter) all accept and return a
  40 dictionary parameter (in addition to the stack and expression) so that
  41 we can implement e.g. a function that adds new functions to the
  42 dictionary. However, there’s no function that does that. Adding a new
  43 function to the dictionary is a meta-interpreter action, you have to do
  44 it in Python, not Joy.</p>
  45 <div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">notebook_preamble</span> <span class="kn">import</span> <span class="n">D</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">V</span>
  46 </pre></div>
  47 </div>
  48 <section id="a-long-trace">
  49 <h2>A long trace<a class="headerlink" href="#a-long-trace" title="Permalink to this headline">¶</a></h2>
  50 <div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span><span class="n">V</span><span class="p">(</span><span class="s1">&#39;[23 18] average&#39;</span><span class="p">)</span>
  51 </pre></div>
  52 </div>
  53 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span>                                  <span class="o">.</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="n">average</span>
  54                           <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="o">.</span> <span class="n">average</span>
  55                           <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="nb">sum</span> <span class="mf">1.0</span> <span class="o">*</span><span class="p">]</span> <span class="p">[</span><span class="n">size</span><span class="p">]</span> <span class="n">cleave</span> <span class="o">/</span>
  56               <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="nb">sum</span> <span class="mf">1.0</span> <span class="o">*</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="n">size</span><span class="p">]</span> <span class="n">cleave</span> <span class="o">/</span>
  57        <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="nb">sum</span> <span class="mf">1.0</span> <span class="o">*</span><span class="p">]</span> <span class="p">[</span><span class="n">size</span><span class="p">]</span> <span class="o">.</span> <span class="n">cleave</span> <span class="o">/</span>
  58        <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="nb">sum</span> <span class="mf">1.0</span> <span class="o">*</span><span class="p">]</span> <span class="p">[</span><span class="n">size</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">app2</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  59    <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="nb">sum</span> <span class="mf">1.0</span> <span class="o">*</span><span class="p">]</span> <span class="p">[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">.</span> <span class="n">app2</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  60 <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[[</span><span class="nb">sum</span> <span class="mf">1.0</span> <span class="o">*</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">.</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  61               <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="nb">sum</span> <span class="mf">1.0</span> <span class="o">*</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="p">[[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  62                           <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="o">.</span> <span class="nb">sum</span> <span class="mf">1.0</span> <span class="o">*</span> <span class="p">[[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  63                                <span class="mi">41</span> <span class="o">.</span> <span class="mf">1.0</span> <span class="o">*</span> <span class="p">[[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  64                            <span class="mi">41</span> <span class="mf">1.0</span> <span class="o">.</span> <span class="o">*</span> <span class="p">[[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  65                              <span class="mf">41.0</span> <span class="o">.</span> <span class="p">[[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  66                    <span class="mf">41.0</span> <span class="p">[[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="o">.</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  67                    <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="mf">41.0</span><span class="p">]</span> <span class="o">.</span> <span class="n">first</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  68                      <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="mf">41.0</span> <span class="o">.</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  69     <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="mf">41.0</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="o">.</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  70 <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="mf">41.0</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">.</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  71                    <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="n">size</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  72                           <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="o">.</span> <span class="n">size</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  73                           <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="o">.</span> <span class="mi">0</span> <span class="n">swap</span> <span class="p">[</span><span class="n">pop</span> <span class="o">++</span><span class="p">]</span> <span class="n">step</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  74                         <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="mi">0</span> <span class="o">.</span> <span class="n">swap</span> <span class="p">[</span><span class="n">pop</span> <span class="o">++</span><span class="p">]</span> <span class="n">step</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  75                         <span class="mi">0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="n">pop</span> <span class="o">++</span><span class="p">]</span> <span class="n">step</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  76                <span class="mi">0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="n">pop</span> <span class="o">++</span><span class="p">]</span> <span class="o">.</span> <span class="n">step</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  77                     <span class="mi">0</span> <span class="mi">23</span> <span class="p">[</span><span class="n">pop</span> <span class="o">++</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="p">[</span><span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="n">pop</span> <span class="o">++</span><span class="p">]</span> <span class="n">step</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  78                              <span class="mi">0</span> <span class="mi">23</span> <span class="o">.</span> <span class="n">pop</span> <span class="o">++</span> <span class="p">[</span><span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="n">pop</span> <span class="o">++</span><span class="p">]</span> <span class="n">step</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  79                                 <span class="mi">0</span> <span class="o">.</span> <span class="o">++</span> <span class="p">[</span><span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="n">pop</span> <span class="o">++</span><span class="p">]</span> <span class="n">step</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  80                                 <span class="mi">1</span> <span class="o">.</span> <span class="p">[</span><span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="n">pop</span> <span class="o">++</span><span class="p">]</span> <span class="n">step</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  81                            <span class="mi">1</span> <span class="p">[</span><span class="mi">18</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="n">pop</span> <span class="o">++</span><span class="p">]</span> <span class="n">step</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  82                   <span class="mi">1</span> <span class="p">[</span><span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="n">pop</span> <span class="o">++</span><span class="p">]</span> <span class="o">.</span> <span class="n">step</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  83                     <span class="mi">1</span> <span class="mi">18</span> <span class="p">[</span><span class="n">pop</span> <span class="o">++</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  84                              <span class="mi">1</span> <span class="mi">18</span> <span class="o">.</span> <span class="n">pop</span> <span class="o">++</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  85                                 <span class="mi">1</span> <span class="o">.</span> <span class="o">++</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  86                                 <span class="mi">2</span> <span class="o">.</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  87                  <span class="mi">2</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="o">.</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  88                  <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="mf">41.0</span> <span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  89                    <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="mf">41.0</span> <span class="mi">2</span> <span class="o">.</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
  90             <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="mf">41.0</span> <span class="mi">2</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="o">.</span> <span class="n">dip</span> <span class="o">/</span>
  91                      <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="mf">41.0</span> <span class="o">.</span> <span class="n">popd</span> <span class="mi">2</span> <span class="o">/</span>
  92                              <span class="mf">41.0</span> <span class="o">.</span> <span class="mi">2</span> <span class="o">/</span>
  93                            <span class="mf">41.0</span> <span class="mi">2</span> <span class="o">.</span> <span class="o">/</span>
  94                              <span class="mf">20.5</span> <span class="o">.</span>
  95 </pre></div>
  96 </div>
  97 </section>
  98 <section id="replacing-size-with-a-python-version">
  99 <h2>Replacing <code class="docutils literal notranslate"><span class="pre">size</span></code> with a Python version<a class="headerlink" href="#replacing-size-with-a-python-version" title="Permalink to this headline">¶</a></h2>
 100 <p>Both <code class="docutils literal notranslate"><span class="pre">sum</span></code> and <code class="docutils literal notranslate"><span class="pre">size</span></code> each convert a sequence to a single value.</p>
 101 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="nb">sum</span> <span class="o">==</span> <span class="mi">0</span> <span class="n">swap</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">step</span>
 102 <span class="n">size</span> <span class="o">==</span> <span class="mi">0</span> <span class="n">swap</span> <span class="p">[</span><span class="n">pop</span> <span class="o">++</span><span class="p">]</span> <span class="n">step</span>
 103 </pre></div>
 104 </div>
 105 <p>An efficient <code class="docutils literal notranslate"><span class="pre">sum</span></code> function is already in the library. But for
 106 <code class="docutils literal notranslate"><span class="pre">size</span></code> we can use a “compiled” version hand-written in Python to speed
 107 up evaluation and make the trace more readable.</p>
 108 <div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">joy.library</span> <span class="kn">import</span> <span class="n">SimpleFunctionWrapper</span>
 109 <span class="kn">from</span> <span class="nn">joy.utils.stack</span> <span class="kn">import</span> <span class="n">iter_stack</span>
 110
 111
 112 <span class="nd">@SimpleFunctionWrapper</span>
 113 <span class="k">def</span> <span class="nf">size</span><span class="p">(</span><span class="n">stack</span><span class="p">):</span>
 114     <span class="sd">&#39;&#39;&#39;Return the size of the sequence on the stack.&#39;&#39;&#39;</span>
 115     <span class="n">sequence</span><span class="p">,</span> <span class="n">stack</span> <span class="o">=</span> <span class="n">stack</span>
 116     <span class="n">n</span> <span class="o">=</span> <span class="mi">0</span>
 117     <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">iter_stack</span><span class="p">(</span><span class="n">sequence</span><span class="p">):</span>
 118         <span class="n">n</span> <span class="o">+=</span> <span class="mi">1</span>
 119     <span class="k">return</span> <span class="n">n</span><span class="p">,</span> <span class="n">stack</span>
 120 </pre></div>
 121 </div>
 122 <p>Now we replace the old version in the dictionary with the new version,
 123 and re-evaluate the expression.</p>
 124 <div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span><span class="n">D</span><span class="p">[</span><span class="s1">&#39;size&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">size</span>
 125 </pre></div>
 126 </div>
 127 </section>
 128 <section id="a-shorter-trace">
 129 <h2>A shorter trace<a class="headerlink" href="#a-shorter-trace" title="Permalink to this headline">¶</a></h2>
 130 <p>You can see that <code class="docutils literal notranslate"><span class="pre">size</span></code> now executes in a single step.</p>
 131 <div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span><span class="n">V</span><span class="p">(</span><span class="s1">&#39;[23 18] average&#39;</span><span class="p">)</span>
 132 </pre></div>
 133 </div>
 134 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span>                                  <span class="o">.</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="n">average</span>
 135                           <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="o">.</span> <span class="n">average</span>
 136                           <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="nb">sum</span> <span class="mf">1.0</span> <span class="o">*</span><span class="p">]</span> <span class="p">[</span><span class="n">size</span><span class="p">]</span> <span class="n">cleave</span> <span class="o">/</span>
 137               <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="nb">sum</span> <span class="mf">1.0</span> <span class="o">*</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="n">size</span><span class="p">]</span> <span class="n">cleave</span> <span class="o">/</span>
 138        <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="nb">sum</span> <span class="mf">1.0</span> <span class="o">*</span><span class="p">]</span> <span class="p">[</span><span class="n">size</span><span class="p">]</span> <span class="o">.</span> <span class="n">cleave</span> <span class="o">/</span>
 139        <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="nb">sum</span> <span class="mf">1.0</span> <span class="o">*</span><span class="p">]</span> <span class="p">[</span><span class="n">size</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">app2</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 140    <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="nb">sum</span> <span class="mf">1.0</span> <span class="o">*</span><span class="p">]</span> <span class="p">[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">.</span> <span class="n">app2</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 141 <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[[</span><span class="nb">sum</span> <span class="mf">1.0</span> <span class="o">*</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">.</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 142               <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="nb">sum</span> <span class="mf">1.0</span> <span class="o">*</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="p">[[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 143                           <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="o">.</span> <span class="nb">sum</span> <span class="mf">1.0</span> <span class="o">*</span> <span class="p">[[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 144                                <span class="mi">41</span> <span class="o">.</span> <span class="mf">1.0</span> <span class="o">*</span> <span class="p">[[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 145                            <span class="mi">41</span> <span class="mf">1.0</span> <span class="o">.</span> <span class="o">*</span> <span class="p">[[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 146                              <span class="mf">41.0</span> <span class="o">.</span> <span class="p">[[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 147                    <span class="mf">41.0</span> <span class="p">[[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="o">.</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 148                    <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="mf">41.0</span><span class="p">]</span> <span class="o">.</span> <span class="n">first</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 149                      <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="mf">41.0</span> <span class="o">.</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 150     <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="mf">41.0</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="o">.</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 151 <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="mf">41.0</span> <span class="p">[[</span><span class="n">size</span><span class="p">]</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">.</span> <span class="n">infra</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 152                    <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="p">[</span><span class="n">size</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 153                           <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="o">.</span> <span class="n">size</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 154                                 <span class="mi">2</span> <span class="o">.</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 155                  <span class="mi">2</span> <span class="p">[</span><span class="mf">41.0</span> <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]]</span> <span class="o">.</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 156                  <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="mf">41.0</span> <span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">first</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 157                    <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="mf">41.0</span> <span class="mi">2</span> <span class="o">.</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="n">dip</span> <span class="o">/</span>
 158             <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="mf">41.0</span> <span class="mi">2</span> <span class="p">[</span><span class="n">popd</span><span class="p">]</span> <span class="o">.</span> <span class="n">dip</span> <span class="o">/</span>
 159                      <span class="p">[</span><span class="mi">23</span> <span class="mi">18</span><span class="p">]</span> <span class="mf">41.0</span> <span class="o">.</span> <span class="n">popd</span> <span class="mi">2</span> <span class="o">/</span>
 160                              <span class="mf">41.0</span> <span class="o">.</span> <span class="mi">2</span> <span class="o">/</span>
 161                            <span class="mf">41.0</span> <span class="mi">2</span> <span class="o">.</span> <span class="o">/</span>
 162                              <span class="mf">20.5</span> <span class="o">.</span>
 163 </pre></div>
 164 </div>
 165 </section>
 166 </section>
 167
 168
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 170
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 176
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 205 <li class="toctree-l2"><a class="reference internal" href="Types.html">The Blissful Elegance of Typing Joy</a></li>
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