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Type inference!

A new treatment of recursion combinator patterns.
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Simon Forman
2018-06-21 21:13:50 -07:00
parent 049cfd22b7
commit ca05ea404a
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@@ -16,7 +16,7 @@
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<link rel="index" title="Index" href="../genindex.html" />
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@@ -257,7 +257,7 @@ generated already and epsilon ε is handy on the stack…</p>
<li><a href="../index.html">Documentation overview</a><ul>
<li><a href="index.html">Essays about Programming in Joy</a><ul>
<li>Previous: <a href="Generator_Programs.html" title="previous chapter">Using <code class="docutils literal notranslate"><span class="pre">x</span></code> to Generate Values</a></li>
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docs/sphinx_docs/_build/html/notebooks/Quadratic.html
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@@ -16,8 +16,8 @@
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<link rel="index" title="Index" href="../genindex.html" />
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@@ -32,156 +32,136 @@
<div class="bodywrapper">
<div class="body" role="main">
<div class="section" id="quadratic-formula">
<h1><a class="reference external" href="https://en.wikipedia.org/wiki/Quadratic_formula">Quadratic formula</a><a class="headerlink" href="#quadratic-formula" title="Permalink to this headline"></a></h1>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">notebook_preamble</span> <span class="k">import</span> <span class="n">J</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="n">define</span>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">notebook_preamble</span> <span class="k">import</span> <span class="n">J</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="n">define</span>
</pre></div>
</div>
<div class="section" id="quadratic-formula">
<h1><a class="reference external" href="https://en.wikipedia.org/wiki/Quadratic_formula">Quadratic formula</a><a class="headerlink" href="#quadratic-formula" title="Permalink to this headline"></a></h1>
<p>Cf.
<a class="reference external" href="http://www.kevinalbrecht.com/code/joy-mirror/jp-quadratic.html">jp-quadratic.html</a></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">-</span><span class="n">b</span> <span class="o">+/-</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">b</span><span class="o">^</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">4</span> <span class="o">*</span> <span class="n">a</span> <span class="o">*</span> <span class="n">c</span><span class="p">)</span>
<span class="o">-----------------------------</span>
<span class="mi">2</span> <span class="o">*</span> <span class="n">a</span>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> -b ± sqrt(b^2 - 4 * a * c)
--------------------------------
2 * a
</pre></div>
</div>
<p><span class="math notranslate nohighlight">\(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)</span></p>
<div class="section" id="write-a-straightforward-program-with-variable-names">
<h2>Write a straightforward program with variable names.<a class="headerlink" href="#write-a-straightforward-program-with-variable-names" title="Permalink to this headline"></a></h2>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">b</span> <span class="n">neg</span> <span class="n">b</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="n">a</span> <span class="n">c</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="p">[</span><span class="o">-</span><span class="p">]</span> <span class="n">cleave</span> <span class="n">a</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<p>This math translates to Joy code in a straightforward manner. We are
going to use named variables to keep track of the arguments, then write
a definition without them.</p>
<div class="section" id="b">
<h3><code class="docutils literal notranslate"><span class="pre">-b</span></code><a class="headerlink" href="#b" title="Permalink to this headline"></a></h3>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">b</span> <span class="n">neg</span>
</pre></div>
</div>
<p>We use <code class="docutils literal notranslate"><span class="pre">cleave</span></code> to compute the sum and difference and then <code class="docutils literal notranslate"><span class="pre">app2</span></code> to
finish computing both roots using a quoted program <code class="docutils literal notranslate"><span class="pre">[2a</span> <span class="pre">truediv]</span></code>
built with <code class="docutils literal notranslate"><span class="pre">cons</span></code>.</p>
<div class="section" id="check-it">
<h3>Check it.<a class="headerlink" href="#check-it" title="Permalink to this headline"></a></h3>
<p>Evaluating by hand:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="n">b</span> <span class="n">neg</span> <span class="n">b</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="n">a</span> <span class="n">c</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="p">[</span><span class="o">-</span><span class="p">]</span> <span class="n">cleave</span> <span class="n">a</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="n">b</span> <span class="n">b</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="n">a</span> <span class="n">c</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="p">[</span><span class="o">-</span><span class="p">]</span> <span class="n">cleave</span> <span class="n">a</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="n">b</span> <span class="n">b</span><span class="o">^</span><span class="mi">2</span> <span class="mi">4</span> <span class="n">a</span> <span class="n">c</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="p">[</span><span class="o">-</span><span class="p">]</span> <span class="n">cleave</span> <span class="n">a</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="n">b</span> <span class="n">b</span><span class="o">^</span><span class="mi">2</span> <span class="mi">4</span><span class="n">ac</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="p">[</span><span class="o">-</span><span class="p">]</span> <span class="n">cleave</span> <span class="n">a</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="n">b</span> <span class="n">b</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="mi">4</span><span class="n">ac</span> <span class="n">sqrt</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="p">[</span><span class="o">-</span><span class="p">]</span> <span class="n">cleave</span> <span class="n">a</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="n">b</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">b</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="mi">4</span><span class="n">ac</span><span class="p">)</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="p">[</span><span class="o">-</span><span class="p">]</span> <span class="n">cleave</span> <span class="n">a</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="n">b</span> <span class="o">-</span><span class="n">b</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="n">b</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="mi">4</span><span class="n">ac</span><span class="p">)</span> <span class="o">-</span><span class="n">b</span><span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="n">b</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="mi">4</span><span class="n">ac</span><span class="p">)</span> <span class="n">a</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="n">b</span> <span class="o">-</span><span class="n">b</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="n">b</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="mi">4</span><span class="n">ac</span><span class="p">)</span> <span class="o">-</span><span class="n">b</span><span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="n">b</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="mi">4</span><span class="n">ac</span><span class="p">)</span> <span class="mi">2</span><span class="n">a</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="n">b</span> <span class="o">-</span><span class="n">b</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="n">b</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="mi">4</span><span class="n">ac</span><span class="p">)</span> <span class="o">-</span><span class="n">b</span><span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="n">b</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="mi">4</span><span class="n">ac</span><span class="p">)</span> <span class="p">[</span><span class="mi">2</span><span class="n">a</span> <span class="n">truediv</span><span class="p">]</span> <span class="n">app2</span>
<span class="o">-</span><span class="n">b</span> <span class="o">-</span><span class="n">b</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="n">b</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="mi">4</span><span class="n">ac</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="n">a</span> <span class="o">-</span><span class="n">b</span><span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="n">b</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="mi">4</span><span class="n">ac</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="n">a</span>
</div>
<div class="section" id="sqrt-b-2-4-a-c">
<h3><code class="docutils literal notranslate"><span class="pre">sqrt(b^2</span> <span class="pre">-</span> <span class="pre">4</span> <span class="pre">*</span> <span class="pre">a</span> <span class="pre">*</span> <span class="pre">c)</span></code><a class="headerlink" href="#sqrt-b-2-4-a-c" title="Permalink to this headline"></a></h3>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">b</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="n">a</span> <span class="n">c</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span>
</pre></div>
</div>
<p>(Eventually we’ll be able to use e.g. Sympy versions of the Joy commands
to do this sort of thing symbolically. This is part of what is meant by
a “categorical” language.)</p>
</div>
<div class="section" id="cleanup">
<h3>Cleanup<a class="headerlink" href="#cleanup" title="Permalink to this headline"></a></h3>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">-</span><span class="n">b</span> <span class="o">-</span><span class="n">b</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="n">b</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="mi">4</span><span class="n">ac</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="n">a</span> <span class="o">-</span><span class="n">b</span><span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="n">b</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="mi">4</span><span class="n">ac</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="n">a</span> <span class="n">roll</span><span class="o">&lt;</span> <span class="n">pop</span>
<span class="o">-</span><span class="n">b</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="n">b</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="mi">4</span><span class="n">ac</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="n">a</span> <span class="o">-</span><span class="n">b</span><span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="n">b</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="mi">4</span><span class="n">ac</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="n">a</span> <span class="o">-</span><span class="n">b</span> <span class="n">pop</span>
<span class="o">-</span><span class="n">b</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="n">b</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="mi">4</span><span class="n">ac</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="n">a</span> <span class="o">-</span><span class="n">b</span><span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="n">b</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="mi">4</span><span class="n">ac</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="n">a</span>
<div class="section" id="a">
<h3><code class="docutils literal notranslate"><span class="pre">/2a</span></code><a class="headerlink" href="#a" title="Permalink to this headline"></a></h3>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">a</span> <span class="mi">2</span> <span class="o">*</span> <span class="o">/</span>
</pre></div>
</div>
</div>
<div class="section" id="id1">
<h3><code class="docutils literal notranslate"><span class="pre">±</span></code><a class="headerlink" href="#id1" title="Permalink to this headline"></a></h3>
<p>There is a function <code class="docutils literal notranslate"><span class="pre">pm</span></code> that accepts two values on the stack and
replaces them with their sum and difference.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">pm</span> <span class="o">==</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="p">[</span><span class="o">-</span><span class="p">]</span> <span class="n">cleave</span> <span class="n">popdd</span>
</pre></div>
</div>
</div>
<div class="section" id="putting-them-together">
<h3>Putting Them Together<a class="headerlink" href="#putting-them-together" title="Permalink to this headline"></a></h3>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">b</span> <span class="n">neg</span> <span class="n">b</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="n">a</span> <span class="n">c</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="n">a</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
</pre></div>
</div>
<p>We use <code class="docutils literal notranslate"><span class="pre">app2</span></code> to compute both roots by using a quoted program
<code class="docutils literal notranslate"><span class="pre">[2a</span> <span class="pre">/]</span></code> built with <code class="docutils literal notranslate"><span class="pre">cons</span></code>.</p>
</div>
</div>
<div class="section" id="derive-a-definition">
<h2>Derive a definition.<a class="headerlink" href="#derive-a-definition" title="Permalink to this headline"></a></h2>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">b</span> <span class="n">neg</span> <span class="n">b</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="n">a</span> <span class="n">c</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="p">[</span><span class="o">-</span><span class="p">]</span> <span class="n">cleave</span> <span class="n">a</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span> <span class="n">roll</span><span class="o">&lt;</span> <span class="n">pop</span>
<span class="n">b</span> <span class="p">[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="n">a</span> <span class="n">c</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="p">[</span><span class="o">-</span><span class="p">]</span> <span class="n">cleave</span> <span class="n">a</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span> <span class="n">roll</span><span class="o">&lt;</span> <span class="n">pop</span>
<span class="n">b</span> <span class="n">a</span> <span class="n">c</span> <span class="p">[[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span><span class="p">]</span> <span class="n">dipd</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="p">[</span><span class="o">-</span><span class="p">]</span> <span class="n">cleave</span> <span class="n">a</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span> <span class="n">roll</span><span class="o">&lt;</span> <span class="n">pop</span>
<span class="n">b</span> <span class="n">a</span> <span class="n">c</span> <span class="n">a</span> <span class="p">[[[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span><span class="p">]</span> <span class="n">dipd</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="p">[</span><span class="o">-</span><span class="p">]</span> <span class="n">cleave</span><span class="p">]</span> <span class="n">dip</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span> <span class="n">roll</span><span class="o">&lt;</span> <span class="n">pop</span>
<span class="n">b</span> <span class="n">a</span> <span class="n">c</span> <span class="n">over</span> <span class="p">[[[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span><span class="p">]</span> <span class="n">dipd</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="p">[</span><span class="o">-</span><span class="p">]</span> <span class="n">cleave</span><span class="p">]</span> <span class="n">dip</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span> <span class="n">roll</span><span class="o">&lt;</span> <span class="n">pop</span>
<p>Working backwards we use <code class="docutils literal notranslate"><span class="pre">dip</span></code> and <code class="docutils literal notranslate"><span class="pre">dipd</span></code> to extract the code from
the variables:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">b</span> <span class="n">neg</span> <span class="n">b</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="n">a</span> <span class="n">c</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="n">a</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="n">b</span> <span class="p">[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="n">a</span> <span class="n">c</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="n">a</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="n">b</span> <span class="n">a</span> <span class="n">c</span> <span class="p">[[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span><span class="p">]</span> <span class="n">dipd</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="n">a</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="n">b</span> <span class="n">a</span> <span class="n">c</span> <span class="n">a</span> <span class="p">[[[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span><span class="p">]</span> <span class="n">dipd</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span><span class="p">]</span> <span class="n">dip</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="n">b</span> <span class="n">a</span> <span class="n">c</span> <span class="n">over</span> <span class="p">[[[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span><span class="p">]</span> <span class="n">dipd</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span><span class="p">]</span> <span class="n">dip</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
</pre></div>
</div>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt [+] [-] cleave] dip 2 * [truediv] cons app2 roll&lt; pop&#39;</span><span class="p">)</span>
<p>The three arguments are to the left, so we can “chop off” everything to
the right and say it’s the definition of the <code class="docutils literal notranslate"><span class="pre">quadratic</span></code> function:</p>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2&#39;</span><span class="p">)</span>
</pre></div>
</div>
<p>Let’s try it out:</p>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;3 1 1 quadratic&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">-</span><span class="mf">0.3819660112501051</span> <span class="o">-</span><span class="mf">2.618033988749895</span>
</pre></div>
</div>
</div>
<div class="section" id="simplify">
<h2>Simplify<a class="headerlink" href="#simplify" title="Permalink to this headline"></a></h2>
<p>We can define a <code class="docutils literal notranslate"><span class="pre">pm</span></code> plus-or-minus function:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">pm</span> <span class="o">==</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="p">[</span><span class="o">-</span><span class="p">]</span> <span class="n">cleave</span> <span class="n">popdd</span>
<p>If you look at the Joy evaluation trace you can see that the first few
lines are the <code class="docutils literal notranslate"><span class="pre">dip</span></code> and <code class="docutils literal notranslate"><span class="pre">dipd</span></code> combinators building the main program
by incorporating the values on the stack. Then that program runs and you
get the results. This is pretty typical of Joy code.</p>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">V</span><span class="p">(</span><span class="s1">&#39;-5 1 4 quadratic&#39;</span><span class="p">)</span>
</pre></div>
</div>
<p>Then <code class="docutils literal notranslate"><span class="pre">quadratic</span></code> becomes:</p>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [truediv] cons app2&#39;</span><span class="p">)</span>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="o">.</span> <span class="o">-</span><span class="mi">5</span> <span class="mi">1</span> <span class="mi">4</span> <span class="n">quadratic</span>
<span class="o">-</span><span class="mi">5</span> <span class="o">.</span> <span class="mi">1</span> <span class="mi">4</span> <span class="n">quadratic</span>
<span class="o">-</span><span class="mi">5</span> <span class="mi">1</span> <span class="o">.</span> <span class="mi">4</span> <span class="n">quadratic</span>
<span class="o">-</span><span class="mi">5</span> <span class="mi">1</span> <span class="mi">4</span> <span class="o">.</span> <span class="n">quadratic</span>
<span class="o">-</span><span class="mi">5</span> <span class="mi">1</span> <span class="mi">4</span> <span class="o">.</span> <span class="n">over</span> <span class="p">[[[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span><span class="p">]</span> <span class="n">dipd</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span><span class="p">]</span> <span class="n">dip</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">5</span> <span class="mi">1</span> <span class="mi">4</span> <span class="mi">1</span> <span class="o">.</span> <span class="p">[[[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span><span class="p">]</span> <span class="n">dipd</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span><span class="p">]</span> <span class="n">dip</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">5</span> <span class="mi">1</span> <span class="mi">4</span> <span class="mi">1</span> <span class="p">[[[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span><span class="p">]</span> <span class="n">dipd</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span><span class="p">]</span> <span class="o">.</span> <span class="n">dip</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">5</span> <span class="mi">1</span> <span class="mi">4</span> <span class="o">.</span> <span class="p">[[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span><span class="p">]</span> <span class="n">dipd</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">5</span> <span class="mi">1</span> <span class="mi">4</span> <span class="p">[[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span><span class="p">]</span> <span class="o">.</span> <span class="n">dipd</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">5</span> <span class="o">.</span> <span class="p">[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">4</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">5</span> <span class="p">[</span><span class="n">neg</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">4</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">5</span> <span class="o">.</span> <span class="n">neg</span> <span class="o">-</span><span class="mi">5</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">4</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">5</span> <span class="o">.</span> <span class="o">-</span><span class="mi">5</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">4</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">5</span> <span class="o">-</span><span class="mi">5</span> <span class="o">.</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">4</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">5</span> <span class="o">-</span><span class="mi">5</span> <span class="o">.</span> <span class="n">dup</span> <span class="n">mul</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">4</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">5</span> <span class="o">-</span><span class="mi">5</span> <span class="o">-</span><span class="mi">5</span> <span class="o">.</span> <span class="n">mul</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">4</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">5</span> <span class="mi">25</span> <span class="o">.</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">4</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">5</span> <span class="mi">25</span> <span class="mi">4</span> <span class="o">.</span> <span class="mi">1</span> <span class="mi">4</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">5</span> <span class="mi">25</span> <span class="mi">4</span> <span class="mi">1</span> <span class="o">.</span> <span class="mi">4</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">5</span> <span class="mi">25</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">4</span> <span class="o">.</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">5</span> <span class="mi">25</span> <span class="mi">4</span> <span class="mi">4</span> <span class="o">.</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">5</span> <span class="mi">25</span> <span class="mi">16</span> <span class="o">.</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">5</span> <span class="mi">9</span> <span class="o">.</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">5</span> <span class="mf">3.0</span> <span class="o">.</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mf">8.0</span> <span class="mf">2.0</span> <span class="o">.</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mf">8.0</span> <span class="mf">2.0</span> <span class="mi">1</span> <span class="o">.</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mf">8.0</span> <span class="mf">2.0</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">.</span> <span class="o">*</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mf">8.0</span> <span class="mf">2.0</span> <span class="mi">2</span> <span class="o">.</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mf">8.0</span> <span class="mf">2.0</span> <span class="mi">2</span> <span class="p">[</span><span class="o">/</span><span class="p">]</span> <span class="o">.</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mf">8.0</span> <span class="mf">2.0</span> <span class="p">[</span><span class="mi">2</span> <span class="o">/</span><span class="p">]</span> <span class="o">.</span> <span class="n">app2</span>
<span class="p">[</span><span class="mf">8.0</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="o">/</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="mf">2.0</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="o">/</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span>
<span class="mf">8.0</span> <span class="o">.</span> <span class="mi">2</span> <span class="o">/</span> <span class="p">[]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="mf">2.0</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="o">/</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span>
<span class="mf">8.0</span> <span class="mi">2</span> <span class="o">.</span> <span class="o">/</span> <span class="p">[]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="mf">2.0</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="o">/</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span>
<span class="mf">4.0</span> <span class="o">.</span> <span class="p">[]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="mf">2.0</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="o">/</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span>
<span class="mf">4.0</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="mf">2.0</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="o">/</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span>
<span class="p">[</span><span class="mf">4.0</span><span class="p">]</span> <span class="o">.</span> <span class="n">first</span> <span class="p">[</span><span class="mf">2.0</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="o">/</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span>
<span class="mf">4.0</span> <span class="o">.</span> <span class="p">[</span><span class="mf">2.0</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="o">/</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span>
<span class="mf">4.0</span> <span class="p">[</span><span class="mf">2.0</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="mi">2</span> <span class="o">/</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span>
<span class="mf">4.0</span> <span class="p">[</span><span class="mf">2.0</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="o">/</span><span class="p">]</span> <span class="o">.</span> <span class="n">infra</span> <span class="n">first</span>
<span class="mf">2.0</span> <span class="o">.</span> <span class="mi">2</span> <span class="o">/</span> <span class="p">[</span><span class="mf">4.0</span><span class="p">]</span> <span class="n">swaack</span> <span class="n">first</span>
<span class="mf">2.0</span> <span class="mi">2</span> <span class="o">.</span> <span class="o">/</span> <span class="p">[</span><span class="mf">4.0</span><span class="p">]</span> <span class="n">swaack</span> <span class="n">first</span>
<span class="mf">1.0</span> <span class="o">.</span> <span class="p">[</span><span class="mf">4.0</span><span class="p">]</span> <span class="n">swaack</span> <span class="n">first</span>
<span class="mf">1.0</span> <span class="p">[</span><span class="mf">4.0</span><span class="p">]</span> <span class="o">.</span> <span class="n">swaack</span> <span class="n">first</span>
<span class="mf">4.0</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">]</span> <span class="o">.</span> <span class="n">first</span>
<span class="mf">4.0</span> <span class="mf">1.0</span> <span class="o">.</span>
</pre></div>
</div>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;3 1 1 quadratic&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">-</span><span class="mf">0.3819660112501051</span> <span class="o">-</span><span class="mf">2.618033988749895</span>
</pre></div>
</div>
<div class="section" id="define-a-native-pm-function">
<h3>Define a “native” <code class="docutils literal notranslate"><span class="pre">pm</span></code> function.<a class="headerlink" href="#define-a-native-pm-function" title="Permalink to this headline"></a></h3>
<p>The definition of <code class="docutils literal notranslate"><span class="pre">pm</span></code> above is pretty elegant, but the implementation
takes a lot of steps relative to what it’s accomplishing. Since we are
likely to use <code class="docutils literal notranslate"><span class="pre">pm</span></code> more than once in the future, let’s write a
primitive in Python and add it to the dictionary. (This has been done
already.)</p>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">pm</span><span class="p">(</span><span class="n">stack</span><span class="p">):</span>
<span class="n">a</span><span class="p">,</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">stack</span><span class="p">)</span> <span class="o">=</span> <span class="n">stack</span>
<span class="n">p</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="o">=</span> <span class="n">b</span> <span class="o">+</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">-</span> <span class="n">a</span>
<span class="k">return</span> <span class="n">m</span><span class="p">,</span> <span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">stack</span><span class="p">)</span>
</pre></div>
</div>
<p>The resulting trace is short enough to fit on a page.</p>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">V</span><span class="p">(</span><span class="s1">&#39;3 1 1 quadratic&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="o">.</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">1</span> <span class="n">quadratic</span>
<span class="mi">3</span> <span class="o">.</span> <span class="mi">1</span> <span class="mi">1</span> <span class="n">quadratic</span>
<span class="mi">3</span> <span class="mi">1</span> <span class="o">.</span> <span class="mi">1</span> <span class="n">quadratic</span>
<span class="mi">3</span> <span class="mi">1</span> <span class="mi">1</span> <span class="o">.</span> <span class="n">quadratic</span>
<span class="mi">3</span> <span class="mi">1</span> <span class="mi">1</span> <span class="o">.</span> <span class="n">over</span> <span class="p">[[[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span><span class="p">]</span> <span class="n">dipd</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span><span class="p">]</span> <span class="n">dip</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">3</span> <span class="mi">1</span> <span class="mi">1</span> <span class="mi">1</span> <span class="o">.</span> <span class="p">[[[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span><span class="p">]</span> <span class="n">dipd</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span><span class="p">]</span> <span class="n">dip</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">3</span> <span class="mi">1</span> <span class="mi">1</span> <span class="mi">1</span> <span class="p">[[[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span><span class="p">]</span> <span class="n">dipd</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span><span class="p">]</span> <span class="o">.</span> <span class="n">dip</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">3</span> <span class="mi">1</span> <span class="mi">1</span> <span class="o">.</span> <span class="p">[[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span><span class="p">]</span> <span class="n">dipd</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">3</span> <span class="mi">1</span> <span class="mi">1</span> <span class="p">[[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span><span class="p">]</span> <span class="o">.</span> <span class="n">dipd</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">3</span> <span class="o">.</span> <span class="p">[</span><span class="n">neg</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">1</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">3</span> <span class="p">[</span><span class="n">neg</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">1</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="mi">3</span> <span class="o">.</span> <span class="n">neg</span> <span class="mi">3</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">1</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">3</span> <span class="o">.</span> <span class="mi">3</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">1</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">3</span> <span class="mi">3</span> <span class="o">.</span> <span class="n">sqr</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">1</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">3</span> <span class="mi">3</span> <span class="o">.</span> <span class="n">dup</span> <span class="n">mul</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">1</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">3</span> <span class="mi">3</span> <span class="mi">3</span> <span class="o">.</span> <span class="n">mul</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">1</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">3</span> <span class="mi">9</span> <span class="o">.</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">1</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">3</span> <span class="mi">9</span> <span class="mi">4</span> <span class="o">.</span> <span class="mi">1</span> <span class="mi">1</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">3</span> <span class="mi">9</span> <span class="mi">4</span> <span class="mi">1</span> <span class="o">.</span> <span class="mi">1</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">3</span> <span class="mi">9</span> <span class="mi">4</span> <span class="mi">1</span> <span class="mi">1</span> <span class="o">.</span> <span class="o">*</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">3</span> <span class="mi">9</span> <span class="mi">4</span> <span class="mi">1</span> <span class="o">.</span> <span class="o">*</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">3</span> <span class="mi">9</span> <span class="mi">4</span> <span class="o">.</span> <span class="o">-</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">3</span> <span class="mi">5</span> <span class="o">.</span> <span class="n">sqrt</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mi">3</span> <span class="mf">2.23606797749979</span> <span class="o">.</span> <span class="n">pm</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mf">0.7639320225002102</span> <span class="o">-</span><span class="mf">5.23606797749979</span> <span class="o">.</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mf">0.7639320225002102</span> <span class="o">-</span><span class="mf">5.23606797749979</span> <span class="mi">1</span> <span class="o">.</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mf">0.7639320225002102</span> <span class="o">-</span><span class="mf">5.23606797749979</span> <span class="mi">1</span> <span class="mi">2</span> <span class="o">.</span> <span class="o">*</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mf">0.7639320225002102</span> <span class="o">-</span><span class="mf">5.23606797749979</span> <span class="mi">2</span> <span class="o">.</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mf">0.7639320225002102</span> <span class="o">-</span><span class="mf">5.23606797749979</span> <span class="mi">2</span> <span class="p">[</span><span class="n">truediv</span><span class="p">]</span> <span class="o">.</span> <span class="n">cons</span> <span class="n">app2</span>
<span class="o">-</span><span class="mf">0.7639320225002102</span> <span class="o">-</span><span class="mf">5.23606797749979</span> <span class="p">[</span><span class="mi">2</span> <span class="n">truediv</span><span class="p">]</span> <span class="o">.</span> <span class="n">app2</span>
<span class="p">[</span><span class="o">-</span><span class="mf">0.7639320225002102</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="n">truediv</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="o">-</span><span class="mf">5.23606797749979</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="n">truediv</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span>
<span class="o">-</span><span class="mf">0.7639320225002102</span> <span class="o">.</span> <span class="mi">2</span> <span class="n">truediv</span> <span class="p">[]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="o">-</span><span class="mf">5.23606797749979</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="n">truediv</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span>
<span class="o">-</span><span class="mf">0.7639320225002102</span> <span class="mi">2</span> <span class="o">.</span> <span class="n">truediv</span> <span class="p">[]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="o">-</span><span class="mf">5.23606797749979</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="n">truediv</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span>
<span class="o">-</span><span class="mf">0.3819660112501051</span> <span class="o">.</span> <span class="p">[]</span> <span class="n">swaack</span> <span class="n">first</span> <span class="p">[</span><span class="o">-</span><span class="mf">5.23606797749979</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="n">truediv</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span>
<span class="o">-</span><span class="mf">0.3819660112501051</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="o">-</span><span class="mf">5.23606797749979</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="n">truediv</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span>
<span class="p">[</span><span class="o">-</span><span class="mf">0.3819660112501051</span><span class="p">]</span> <span class="o">.</span> <span class="n">first</span> <span class="p">[</span><span class="o">-</span><span class="mf">5.23606797749979</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="n">truediv</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span>
<span class="o">-</span><span class="mf">0.3819660112501051</span> <span class="o">.</span> <span class="p">[</span><span class="o">-</span><span class="mf">5.23606797749979</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="n">truediv</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span>
<span class="o">-</span><span class="mf">0.3819660112501051</span> <span class="p">[</span><span class="o">-</span><span class="mf">5.23606797749979</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="mi">2</span> <span class="n">truediv</span><span class="p">]</span> <span class="n">infra</span> <span class="n">first</span>
<span class="o">-</span><span class="mf">0.3819660112501051</span> <span class="p">[</span><span class="o">-</span><span class="mf">5.23606797749979</span><span class="p">]</span> <span class="p">[</span><span class="mi">2</span> <span class="n">truediv</span><span class="p">]</span> <span class="o">.</span> <span class="n">infra</span> <span class="n">first</span>
<span class="o">-</span><span class="mf">5.23606797749979</span> <span class="o">.</span> <span class="mi">2</span> <span class="n">truediv</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.3819660112501051</span><span class="p">]</span> <span class="n">swaack</span> <span class="n">first</span>
<span class="o">-</span><span class="mf">5.23606797749979</span> <span class="mi">2</span> <span class="o">.</span> <span class="n">truediv</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.3819660112501051</span><span class="p">]</span> <span class="n">swaack</span> <span class="n">first</span>
<span class="o">-</span><span class="mf">2.618033988749895</span> <span class="o">.</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.3819660112501051</span><span class="p">]</span> <span class="n">swaack</span> <span class="n">first</span>
<span class="o">-</span><span class="mf">2.618033988749895</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.3819660112501051</span><span class="p">]</span> <span class="o">.</span> <span class="n">swaack</span> <span class="n">first</span>
<span class="o">-</span><span class="mf">0.3819660112501051</span> <span class="p">[</span><span class="o">-</span><span class="mf">2.618033988749895</span><span class="p">]</span> <span class="o">.</span> <span class="n">first</span>
<span class="o">-</span><span class="mf">0.3819660112501051</span> <span class="o">-</span><span class="mf">2.618033988749895</span> <span class="o">.</span>
</pre></div>
</div>
</div>
</div>
</div>
@@ -195,15 +175,14 @@ already.)</p>
<ul>
<li><a class="reference internal" href="#">Quadratic formula</a><ul>
<li><a class="reference internal" href="#write-a-straightforward-program-with-variable-names">Write a straightforward program with variable names.</a><ul>
<li><a class="reference internal" href="#check-it">Check it.</a></li>
<li><a class="reference internal" href="#cleanup">Cleanup</a></li>
<li><a class="reference internal" href="#b"><code class="docutils literal notranslate"><span class="pre">-b</span></code></a></li>
<li><a class="reference internal" href="#sqrt-b-2-4-a-c"><code class="docutils literal notranslate"><span class="pre">sqrt(b^2</span> <span class="pre">-</span> <span class="pre">4</span> <span class="pre">*</span> <span class="pre">a</span> <span class="pre">*</span> <span class="pre">c)</span></code></a></li>
<li><a class="reference internal" href="#a"><code class="docutils literal notranslate"><span class="pre">/2a</span></code></a></li>
<li><a class="reference internal" href="#id1"><code class="docutils literal notranslate"><span class="pre">±</span></code></a></li>
<li><a class="reference internal" href="#putting-them-together">Putting Them Together</a></li>
</ul>
</li>
<li><a class="reference internal" href="#derive-a-definition">Derive a definition.</a></li>
<li><a class="reference internal" href="#simplify">Simplify</a><ul>
<li><a class="reference internal" href="#define-a-native-pm-function">Define a “native” <code class="docutils literal notranslate"><span class="pre">pm</span></code> function.</a></li>
</ul>
</li>
</ul>
</li>
</ul>
@@ -212,8 +191,8 @@ already.)</p>
<ul>
<li><a href="../index.html">Documentation overview</a><ul>
<li><a href="index.html">Essays about Programming in Joy</a><ul>
<li>Previous: <a href="Newton-Raphson.html" title="previous chapter">Newton’s method</a></li>
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<li>Previous: <a href="Developing.html" title="previous chapter">Developing a Program in Joy</a></li>
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</ul></li>
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</ul>
docs/sphinx_docs/_build/html/notebooks/Zipper.html
+4 -4
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@@ -16,8 +16,8 @@
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<link rel="index" title="Index" href="../genindex.html" />
<link rel="search" title="Search" href="../search.html" />
<link rel="next" title="No Updates" href="NoUpdates.html" />
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<link rel="stylesheet" href="../_static/custom.css" type="text/css" />
@@ -326,8 +326,8 @@ i d i d i d d Bingo!
<ul>
<li><a href="../index.html">Documentation overview</a><ul>
<li><a href="index.html">Essays about Programming in Joy</a><ul>
<li>Previous: <a href="Quadratic.html" title="previous chapter">Quadratic formula</a></li>
<li>Next: <a href="NoUpdates.html" title="next chapter">No Updates</a></li>
<li>Previous: <a href="Newton-Raphson.html" title="previous chapter">Newton’s method</a></li>
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</ul>
docs/sphinx_docs/_build/html/notebooks/index.html
+34 -6
View File
@@ -44,12 +44,32 @@
<li class="toctree-l2"><a class="reference internal" href="Developing.html#the-simplest-program">The Simplest Program</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="Quadratic.html">Quadratic formula</a><ul>
<li class="toctree-l2"><a class="reference internal" href="Quadratic.html#write-a-straightforward-program-with-variable-names">Write a straightforward program with variable names.</a></li>
<li class="toctree-l2"><a class="reference internal" href="Quadratic.html#derive-a-definition">Derive a definition.</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="Replacing.html">Replacing Functions in the Dictionary</a><ul>
<li class="toctree-l2"><a class="reference internal" href="Replacing.html#a-long-trace">A long trace</a></li>
<li class="toctree-l2"><a class="reference internal" href="Replacing.html#replacing-size-with-a-python-version">Replacing <code class="docutils literal notranslate"><span class="pre">size</span></code> with a Python version</a></li>
<li class="toctree-l2"><a class="reference internal" href="Replacing.html#a-shorter-trace">A shorter trace</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="Recursion_Combinators.html">Recursive Combinators</a><ul>
<li class="toctree-l2"><a class="reference internal" href="Recursion_Combinators.html#designing-recursive-functions">Designing Recursive Functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="Recursion_Combinators.html#primitive-recursive-functions">Primitive Recursive Functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="Recursion_Combinators.html#hylomorphism">Hylomorphism</a></li>
<li class="toctree-l2"><a class="reference internal" href="Recursion_Combinators.html#hylomorphism-in-joy">Hylomorphism in Joy</a></li>
<li class="toctree-l2"><a class="reference internal" href="Recursion_Combinators.html#derivation-of-hylomorphism-combinator">Derivation of <code class="docutils literal notranslate"><span class="pre">hylomorphism</span></code> combinator</a></li>
<li class="toctree-l2"><a class="reference internal" href="Recursion_Combinators.html#four-specializations">Four Specializations</a></li>
<li class="toctree-l2"><a class="reference internal" href="Recursion_Combinators.html#anamorphism">Anamorphism</a></li>
<li class="toctree-l2"><a class="reference internal" href="Recursion_Combinators.html#catamorphism">Catamorphism</a></li>
<li class="toctree-l2"><a class="reference internal" href="Recursion_Combinators.html#example-factorial-function">Example: Factorial Function</a></li>
<li class="toctree-l2"><a class="reference internal" href="Recursion_Combinators.html#example-tails">Example: <code class="docutils literal notranslate"><span class="pre">tails</span></code></a></li>
<li class="toctree-l2"><a class="reference internal" href="Recursion_Combinators.html#conclusion-patterns-of-recursion">Conclusion: Patterns of Recursion</a></li>
<li class="toctree-l2"><a class="reference internal" href="Recursion_Combinators.html#appendix-fun-with-symbols">Appendix: Fun with Symbols</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="Ordered_Binary_Trees.html">Treating Trees I: Ordered Binary Trees</a><ul>
<li class="toctree-l2"><a class="reference internal" href="Ordered_Binary_Trees.html#adding-nodes-to-the-tree">Adding Nodes to the Tree</a></li>
<li class="toctree-l2"><a class="reference internal" href="Ordered_Binary_Trees.html#interlude-cmp-combinator">Interlude: <code class="docutils literal notranslate"><span class="pre">cmp</span></code> combinator</a></li>
@@ -89,12 +109,6 @@
<li class="toctree-l2"><a class="reference internal" href="Newton-Raphson.html#finding-consecutive-approximations-within-a-tolerance">Finding Consecutive Approximations within a Tolerance</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="Quadratic.html">Quadratic formula</a><ul>
<li class="toctree-l2"><a class="reference internal" href="Quadratic.html#write-a-straightforward-program-with-variable-names">Write a straightforward program with variable names.</a></li>
<li class="toctree-l2"><a class="reference internal" href="Quadratic.html#derive-a-definition">Derive a definition.</a></li>
<li class="toctree-l2"><a class="reference internal" href="Quadratic.html#simplify">Simplify</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="Zipper.html">Traversing Datastructures with Zippers</a><ul>
<li class="toctree-l2"><a class="reference internal" href="Zipper.html#trees">Trees</a></li>
<li class="toctree-l2"><a class="reference internal" href="Zipper.html#zipper-in-joy">Zipper in Joy</a></li>
@@ -104,6 +118,20 @@
<li class="toctree-l2"><a class="reference internal" href="Zipper.html#determining-the-right-path-for-an-item-in-a-tree">Determining the right “path” for an item in a tree.</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="Types.html">Type Inference</a><ul>
<li class="toctree-l2"><a class="reference internal" href="Types.html#poial-s-rules">Pöial’s Rules</a></li>
<li class="toctree-l2"><a class="reference internal" href="Types.html#examples">Examples</a></li>
<li class="toctree-l2"><a class="reference internal" href="Types.html#implementation">Implementation</a></li>
<li class="toctree-l2"><a class="reference internal" href="Types.html#compiling">Compiling</a></li>
<li class="toctree-l2"><a class="reference internal" href="Types.html#types-and-subtypes-of-arguments">Types and Subtypes of Arguments</a></li>
<li class="toctree-l2"><a class="reference internal" href="Types.html#functions-that-use-the-stack">Functions that use the Stack</a></li>
<li class="toctree-l2"><a class="reference internal" href="Types.html#sets-of-stack-effects">Sets of Stack Effects</a></li>
<li class="toctree-l2"><a class="reference internal" href="Types.html#concat"><code class="docutils literal notranslate"><span class="pre">concat</span></code></a></li>
<li class="toctree-l2"><a class="reference internal" href="Types.html#typing-combinators">Typing Combinators</a></li>
<li class="toctree-l2"><a class="reference internal" href="Types.html#abstract-interpretation">Abstract Interpretation</a></li>
<li class="toctree-l2"><a class="reference internal" href="Types.html#something-else">Something else…</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="NoUpdates.html">No Updates</a></li>
<li class="toctree-l1"><a class="reference internal" href="Categorical.html">Categorical Programming</a></li>
</ul>