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Recover the square spiral example code.
[joypy/Thun.git] / docs / sphinx_docs / _build / html / notebooks / Quadratic.html
diff --git a/docs/sphinx_docs/_build/html/notebooks/Quadratic.html b/docs/sphinx_docs/_build/html/notebooks/Quadratic.html
index 68867b3..93bb733 100644 (file)
--- a/docs/sphinx_docs/_build/html/notebooks/Quadratic.html
+++ b/docs/sphinx_docs/_build/html/notebooks/Quadratic.html
@@ -1,23 +1,23 @@
 
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
+<!DOCTYPE html>
 
-<html xmlns="http://www.w3.org/1999/xhtml">
+<html>
   <head>
-    <meta http-equiv="X-UA-Compatible" content="IE=Edge" />
-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
-    <title>Quadratic formula &#8212; Thun 0.2.0 documentation</title>
-    <link rel="stylesheet" href="../_static/alabaster.css" type="text/css" />
-    <link rel="stylesheet" href="../_static/pygments.css" type="text/css" />
-    <script type="text/javascript" src="../_static/documentation_options.js"></script>
-    <script type="text/javascript" src="../_static/jquery.js"></script>
-    <script type="text/javascript" src="../_static/underscore.js"></script>
-    <script type="text/javascript" src="../_static/doctools.js"></script>
-    <script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
+    <meta charset="utf-8" />
+    <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
+
+    <title>Quadratic formula &#8212; Thun 0.4.1 documentation</title>
+    <link rel="stylesheet" type="text/css" href="../_static/pygments.css" />
+    <link rel="stylesheet" type="text/css" href="../_static/alabaster.css" />
+    <script data-url_root="../" id="documentation_options" src="../_static/documentation_options.js"></script>
+    <script src="../_static/jquery.js"></script>
+    <script src="../_static/underscore.js"></script>
+    <script src="../_static/doctools.js"></script>
+    <script defer="defer" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></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" />
-    <link rel="prev" title="Newton’s method" href="Newton-Raphson.html" />
+    <link rel="next" title="Replacing Functions in the Dictionary" href="Replacing.html" />
+    <link rel="prev" title="Developing a Program in Joy" href="Developing.html" />
    
   <link rel="stylesheet" href="../_static/custom.css" type="text/css" />
   
@@ -30,213 +30,218 @@
     <div class="document">
       <div class="documentwrapper">
         <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="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">J</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="n">define</span>
 </pre></div>
 </div>
+<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">
+<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>
-</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>
+<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>
+<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>(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>
-</pre></div>
-</div>
-</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>
+</section>
+<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>
-<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>
+</section>
+<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 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>
+</section>
+<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 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>
+</section>
+<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>
-</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>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>
+</section>
+</section>
+<section id="derive-a-definition">
+<h2>Derive a definition.<a class="headerlink" href="#derive-a-definition" title="Permalink to this headline">¶</a></h2>
+<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>
-<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>
+<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="highlight-ipython2 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>
-<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>
+<p>Let’s try it out:</p>
+<div class="highlight-ipython2 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>
+<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="highlight-ipython2 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>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>
+<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="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>
+</section>
+</section>
 
 
           </div>
+          
         </div>
       </div>
       <div class="sphinxsidebar" role="navigation" aria-label="main navigation">
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-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <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>
-</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>
+<h1 class="logo"><a href="../index.html">Thun</a></h1>
+
+
+
+
+
+
+
+
+<h3>Navigation</h3>
+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="Intro.html">Thun: Joy in Python</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../joy.html">Joy Interpreter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../stack.html">Stack or Quote or Sequence or List…</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../parser.html">Parsing Text into Joy Expressions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../pretty.html">Tracing Joy Execution</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../library.html">Function Reference</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../lib.html">Functions Grouped by, er, Function with Examples</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../types.html">Type Inference of Joy Expressions</a></li>
+<li class="toctree-l1 current"><a class="reference internal" href="index.html">Essays about Programming in Joy</a><ul class="current">
+<li class="toctree-l2"><a class="reference internal" href="Developing.html">Developing a Program in Joy</a></li>
+<li class="toctree-l2 current"><a class="current reference internal" href="#">Quadratic formula</a></li>
+<li class="toctree-l2"><a class="reference internal" href="Replacing.html">Replacing Functions in the Dictionary</a></li>
+<li class="toctree-l2"><a class="reference internal" href="Recursion_Combinators.html">Recursion Combinators</a></li>
+<li class="toctree-l2"><a class="reference internal" href="Ordered_Binary_Trees.html">Treating Trees I: Ordered Binary Trees</a></li>
+<li class="toctree-l2"><a class="reference internal" href="Treestep.html">Treating Trees II: <code class="docutils literal notranslate"><span class="pre">treestep</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="Generator_Programs.html">Using <code class="docutils literal notranslate"><span class="pre">x</span></code> to Generate Values</a></li>
+<li class="toctree-l2"><a class="reference internal" href="Newton-Raphson.html">Newton’s method</a></li>
+<li class="toctree-l2"><a class="reference internal" href="Zipper.html">Traversing Datastructures with Zippers</a></li>
+<li class="toctree-l2"><a class="reference internal" href="Types.html">The Blissful Elegance of Typing Joy</a></li>
+<li class="toctree-l2"><a class="reference internal" href="TypeChecking.html">Type Checking</a></li>
+<li class="toctree-l2"><a class="reference internal" href="NoUpdates.html">No Updates</a></li>
+<li class="toctree-l2"><a class="reference internal" href="Categorical.html">Categorical Programming</a></li>
+<li class="toctree-l2"><a class="reference internal" href="The_Four_Operations.html">The Four Fundamental Operations of Definite Action</a></li>
+<li class="toctree-l2"><a class="reference internal" href="Derivatives_of_Regular_Expressions.html">∂RE</a></li>
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+
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   <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>
-      <li>Next: <a href="NoUpdates.html" title="next chapter">No Updates</a></li>
+      <li>Previous: <a href="Developing.html" title="previous chapter">Developing a Program in Joy</a></li>
+      <li>Next: <a href="Replacing.html" title="next chapter">Replacing Functions in the Dictionary</a></li>
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-  <div role="note" aria-label="source link">
-    <h3>This Page</h3>
-    <ul class="this-page-menu">
-      <li><a href="../_sources/notebooks/Quadratic.rst.txt"
-            rel="nofollow">Show Source</a></li>
-    </ul>
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-  <h3>Quick search</h3>
+  <h3 id="searchlabel">Quick search</h3>
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     <form class="search" action="../search.html" method="get">
-      <input type="text" name="q" />
+      <input type="text" name="q" aria-labelledby="searchlabel" autocomplete="off" autocorrect="off" autocapitalize="off" spellcheck="false"/>
       <input type="submit" value="Go" />
-      <input type="hidden" name="check_keywords" value="yes" />
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+<script>$('#searchbox').show(0);</script>
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@@ -247,7 +252,7 @@ already.)</p>
 </a>
 <br />
 <span xmlns:dct="http://purl.org/dc/terms/" property="dct:title">Thun Documentation</span> by <a xmlns:cc="http://creativecommons.org/ns#" href="https://joypy.osdn.io/" property="cc:attributionName" rel="cc:attributionURL">Simon Forman</a> is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License</a>.<br />Based on a work at <a xmlns:dct="http://purl.org/dc/terms/" href="https://osdn.net/projects/joypy/" rel="dct:source">https://osdn.net/projects/joypy/</a>.
-      Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.7.3.
+      Created using <a href="http://sphinx-doc.org/">Sphinx</a> 4.3.0.
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