Bunches of new docs.
Type inference! A new treatment of recursion combinator patterns.
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Simon Forman
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<div class="bodywrapper">
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<div class="body" role="main">
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<div class="section" id="quadratic-formula">
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<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>
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<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>
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<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>
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</pre></div>
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</div>
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<div class="section" id="quadratic-formula">
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<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>
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<p>Cf.
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<a class="reference external" href="http://www.kevinalbrecht.com/code/joy-mirror/jp-quadratic.html">jp-quadratic.html</a></p>
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<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>
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<span class="o">-----------------------------</span>
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<span class="mi">2</span> <span class="o">*</span> <span class="n">a</span>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> -b ± sqrt(b^2 - 4 * a * c)
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--------------------------------
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2 * a
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</pre></div>
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</div>
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<p><span class="math notranslate nohighlight">\(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)</span></p>
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<div class="section" id="write-a-straightforward-program-with-variable-names">
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<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>
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<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>
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<p>This math translates to Joy code in a straightforward manner. We are
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going to use named variables to keep track of the arguments, then write
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a definition without them.</p>
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<div class="section" id="b">
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<h3><code class="docutils literal notranslate"><span class="pre">-b</span></code><a class="headerlink" href="#b" title="Permalink to this headline">¶</a></h3>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">b</span> <span class="n">neg</span>
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</pre></div>
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</div>
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<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
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finish computing both roots using a quoted program <code class="docutils literal notranslate"><span class="pre">[2a</span> <span class="pre">truediv]</span></code>
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built with <code class="docutils literal notranslate"><span class="pre">cons</span></code>.</p>
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<div class="section" id="check-it">
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<h3>Check it.<a class="headerlink" href="#check-it" title="Permalink to this headline">¶</a></h3>
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<p>Evaluating by hand:</p>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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</div>
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<div class="section" id="sqrt-b-2-4-a-c">
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<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>
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<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>
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</pre></div>
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</div>
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<p>(Eventually we’ll be able to use e.g. Sympy versions of the Joy commands
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to do this sort of thing symbolically. This is part of what is meant by
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a “categorical” language.)</p>
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</div>
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<div class="section" id="cleanup">
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<h3>Cleanup<a class="headerlink" href="#cleanup" title="Permalink to this headline">¶</a></h3>
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<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"><</span> <span class="n">pop</span>
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<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>
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<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>
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<div class="section" id="a">
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<h3><code class="docutils literal notranslate"><span class="pre">/2a</span></code><a class="headerlink" href="#a" title="Permalink to this headline">¶</a></h3>
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<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>
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</pre></div>
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</div>
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</div>
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<div class="section" id="id1">
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<h3><code class="docutils literal notranslate"><span class="pre">±</span></code><a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h3>
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<p>There is a function <code class="docutils literal notranslate"><span class="pre">pm</span></code> that accepts two values on the stack and
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replaces them with their sum and difference.</p>
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<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>
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</pre></div>
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</div>
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</div>
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<div class="section" id="putting-them-together">
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<h3>Putting Them Together<a class="headerlink" href="#putting-them-together" title="Permalink to this headline">¶</a></h3>
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<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>
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</pre></div>
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</div>
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<p>We use <code class="docutils literal notranslate"><span class="pre">app2</span></code> to compute both roots by using a quoted program
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<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>
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</div>
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</div>
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<div class="section" id="derive-a-definition">
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<h2>Derive a definition.<a class="headerlink" href="#derive-a-definition" title="Permalink to this headline">¶</a></h2>
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<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"><</span> <span class="n">pop</span>
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<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"><</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"><</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"><</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"><</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">'quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt [+] [-] cleave] dip 2 * [truediv] cons app2 roll< pop'</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">'quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2'</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">'3 1 1 quadratic'</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">'-5 1 4 quadratic'</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">'quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [truediv] cons app2'</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">'3 1 1 quadratic'</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">'3 1 1 quadratic'</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>
|
||||
<li>Next: <a href="Zipper.html" title="next chapter">Traversing Datastructures with Zippers</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>
|
||||
</ul></li>
|
||||
</ul></li>
|
||||
</ul>
|
||||
|
||||
Reference in New Issue
Block a user