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<div class="section" id="developing-a-program-in-joy">
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<h1>Developing a Program in Joy<a class="headerlink" href="#developing-a-program-in-joy" title="Permalink to this headline">¶</a></h1>
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<p>As an example of developing a program in Joy let’s take the first problem from the Project Euler website.</p>
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<div class="section" id="project-euler-first-problem-multiples-of-3-and-5">
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<h2><a class="reference external" href="https://projecteuler.net/problem=1">Project Euler, first problem: “Multiples of 3 and 5”</a><a class="headerlink" href="#project-euler-first-problem-multiples-of-3-and-5" title="Permalink to this headline">¶</a></h2>
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<blockquote>
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<div><p>If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.</p>
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<p>Find the sum of all the multiples of 3 or 5 below 1000.</p>
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</div></blockquote>
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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="sum-a-range-filtered-by-a-predicate">
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<h3>Sum a range filtered by a predicate<a class="headerlink" href="#sum-a-range-filtered-by-a-predicate" title="Permalink to this headline">¶</a></h3>
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<p>Let’s create a predicate that returns <code class="docutils literal notranslate"><span class="pre">True</span></code> if a number is a multiple
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of 3 or 5 and <code class="docutils literal notranslate"><span class="pre">False</span></code> otherwise.</p>
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<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">'P == [3 % not] dupdip 5 % not or'</span><span class="p">)</span>
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</pre></div>
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</div>
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<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">'80 P'</span><span class="p">)</span>
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</pre></div>
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</div>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="o">.</span> <span class="mi">80</span> <span class="n">P</span>
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<span class="mi">80</span> <span class="o">.</span> <span class="n">P</span>
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<span class="mi">80</span> <span class="o">.</span> <span class="p">[</span><span class="mi">3</span> <span class="o">%</span> <span class="ow">not</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">5</span> <span class="o">%</span> <span class="ow">not</span> <span class="ow">or</span>
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<span class="mi">80</span> <span class="p">[</span><span class="mi">3</span> <span class="o">%</span> <span class="ow">not</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">5</span> <span class="o">%</span> <span class="ow">not</span> <span class="ow">or</span>
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<span class="mi">80</span> <span class="o">.</span> <span class="mi">3</span> <span class="o">%</span> <span class="ow">not</span> <span class="mi">80</span> <span class="mi">5</span> <span class="o">%</span> <span class="ow">not</span> <span class="ow">or</span>
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<span class="mi">80</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">%</span> <span class="ow">not</span> <span class="mi">80</span> <span class="mi">5</span> <span class="o">%</span> <span class="ow">not</span> <span class="ow">or</span>
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<span class="mi">2</span> <span class="o">.</span> <span class="ow">not</span> <span class="mi">80</span> <span class="mi">5</span> <span class="o">%</span> <span class="ow">not</span> <span class="ow">or</span>
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<span class="kc">False</span> <span class="o">.</span> <span class="mi">80</span> <span class="mi">5</span> <span class="o">%</span> <span class="ow">not</span> <span class="ow">or</span>
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<span class="kc">False</span> <span class="mi">80</span> <span class="o">.</span> <span class="mi">5</span> <span class="o">%</span> <span class="ow">not</span> <span class="ow">or</span>
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<span class="kc">False</span> <span class="mi">80</span> <span class="mi">5</span> <span class="o">.</span> <span class="o">%</span> <span class="ow">not</span> <span class="ow">or</span>
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<span class="kc">False</span> <span class="mi">0</span> <span class="o">.</span> <span class="ow">not</span> <span class="ow">or</span>
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<span class="kc">False</span> <span class="kc">True</span> <span class="o">.</span> <span class="ow">or</span>
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<span class="kc">True</span> <span class="o">.</span>
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</pre></div>
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</div>
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<p>Given the predicate function <code class="docutils literal notranslate"><span class="pre">P</span></code> a suitable program is:</p>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">PE1</span> <span class="o">==</span> <span class="mi">1000</span> <span class="nb">range</span> <span class="p">[</span><span class="n">P</span><span class="p">]</span> <span class="nb">filter</span> <span class="nb">sum</span>
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</pre></div>
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</div>
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<p>This function generates a list of the integers from 0 to 999, filters
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that list by <code class="docutils literal notranslate"><span class="pre">P</span></code>, and then sums the result.</p>
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<p>Logically this is fine, but pragmatically we are doing more work than we
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should be; we generate one thousand integers but actually use less than
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half of them. A better solution would be to generate just the multiples
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we want to sum, and to add them as we go rather than storing them and
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and summing them at the end.</p>
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</div>
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<div class="section" id="generate-just-the-multiples">
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<h3>Generate just the multiples<a class="headerlink" href="#generate-just-the-multiples" title="Permalink to this headline">¶</a></h3>
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<p>At first I had the idea to use two counters and increase them by three
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and five, respectively. This way we only generate the terms that we
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actually want to sum. We have to proceed by incrementing the counter
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that is lower, or if they are equal, the three counter, and we have to
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take care not to double add numbers like 15 that are multiples of both
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three and five.</p>
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<p>This seemed a little clunky, so I tried a different approach.</p>
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<p>Consider the first few terms in the series:</p>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">3</span> <span class="mi">5</span> <span class="mi">6</span> <span class="mi">9</span> <span class="mi">10</span> <span class="mi">12</span> <span class="mi">15</span> <span class="mi">18</span> <span class="mi">20</span> <span class="mi">21</span> <span class="o">...</span>
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</pre></div>
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</div>
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<p>Subtract each number from the one after it (subtracting 0 from 3):</p>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">3</span> <span class="mi">5</span> <span class="mi">6</span> <span class="mi">9</span> <span class="mi">10</span> <span class="mi">12</span> <span class="mi">15</span> <span class="mi">18</span> <span class="mi">20</span> <span class="mi">21</span> <span class="mi">24</span> <span class="mi">25</span> <span class="mi">27</span> <span class="mi">30</span> <span class="o">...</span>
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<span class="mi">0</span> <span class="mi">3</span> <span class="mi">5</span> <span class="mi">6</span> <span class="mi">9</span> <span class="mi">10</span> <span class="mi">12</span> <span class="mi">15</span> <span class="mi">18</span> <span class="mi">20</span> <span class="mi">21</span> <span class="mi">24</span> <span class="mi">25</span> <span class="mi">27</span> <span class="o">...</span>
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<span class="o">-------------------------------------------</span>
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<span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="o">...</span>
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</pre></div>
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</div>
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<p>You get this lovely repeating palindromic sequence:</p>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span>
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</pre></div>
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</div>
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<p>To make a counter that increments by factors of 3 and 5 you just add
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these differences to the counter one-by-one in a loop.</p>
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<p>To make use of this sequence to increment a counter and sum terms as we
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go we need a function that will accept the sum, the counter, and the
|
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next term to add, and that adds the term to the counter and a copy of
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the counter to the running sum. This function will do that:</p>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="o">==</span> <span class="o">+</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span>
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</pre></div>
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</div>
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<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">'PE1.1 == + [+] dupdip'</span><span class="p">)</span>
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</pre></div>
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</div>
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<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">'0 0 3 PE1.1'</span><span class="p">)</span>
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</pre></div>
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</div>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="o">.</span> <span class="mi">0</span> <span class="mi">0</span> <span class="mi">3</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span>
|
||
<span class="mi">0</span> <span class="o">.</span> <span class="mi">0</span> <span class="mi">3</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="o">.</span> <span class="mi">3</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">3</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">+</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span>
|
||
<span class="mi">0</span> <span class="mi">3</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span>
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<span class="mi">0</span> <span class="mi">3</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span>
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<span class="mi">0</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">3</span>
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<span class="mi">3</span> <span class="o">.</span> <span class="mi">3</span>
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<span class="mi">3</span> <span class="mi">3</span> <span class="o">.</span>
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</pre></div>
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</div>
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<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">'0 0 [3 2 1 3 1 2 3] [PE1.1] step'</span><span class="p">)</span>
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</pre></div>
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</div>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="o">.</span> <span class="mi">0</span> <span class="mi">0</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">0</span> <span class="o">.</span> <span class="mi">0</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="o">.</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">step</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">3</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">+</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">0</span> <span class="mi">3</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">0</span> <span class="mi">3</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">0</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">3</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
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||
<span class="mi">3</span> <span class="o">.</span> <span class="mi">3</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
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||
<span class="mi">3</span> <span class="mi">3</span> <span class="o">.</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">step</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">3</span> <span class="mi">3</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">dupdip</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">3</span> <span class="mi">5</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">3</span> <span class="mi">5</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">3</span> <span class="mi">5</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">5</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">8</span> <span class="o">.</span> <span class="mi">5</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="o">.</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">step</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">1</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">1</span> <span class="o">.</span> <span class="o">+</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">8</span> <span class="mi">6</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">8</span> <span class="mi">6</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">8</span> <span class="mi">6</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">6</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">14</span> <span class="o">.</span> <span class="mi">6</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="o">.</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">step</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="mi">3</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">+</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">14</span> <span class="mi">9</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">14</span> <span class="mi">9</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">14</span> <span class="mi">9</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">9</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">23</span> <span class="o">.</span> <span class="mi">9</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="o">.</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">step</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="mi">1</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="mi">1</span> <span class="o">.</span> <span class="o">+</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">23</span> <span class="mi">10</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">23</span> <span class="mi">10</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">23</span> <span class="mi">10</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">10</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">33</span> <span class="o">.</span> <span class="mi">10</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="o">.</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">step</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="mi">2</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">33</span> <span class="mi">10</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">dupdip</span> <span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">33</span> <span class="mi">12</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">33</span> <span class="mi">12</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">33</span> <span class="mi">12</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">12</span> <span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">45</span> <span class="o">.</span> <span class="mi">12</span> <span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="o">.</span> <span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">step</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">step</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="mi">3</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">+</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span>
|
||
<span class="mi">45</span> <span class="mi">15</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span>
|
||
<span class="mi">45</span> <span class="mi">15</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span>
|
||
<span class="mi">45</span> <span class="mi">15</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">15</span>
|
||
<span class="mi">60</span> <span class="o">.</span> <span class="mi">15</span>
|
||
<span class="mi">60</span> <span class="mi">15</span> <span class="o">.</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>So one <code class="docutils literal notranslate"><span class="pre">step</span></code> through all seven terms brings the counter to 15 and the
|
||
total to 60.</p>
|
||
</div>
|
||
<div class="section" id="how-many-multiples-to-sum">
|
||
<h3>How many multiples to sum?<a class="headerlink" href="#how-many-multiples-to-sum" title="Permalink to this headline">¶</a></h3>
|
||
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">1000</span> <span class="o">/</span> <span class="mi">15</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">66</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">66</span> <span class="o">*</span> <span class="mi">15</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">990</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">1000</span> <span class="o">-</span> <span class="mi">990</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">10</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>We only want the terms <em>less than</em> 1000.</p>
|
||
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">999</span> <span class="o">-</span> <span class="mi">990</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">9</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>That means we want to run the full list of numbers sixty-six times to
|
||
get to 990 and then the first four numbers 3 2 1 3 to get to 999.</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">'PE1 == 0 0 66 [[3 2 1 3 1 2 3] [PE1.1] step] times [3 2 1 3] [PE1.1] step pop'</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">'PE1'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">233168</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="packing-the-terms-into-an-integer">
|
||
<h3>Packing the terms into an integer<a class="headerlink" href="#packing-the-terms-into-an-integer" title="Permalink to this headline">¶</a></h3>
|
||
<p>This form uses no extra storage and produces no unused summands. It’s
|
||
good but there’s one more trick we can apply. The list of seven terms
|
||
takes up at least seven bytes. But notice that all of the terms are less
|
||
than four, and so each can fit in just two bits. We could store all
|
||
seven terms in just fourteen bits and use masking and shifts to pick out
|
||
each term as we go. This will use less space and save time loading whole
|
||
integer terms from the list.</p>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span>
|
||
<span class="mi">0</span><span class="n">b</span> <span class="mi">11</span> <span class="mi">10</span> <span class="mi">01</span> <span class="mi">11</span> <span class="mi">01</span> <span class="mi">10</span> <span class="mi">11</span> <span class="o">==</span> <span class="mi">14811</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mb">0b11100111011011</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">14811</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">'PE1.2 == [3 & PE1.1] dupdip 2 >>'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<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">'0 0 14811 PE1.2'</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">0</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">2</span>
|
||
<span class="mi">0</span> <span class="o">.</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">2</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="o">.</span> <span class="mi">14811</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">2</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">2</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="o">.</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="o">.</span> <span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">3</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">+</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">0</span> <span class="mi">3</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">0</span> <span class="mi">3</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">0</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">3</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">3</span> <span class="o">.</span> <span class="mi">3</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="o">.</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">14811</span> <span class="o">.</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">.</span> <span class="o">>></span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">3702</span> <span class="o">.</span>
|
||
</pre></div>
|
||
</div>
|
||
<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 3 3702 PE1.2'</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">3</span> <span class="mi">3702</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">2</span>
|
||
<span class="mi">3</span> <span class="o">.</span> <span class="mi">3</span> <span class="mi">3702</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">2</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="o">.</span> <span class="mi">3702</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">2</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">3702</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">2</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">3702</span> <span class="o">.</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">3702</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">3702</span> <span class="o">.</span> <span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">3702</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">3</span> <span class="mi">3</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">dupdip</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">3</span> <span class="mi">5</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">3</span> <span class="mi">5</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">3</span> <span class="mi">5</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">5</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">8</span> <span class="o">.</span> <span class="mi">5</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="o">.</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">3702</span> <span class="o">.</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">.</span> <span class="o">>></span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">925</span> <span class="o">.</span>
|
||
</pre></div>
|
||
</div>
|
||
<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">'0 0 14811 7 [PE1.2] times pop'</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">0</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="mi">7</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">0</span> <span class="o">.</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="mi">7</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="o">.</span> <span class="mi">14811</span> <span class="mi">7</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="o">.</span> <span class="mi">7</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="mi">7</span> <span class="o">.</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="mi">7</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">2</span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="o">.</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="o">.</span> <span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">14811</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">3</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">+</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">0</span> <span class="mi">3</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">0</span> <span class="mi">3</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">0</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">3</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="o">.</span> <span class="mi">3</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="o">.</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">14811</span> <span class="o">.</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">14811</span> <span class="mi">2</span> <span class="o">.</span> <span class="o">>></span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">3702</span> <span class="o">.</span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">3702</span> <span class="mi">6</span> <span class="o">.</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">3702</span> <span class="mi">6</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">3702</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">3702</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">2</span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">3702</span> <span class="o">.</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">3702</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">3702</span> <span class="o">.</span> <span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">3702</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">3</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">dupdip</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">5</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">5</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">3</span> <span class="mi">5</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">5</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="o">.</span> <span class="mi">5</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="o">.</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">3702</span> <span class="o">.</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">3702</span> <span class="mi">2</span> <span class="o">.</span> <span class="o">>></span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">925</span> <span class="o">.</span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">925</span> <span class="mi">5</span> <span class="o">.</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">925</span> <span class="mi">5</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">925</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">925</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">2</span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">925</span> <span class="o">.</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">925</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">925</span> <span class="o">.</span> <span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">925</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">925</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">925</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">1</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">925</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">5</span> <span class="mi">1</span> <span class="o">.</span> <span class="o">+</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">925</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">6</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">925</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">6</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">925</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">8</span> <span class="mi">6</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">6</span> <span class="mi">925</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="o">.</span> <span class="mi">6</span> <span class="mi">925</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="o">.</span> <span class="mi">925</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="mi">925</span> <span class="o">.</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="mi">925</span> <span class="mi">2</span> <span class="o">.</span> <span class="o">>></span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="mi">231</span> <span class="o">.</span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="mi">231</span> <span class="mi">4</span> <span class="o">.</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="mi">231</span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="mi">231</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="mi">231</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">2</span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="mi">231</span> <span class="o">.</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="mi">231</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="mi">231</span> <span class="o">.</span> <span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">231</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="mi">231</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">231</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="mi">3</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">231</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">6</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">+</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">231</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">9</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">231</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">9</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">231</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14</span> <span class="mi">9</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">9</span> <span class="mi">231</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="o">.</span> <span class="mi">9</span> <span class="mi">231</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="o">.</span> <span class="mi">231</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="mi">231</span> <span class="o">.</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="mi">231</span> <span class="mi">2</span> <span class="o">.</span> <span class="o">>></span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="mi">57</span> <span class="o">.</span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="mi">57</span> <span class="mi">3</span> <span class="o">.</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="mi">57</span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="mi">57</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="mi">57</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">2</span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="mi">57</span> <span class="o">.</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="mi">57</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="mi">57</span> <span class="o">.</span> <span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">57</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="mi">57</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">57</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="mi">1</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">57</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">9</span> <span class="mi">1</span> <span class="o">.</span> <span class="o">+</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">57</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">10</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">57</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">10</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">57</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">23</span> <span class="mi">10</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">10</span> <span class="mi">57</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="o">.</span> <span class="mi">10</span> <span class="mi">57</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="o">.</span> <span class="mi">57</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="mi">57</span> <span class="o">.</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="mi">57</span> <span class="mi">2</span> <span class="o">.</span> <span class="o">>></span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="mi">14</span> <span class="o">.</span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="mi">14</span> <span class="mi">2</span> <span class="o">.</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="mi">14</span> <span class="mi">2</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="mi">14</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="mi">14</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">2</span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="mi">14</span> <span class="o">.</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="mi">14</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="mi">14</span> <span class="o">.</span> <span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">14</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="mi">14</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">14</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">10</span> <span class="mi">2</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">14</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">10</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">dupdip</span> <span class="mi">14</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">12</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">14</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">12</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">14</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">33</span> <span class="mi">12</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">12</span> <span class="mi">14</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="o">.</span> <span class="mi">12</span> <span class="mi">14</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="o">.</span> <span class="mi">14</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="mi">14</span> <span class="o">.</span> <span class="mi">2</span> <span class="o">>></span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="mi">14</span> <span class="mi">2</span> <span class="o">.</span> <span class="o">>></span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="mi">3</span> <span class="o">.</span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="mi">3</span> <span class="mi">1</span> <span class="o">.</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="mi">3</span> <span class="mi">1</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="mi">3</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="mi">3</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">2</span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="mi">3</span> <span class="o">.</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="mi">3</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="mi">3</span> <span class="o">.</span> <span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">3</span> <span class="mi">2</span> <span class="o">>></span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="mi">3</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">3</span> <span class="mi">2</span> <span class="o">>></span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="mi">3</span> <span class="o">.</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="mi">3</span> <span class="mi">2</span> <span class="o">>></span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">12</span> <span class="mi">3</span> <span class="o">.</span> <span class="o">+</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">3</span> <span class="mi">2</span> <span class="o">>></span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">15</span> <span class="o">.</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">3</span> <span class="mi">2</span> <span class="o">>></span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">15</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="o">.</span> <span class="n">dupdip</span> <span class="mi">3</span> <span class="mi">2</span> <span class="o">>></span> <span class="n">pop</span>
|
||
<span class="mi">45</span> <span class="mi">15</span> <span class="o">.</span> <span class="o">+</span> <span class="mi">15</span> <span class="mi">3</span> <span class="mi">2</span> <span class="o">>></span> <span class="n">pop</span>
|
||
<span class="mi">60</span> <span class="o">.</span> <span class="mi">15</span> <span class="mi">3</span> <span class="mi">2</span> <span class="o">>></span> <span class="n">pop</span>
|
||
<span class="mi">60</span> <span class="mi">15</span> <span class="o">.</span> <span class="mi">3</span> <span class="mi">2</span> <span class="o">>></span> <span class="n">pop</span>
|
||
<span class="mi">60</span> <span class="mi">15</span> <span class="mi">3</span> <span class="o">.</span> <span class="mi">2</span> <span class="o">>></span> <span class="n">pop</span>
|
||
<span class="mi">60</span> <span class="mi">15</span> <span class="mi">3</span> <span class="mi">2</span> <span class="o">.</span> <span class="o">>></span> <span class="n">pop</span>
|
||
<span class="mi">60</span> <span class="mi">15</span> <span class="mi">0</span> <span class="o">.</span> <span class="n">pop</span>
|
||
<span class="mi">60</span> <span class="mi">15</span> <span class="o">.</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>And so we have at last:</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">'PE1 == 0 0 66 [14811 7 [PE1.2] times pop] times 14811 4 [PE1.2] times popop'</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">'PE1'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">233168</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="let-s-refactor">
|
||
<h3>Let’s refactor<a class="headerlink" href="#let-s-refactor" title="Permalink to this headline">¶</a></h3>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="mi">14811</span> <span class="mi">7</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14811</span> <span class="mi">4</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="mi">14811</span> <span class="n">n</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="n">n</span> <span class="mi">14811</span> <span class="n">swap</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</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">'PE1.3 == 14811 swap [PE1.2] times pop'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>Now we can simplify the definition above:</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">'PE1 == 0 0 66 [7 PE1.3] times 4 PE1.3 pop'</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">'PE1'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">233168</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>Here’s our joy program all in one place. It doesn’t make so much sense,
|
||
but if you have read through the above description of how it was derived
|
||
I hope it’s clear.</p>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span> <span class="o">==</span> <span class="o">+</span> <span class="p">[</span><span class="o">+</span><span class="p">]</span> <span class="n">dupdip</span>
|
||
<span class="n">PE1</span><span class="o">.</span><span class="mi">2</span> <span class="o">==</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span>
|
||
<span class="n">PE1</span><span class="o">.</span><span class="mi">3</span> <span class="o">==</span> <span class="mi">14811</span> <span class="n">swap</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">2</span><span class="p">]</span> <span class="n">times</span> <span class="n">pop</span>
|
||
<span class="n">PE1</span> <span class="o">==</span> <span class="mi">0</span> <span class="mi">0</span> <span class="mi">66</span> <span class="p">[</span><span class="mi">7</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">3</span><span class="p">]</span> <span class="n">times</span> <span class="mi">4</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">3</span> <span class="n">pop</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="generator-version">
|
||
<h2>Generator Version<a class="headerlink" href="#generator-version" title="Permalink to this headline">¶</a></h2>
|
||
<p>It’s a little clunky iterating sixty-six times though the seven numbers
|
||
then four more. In the <em>Generator Programs</em> notebook we derive a
|
||
generator that can be repeatedly driven by the <code class="docutils literal notranslate"><span class="pre">x</span></code> combinator to
|
||
produce a stream of the seven numbers repeating over and over again.</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">'PE1.terms == [0 swap [dup [pop 14811] [] branch [3 &] dupdip 2 >>] dip rest cons]'</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">'PE1.terms 21 [x] times'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="p">[</span><span class="mi">0</span> <span class="n">swap</span> <span class="p">[</span><span class="n">dup</span> <span class="p">[</span><span class="n">pop</span> <span class="mi">14811</span><span class="p">]</span> <span class="p">[]</span> <span class="n">branch</span> <span class="p">[</span><span class="mi">3</span> <span class="o">&</span><span class="p">]</span> <span class="n">dupdip</span> <span class="mi">2</span> <span class="o">>></span><span class="p">]</span> <span class="n">dip</span> <span class="n">rest</span> <span class="n">cons</span><span class="p">]</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>We know from above that we need sixty-six times seven then four more
|
||
terms to reach up to but not over one thousand.</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">'7 66 * 4 +'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">466</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">'PE1.terms 466 [x] times pop'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</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">'[PE1.terms 466 [x] times pop] run sum'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">999</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>Now we can use <code class="docutils literal notranslate"><span class="pre">PE1.1</span></code> to accumulate the terms as we go, and then
|
||
<code class="docutils literal notranslate"><span class="pre">pop</span></code> the generator and the counter from the stack when we’re done,
|
||
leaving just the sum.</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">'0 0 PE1.terms 466 [x [PE1.1] dip] times popop'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">233168</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="a-little-further-analysis-renders-iteration-unnecessary">
|
||
<h2>A little further analysis renders iteration unnecessary.<a class="headerlink" href="#a-little-further-analysis-renders-iteration-unnecessary" title="Permalink to this headline">¶</a></h2>
|
||
<p>Consider finding the sum of the positive integers less than or equal to
|
||
ten.</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">'[10 9 8 7 6 5 4 3 2 1] sum'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">55</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>Instead of summing them,
|
||
<a class="reference external" href="https://en.wikipedia.org/wiki/File:Animated_proof_for_the_formula_giving_the_sum_of_the_first_integers_1%2B2%2B...%2Bn.gif">observe</a>:</p>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="mi">10</span> <span class="mi">9</span> <span class="mi">8</span> <span class="mi">7</span> <span class="mi">6</span>
|
||
<span class="o">+</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">4</span> <span class="mi">5</span>
|
||
<span class="o">----</span> <span class="o">--</span> <span class="o">--</span> <span class="o">--</span> <span class="o">--</span>
|
||
<span class="mi">11</span> <span class="mi">11</span> <span class="mi">11</span> <span class="mi">11</span> <span class="mi">11</span>
|
||
|
||
<span class="mi">11</span> <span class="o">*</span> <span class="mi">5</span> <span class="o">=</span> <span class="mi">55</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>From the above example we can deduce that the sum of the first N
|
||
positive integers is:</p>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="n">N</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">N</span> <span class="o">/</span> <span class="mi">2</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>(The formula also works for odd values of N, I’ll leave that to you if
|
||
you want to work it out or you can take my word for it.)</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">'F == dup ++ * 2 floordiv'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<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">'10 F'</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">10</span> <span class="n">F</span>
|
||
<span class="mi">10</span> <span class="o">.</span> <span class="n">F</span>
|
||
<span class="mi">10</span> <span class="o">.</span> <span class="n">dup</span> <span class="o">++</span> <span class="o">*</span> <span class="mi">2</span> <span class="n">floordiv</span>
|
||
<span class="mi">10</span> <span class="mi">10</span> <span class="o">.</span> <span class="o">++</span> <span class="o">*</span> <span class="mi">2</span> <span class="n">floordiv</span>
|
||
<span class="mi">10</span> <span class="mi">11</span> <span class="o">.</span> <span class="o">*</span> <span class="mi">2</span> <span class="n">floordiv</span>
|
||
<span class="mi">110</span> <span class="o">.</span> <span class="mi">2</span> <span class="n">floordiv</span>
|
||
<span class="mi">110</span> <span class="mi">2</span> <span class="o">.</span> <span class="n">floordiv</span>
|
||
<span class="mi">55</span> <span class="o">.</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>We can apply the same reasoning to the PE1 problem.</p>
|
||
<p>Between 0 and 990 inclusive there are sixty-six “blocks” of seven terms
|
||
each, starting with:</p>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="mi">3</span> <span class="mi">5</span> <span class="mi">6</span> <span class="mi">9</span> <span class="mi">10</span> <span class="mi">12</span> <span class="mi">15</span><span class="p">]</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>And ending with:</p>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="mi">978</span> <span class="mi">980</span> <span class="mi">981</span> <span class="mi">984</span> <span class="mi">985</span> <span class="mi">987</span> <span class="mi">990</span><span class="p">]</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>If we reverse one of these two blocks and sum pairs…</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 5 6 9 10 12 15] reverse [978 980 981 984 985 987 990] zip'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[[</span><span class="mi">978</span> <span class="mi">15</span><span class="p">]</span> <span class="p">[</span><span class="mi">980</span> <span class="mi">12</span><span class="p">]</span> <span class="p">[</span><span class="mi">981</span> <span class="mi">10</span><span class="p">]</span> <span class="p">[</span><span class="mi">984</span> <span class="mi">9</span><span class="p">]</span> <span class="p">[</span><span class="mi">985</span> <span class="mi">6</span><span class="p">]</span> <span class="p">[</span><span class="mi">987</span> <span class="mi">5</span><span class="p">]</span> <span class="p">[</span><span class="mi">990</span> <span class="mi">3</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">'[3 5 6 9 10 12 15] reverse [978 980 981 984 985 987 990] zip [sum] map'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="mi">993</span> <span class="mi">992</span> <span class="mi">991</span> <span class="mi">993</span> <span class="mi">991</span> <span class="mi">992</span> <span class="mi">993</span><span class="p">]</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>(Interesting that the sequence of seven numbers appears again in the
|
||
rightmost digit of each term.)</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 5 6 9 10 12 15] reverse [978 980 981 984 985 987 990] zip [sum] map sum'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">6945</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>Since there are sixty-six blocks and we are pairing them up, there must
|
||
be thirty-three pairs, each of which sums to 6945. We also have these
|
||
additional unpaired terms between 990 and 1000:</p>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">993</span> <span class="mi">995</span> <span class="mi">996</span> <span class="mi">999</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>So we can give the “sum of all the multiples of 3 or 5 below 1000” like
|
||
so:</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">'6945 33 * [993 995 996 999] cons sum'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">233168</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>It’s worth noting, I think, that this same reasoning holds for any two
|
||
numbers <span class="math notranslate nohighlight">\(n\)</span> and <span class="math notranslate nohighlight">\(m\)</span> the multiples of which we hope to sum.
|
||
The multiples would have a cycle of differences of length <span class="math notranslate nohighlight">\(k\)</span> and
|
||
so we could compute the sum of <span class="math notranslate nohighlight">\(Nk\)</span> multiples as above.</p>
|
||
<p>The sequence of differences will always be a palidrome. Consider an
|
||
interval spanning the least common multiple of <span class="math notranslate nohighlight">\(n\)</span> and <span class="math notranslate nohighlight">\(m\)</span>:</p>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">|</span> <span class="o">|</span> <span class="o">|</span> <span class="o">|</span> <span class="o">|</span> <span class="o">|</span> <span class="o">|</span> <span class="o">|</span>
|
||
<span class="o">|</span> <span class="o">|</span> <span class="o">|</span> <span class="o">|</span> <span class="o">|</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>Here we have 4 and 7, and you can read off the sequence of differences
|
||
directly from the diagram: 4 3 1 4 2 2 4 1 3 4.</p>
|
||
<p>Geometrically, the actual values of <span class="math notranslate nohighlight">\(n\)</span> and <span class="math notranslate nohighlight">\(m\)</span> and their
|
||
<em>lcm</em> don’t matter, the pattern they make will always be symmetrical
|
||
around its midpoint. The same reasoning holds for multiples of more than
|
||
two numbers.</p>
|
||
</div>
|
||
<div class="section" id="the-simplest-program">
|
||
<h2>The Simplest Program<a class="headerlink" href="#the-simplest-program" title="Permalink to this headline">¶</a></h2>
|
||
<p>Of course, the simplest joy program for the first Project Euler problem
|
||
is just:</p>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">PE1</span> <span class="o">==</span> <span class="mi">233168</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>Fin.</p>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
</div>
|
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<div class="sphinxsidebar" role="navigation" aria-label="main navigation">
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<div class="sphinxsidebarwrapper">
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<h3><a href="../index.html">Table Of Contents</a></h3>
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||
<ul>
|
||
<li><a class="reference internal" href="#">Developing a Program in Joy</a><ul>
|
||
<li><a class="reference internal" href="#project-euler-first-problem-multiples-of-3-and-5">Project Euler, first problem: “Multiples of 3 and 5”</a><ul>
|
||
<li><a class="reference internal" href="#sum-a-range-filtered-by-a-predicate">Sum a range filtered by a predicate</a></li>
|
||
<li><a class="reference internal" href="#generate-just-the-multiples">Generate just the multiples</a></li>
|
||
<li><a class="reference internal" href="#how-many-multiples-to-sum">How many multiples to sum?</a></li>
|
||
<li><a class="reference internal" href="#packing-the-terms-into-an-integer">Packing the terms into an integer</a></li>
|
||
<li><a class="reference internal" href="#let-s-refactor">Let’s refactor</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a class="reference internal" href="#generator-version">Generator Version</a></li>
|
||
<li><a class="reference internal" href="#a-little-further-analysis-renders-iteration-unnecessary">A little further analysis renders iteration unnecessary.</a></li>
|
||
<li><a class="reference internal" href="#the-simplest-program">The Simplest Program</a></li>
|
||
</ul>
|
||
</li>
|
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</ul>
|
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<h3>Related Topics</h3>
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<li><a href="../index.html">Documentation overview</a><ul>
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<li><a href="index.html">Essays about Programming in Joy</a><ul>
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<li>Next: <a href="Quadratic.html" title="next chapter">Quadratic formula</a></li>
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<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>.
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