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<div class="section" id="developing-a-program-in-joy">
<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>
<p>Find the sum of all the multiples of 3 or 5 below 1000.</p>
</div></blockquote>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">notebook_preamble</span> <span class="k">import</span> <span class="n">J</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="n">define</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">notebook_preamble</span> <span class="kn">import</span> <span class="n">J</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="n">define</span>
</pre></div>
</div>
<div class="section" id="sum-a-range-filtered-by-a-predicate">
<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>
<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
of 3 or 5 and <code class="docutils literal notranslate"><span class="pre">False</span></code> otherwise.</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">'P == [3 % not] dupdip 5 % not or'</span><span class="p">)</span>
+<div class="highlight-ipython2 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>
</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">'80 P'</span><span class="p">)</span>
+<div class="highlight-ipython2 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>
</pre></div>
</div>
<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>
<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>
</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">'PE1.1 == + [+] dupdip'</span><span class="p">)</span>
+<div class="highlight-ipython2 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>
</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 3 PE1.1'</span><span class="p">)</span>
+<div class="highlight-ipython2 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>
</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">3</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span>
<span class="mi">3</span> <span class="mi">3</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 [3 2 1 3 1 2 3] [PE1.1] step'</span><span class="p">)</span>
+<div class="highlight-ipython2 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>
</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="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>
</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>
+<div class="highlight-ipython2 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>
+<div class="highlight-ipython2 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>
+<div class="highlight-ipython2 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>
+<div class="highlight-ipython2 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>
</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>
+<div class="highlight-ipython2 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>
+<div class="highlight-ipython2 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>
<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>
+<div class="highlight-ipython2 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>
+<div class="highlight-ipython2 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>
+<div class="highlight-ipython2 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">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>
+<div class="highlight-ipython2 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">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>
+<div class="highlight-ipython2 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>
</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>
+<div class="highlight-ipython2 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>
+<div class="highlight-ipython2 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>
<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>
+<div class="highlight-ipython2 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>
+<div class="highlight-ipython2 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>
+<div class="highlight-ipython2 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>
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>
+<div class="highlight-ipython2 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>
+<div class="highlight-ipython2 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>
</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>
+<div class="highlight-ipython2 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>
+<div class="highlight-ipython2 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>
+<div class="highlight-ipython2 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>
<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>
+<div class="highlight-ipython2 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>
<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>
+<div class="highlight-ipython2 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>
</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>
+<div class="highlight-ipython2 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>
+<div class="highlight-ipython2 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>
</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>
+<div class="highlight-ipython2 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>
+<div class="highlight-ipython2 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>
</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>
+<div class="highlight-ipython2 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>
</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>
+<div class="highlight-ipython2 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>
</div>
+
</div>
</div>
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- <h3><a href="../index.html">Table Of Contents</a></h3>
- <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>
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