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    <p>This is just an initial stab at your questions, since nobody has replied so far:</p> <p>Adverbs have long left scope:</p> <pre><code>(2&amp;*~) &lt;-&gt; ((2&amp;*)~) </code></pre> <p>The dyadic case of the function returned by bond is <em>powered</em>:</p> <pre><code>(x m&amp;v y) &lt;-&gt; ((m&amp;v)^:x y) </code></pre> <p><code>2&amp;*</code> doubles its argument every time it's applied, so powered <code>2&amp;*</code> multiplies its argument by a power of <code>2</code>.</p> <p><code>(1&gt;.&lt;.@-:)^: 27</code> defines a verb (since <code>^:</code> is a conjunction), but does not run it. In the original phrase <code>foo^:a:</code> was the verb, and <code>27</code> the argument. When <code>a:</code> is the right argument of <code>^:</code> it just runs until it converges.</p> <p>Monadic <code>+:</code> doubles its argument, but the dyadic cases of <code>2&amp;*</code> and <code>+:</code> aren't related in any way, and can't be interchanged. Note that <code>double~</code> is the left side of a hook, which is always called dyadically.</p> <blockquote> <p>The only good thing is that the J verb odd has nothing to do with testing for an odd number.</p> </blockquote> <p>Actually, <code>2&amp;| y</code> is <code>1</code> if <code>y</code> is an odd integer and <code>0</code> if it is an even integer, so it is a test for odd numbers.</p> <p>Now, to try to explain what the algorithm is doing in plain English, let's look at it phrase by phrase, using <code>12 ethiop 27</code> as an example:</p> <pre><code>+/@(2&amp;|@] # (2&amp;*~ &lt;@#)) (1&gt;.&lt;.@-:)^:a: </code></pre> <p>is a hook. In hooks the right part is always a monad and the left part is always a dyad.</p> <pre><code>(1 &gt;. (&lt;. @ -:)) ^: a: </code></pre> <p>Monad <code>(&lt;. @ -:)</code> = <code>floor @ half</code>, <em>halves its argument and rounds down</em>, and then <code>1 &gt;. (&lt;. @ -:</code> returns the minimum of that and <code>1</code>. <code>^:a:</code> keeps going until convergence and makes a list of the result. Thus <code>(1 &gt;. (&lt;. @ -:)) ^: a: 27</code> is <code>27</code> halved repeatedly until it reaches <code>1</code>, yielding <code>27 13 6 3 1</code></p> <p>Now let's look at the left half of the hook. <code>(2&amp;|@] # (2&amp;*~ &lt;@#))</code> is a dyadic fork, with the arguments being the left argument to <code>ethiop</code> and the result of the right part of the aforementioned hook.</p> <p><code>2&amp;|@]</code> is <code>1</code> if its right argument is odd, <code>0</code> otherwise. For the argument <code>27 13 6 3 1</code> the result is <code>1 1 0 1 1</code></p> <p><code>(2&amp;*~ &lt;@#)</code> is a hook again. &lt;@# is applied monadically to <code>27 13 6 3 1</code>, and boxes the length, returning <code>(&lt;5)</code>. Then we get to <code>2&amp;*~</code>, which as we saw, is powered in the dyadic, so this is (after <code>~</code> switches the arguments) <code>+:^:(&lt;5) 12</code>.</p> <p><code>f^:m</code>, when <code>m</code> is boxed, does <code>f</code> <code>&lt;: &gt;m</code> times, yielding an <code>m</code>-length list of results (unless <code>m</code> is empty, in which case it runs until convergence). So this will yield <code>12 * 2 ^ i.5</code></p> <p>Then the middle part of the fork is simply <code>#</code>, which with a boolean left argument simply filters its right argument, in this case leaving 12 * those powers of 2 for which 27 is odd.</p> <pre><code> 1 1 0 1 1 # 12 24 48 96 192 12 24 96 192 </code></pre> <p>Finally <code>+/</code> is <code>sum</code></p> <pre><code> +/12 24 96 192 324 </code></pre>
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