Working on the notebooks.
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@ -244,103 +244,160 @@ b [b+a b F]
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</code></pre>
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</code></pre>
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<p>Putting it all together:</p>
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<p>Putting it all together:</p>
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<pre><code>F == + [popdd over] cons infra uncons
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<pre><code>F == + [popdd over] cons infra uncons
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fib_gen == [1 1 F]
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</code></pre>
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</code></pre>
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<p>Let's call <code>F</code> <code>fib_gen</code>:</p>
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<h3><code>fib-gen</code></h3>
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<pre><code>[fib_gen + [popdd over] cons infra uncons] inscribe
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<p>Let's call <code>F</code> <code>fib-gen</code>:</p>
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<pre><code>[fib-gen + [popdd over] cons infra uncons] inscribe
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</code></pre>
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</code></pre>
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<p>We can just write the initial quote and then "force" it with <code>x</code>:</p>
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<p>We can just write the initial quote and then "force" it with <code>x</code>:</p>
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<pre><code>joy? [1 1 fib_gen] 10 [x] times
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<pre><code>joy? [1 1 fib-gen] 10 [x] times
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1 2 3 5 8 13 21 34 55 89 [144 89 fib_gen]
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1 2 3 5 8 13 21 34 55 89 [144 89 fib-gen]
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</code></pre>
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</code></pre>
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<p>It skips the first term (1) but if that bothers you you can just prepend it to the program:</p>
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<p>It skips the first term (1) but if that bothers you you can just prepend it to the program:</p>
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<pre><code>1 [1 1 fib_gen] 10 [x] times
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<pre><code>1 [1 1 fib-gen] 10 [x] times
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</code></pre>
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</code></pre>
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<h2>Project Euler Problem Two</h2>
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<h2>Project Euler Problem Two</h2>
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<blockquote>
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<blockquote>
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<p>By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.</p>
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<p>By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.</p>
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</blockquote>
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</blockquote>
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<p>Now that we have a generator for the Fibonacci sequence, we need a function that adds a term in the sequence to a sum if it is even, and <code>pop</code>s it otherwise.</p>
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<p>Now that we have a generator for the Fibonacci sequence, we need a function that adds
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<p><code>python
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a term in the sequence to a sum if it is even, and <code>pop</code>s it otherwise.</p>
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define('PE2.1 == dup 2 % [+] [pop] branch')</code></p>
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<h3><code>even</code></h3>
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<pre><code>[even 2 % bool] inscribe
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</code></pre>
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<h3><code>PE2.1</code></h3>
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<pre><code>[PE2.1 dup even [+] [pop] branch] inscribe
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</code></pre>
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<p>And a predicate function that detects when the terms in the series "exceed four million".</p>
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<p>And a predicate function that detects when the terms in the series "exceed four million".</p>
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<p><code>python
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<h3><code>>4M</code></h3>
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define('>4M == 4000000 >')</code></p>
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<pre><code>[>4M 4000000 >] inscribe
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<p>Now it's straightforward to define <code>PE2</code> as a recursive function that generates terms in the Fibonacci sequence until they exceed four million and sums the even ones.</p>
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<p><code>python
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define('PE2 == 0 fib_gen x [pop >4M] [popop] [[PE2.1] dip x] primrec')</code></p>
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<p><code>python
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J('PE2')</code></p>
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<pre><code>4613732
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</code></pre>
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</code></pre>
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<p>Here's the collected program definitions:</p>
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<p>Now it's straightforward to define <code>PE2</code> as a recursive function that generates terms
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<pre><code>fib == + swons [popdd over] infra uncons
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in the Fibonacci sequence until they exceed four million and sums the even ones.</p>
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fib_gen == [1 1 fib]
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<pre><code>joy? 0 [1 1 fib-gen] x [pop >4M] [popop] [[PE2.1] dip x] tailrec
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4613732
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even == dup 2 %
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>4M == 4000000 >
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PE2.1 == even [+] [pop] branch
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PE2 == 0 fib_gen x [pop >4M] [popop] [[PE2.1] dip x] primrec
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</code></pre>
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</code></pre>
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<h3>Even-valued Fibonacci Terms</h3>
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<h3><code>PE2</code></h3>
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<pre><code>[PE2 0 [1 1 fib-gen] x [pop >4M] [popop] [[PE2.1] dip x] tailrec] inscribe
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</code></pre>
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<p>Here's the collected program definitions (with a little editorializing):</p>
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<pre><code>fib-gen + [popdd over] cons infra uncons
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even 2 % bool
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>4M 4000000 >
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PE2.1 dup even [+] [pop] branch
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PE2.2 [PE2.1] dip x
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PE2.init 0 [1 1 fib-gen] x
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PE2.rec [pop >4M] [popop] [PE2.2] tailrec
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PE2 PE2.init PE2.rec
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</code></pre>
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<h3>Hmm...</h3>
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<pre><code>fib-gen + swons [popdd over] infra uncons
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</code></pre>
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<h2>Even-valued Fibonacci Terms</h2>
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<p>Using <code>o</code> for odd and <code>e</code> for even:</p>
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<p>Using <code>o</code> for odd and <code>e</code> for even:</p>
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<pre><code>o + o = e
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<pre><code>o + o = e
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e + e = e
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e + e = e
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o + e = o
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o + e = o
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</code></pre>
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</code></pre>
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<p>So the Fibonacci sequence considered in terms of just parity would be:</p>
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<p>So the Fibonacci sequence considered in terms of just parity would be:</p>
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<pre><code>o o e o o e o o e o o e o o e o o e
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<pre><code>o o e o o e o o e o o e . . .
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1 1 2 3 5 8 . . .
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1 1 2 3 5 8 13 21 34 55 89 144 . . .
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</code></pre>
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</code></pre>
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<p>Every third term is even.</p>
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<p>Every third term is even.</p>
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<p><code>python
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<p>So what if we drive the generator three times and discard the odd terms?
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J('[1 0 fib] x x x') # To start the sequence with 1 1 2 3 instead of 1 2 3.</code></p>
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We would have to initialize our <code>fib</code> generator with 1 0:</p>
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<pre><code>1 1 2 [3 2 fib]
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<pre><code>[1 0 fib-gen]
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</code></pre>
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</code></pre>
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<p>Drive the generator three times and <code>popop</code> the two odd terms.</p>
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<h3><code>third-term</code></h3>
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<p><code>python
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<pre><code>[third-term x x x [popop] dipd] inscribe
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J('[1 0 fib] x x x [popop] dipd')</code></p>
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<pre><code>2 [3 2 fib]
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</code></pre>
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</code></pre>
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<p><code>python
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<p>So:</p>
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define('PE2.2 == x x x [popop] dipd')</code></p>
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<pre><code>joy? [1 0 fib-gen]
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<p><code>python
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[1 0 fib-gen]
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J('[1 0 fib] 10 [PE2.2] times')</code></p>
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<pre><code>2 8 34 144 610 2584 10946 46368 196418 832040 [1346269 832040 fib]
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joy? third-term
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2 [3 2 fib-gen]
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joy? third-term
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2 8 [13 8 fib-gen]
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joy? third-term
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2 8 34 [55 34 fib-gen]
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joy? third-term
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2 8 34 144 [233 144 fib-gen]
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</code></pre>
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</code></pre>
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<p>Replace <code>x</code> with our new driver function <code>PE2.2</code> and start our <code>fib</code> generator at <code>1 0</code>.</p>
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<p>So now we need a sum:</p>
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<p><code>python
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<pre><code>joy? 0
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J('0 [1 0 fib] PE2.2 [pop >4M] [popop] [[PE2.1] dip PE2.2] primrec')</code></p>
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0
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<pre><code>4613732
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</code></pre>
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</code></pre>
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<h2>How to compile these?</h2>
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<p>And our Fibonacci generator:</p>
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<p>You would probably start with a special version of <code>G</code>, and perhaps modifications to the default <code>x</code>?</p>
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<pre><code>joy? [1 0 fib-gen]
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<h2>An Interesting Variation</h2>
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0 [1 0 fib-gen]
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<p><code>python
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define('codireco == cons dip rest cons')</code></p>
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<p><code>python
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V('[0 [dup ++] codireco] x')</code></p>
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<pre><code> . [0 [dup ++] codireco] x
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[0 [dup ++] codireco] . x
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[0 [dup ++] codireco] . 0 [dup ++] codireco
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[0 [dup ++] codireco] 0 . [dup ++] codireco
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[0 [dup ++] codireco] 0 [dup ++] . codireco
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[0 [dup ++] codireco] 0 [dup ++] . cons dip rest cons
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[0 [dup ++] codireco] [0 dup ++] . dip rest cons
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. 0 dup ++ [0 [dup ++] codireco] rest cons
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0 . dup ++ [0 [dup ++] codireco] rest cons
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0 0 . ++ [0 [dup ++] codireco] rest cons
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0 1 . [0 [dup ++] codireco] rest cons
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0 1 [0 [dup ++] codireco] . rest cons
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0 1 [[dup ++] codireco] . cons
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0 [1 [dup ++] codireco] .
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</code></pre>
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</code></pre>
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<p><code>python
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<p>We want to generate the initial term:</p>
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define('G == [codireco] cons cons')</code></p>
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<pre><code>joy? third-term
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<p><code>python
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0 2 [3 2 fib-gen]
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J('230 [dup ++] G 5 [x] times pop')</code></p>
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<pre><code>230 231 232 233 234
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</code></pre>
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</code></pre>
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<p>Now we check if the term is less than four million,
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if so we add it and recur,
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otherwise we discard the term and the generator leaving the sum on the stack:</p>
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<pre><code>joy? [pop >4M] [popop] [[PE2.1] dip third-term] tailrec
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4613732
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</code></pre>
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<h2>Math</h2>
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<pre><code>a b
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b a+b
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a+b a+b+b
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a+b+b a+a+b+b+b
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</code></pre>
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<p>So if (a,b) and a is even then the next even term pair is (a+2b, 2a+3b)</p>
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<p>Reconsider:</p>
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<pre><code>[b a F] x
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[b a F] b a F
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</code></pre>
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<p>From here we want to arrive at:</p>
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<pre><code>(a+2b) [(2a+3b) (a+2b) F]
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b a F
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b a [F0] [F1] fork
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b a over [+] ii
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---------------------
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a+2b
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</code></pre>
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<p>And:</p>
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<pre><code> b a over [dup + +] ii
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---------------------------
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2a+3b
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[over [dup + +] ii] [over [+] ii] clop
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roll< rrest [tuck] dip ccons
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[b a F] b a F
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[b a F] (2a+3b) (a+2b) roll<
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(2a+3b) (a+2b) [b a F] rrest
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(2a+3b) (a+2b) [F] [tuck] dip ccons
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joy? [1 0 [over [dup + +] ii] [over [+] ii] clop roll< rrest [tuck] dip ccons]
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[1 0 [over [dup + +] ii] [over [+] ii] clop roll< rrest [tuck] dip ccons]
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joy? x
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2 [3 2 [over [dup + +] ii] [over [+] ii] clop roll< rrest [tuck] dip ccons]
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joy? x
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2 8 [13 8 [over [dup + +] ii] [over [+] ii] clop roll< rrest [tuck] dip ccons]
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joy? x
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2 8 34 [55 34 [over [dup + +] ii] [over [+] ii] clop roll< rrest [tuck] dip ccons]
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joy? x
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2 8 34 144 [233 144 [over [dup + +] ii] [over [+] ii] clop roll< rrest [tuck] dip ccons]
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</code></pre>
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<p>And so it goes...</p>
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</body>
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</body>
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</html>
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</html>
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@ -93,3 +93,43 @@ them in host language for greater efficiency if you like.)
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| NOT | `not` | | |
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| NOT | `not` | | |
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## An Interesting Variation
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codireco == cons dip rest cons')
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[0 [dup ++] codireco] x
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. [0 [dup ++] codireco] x
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[0 [dup ++] codireco] . x
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[0 [dup ++] codireco] . 0 [dup ++] codireco
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[0 [dup ++] codireco] 0 . [dup ++] codireco
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[0 [dup ++] codireco] 0 [dup ++] . codireco
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[0 [dup ++] codireco] 0 [dup ++] . cons dip rest cons
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[0 [dup ++] codireco] [0 dup ++] . dip rest cons
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. 0 dup ++ [0 [dup ++] codireco] rest cons
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0 . dup ++ [0 [dup ++] codireco] rest cons
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0 0 . ++ [0 [dup ++] codireco] rest cons
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0 1 . [0 [dup ++] codireco] rest cons
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0 1 [0 [dup ++] codireco] . rest cons
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0 1 [[dup ++] codireco] . cons
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0 [1 [dup ++] codireco] .
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```python
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define('G == [codireco] cons cons')
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```
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```python
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J('230 [dup ++] G 5 [x] times pop')
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```
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230 231 232 233 234
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File diff suppressed because it is too large
Load Diff
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@ -294,66 +294,72 @@ Lastly:
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Putting it all together:
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Putting it all together:
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F == + [popdd over] cons infra uncons
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F == + [popdd over] cons infra uncons
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fib_gen == [1 1 F]
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Let's call `F` `fib_gen`:
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### `fib-gen`
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[fib_gen + [popdd over] cons infra uncons] inscribe
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Let's call `F` `fib-gen`:
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|
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|
[fib-gen + [popdd over] cons infra uncons] inscribe
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|
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We can just write the initial quote and then "force" it with `x`:
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We can just write the initial quote and then "force" it with `x`:
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|
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joy? [1 1 fib_gen] 10 [x] times
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joy? [1 1 fib-gen] 10 [x] times
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1 2 3 5 8 13 21 34 55 89 [144 89 fib_gen]
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1 2 3 5 8 13 21 34 55 89 [144 89 fib-gen]
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It skips the first term (1) but if that bothers you you can just prepend it to the program:
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It skips the first term (1) but if that bothers you you can just prepend it to the program:
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|
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1 [1 1 fib_gen] 10 [x] times
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1 [1 1 fib-gen] 10 [x] times
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## Project Euler Problem Two
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## Project Euler Problem Two
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|
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> By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.
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> By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.
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|
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Now that we have a generator for the Fibonacci sequence, we need a function that adds a term in the sequence to a sum if it is even, and `pop`s it otherwise.
|
Now that we have a generator for the Fibonacci sequence, we need a function that adds
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a term in the sequence to a sum if it is even, and `pop`s it otherwise.
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|
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### `even`
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|
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```python
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[even 2 % bool] inscribe
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define('PE2.1 == dup 2 % [+] [pop] branch')
|
|
||||||
```
|
### `PE2.1`
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|
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[PE2.1 dup even [+] [pop] branch] inscribe
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|
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And a predicate function that detects when the terms in the series "exceed four million".
|
And a predicate function that detects when the terms in the series "exceed four million".
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|
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|
### `>4M`
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|
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```python
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[>4M 4000000 >] inscribe
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define('>4M == 4000000 >')
|
|
||||||
```
|
|
||||||
|
|
||||||
Now it's straightforward to define `PE2` as a recursive function that generates terms in the Fibonacci sequence until they exceed four million and sums the even ones.
|
Now it's straightforward to define `PE2` as a recursive function that generates terms
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|
in the Fibonacci sequence until they exceed four million and sums the even ones.
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|
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```python
|
|
||||||
define('PE2 == 0 fib_gen x [pop >4M] [popop] [[PE2.1] dip x] primrec')
|
|
||||||
```
|
|
||||||
|
|
||||||
|
|
||||||
```python
|
|
||||||
J('PE2')
|
|
||||||
```
|
|
||||||
|
|
||||||
|
joy? 0 [1 1 fib-gen] x [pop >4M] [popop] [[PE2.1] dip x] tailrec
|
||||||
4613732
|
4613732
|
||||||
|
|
||||||
|
### `PE2`
|
||||||
|
|
||||||
Here's the collected program definitions:
|
[PE2 0 [1 1 fib-gen] x [pop >4M] [popop] [[PE2.1] dip x] tailrec] inscribe
|
||||||
|
|
||||||
fib == + swons [popdd over] infra uncons
|
Here's the collected program definitions (with a little editorializing):
|
||||||
fib_gen == [1 1 fib]
|
|
||||||
|
|
||||||
even == dup 2 %
|
fib-gen + [popdd over] cons infra uncons
|
||||||
>4M == 4000000 >
|
even 2 % bool
|
||||||
|
>4M 4000000 >
|
||||||
|
PE2.1 dup even [+] [pop] branch
|
||||||
|
PE2.2 [PE2.1] dip x
|
||||||
|
PE2.init 0 [1 1 fib-gen] x
|
||||||
|
PE2.rec [pop >4M] [popop] [PE2.2] tailrec
|
||||||
|
PE2 PE2.init PE2.rec
|
||||||
|
|
||||||
PE2.1 == even [+] [pop] branch
|
|
||||||
PE2 == 0 fib_gen x [pop >4M] [popop] [[PE2.1] dip x] primrec
|
|
||||||
|
|
||||||
### Even-valued Fibonacci Terms
|
### Hmm...
|
||||||
|
|
||||||
|
fib-gen + swons [popdd over] infra uncons
|
||||||
|
|
||||||
|
|
||||||
|
## Even-valued Fibonacci Terms
|
||||||
|
|
||||||
Using `o` for odd and `e` for even:
|
Using `o` for odd and `e` for even:
|
||||||
|
|
||||||
|
|
@ -363,93 +369,116 @@ Using `o` for odd and `e` for even:
|
||||||
|
|
||||||
So the Fibonacci sequence considered in terms of just parity would be:
|
So the Fibonacci sequence considered in terms of just parity would be:
|
||||||
|
|
||||||
o o e o o e o o e o o e o o e o o e
|
o o e o o e o o e o o e . . .
|
||||||
1 1 2 3 5 8 . . .
|
1 1 2 3 5 8 13 21 34 55 89 144 . . .
|
||||||
|
|
||||||
Every third term is even.
|
Every third term is even.
|
||||||
|
|
||||||
|
So what if we drive the generator three times and discard the odd terms?
|
||||||
|
We would have to initialize our `fib` generator with 1 0:
|
||||||
|
|
||||||
|
[1 0 fib-gen]
|
||||||
|
|
||||||
```python
|
### `third-term`
|
||||||
J('[1 0 fib] x x x') # To start the sequence with 1 1 2 3 instead of 1 2 3.
|
|
||||||
```
|
|
||||||
|
|
||||||
1 1 2 [3 2 fib]
|
[third-term x x x [popop] dipd] inscribe
|
||||||
|
|
||||||
|
So:
|
||||||
|
|
||||||
Drive the generator three times and `popop` the two odd terms.
|
joy? [1 0 fib-gen]
|
||||||
|
[1 0 fib-gen]
|
||||||
|
|
||||||
|
joy? third-term
|
||||||
|
2 [3 2 fib-gen]
|
||||||
|
|
||||||
|
joy? third-term
|
||||||
|
2 8 [13 8 fib-gen]
|
||||||
|
|
||||||
|
joy? third-term
|
||||||
|
2 8 34 [55 34 fib-gen]
|
||||||
|
|
||||||
|
joy? third-term
|
||||||
|
2 8 34 144 [233 144 fib-gen]
|
||||||
|
|
||||||
|
So now we need a sum:
|
||||||
|
|
||||||
```python
|
joy? 0
|
||||||
J('[1 0 fib] x x x [popop] dipd')
|
0
|
||||||
```
|
|
||||||
|
|
||||||
2 [3 2 fib]
|
And our Fibonacci generator:
|
||||||
|
|
||||||
|
joy? [1 0 fib-gen]
|
||||||
|
0 [1 0 fib-gen]
|
||||||
|
|
||||||
|
We want to generate the initial term:
|
||||||
|
|
||||||
```python
|
joy? third-term
|
||||||
define('PE2.2 == x x x [popop] dipd')
|
0 2 [3 2 fib-gen]
|
||||||
```
|
|
||||||
|
|
||||||
|
Now we check if the term is less than four million,
|
||||||
|
if so we add it and recur,
|
||||||
|
otherwise we discard the term and the generator leaving the sum on the stack:
|
||||||
|
|
||||||
```python
|
joy? [pop >4M] [popop] [[PE2.1] dip third-term] tailrec
|
||||||
J('[1 0 fib] 10 [PE2.2] times')
|
|
||||||
```
|
|
||||||
|
|
||||||
2 8 34 144 610 2584 10946 46368 196418 832040 [1346269 832040 fib]
|
|
||||||
|
|
||||||
|
|
||||||
Replace `x` with our new driver function `PE2.2` and start our `fib` generator at `1 0`.
|
|
||||||
|
|
||||||
|
|
||||||
```python
|
|
||||||
J('0 [1 0 fib] PE2.2 [pop >4M] [popop] [[PE2.1] dip PE2.2] primrec')
|
|
||||||
```
|
|
||||||
|
|
||||||
4613732
|
4613732
|
||||||
|
|
||||||
|
## Math
|
||||||
|
|
||||||
## How to compile these?
|
a b
|
||||||
You would probably start with a special version of `G`, and perhaps modifications to the default `x`?
|
b a+b
|
||||||
|
a+b a+b+b
|
||||||
|
a+b+b a+a+b+b+b
|
||||||
|
|
||||||
## An Interesting Variation
|
So if (a,b) and a is even then the next even term pair is (a+2b, 2a+3b)
|
||||||
|
|
||||||
|
Reconsider:
|
||||||
|
|
||||||
|
[b a F] x
|
||||||
|
[b a F] b a F
|
||||||
|
|
||||||
|
From here we want to arrive at:
|
||||||
|
|
||||||
|
(a+2b) [(2a+3b) (a+2b) F]
|
||||||
|
|
||||||
|
b a F
|
||||||
|
b a [F0] [F1] fork
|
||||||
|
|
||||||
|
b a over [+] ii
|
||||||
|
---------------------
|
||||||
|
a+2b
|
||||||
|
|
||||||
|
And:
|
||||||
|
|
||||||
|
b a over [dup + +] ii
|
||||||
|
---------------------------
|
||||||
|
2a+3b
|
||||||
|
|
||||||
|
|
||||||
```python
|
[over [dup + +] ii] [over [+] ii] clop
|
||||||
define('codireco == cons dip rest cons')
|
roll< rrest [tuck] dip ccons
|
||||||
```
|
|
||||||
|
|
||||||
|
|
||||||
```python
|
[b a F] b a F
|
||||||
V('[0 [dup ++] codireco] x')
|
|
||||||
```
|
|
||||||
|
|
||||||
. [0 [dup ++] codireco] x
|
[b a F] (2a+3b) (a+2b) roll<
|
||||||
[0 [dup ++] codireco] . x
|
(2a+3b) (a+2b) [b a F] rrest
|
||||||
[0 [dup ++] codireco] . 0 [dup ++] codireco
|
(2a+3b) (a+2b) [F] [tuck] dip ccons
|
||||||
[0 [dup ++] codireco] 0 . [dup ++] codireco
|
|
||||||
[0 [dup ++] codireco] 0 [dup ++] . codireco
|
|
||||||
[0 [dup ++] codireco] 0 [dup ++] . cons dip rest cons
|
|
||||||
[0 [dup ++] codireco] [0 dup ++] . dip rest cons
|
|
||||||
. 0 dup ++ [0 [dup ++] codireco] rest cons
|
|
||||||
0 . dup ++ [0 [dup ++] codireco] rest cons
|
|
||||||
0 0 . ++ [0 [dup ++] codireco] rest cons
|
|
||||||
0 1 . [0 [dup ++] codireco] rest cons
|
|
||||||
0 1 [0 [dup ++] codireco] . rest cons
|
|
||||||
0 1 [[dup ++] codireco] . cons
|
|
||||||
0 [1 [dup ++] codireco] .
|
|
||||||
|
|
||||||
|
|
||||||
|
joy? [1 0 [over [dup + +] ii] [over [+] ii] clop roll< rrest [tuck] dip ccons]
|
||||||
|
[1 0 [over [dup + +] ii] [over [+] ii] clop roll< rrest [tuck] dip ccons]
|
||||||
|
|
||||||
|
joy? x
|
||||||
|
2 [3 2 [over [dup + +] ii] [over [+] ii] clop roll< rrest [tuck] dip ccons]
|
||||||
|
|
||||||
|
joy? x
|
||||||
|
2 8 [13 8 [over [dup + +] ii] [over [+] ii] clop roll< rrest [tuck] dip ccons]
|
||||||
|
|
||||||
|
joy? x
|
||||||
|
2 8 34 [55 34 [over [dup + +] ii] [over [+] ii] clop roll< rrest [tuck] dip ccons]
|
||||||
|
|
||||||
|
joy? x
|
||||||
|
2 8 34 144 [233 144 [over [dup + +] ii] [over [+] ii] clop roll< rrest [tuck] dip ccons]
|
||||||
|
|
||||||
```python
|
And so it goes...
|
||||||
define('G == [codireco] cons cons')
|
|
||||||
```
|
|
||||||
|
|
||||||
|
|
||||||
```python
|
|
||||||
J('230 [dup ++] G 5 [x] times pop')
|
|
||||||
```
|
|
||||||
|
|
||||||
230 231 232 233 234
|
|
||||||
|
|
||||||
|
|
|
||||||
Loading…
Reference in New Issue