159 lines
6.3 KiB
ReStructuredText
159 lines
6.3 KiB
ReStructuredText
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.. code:: ipython2
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from notebook_preamble import J, V, define
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`Quadratic formula <https://en.wikipedia.org/wiki/Quadratic_formula>`__
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=======================================================================
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Cf.
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`jp-quadratic.html <http://www.kevinalbrecht.com/code/joy-mirror/jp-quadratic.html>`__
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::
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-b ± sqrt(b^2 - 4 * a * c)
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--------------------------------
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2 * a
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:math:`\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}`
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Write a straightforward program with variable names.
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----------------------------------------------------
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This math translates to Joy code in a straightforward manner. We are
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going to use named variables to keep track of the arguments, then write
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a definition without them.
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``-b``
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~~~~~~
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::
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b neg
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``sqrt(b^2 - 4 * a * c)``
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~~~~~~~~~~~~~~~~~~~~~~~~~
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::
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b sqr 4 a c * * - sqrt
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``/2a``
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~~~~~~~
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::
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a 2 * /
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``±``
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~~~~~
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There is a function ``pm`` that accepts two values on the stack and
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replaces them with their sum and difference.
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::
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pm == [+] [-] cleave popdd
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Putting Them Together
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~~~~~~~~~~~~~~~~~~~~~
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::
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b neg b sqr 4 a c * * - sqrt pm a 2 * [/] cons app2
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We use ``app2`` to compute both roots by using a quoted program
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``[2a /]`` built with ``cons``.
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Derive a definition.
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--------------------
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Working backwards we use ``dip`` and ``dipd`` to extract the code from
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the variables:
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::
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b neg b sqr 4 a c * * - sqrt pm a 2 * [/] cons app2
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b [neg] dupdip sqr 4 a c * * - sqrt pm a 2 * [/] cons app2
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b a c [[neg] dupdip sqr 4] dipd * * - sqrt pm a 2 * [/] cons app2
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b a c a [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2
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b a c over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2
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The three arguments are to the left, so we can "chop off" everything to
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the right and say it's the definition of the ``quadratic`` function:
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.. code:: ipython2
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define('quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2')
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Let's try it out:
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.. code:: ipython2
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J('3 1 1 quadratic')
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.. parsed-literal::
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-0.3819660112501051 -2.618033988749895
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If you look at the Joy evaluation trace you can see that the first few
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lines are the ``dip`` and ``dipd`` combinators building the main program
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by incorporating the values on the stack. Then that program runs and you
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get the results. This is pretty typical of Joy code.
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.. code:: ipython2
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V('-5 1 4 quadratic')
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.. parsed-literal::
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. -5 1 4 quadratic
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-5 . 1 4 quadratic
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-5 1 . 4 quadratic
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-5 1 4 . quadratic
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-5 1 4 . over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2
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-5 1 4 1 . [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2
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-5 1 4 1 [[[neg] dupdip sqr 4] dipd * * - sqrt pm] . dip 2 * [/] cons app2
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-5 1 4 . [[neg] dupdip sqr 4] dipd * * - sqrt pm 1 2 * [/] cons app2
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-5 1 4 [[neg] dupdip sqr 4] . dipd * * - sqrt pm 1 2 * [/] cons app2
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-5 . [neg] dupdip sqr 4 1 4 * * - sqrt pm 1 2 * [/] cons app2
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-5 [neg] . dupdip sqr 4 1 4 * * - sqrt pm 1 2 * [/] cons app2
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-5 . neg -5 sqr 4 1 4 * * - sqrt pm 1 2 * [/] cons app2
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5 . -5 sqr 4 1 4 * * - sqrt pm 1 2 * [/] cons app2
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5 -5 . sqr 4 1 4 * * - sqrt pm 1 2 * [/] cons app2
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5 -5 . dup mul 4 1 4 * * - sqrt pm 1 2 * [/] cons app2
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5 -5 -5 . mul 4 1 4 * * - sqrt pm 1 2 * [/] cons app2
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5 25 . 4 1 4 * * - sqrt pm 1 2 * [/] cons app2
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5 25 4 . 1 4 * * - sqrt pm 1 2 * [/] cons app2
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5 25 4 1 . 4 * * - sqrt pm 1 2 * [/] cons app2
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5 25 4 1 4 . * * - sqrt pm 1 2 * [/] cons app2
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5 25 4 4 . * - sqrt pm 1 2 * [/] cons app2
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5 25 16 . - sqrt pm 1 2 * [/] cons app2
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5 9 . sqrt pm 1 2 * [/] cons app2
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5 3.0 . pm 1 2 * [/] cons app2
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8.0 2.0 . 1 2 * [/] cons app2
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8.0 2.0 1 . 2 * [/] cons app2
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8.0 2.0 1 2 . * [/] cons app2
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8.0 2.0 2 . [/] cons app2
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8.0 2.0 2 [/] . cons app2
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8.0 2.0 [2 /] . app2
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[8.0] [2 /] . infra first [2.0] [2 /] infra first
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8.0 . 2 / [] swaack first [2.0] [2 /] infra first
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8.0 2 . / [] swaack first [2.0] [2 /] infra first
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4.0 . [] swaack first [2.0] [2 /] infra first
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4.0 [] . swaack first [2.0] [2 /] infra first
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[4.0] . first [2.0] [2 /] infra first
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4.0 . [2.0] [2 /] infra first
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4.0 [2.0] . [2 /] infra first
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4.0 [2.0] [2 /] . infra first
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2.0 . 2 / [4.0] swaack first
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2.0 2 . / [4.0] swaack first
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1.0 . [4.0] swaack first
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1.0 [4.0] . swaack first
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4.0 [1.0] . first
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4.0 1.0 .
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