Bunches of new docs.
Type inference! A new treatment of recursion combinator patterns.
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Simon Forman
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# [Quadratic formula](https://en.wikipedia.org/wiki/Quadratic_formula)
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```python
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from notebook_preamble import J, V, define
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```
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# [Quadratic formula](https://en.wikipedia.org/wiki/Quadratic_formula)
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Cf. [jp-quadratic.html](http://www.kevinalbrecht.com/code/joy-mirror/jp-quadratic.html)
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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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-b ± sqrt(b^2 - 4 * a * c)
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--------------------------------
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2 * a
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$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
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## Write a straightforward program with variable names.
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This math translates to Joy code in a straightforward manner. We are going to use named variables to keep track of the arguments, then write a definition without them.
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b neg b sqr 4 a c * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2
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### `-b`
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b neg
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We use `cleave` to compute the sum and difference and then `app2` to finish computing both roots using a quoted program `[2a truediv]` built with `cons`.
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### `sqrt(b^2 - 4 * a * c)`
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b sqr 4 a c * * - sqrt
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### Check it.
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Evaluating by hand:
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### `/2a`
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a 2 * /
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b neg b sqr 4 a c * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2
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-b b sqr 4 a c * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2
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-b b^2 4 a c * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2
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-b b^2 4ac - sqrt [+] [-] cleave a 2 * [truediv] cons app2
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-b b^2-4ac sqrt [+] [-] cleave a 2 * [truediv] cons app2
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-b sqrt(b^2-4ac) [+] [-] cleave a 2 * [truediv] cons app2
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-b -b+sqrt(b^2-4ac) -b-sqrt(b^2-4ac) a 2 * [truediv] cons app2
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-b -b+sqrt(b^2-4ac) -b-sqrt(b^2-4ac) 2a [truediv] cons app2
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-b -b+sqrt(b^2-4ac) -b-sqrt(b^2-4ac) [2a truediv] app2
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-b -b+sqrt(b^2-4ac)/2a -b-sqrt(b^2-4ac)/2a
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(Eventually we’ll be able to use e.g. Sympy versions of the Joy commands to do this sort of thing symbolically. This is part of what is meant by a “categorical” language.)
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### Cleanup
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-b -b+sqrt(b^2-4ac)/2a -b-sqrt(b^2-4ac)/2a roll< pop
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-b+sqrt(b^2-4ac)/2a -b-sqrt(b^2-4ac)/2a -b pop
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-b+sqrt(b^2-4ac)/2a -b-sqrt(b^2-4ac)/2a
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## Derive a definition.
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b neg b sqr 4 a c * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2 roll< pop
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b [neg] dupdip sqr 4 a c * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2 roll< pop
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b a c [[neg] dupdip sqr 4] dipd * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2 roll< pop
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b a c a [[[neg] dupdip sqr 4] dipd * * - sqrt [+] [-] cleave] dip 2 * [truediv] cons app2 roll< pop
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b a c over [[[neg] dupdip sqr 4] dipd * * - sqrt [+] [-] cleave] dip 2 * [truediv] cons app2 roll< pop
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```python
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define('quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt [+] [-] cleave] dip 2 * [truediv] cons app2 roll< pop')
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```
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```python
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J('3 1 1 quadratic')
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```
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-0.3819660112501051 -2.618033988749895
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## Simplify
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We can define a `pm` plus-or-minus function:
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### `±`
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There is a function `pm` that accepts two values on the stack and replaces them with their sum and difference.
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pm == [+] [-] cleave popdd
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Then `quadratic` becomes:
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### Putting Them Together
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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 `[2a /]` built with `cons`.
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## Derive a definition.
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Working backwards we use `dip` and `dipd` to extract the code from the variables:
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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 the right and say it's the definition of the `quadratic` function:
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```python
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define('quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [truediv] cons app2')
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define('quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2')
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```
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Let's try it out:
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```python
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J('3 1 1 quadratic')
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@@ -83,67 +63,56 @@ J('3 1 1 quadratic')
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-0.3819660112501051 -2.618033988749895
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### Define a "native" `pm` function.
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The definition of `pm` above is pretty elegant, but the implementation takes a lot of steps relative to what it's accomplishing. Since we are likely to use `pm` more than once in the future, let's write a primitive in Python and add it to the dictionary. (This has been done already.)
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If you look at the Joy evaluation trace you can see that the first few lines are the `dip` and `dipd` combinators building the main program by incorporating the values on the stack. Then that program runs and you get the results. This is pretty typical of Joy code.
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```python
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def pm(stack):
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a, (b, stack) = stack
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p, m, = b + a, b - a
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return m, (p, stack)
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V('-5 1 4 quadratic')
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```
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The resulting trace is short enough to fit on a page.
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```python
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V('3 1 1 quadratic')
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```
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. 3 1 1 quadratic
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3 . 1 1 quadratic
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3 1 . 1 quadratic
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3 1 1 . quadratic
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3 1 1 . over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [truediv] cons app2
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3 1 1 1 . [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [truediv] cons app2
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3 1 1 1 [[[neg] dupdip sqr 4] dipd * * - sqrt pm] . dip 2 * [truediv] cons app2
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3 1 1 . [[neg] dupdip sqr 4] dipd * * - sqrt pm 1 2 * [truediv] cons app2
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3 1 1 [[neg] dupdip sqr 4] . dipd * * - sqrt pm 1 2 * [truediv] cons app2
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3 . [neg] dupdip sqr 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
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3 [neg] . dupdip sqr 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
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3 . neg 3 sqr 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
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-3 . 3 sqr 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
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-3 3 . sqr 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
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-3 3 . dup mul 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
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-3 3 3 . mul 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
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-3 9 . 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
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-3 9 4 . 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
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-3 9 4 1 . 1 * * - sqrt pm 1 2 * [truediv] cons app2
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-3 9 4 1 1 . * * - sqrt pm 1 2 * [truediv] cons app2
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-3 9 4 1 . * - sqrt pm 1 2 * [truediv] cons app2
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-3 9 4 . - sqrt pm 1 2 * [truediv] cons app2
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-3 5 . sqrt pm 1 2 * [truediv] cons app2
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-3 2.23606797749979 . pm 1 2 * [truediv] cons app2
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-0.7639320225002102 -5.23606797749979 . 1 2 * [truediv] cons app2
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-0.7639320225002102 -5.23606797749979 1 . 2 * [truediv] cons app2
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-0.7639320225002102 -5.23606797749979 1 2 . * [truediv] cons app2
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-0.7639320225002102 -5.23606797749979 2 . [truediv] cons app2
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-0.7639320225002102 -5.23606797749979 2 [truediv] . cons app2
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-0.7639320225002102 -5.23606797749979 [2 truediv] . app2
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[-0.7639320225002102] [2 truediv] . infra first [-5.23606797749979] [2 truediv] infra first
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-0.7639320225002102 . 2 truediv [] swaack first [-5.23606797749979] [2 truediv] infra first
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-0.7639320225002102 2 . truediv [] swaack first [-5.23606797749979] [2 truediv] infra first
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-0.3819660112501051 . [] swaack first [-5.23606797749979] [2 truediv] infra first
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-0.3819660112501051 [] . swaack first [-5.23606797749979] [2 truediv] infra first
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[-0.3819660112501051] . first [-5.23606797749979] [2 truediv] infra first
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-0.3819660112501051 . [-5.23606797749979] [2 truediv] infra first
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-0.3819660112501051 [-5.23606797749979] . [2 truediv] infra first
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-0.3819660112501051 [-5.23606797749979] [2 truediv] . infra first
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-5.23606797749979 . 2 truediv [-0.3819660112501051] swaack first
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-5.23606797749979 2 . truediv [-0.3819660112501051] swaack first
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-2.618033988749895 . [-0.3819660112501051] swaack first
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-2.618033988749895 [-0.3819660112501051] . swaack first
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-0.3819660112501051 [-2.618033988749895] . first
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-0.3819660112501051 -2.618033988749895 .
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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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