Bunches of new docs.
Type inference! A new treatment of recursion combinator patterns.
This commit is contained in:
Simon Forman
+109
-150
@@ -1,12 +1,5 @@
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# [Quadratic formula](https://en.wikipedia.org/wiki/Quadratic_formula)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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@@ -16,6 +9,13 @@
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"from notebook_preamble import J, V, define"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# [Quadratic formula](https://en.wikipedia.org/wiki/Quadratic_formula)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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@@ -27,9 +27,9 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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" -b +/- sqrt(b^2 - 4 * a * c)\n",
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" -----------------------------\n",
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" 2 * a"
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" -b ± sqrt(b^2 - 4 * a * c)\n",
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" --------------------------------\n",
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" 2 * a"
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]
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},
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{
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@@ -44,42 +44,52 @@
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"metadata": {},
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"source": [
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"## Write a straightforward program with variable names.\n",
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"\n",
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" b neg b sqr 4 a c * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2\n",
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"\n",
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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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"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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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Check it.\n",
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"Evaluating by hand:\n",
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"\n",
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" b neg b sqr 4 a c * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2\n",
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" -b b sqr 4 a c * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2\n",
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" -b b^2 4 a c * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2\n",
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" -b b^2 4ac - sqrt [+] [-] cleave a 2 * [truediv] cons app2\n",
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" -b b^2-4ac sqrt [+] [-] cleave a 2 * [truediv] cons app2\n",
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" -b sqrt(b^2-4ac) [+] [-] cleave a 2 * [truediv] cons app2\n",
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"\n",
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" -b -b+sqrt(b^2-4ac) -b-sqrt(b^2-4ac) a 2 * [truediv] cons app2\n",
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" -b -b+sqrt(b^2-4ac) -b-sqrt(b^2-4ac) 2a [truediv] cons app2\n",
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" -b -b+sqrt(b^2-4ac) -b-sqrt(b^2-4ac) [2a truediv] app2\n",
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" -b -b+sqrt(b^2-4ac)/2a -b-sqrt(b^2-4ac)/2a\n",
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"\n",
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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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"### `-b`\n",
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" b neg"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Cleanup\n",
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" -b -b+sqrt(b^2-4ac)/2a -b-sqrt(b^2-4ac)/2a roll< pop\n",
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" -b+sqrt(b^2-4ac)/2a -b-sqrt(b^2-4ac)/2a -b pop\n",
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" -b+sqrt(b^2-4ac)/2a -b-sqrt(b^2-4ac)/2a"
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"### `sqrt(b^2 - 4 * a * c)`\n",
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" b sqr 4 a c * * - sqrt"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### `/2a`\n",
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" a 2 * /"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### `±`\n",
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"There is a function `pm` that accepts two values on the stack and replaces them with their sum and difference.\n",
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"\n",
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" pm == [+] [-] cleave popdd"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Putting Them Together\n",
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"\n",
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" b neg b sqr 4 a c * * - sqrt pm a 2 * [/] cons app2\n",
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"\n",
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"We use `app2` to compute both roots by using a quoted program `[2a /]` built with `cons`."
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]
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},
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{
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@@ -87,12 +97,15 @@
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"metadata": {},
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"source": [
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"## Derive a definition.\n",
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"Working backwards we use `dip` and `dipd` to extract the code from the variables:\n",
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"\n",
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" b neg b sqr 4 a c * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2 roll< pop\n",
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" b [neg] dupdip sqr 4 a c * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2 roll< pop\n",
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" b a c [[neg] dupdip sqr 4] dipd * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2 roll< pop\n",
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" b a c a [[[neg] dupdip sqr 4] dipd * * - sqrt [+] [-] cleave] dip 2 * [truediv] cons app2 roll< pop\n",
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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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" b neg b sqr 4 a c * * - sqrt pm a 2 * [/] cons app2\n",
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" b [neg] dupdip sqr 4 a c * * - sqrt pm a 2 * [/] cons app2\n",
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" b a c [[neg] dupdip sqr 4] dipd * * - sqrt pm a 2 * [/] cons app2\n",
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" b a c a [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2\n",
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" b a c over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2\n",
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"\n",
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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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]
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},
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{
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@@ -101,7 +114,14 @@
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"metadata": {},
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"outputs": [],
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"source": [
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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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"define('quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2')"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Let's try it out:"
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]
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},
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{
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@@ -125,137 +145,76 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Simplify\n",
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"We can define a `pm` plus-or-minus function:"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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" pm == [+] [-] cleave popdd"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Then `quadratic` becomes:"
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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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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [],
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"source": [
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"define('quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [truediv] cons app2')"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"-0.3819660112501051 -2.618033988749895\n"
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" . -5 1 4 quadratic\n",
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" -5 . 1 4 quadratic\n",
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" -5 1 . 4 quadratic\n",
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" -5 1 4 . quadratic\n",
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" -5 1 4 . over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2\n",
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" -5 1 4 1 . [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2\n",
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"-5 1 4 1 [[[neg] dupdip sqr 4] dipd * * - sqrt pm] . dip 2 * [/] cons app2\n",
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" -5 1 4 . [[neg] dupdip sqr 4] dipd * * - sqrt pm 1 2 * [/] cons app2\n",
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" -5 1 4 [[neg] dupdip sqr 4] . dipd * * - sqrt pm 1 2 * [/] cons app2\n",
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" -5 . [neg] dupdip sqr 4 1 4 * * - sqrt pm 1 2 * [/] cons app2\n",
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" -5 [neg] . dupdip sqr 4 1 4 * * - sqrt pm 1 2 * [/] cons app2\n",
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" -5 . neg -5 sqr 4 1 4 * * - sqrt pm 1 2 * [/] cons app2\n",
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" 5 . -5 sqr 4 1 4 * * - sqrt pm 1 2 * [/] cons app2\n",
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" 5 -5 . sqr 4 1 4 * * - sqrt pm 1 2 * [/] cons app2\n",
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" 5 -5 . dup mul 4 1 4 * * - sqrt pm 1 2 * [/] cons app2\n",
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" 5 -5 -5 . mul 4 1 4 * * - sqrt pm 1 2 * [/] cons app2\n",
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" 5 25 . 4 1 4 * * - sqrt pm 1 2 * [/] cons app2\n",
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" 5 25 4 . 1 4 * * - sqrt pm 1 2 * [/] cons app2\n",
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" 5 25 4 1 . 4 * * - sqrt pm 1 2 * [/] cons app2\n",
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" 5 25 4 1 4 . * * - sqrt pm 1 2 * [/] cons app2\n",
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" 5 25 4 4 . * - sqrt pm 1 2 * [/] cons app2\n",
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" 5 25 16 . - sqrt pm 1 2 * [/] cons app2\n",
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" 5 9 . sqrt pm 1 2 * [/] cons app2\n",
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" 5 3.0 . pm 1 2 * [/] cons app2\n",
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" 8.0 2.0 . 1 2 * [/] cons app2\n",
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" 8.0 2.0 1 . 2 * [/] cons app2\n",
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" 8.0 2.0 1 2 . * [/] cons app2\n",
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" 8.0 2.0 2 . [/] cons app2\n",
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" 8.0 2.0 2 [/] . cons app2\n",
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" 8.0 2.0 [2 /] . app2\n",
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" [8.0] [2 /] . infra first [2.0] [2 /] infra first\n",
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" 8.0 . 2 / [] swaack first [2.0] [2 /] infra first\n",
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" 8.0 2 . / [] swaack first [2.0] [2 /] infra first\n",
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" 4.0 . [] swaack first [2.0] [2 /] infra first\n",
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" 4.0 [] . swaack first [2.0] [2 /] infra first\n",
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" [4.0] . first [2.0] [2 /] infra first\n",
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" 4.0 . [2.0] [2 /] infra first\n",
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" 4.0 [2.0] . [2 /] infra first\n",
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" 4.0 [2.0] [2 /] . infra first\n",
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" 2.0 . 2 / [4.0] swaack first\n",
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" 2.0 2 . / [4.0] swaack first\n",
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" 1.0 . [4.0] swaack first\n",
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" 1.0 [4.0] . swaack first\n",
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" 4.0 [1.0] . first\n",
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" 4.0 1.0 . \n"
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]
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}
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],
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"source": [
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"J('3 1 1 quadratic')"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Define a \"native\" `pm` function.\n",
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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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"V('-5 1 4 quadratic')"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"def pm(stack):\n",
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" a, (b, stack) = stack\n",
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" p, m, = b + a, b - a\n",
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" return m, (p, stack)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The resulting trace is short enough to fit on a page."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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" . 3 1 1 quadratic\n",
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" 3 . 1 1 quadratic\n",
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" 3 1 . 1 quadratic\n",
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" 3 1 1 . quadratic\n",
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" 3 1 1 . over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [truediv] cons app2\n",
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" 3 1 1 1 . [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [truediv] cons app2\n",
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" 3 1 1 1 [[[neg] dupdip sqr 4] dipd * * - sqrt pm] . dip 2 * [truediv] cons app2\n",
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" 3 1 1 . [[neg] dupdip sqr 4] dipd * * - sqrt pm 1 2 * [truediv] cons app2\n",
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" 3 1 1 [[neg] dupdip sqr 4] . dipd * * - sqrt pm 1 2 * [truediv] cons app2\n",
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" 3 . [neg] dupdip sqr 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2\n",
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" 3 [neg] . dupdip sqr 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2\n",
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" 3 . neg 3 sqr 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2\n",
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" -3 . 3 sqr 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2\n",
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" -3 3 . sqr 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2\n",
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" -3 3 . dup mul 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2\n",
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" -3 3 3 . mul 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2\n",
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" -3 9 . 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2\n",
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" -3 9 4 . 1 1 * * - sqrt pm 1 2 * [truediv] cons app2\n",
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" -3 9 4 1 . 1 * * - sqrt pm 1 2 * [truediv] cons app2\n",
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" -3 9 4 1 1 . * * - sqrt pm 1 2 * [truediv] cons app2\n",
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" -3 9 4 1 . * - sqrt pm 1 2 * [truediv] cons app2\n",
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" -3 9 4 . - sqrt pm 1 2 * [truediv] cons app2\n",
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" -3 5 . sqrt pm 1 2 * [truediv] cons app2\n",
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" -3 2.23606797749979 . pm 1 2 * [truediv] cons app2\n",
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" -0.7639320225002102 -5.23606797749979 . 1 2 * [truediv] cons app2\n",
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" -0.7639320225002102 -5.23606797749979 1 . 2 * [truediv] cons app2\n",
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" -0.7639320225002102 -5.23606797749979 1 2 . * [truediv] cons app2\n",
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" -0.7639320225002102 -5.23606797749979 2 . [truediv] cons app2\n",
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" -0.7639320225002102 -5.23606797749979 2 [truediv] . cons app2\n",
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" -0.7639320225002102 -5.23606797749979 [2 truediv] . app2\n",
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" [-0.7639320225002102] [2 truediv] . infra first [-5.23606797749979] [2 truediv] infra first\n",
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" -0.7639320225002102 . 2 truediv [] swaack first [-5.23606797749979] [2 truediv] infra first\n",
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" -0.7639320225002102 2 . truediv [] swaack first [-5.23606797749979] [2 truediv] infra first\n",
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" -0.3819660112501051 . [] swaack first [-5.23606797749979] [2 truediv] infra first\n",
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" -0.3819660112501051 [] . swaack first [-5.23606797749979] [2 truediv] infra first\n",
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" [-0.3819660112501051] . first [-5.23606797749979] [2 truediv] infra first\n",
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" -0.3819660112501051 . [-5.23606797749979] [2 truediv] infra first\n",
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" -0.3819660112501051 [-5.23606797749979] . [2 truediv] infra first\n",
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"-0.3819660112501051 [-5.23606797749979] [2 truediv] . infra first\n",
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" -5.23606797749979 . 2 truediv [-0.3819660112501051] swaack first\n",
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" -5.23606797749979 2 . truediv [-0.3819660112501051] swaack first\n",
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" -2.618033988749895 . [-0.3819660112501051] swaack first\n",
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" -2.618033988749895 [-0.3819660112501051] . swaack first\n",
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" -0.3819660112501051 [-2.618033988749895] . first\n",
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" -0.3819660112501051 -2.618033988749895 . \n"
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]
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}
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],
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"source": [
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"V('3 1 1 quadratic')"
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]
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"source": []
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||||
}
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],
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"metadata": {
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