-
-

Quadratic formula

-
from notebook_preamble import J, V, define
+  
from notebook_preamble import J, V, define
 

 

+

+
Quadratic formula

 
Cf.
 jp-quadratic.html

-
-b  +/- sqrt(b^2 - 4 * a * c)
------------------------------
-           2 * a
+
   -b ± sqrt(b^2 - 4 * a * c)
+--------------------------------
+            2 * a
 

 

 
\(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

 

 
Write a straightforward program with variable names.

-
b neg b sqr 4 a c * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2
+
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.

+

+
-b

+
b neg
 

 

-
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.

-

-
Check it.

-
Evaluating by hand:

-
 b neg b sqr 4 a c * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2
--b     b sqr 4 a c * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2
--b     b^2   4 a c * * - sqrt [+] [-] cleave a 2 * [truediv] cons app2
--b     b^2 4ac         - sqrt [+] [-] cleave a 2 * [truediv] cons app2
--b     b^2-4ac           sqrt [+] [-] cleave a 2 * [truediv] cons app2
--b sqrt(b^2-4ac)              [+] [-] cleave a 2 * [truediv] cons app2
-
--b -b+sqrt(b^2-4ac)    -b-sqrt(b^2-4ac)    a 2 * [truediv] cons app2
--b -b+sqrt(b^2-4ac)    -b-sqrt(b^2-4ac)    2a    [truediv] cons app2
--b -b+sqrt(b^2-4ac)    -b-sqrt(b^2-4ac)    [2a truediv]         app2
--b -b+sqrt(b^2-4ac)/2a -b-sqrt(b^2-4ac)/2a
+

+

+
sqrt(b^2 - 4 * a * c)

+
b sqr 4 a c * * - sqrt
 

 

-
(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.)

 

-

-
Cleanup

-
-b -b+sqrt(b^2-4ac)/2a -b-sqrt(b^2-4ac)/2a                          roll< pop
-   -b+sqrt(b^2-4ac)/2a -b-sqrt(b^2-4ac)/2a -b                             pop
-   -b+sqrt(b^2-4ac)/2a -b-sqrt(b^2-4ac)/2a
+

+
/2a

+
a 2 * /
 

 

 

+

+
±

+
There is a function pm that accepts two values on the stack and
+replaces them with their sum and difference.

+
pm == [+] [-] cleave popdd
+

+

+

+

+
Putting Them Together

+
b neg b sqr 4 a c * * - sqrt pm a 2 * [/] cons app2
+

+

+
We use app2 to compute both roots by using a quoted program
+[2a /] built with cons.

+

 

 

 
Derive a definition.

-
b neg b           sqr 4 a c        * * - sqrt [+] [-] cleave a       2 * [truediv] cons app2 roll< pop
-b    [neg] dupdip sqr 4 a c        * * - sqrt [+] [-] cleave a       2 * [truediv] cons app2 roll< pop
-b a c    [[neg] dupdip sqr 4] dipd * * - sqrt [+] [-] cleave a       2 * [truediv] cons app2 roll< pop
-b a c a    [[[neg] dupdip sqr 4] dipd * * - sqrt [+] [-] cleave] dip 2 * [truediv] cons app2 roll< pop
-b a c over [[[neg] dupdip sqr 4] dipd * * - sqrt [+] [-] cleave] dip 2 * [truediv] cons app2 roll< pop
+
Working backwards we use dip and dipd to extract the code from
+the variables:

+
b             neg  b      sqr 4 a c   * * - sqrt pm a    2 * [/] cons app2
+b            [neg] dupdip sqr 4 a c   * * - sqrt pm a    2 * [/] cons app2
+b a c       [[neg] dupdip sqr 4] dipd * * - sqrt pm a    2 * [/] cons app2
+b a c a    [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2
+b a c over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2
 

 

-
define('quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt [+] [-] cleave] dip 2 * [truediv] cons app2 roll< pop')
+
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:

+
define('quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2')
 

 

+
Let’s try it out:

 
J('3 1 1 quadratic')
 

 

 
-0.3819660112501051 -2.618033988749895
 

 

-

-

-
Simplify

-
We can define a pm plus-or-minus function:

-
pm == [+] [-] cleave popdd
+
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.

+
V('-5 1 4 quadratic')
 

 

-
Then quadratic becomes:

-
define('quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [truediv] cons app2')
+
                                                   . -5 1 4 quadratic
+                                                -5 . 1 4 quadratic
+                                              -5 1 . 4 quadratic
+                                            -5 1 4 . quadratic
+                                            -5 1 4 . over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2
+                                          -5 1 4 1 . [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [/] cons app2
+-5 1 4 1 [[[neg] dupdip sqr 4] dipd * * - sqrt pm] . dip 2 * [/] cons app2
+                                            -5 1 4 . [[neg] dupdip sqr 4] dipd * * - sqrt pm 1 2 * [/] cons app2
+                       -5 1 4 [[neg] dupdip sqr 4] . dipd * * - sqrt pm 1 2 * [/] cons app2
+                                                -5 . [neg] dupdip sqr 4 1 4 * * - sqrt pm 1 2 * [/] cons app2
+                                          -5 [neg] . dupdip sqr 4 1 4 * * - sqrt pm 1 2 * [/] cons app2
+                                                -5 . neg -5 sqr 4 1 4 * * - sqrt pm 1 2 * [/] cons app2
+                                                 5 . -5 sqr 4 1 4 * * - sqrt pm 1 2 * [/] cons app2
+                                              5 -5 . sqr 4 1 4 * * - sqrt pm 1 2 * [/] cons app2
+                                              5 -5 . dup mul 4 1 4 * * - sqrt pm 1 2 * [/] cons app2
+                                           5 -5 -5 . mul 4 1 4 * * - sqrt pm 1 2 * [/] cons app2
+                                              5 25 . 4 1 4 * * - sqrt pm 1 2 * [/] cons app2
+                                            5 25 4 . 1 4 * * - sqrt pm 1 2 * [/] cons app2
+                                          5 25 4 1 . 4 * * - sqrt pm 1 2 * [/] cons app2
+                                        5 25 4 1 4 . * * - sqrt pm 1 2 * [/] cons app2
+                                          5 25 4 4 . * - sqrt pm 1 2 * [/] cons app2
+                                           5 25 16 . - sqrt pm 1 2 * [/] cons app2
+                                               5 9 . sqrt pm 1 2 * [/] cons app2
+                                             5 3.0 . pm 1 2 * [/] cons app2
+                                           8.0 2.0 . 1 2 * [/] cons app2
+                                         8.0 2.0 1 . 2 * [/] cons app2
+                                       8.0 2.0 1 2 . * [/] cons app2
+                                         8.0 2.0 2 . [/] cons app2
+                                     8.0 2.0 2 [/] . cons app2
+                                     8.0 2.0 [2 /] . app2
+                                       [8.0] [2 /] . infra first [2.0] [2 /] infra first
+                                               8.0 . 2 / [] swaack first [2.0] [2 /] infra first
+                                             8.0 2 . / [] swaack first [2.0] [2 /] infra first
+                                               4.0 . [] swaack first [2.0] [2 /] infra first
+                                            4.0 [] . swaack first [2.0] [2 /] infra first
+                                             [4.0] . first [2.0] [2 /] infra first
+                                               4.0 . [2.0] [2 /] infra first
+                                         4.0 [2.0] . [2 /] infra first
+                                   4.0 [2.0] [2 /] . infra first
+                                               2.0 . 2 / [4.0] swaack first
+                                             2.0 2 . / [4.0] swaack first
+                                               1.0 . [4.0] swaack first
+                                         1.0 [4.0] . swaack first
+                                         4.0 [1.0] . first
+                                           4.0 1.0 .
 

 

-
J('3 1 1 quadratic')
-

-

-
-0.3819660112501051 -2.618033988749895
-

-

-

-
Define a “native” pm function.

-
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.)

-
def pm(stack):
-    a, (b, stack) = stack
-    p, m, = b + a, b - a
-    return m, (p, stack)
-

-

-
The resulting trace is short enough to fit on a page.

-
V('3 1 1 quadratic')
-

-

-
                                                    . 3 1 1 quadratic
-                                                  3 . 1 1 quadratic
-                                                3 1 . 1 quadratic
-                                              3 1 1 . quadratic
-                                              3 1 1 . over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [truediv] cons app2
-                                            3 1 1 1 . [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [truediv] cons app2
-  3 1 1 1 [[[neg] dupdip sqr 4] dipd * * - sqrt pm] . dip 2 * [truediv] cons app2
-                                              3 1 1 . [[neg] dupdip sqr 4] dipd * * - sqrt pm 1 2 * [truediv] cons app2
-                         3 1 1 [[neg] dupdip sqr 4] . dipd * * - sqrt pm 1 2 * [truediv] cons app2
-                                                  3 . [neg] dupdip sqr 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
-                                            3 [neg] . dupdip sqr 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
-                                                  3 . neg 3 sqr 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
-                                                 -3 . 3 sqr 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
-                                               -3 3 . sqr 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
-                                               -3 3 . dup mul 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
-                                             -3 3 3 . mul 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
-                                               -3 9 . 4 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
-                                             -3 9 4 . 1 1 * * - sqrt pm 1 2 * [truediv] cons app2
-                                           -3 9 4 1 . 1 * * - sqrt pm 1 2 * [truediv] cons app2
-                                         -3 9 4 1 1 . * * - sqrt pm 1 2 * [truediv] cons app2
-                                           -3 9 4 1 . * - sqrt pm 1 2 * [truediv] cons app2
-                                             -3 9 4 . - sqrt pm 1 2 * [truediv] cons app2
-                                               -3 5 . sqrt pm 1 2 * [truediv] cons app2
-                                -3 2.23606797749979 . pm 1 2 * [truediv] cons app2
-              -0.7639320225002102 -5.23606797749979 . 1 2 * [truediv] cons app2
-            -0.7639320225002102 -5.23606797749979 1 . 2 * [truediv] cons app2
-          -0.7639320225002102 -5.23606797749979 1 2 . * [truediv] cons app2
-            -0.7639320225002102 -5.23606797749979 2 . [truediv] cons app2
-  -0.7639320225002102 -5.23606797749979 2 [truediv] . cons app2
-  -0.7639320225002102 -5.23606797749979 [2 truediv] . app2
-                  [-0.7639320225002102] [2 truediv] . infra first [-5.23606797749979] [2 truediv] infra first
-                                -0.7639320225002102 . 2 truediv [] swaack first [-5.23606797749979] [2 truediv] infra first
-                              -0.7639320225002102 2 . truediv [] swaack first [-5.23606797749979] [2 truediv] infra first
-                                -0.3819660112501051 . [] swaack first [-5.23606797749979] [2 truediv] infra first
-                             -0.3819660112501051 [] . swaack first [-5.23606797749979] [2 truediv] infra first
-                              [-0.3819660112501051] . first [-5.23606797749979] [2 truediv] infra first
-                                -0.3819660112501051 . [-5.23606797749979] [2 truediv] infra first
-            -0.3819660112501051 [-5.23606797749979] . [2 truediv] infra first
--0.3819660112501051 [-5.23606797749979] [2 truediv] . infra first
-                                  -5.23606797749979 . 2 truediv [-0.3819660112501051] swaack first
-                                -5.23606797749979 2 . truediv [-0.3819660112501051] swaack first
-                                 -2.618033988749895 . [-0.3819660112501051] swaack first
-           -2.618033988749895 [-0.3819660112501051] . swaack first
-           -0.3819660112501051 [-2.618033988749895] . first
-             -0.3819660112501051 -2.618033988749895 .
-

-

-

 

 

 
@@ -195,15 +175,14 @@ already.)

   
@@ -212,8 +191,8 @@ already.)

 
diff --git a/docs/sphinx_docs/_build/html/notebooks/Zipper.html b/docs/sphinx_docs/_build/html/notebooks/Zipper.html
index d424e17..22d2d17 100644
--- a/docs/sphinx_docs/_build/html/notebooks/Zipper.html
+++ b/docs/sphinx_docs/_build/html/notebooks/Zipper.html
@@ -16,8 +16,8 @@
     
     
     
-    
-    
+    
+    
    
   
   
@@ -326,8 +326,8 @@ i d i d i d d Bingo!
 
diff --git a/docs/sphinx_docs/_build/html/notebooks/index.html b/docs/sphinx_docs/_build/html/notebooks/index.html
index f5213e5..4c74e30 100644
--- a/docs/sphinx_docs/_build/html/notebooks/index.html
+++ b/docs/sphinx_docs/_build/html/notebooks/index.html
@@ -44,12 +44,32 @@
 
  • The Simplest Program
    •