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Quadratic formula





In [1]:



    


from notebook_preamble import J, V, define

Cf. jp-quadratic.html

         -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

Check it.

 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

Codicil

-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

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




In [2]:



    


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



    











In [3]:



    


J('3 1 1 quadratic')



    


















    






-0.3819660112501051 -2.618033988749895

Simplify

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





In [4]:



    


define('pm == [+] [-] cleave popdd')

Then quadratic becomes:





In [5]:



    


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



    











In [6]:



    


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.





In [7]:



    


from joy.library import SimpleFunctionWrapper
from notebook_preamble import D


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


D['pm'] = pm

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





In [8]:



    


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 .

Content source: calroc/joypy

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