7.3 KiB
7.3 KiB
Quadratic formula
from notebook_preamble import J, V, define
-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
define('quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt [+] [-] cleave] dip 2 * [truediv] cons app2 roll< pop')
J('3 1 1 quadratic')
-0.3819660112501051 -2.618033988749895
Simplify
We can define a pm plus-or-minus function:
define('pm == [+] [-] cleave popdd')
Then quadratic becomes:
define('quadratic == over [[[neg] dupdip sqr 4] dipd * * - sqrt pm] dip 2 * [truediv] cons app2')
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.
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.
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 .