```
from notebook_preamble import J, V, define
```

# Quadratic formula¶

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

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

## Write a straightforward program with variable names.¶

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
```

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

¶

```
b sqr 4 a c * * - sqrt
```

`/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.¶

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
```

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
```

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')
```

```
. -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 .
```