    from notebook_preamble import J, V, define

# Square Spiral Example Joy Code

Here is the example of Joy code from the `README` file:

    [[[abs]ii <=][[<>][pop !-]||]&&][[!-][[++]][[--]]ifte dip][[pop !-][--][++]ifte]ifte

It might seem unreadable but with a little familiarity it becomes just as legible as any other notation. Some layout helps:

    [   [[abs] ii <=]
        [
            [<>] [pop !-] ||
        ] &&
    ]
    [[    !-] [[++]] [[--]] ifte dip]
    [[pop !-]  [--]   [++]  ifte    ]
    ifte

This function accepts two integers on the stack and increments or decrements one of them such that the new pair of numbers is the next coordinate pair in a square spiral (like the kind used to construct an Ulam Spiral).

## Original Form

It’s adapted from the original code on StackOverflow:

> If all you’re trying to do is generate the first N points in the spiral (without the original problem’s constraint of masking to an N x M region), the code becomes very simple:

    void spiral(const int N)
    {
        int x = 0;
        int y = 0;
        for(int i = 0; i < N; ++i)
        {
            cout << x << '\t' << y << '\n';
            if(abs(x) <= abs(y) && (x != y || x >= 0))
                x += ((y >= 0) ? 1 : -1);
            else
                y += ((x >= 0) ? -1 : 1);
        }
    }

## Translation to Joy

I’m going to make a function that take two ints (`x` and `y`) and generates the next pair, we’ll turn it into a generator later using the `x` combinator.

### First Boolean Predicate

We need a function that computes `abs(x) <= abs(y)`, we can use `ii` to apply `abs` to both values and then compare them with `<=`:

    [abs] ii <=

I’ve defined two short-circuiting Boolean combinators `&&` and `||` that each accept two quoted predicate programs, run the first, and conditionally run the second only if required (to compute the final Boolean value). They run their predicate arguments `nullary`.

    define('&& [nullary] cons [nullary [0]] dip branch')
    define('|| [nullary] cons [nullary] dip [1] branch')

Given those, we can define `x != y || x >= 0` as:

    [<>] [pop 0 >=] ||

And `(abs(x) <= abs(y) && (x != y || x >= 0))` as:

    [[abs] ii <=] [[<>] [pop 0 >=] ||] &&

It’s a little rough, but, as I say, with a little familiarity it becomes legible.

### The Increment / Decrement Branches

Turning to the branches of the main `if` statement:

    x += ((y >= 0) ? 1 : -1);

Rewrite as a hybrid (pseudo-code) `ifte` expression:

    [y >= 0] [x += 1] [X -= 1] ifte

Change each C phrase to Joy code:

    [0 >=] [[++] dip] [[--] dip] ifte

Factor out the dip from each branch:

    [0 >=] [[++]] [[--]] ifte dip

Similar logic applies to the other branch:

    y += ((x >= 0) ? -1 : 1);

    [x >= 0] [y -= 1] [y += 1] ifte

    [pop 0 >=] [--] [++] ifte

### “Not Negative”

    define('!- 0 >=')

## Putting the Pieces Together

We can assemble the three functions we just defined in quotes and give them them to the `ifte` combinator. With some arrangement to show off the symmetry of the two branches, we have:

    [[[abs] ii <=] [[<>] [pop !-] ||] &&]
    [[    !-] [[++]] [[--]] ifte dip]
    [[pop !-]  [--]   [++]  ifte    ]
    ifte

As I was writing this up I realized that, since the `&&` combinator doesn’t consume the stack (below its quoted args), I can unquote the predicate, swap the branches, and use the `branch` combinator instead of `ifte`:

    [[abs] ii <=] [[<>] [pop !-] ||] &&
    [[pop !-]  [--]   [++]  ifte    ]
    [[    !-] [[++]] [[--]] ifte dip]
    branch

    define('spiral_next [[[abs] ii <=] [[<>] [pop !-] ||] &&] [[!-] [[++]] [[--]] ifte dip] [[pop !-] [--] [++] ifte] ifte')

Let’s try it out:

    J('0 0 spiral_next')

    1 0

    J('1 0 spiral_next')

    1 -1

    J('1 -1 spiral_next')

    0 -1

    J('0 -1 spiral_next')

    -1 -1

## Turning it into a Generator with `x`

It can be used with the x combinator to make a kind of generator for spiral square coordinates.

We can use `codireco` to make a generator

    codireco ::= cons dip rest cons

It will look like this:

    [value [F] codireco]

Here’s a trace of how it works:

               [0 [dup ++] codireco] . x
               [0 [dup ++] codireco] . 0 [dup ++] codireco
             [0 [dup ++] codireco] 0 . [dup ++] codireco
    [0 [dup ++] codireco] 0 [dup ++] . codireco
    [0 [dup ++] codireco] 0 [dup ++] . cons dip rest cons
    [0 [dup ++] codireco] [0 dup ++] . dip rest cons
                                     . 0 dup ++ [0 [dup ++] codireco] rest cons
                                   0 . dup ++ [0 [dup ++] codireco] rest cons
                                 0 0 . ++ [0 [dup ++] codireco] rest cons
                                 0 1 . [0 [dup ++] codireco] rest cons
           0 1 [0 [dup ++] codireco] . rest cons
             0 1 [[dup ++] codireco] . cons
             0 [1 [dup ++] codireco] .

But first we have to change the `spiral_next` function to work on a quoted pair of integers, and leave a copy of the pair on the stack. From:

       y x spiral_next
    ---------------------
            y' x'

to:

       [x y] [spiral_next] infra
    -------------------------------
               [x' y']

    J('[0 0] [spiral_next] infra')

    [0 1]

So our generator is:

    [[x y] [dup [spiral_next] infra] codireco]

Or rather:

    [[0 0] [dup [spiral_next] infra] codireco]

There is a function `make_generator` that will build the generator for us out of the value and stepper function:

       [0 0] [dup [spiral_next] infra] make_generator
    ----------------------------------------------------
         [[0 0] [dup [spiral_next] infra] codireco]

Here it is in action:

    J('[0 0] [dup [spiral_next] infra] make_generator x x x x pop')

    [0 0] [0 1] [-1 1] [-1 0]

Four `x` combinators, four pairs of coordinates.

## Conclusion

So that’s an example of Joy code. It’s a straightforward translation of the original. It’s a little long for a single definition, you might break it up like so:

         _spn_P ::= [[abs] ii <=] [[<>] [pop !-] ||] &&

         _spn_T ::= [    !-] [[++]] [[--]] ifte dip
         _spn_E ::= [pop !-]  [--]   [++]  ifte

    spiral_next ::= _spn_P [_spn_E] [_spn_T] branch

This way it’s easy to see that the function is a branch with two quasi-symmetrical paths.

We then used this function to make a simple generator of coordinate pairs, where the next pair in the series can be generated at any time by using the `x` combinator on the generator (which is just a quoted expression containing a copy of the current pair and the “stepper function” to generate the next pair from that.)

    define('_spn_P [[abs] ii <=] [[<>] [pop !-] ||] &&')
    define('_spn_T [!-] [[++]] [[--]] ifte dip')
    define('_spn_E [pop !-] [--] [++] ifte')
    define('spiral_next _spn_P [_spn_E] [_spn_T] branch')

    V('23 18 spiral_next')

                                                                   . 23 18 spiral_next
                                                                23 . 18 spiral_next
                                                             23 18 . spiral_next
                                                             23 18 . _spn_P [_spn_E] [_spn_T] branch
                                                             23 18 . [[abs] ii <=] [[<>] [pop !-] ||] && [_spn_E] [_spn_T] branch
                                               23 18 [[abs] ii <=] . [[<>] [pop !-] ||] && [_spn_E] [_spn_T] branch
                            23 18 [[abs] ii <=] [[<>] [pop !-] ||] . && [_spn_E] [_spn_T] branch
                            23 18 [[abs] ii <=] [[<>] [pop !-] ||] . [nullary] cons [nullary [0]] dip branch [_spn_E] [_spn_T] branch
                  23 18 [[abs] ii <=] [[<>] [pop !-] ||] [nullary] . cons [nullary [0]] dip branch [_spn_E] [_spn_T] branch
                  23 18 [[abs] ii <=] [[[<>] [pop !-] ||] nullary] . [nullary [0]] dip branch [_spn_E] [_spn_T] branch
    23 18 [[abs] ii <=] [[[<>] [pop !-] ||] nullary] [nullary [0]] . dip branch [_spn_E] [_spn_T] branch
                                               23 18 [[abs] ii <=] . nullary [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                               23 18 [[abs] ii <=] . [stack] dinfrirst [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                       23 18 [[abs] ii <=] [stack] . dinfrirst [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                       23 18 [[abs] ii <=] [stack] . dip infra first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                             23 18 . stack [[abs] ii <=] infra first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                     23 18 [18 23] . [[abs] ii <=] infra first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                       23 18 [18 23] [[abs] ii <=] . infra first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                             23 18 . [abs] ii <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                       23 18 [abs] . ii <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                       23 18 [abs] . [dip] dupdip i <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                 23 18 [abs] [dip] . dupdip i <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                       23 18 [abs] . dip [abs] i <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                                23 . abs 18 [abs] i <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                                23 . 18 [abs] i <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                             23 18 . [abs] i <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                       23 18 [abs] . i <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                             23 18 . abs <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                             23 18 . <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                             False . [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                     False [18 23] . swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                     23 18 [False] . first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                       23 18 False . [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                                                   23 18 False [0] . [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch
                      23 18 False [0] [[[<>] [pop !-] ||] nullary] . branch [_spn_E] [_spn_T] branch
                                                             23 18 . 0 [_spn_E] [_spn_T] branch
                                                           23 18 0 . [_spn_E] [_spn_T] branch
                                                  23 18 0 [_spn_E] . [_spn_T] branch
                                         23 18 0 [_spn_E] [_spn_T] . branch
                                                             23 18 . _spn_E
                                                             23 18 . [pop !-] [--] [++] ifte
                                                    23 18 [pop !-] . [--] [++] ifte
                                               23 18 [pop !-] [--] . [++] ifte
                                          23 18 [pop !-] [--] [++] . ifte
                                          23 18 [pop !-] [--] [++] . [nullary not] dipd branch
                            23 18 [pop !-] [--] [++] [nullary not] . dipd branch
                                                    23 18 [pop !-] . nullary not [--] [++] branch
                                                    23 18 [pop !-] . [stack] dinfrirst not [--] [++] branch
                                            23 18 [pop !-] [stack] . dinfrirst not [--] [++] branch
                                            23 18 [pop !-] [stack] . dip infra first not [--] [++] branch
                                                             23 18 . stack [pop !-] infra first not [--] [++] branch
                                                     23 18 [18 23] . [pop !-] infra first not [--] [++] branch
                                            23 18 [18 23] [pop !-] . infra first not [--] [++] branch
                                                             23 18 . pop !- [18 23] swaack first not [--] [++] branch
                                                                23 . !- [18 23] swaack first not [--] [++] branch
                                                                23 . 0 >= [18 23] swaack first not [--] [++] branch
                                                              23 0 . >= [18 23] swaack first not [--] [++] branch
                                                              True . [18 23] swaack first not [--] [++] branch
                                                      True [18 23] . swaack first not [--] [++] branch
                                                      23 18 [True] . first not [--] [++] branch
                                                        23 18 True . not [--] [++] branch
                                                       23 18 False . [--] [++] branch
                                                  23 18 False [--] . [++] branch
                                             23 18 False [--] [++] . branch
                                                             23 18 . --
                                                             23 17 .