```python 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](https://stackoverflow.com/questions/398299/looping-in-a-spiral/31864777#31864777): > 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`. ```python 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" ```python 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 ```python define('spiral_next [[[abs] ii <=] [[<>] [pop !-] ||] &&] [[!-] [[++]] [[--]] ifte dip] [[pop !-] [--] [++] ifte] ifte') ``` Let's try it out: ```python J('0 0 spiral_next') ``` 1 0 ```python J('1 0 spiral_next') ``` 1 -1 ```python J('1 -1 spiral_next') ``` 0 -1 ```python 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'] ```python 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: ```python 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.) ```python 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') ``` ```python 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 .