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 <= .. code:: Joy [_p [abs] ii <=] inscribe .. parsed-literal:: .. code:: Joy clear 23 -18 .. parsed-literal:: 23 -18 .. code:: Joy [_p] trace .. parsed-literal:: 23 -18 • _p 23 -18 • [abs] ii <= 23 -18 [abs] • ii <= 23 • abs -18 abs <= 23 • -18 abs <= 23 -18 • abs <= 23 18 • <= false • false .. code:: Joy clear .. parsed-literal:: Short-Circuiting Boolean Combinators ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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``. .. code:: Joy [&& [nullary] cons [nullary [false]] dip branch] inscribe [|| [nullary] cons [nullary] dip [true] branch] inscribe .. parsed-literal:: .. code:: Joy clear [true] [false] && .. parsed-literal:: false .. code:: Joy clear [false] [true] && .. parsed-literal:: false .. code:: Joy clear [true] [false] || .. parsed-literal:: true .. code:: Joy clear [false] [true] || .. parsed-literal:: true .. code:: Joy clear .. parsed-literal:: Translating the Conditionals ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Given those, we can define ``x != y || x >= 0`` as: :: _a == [!=] [pop 0 >=] || .. code:: Joy [_a [!=] [pop 0 >=] ||] inscribe .. parsed-literal:: And ``(abs(x) <= abs(y) && (x != y || x >= 0))`` as: :: _b == [_p] [_a] && .. code:: Joy [_b [_p] [_a] &&] inscribe .. parsed-literal:: It’s a little rough, but, as I say, with a little familiarity it becomes legible. .. code:: Joy clear 23 -18 .. parsed-literal:: 23 -18 .. code:: Joy [_b] trace .. parsed-literal:: 23 -18 • _b 23 -18 • [_p] [_a] && 23 -18 [_p] • [_a] && 23 -18 [_p] [_a] • && 23 -18 [_p] [_a] • [nullary] cons [nullary [false]] dip branch 23 -18 [_p] [_a] [nullary] • cons [nullary [false]] dip branch 23 -18 [_p] [[_a] nullary] • [nullary [false]] dip branch 23 -18 [_p] [[_a] nullary] [nullary [false]] • dip branch 23 -18 [_p] • nullary [false] [[_a] nullary] branch 23 -18 [_p] • [stack] dinfrirst [false] [[_a] nullary] branch 23 -18 [_p] [stack] • dinfrirst [false] [[_a] nullary] branch 23 -18 [_p] [stack] • dip infrst [false] [[_a] nullary] branch 23 -18 • stack [_p] infrst [false] [[_a] nullary] branch 23 -18 [-18 23] • [_p] infrst [false] [[_a] nullary] branch 23 -18 [-18 23] [_p] • infrst [false] [[_a] nullary] branch 23 -18 [-18 23] [_p] • infra first [false] [[_a] nullary] branch 23 -18 • _p [-18 23] swaack first [false] [[_a] nullary] branch 23 -18 • [abs] ii <= [-18 23] swaack first [false] [[_a] nullary] branch 23 -18 [abs] • ii <= [-18 23] swaack first [false] [[_a] nullary] branch 23 • abs -18 abs <= [-18 23] swaack first [false] [[_a] nullary] branch 23 • -18 abs <= [-18 23] swaack first [false] [[_a] nullary] branch 23 -18 • abs <= [-18 23] swaack first [false] [[_a] nullary] branch 23 18 • <= [-18 23] swaack first [false] [[_a] nullary] branch false • [-18 23] swaack first [false] [[_a] nullary] branch false [-18 23] • swaack first [false] [[_a] nullary] branch 23 -18 [false] • first [false] [[_a] nullary] branch 23 -18 false • [false] [[_a] nullary] branch 23 -18 false [false] • [[_a] nullary] branch 23 -18 false [false] [[_a] nullary] • branch 23 -18 • false 23 -18 false • 23 -18 false .. code:: Joy clear .. parsed-literal:: 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 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 .. code:: Joy [spiral_next [_b] [[ !-] [[++]] [[--]] ifte dip] [[pop !-] [--] [++] ifte ] ifte ] inscribe .. parsed-literal:: 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 Let’s try it out: .. code:: Joy clear 0 0 .. parsed-literal:: 0 0 .. code:: Joy spiral_next .. parsed-literal:: 1 0 .. code:: Joy spiral_next .. parsed-literal:: 1 -1 .. code:: Joy spiral_next .. parsed-literal:: 0 -1 .. code:: Joy spiral_next .. parsed-literal:: -1 -1 .. code:: Joy spiral_next .. parsed-literal:: -1 0 .. code:: Joy spiral_next .. parsed-literal:: -1 1 .. code:: Joy spiral_next .. parsed-literal:: 0 1 .. code:: Joy spiral_next .. parsed-literal:: 1 1 .. code:: Joy spiral_next .. parsed-literal:: 2 1 .. code:: Joy spiral_next .. parsed-literal:: 2 0 .. code:: Joy spiral_next .. parsed-literal:: 2 -1 .. code:: Joy spiral_next .. parsed-literal:: 2 -2 .. code:: Joy spiral_next .. parsed-literal:: 1 -2 .. code:: Joy spiral_next .. parsed-literal:: 0 -2 .. code:: Joy spiral_next .. parsed-literal:: -1 -2 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: .. code:: Joy clear [0 [dup ++] codireco] [x] trace .. parsed-literal:: [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 ++] • codi reco [0 [dup ++] codireco] 0 [dup ++] • cons dip reco [0 [dup ++] codireco] [0 dup ++] • dip reco • 0 dup ++ [0 [dup ++] codireco] reco 0 • dup ++ [0 [dup ++] codireco] reco 0 0 • ++ [0 [dup ++] codireco] reco 0 1 • [0 [dup ++] codireco] reco 0 1 [0 [dup ++] codireco] • reco 0 1 [0 [dup ++] codireco] • rest cons 0 1 [[dup ++] codireco] • cons 0 [1 [dup ++] codireco] • 0 [1 [dup ++] codireco] .. code:: Joy clear .. parsed-literal:: 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'] .. code:: Joy [0 0] [spiral_next] infra .. parsed-literal:: [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] .. code:: Joy clear .. parsed-literal:: Here it is in action: .. code:: Joy [0 0] [dup [spiral_next] infra] make_generator x x x x pop .. parsed-literal:: [0 0] [0 1] [-1 1] [-1 0] Four ``x`` combinators, four pairs of coordinates. Or you can leave out ``dup`` and let the value stay in the generator until you want it: .. code:: Joy clear [0 0] [[spiral_next] infra] make_generator 50 [x] times first .. parsed-literal:: [2 4] 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_Pa == [abs] ii <= _spn_Pb == [!=] [pop 0 >=] || _spn_P == [_spn_Pa] [_spn_Pb] && _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.)