How to express a periodic piecewise linear function in terms of Mod function? or linear combination of Mod?

Question

I am programming a small game (with other programming language) in which I have to use Mod function to express piecewise linear period functions.

In my case, two hard parts I think exist.

(1) The periodic function is linear in the piecewise sence, like the function below. In a full period it is NOT linear.

(2) There are NO powerful function like Piecewise function in Mathematica, and all I can use are some very basic functions including Mod, +,-, *, \, Abs, Floor, Ceiling,Exp,10^,Sin,Cos,Tan,ASin,ACos, ATan,Log and logic control commands If, If Then Else.

(3) After many times trying, I also found another challenging part is related to Mod of a negative number .My piecewise linear periodic function can thake negative integers whose Mod is so confusing to me. I can calculate the Mod of a negative number by hand, but I have no clear idea how it relates to a repeat pattern.

So is it possible to use Mathematica to find something like linear combinations of Mod functions for any periodic piecewise linear function?

For example, the periodic piecewise linear period function like this:

f[x_?IntegerQ] := 
 Piecewise[{{-10 x + 2, 1 <= x < 4}, {-5 x + 8, 4 <= x <= 10}}, True]
f[x_?IntegerQ] := f[x + 10] /; x < 1
f[x_?IntegerQ] := f[x - 10] /; x > 10
ListPlot[Table[{x, f[x]}, {x, -9, 20}]]

Any advices are highly appreciated! Thanks :)

Answer 1

Here's a way to take advantage of Mathematica's code generation capabilities. In particular we take advantage of two internal workhorses, Simplify`PWToUnitStep and Experimental`OptimizeExpression. You should check to see if your library has a unit step function, and if so, you can just use the output of Simplify`PWToUnitStep, inserting x = Mod[y, 10] for input y, and avoid the rest of the trickery.

(* Auxiliary functions *)
ifArithmeticRules = {
   Times[z_, If[c_, a_, b_]] :> With[{i = If @@ {c, a*z, b*z}}, i /; True], 
   Plus[z_,  If[c_, a_, b_]] :> With[{i = If @@ {c, a+z, b+z}}, i /; True]};
compileToTempVars = 
  With[{c = #[[1, 1]]}, # /. 
     Thread[c -> Block[{nnn = 1}, Symbol["tmp" <> ToString[nnn++]] & /@ c]]
   ] &;

Simplify`PWToUnitStep[
      Piecewise[{{-10 x + 2, 1 <= x < 4}, {-5 x + 8, 4 <= x <= 10}}]
      ] // ReplaceAll[{
       x -> Mod[x, 10],             (* adjust if Mod[] != mod() *)
       UnitStep -> (Inactive[If][#>=0,1,0]&)  (* is translating UnitStep[] necessary? *)
       }] // 
    Experimental`OptimizeExpression //
   Activate //     (* If had to be Inactive[] for condition to be optimized *)
  compileToTempVars //              (* optional: change variable names *)
 ReplaceRepeated@ifArithmeticRules  (* not needed if you can use UnitStep[] *)

(*
  Experimental`OptimizedExpression[Block[{tmp1, tmp2, tmp3, tmp4},
    tmp1 = Mod[x, 10];
    tmp2 = -4 + tmp1;
    tmp3 = tmp2 >= 0;
    tmp4 = If[tmp3, 1, 0];
    If[10 - tmp1 >= 0,
     If[-1 + tmp1 >= 0,
      -2 (-1 + 5 tmp1) (1 - tmp4) + (8 - 5 tmp1) tmp4,
      (8 - 5 tmp1) tmp4],
     If[-1 + tmp1 >= 0, -2 (-1 + 5 tmp1) (1 - tmp4),
      0
      ]]]]
*)