graphics - How to compile Heike's winding number function?

Heike gave the following function for winding number:

winding[poly_, pt_] :=  
Round[(Total@Mod[(# - RotateRight[#]) &@(ArcTan @@ (pt - #) & /@ poly), 
   2 Pi, -Pi]/2./Pi)]

I attempted to compile it as follows:

winding2 := Compile[{{poly, _Real, 2}, {pt, _Real, 1}},   
Round[(Total@Mod[(# - RotateRight[#]) &@(ArcTan @@ (pt - #) & /@ poly), 

    2 Pi, -Pi]/2./Pi)]]

Applied to the following simple problem, the compiled version gives error messages:

poly = {{0., 0.}, {10., 0.}, {10., 6.}, {0, 6}, {0., 0.}};
pt = {5., 3.}; 
winding2[poly, pt]

The error messages include:

Compile::cpapot: Compilation of ArcTan@@(ptCompile`GetElement[poly,System`Private`CompileSymbol[0]]) 
is not supported for the function argument ArcTan. The only function arguments supported are 

Times, Plus, or List. Evaluation will use the uncompiled function. >>
CompiledFunction::cfse: Compiled expression 6.283185307179586` should be a machine-size integer. >>
CompiledFunction::cfex: Could not complete external evaluation at instruction 1;
proceeding with uncompiled evaluation. >>

Where am I going wrong?

Answer

First, if you use := in your assignment, then the compilation is not done instantly but every time you call winding2. That's btw the reason why you get the error message when you try to call the function because it is not compiled until then and the error is a compilation error.

Secondly, as the error messages sais, @@ can only be used with Times, Plus or List, so you have to expand this part:

winding2 = 

 Compile[{{poly, _Real, 2}, {pt, _Real, 1}}, 
  Round[Total@Mod[# - RotateRight[#] &@(ArcTan[#[[1]], #[[2]]] &@
    (Transpose@poly - pt)), 2 Pi, -Pi]/(2. Pi)]
 ]

Seems to work pretty smoothly

And here the code:

winding2 = 
  Compile[{{poly, _Real, 2}, {pt, _Real, 1}}, 

   0 != Round[
     Total@Mod[# - 
           RotateRight[#] &@(ArcTan[#[[1]], #[[2]]] &@(Transpose@
              poly - pt)), 2 Pi, -Pi]/(2. Pi)], 
   CompilationTarget -> "C", RuntimeOptions -> "Speed"];

With[{gr = 
   RegionPlot[x^2 + y^3 < 2, {x, -2, 2}, {y, -2, 2}, Mesh -> All, 
    FrameTicks -> None, PlotPoints -> 3, MaxRecursion -> 4, 
    PlotStyle -> RGBColor[14/15, 232/255, 71/85], 

    MeshStyle -> RGBColor[88/255, 22/51, 39/85]]}, 
 DynamicModule[{pt = {0, 1}, polyPts}, 
  polyPts = 
   gr[[1, 1, #]] & /@ 
    Flatten[Cases[gr, Polygon[pts__] :> Sequence[pts], Infinity], 1];
  LocatorPane[Dynamic[pt], 
   Dynamic@Show[gr, 
     Graphics[{Opacity[.5], RGBColor[38/255, 139/255, 14/17], 
       Polygon[Pick[polyPts, winding2[#, pt] & /@ polyPts]]}]]]]]