Correct use of Solve inside Compile

Question

I have a function that behaves itself, but I want to make it much faster so I tried to put it into a compilable form After much fiddling I can't get it to work. Here's one of my failed attempts:

DiracPoint = Compile[{{angle, _Complex}},
   Module[{c1, c2, c3, Phi, cond1, cond2, cond3, sols, sol, theta, 
     phi},
    theta = Re[angle];
    phi = Im[angle];
    {c1, c2, c3} = 
     1 - 3 Sin[theta]^2 Cos[phi - 2 Pi (# - 1)/3]^2 & /@ {1, 2, 3};
    If[0 <= ((c2 + c3)^2 - c1^2)/(4 c2 c3) <= 1,
     Phi = Sqrt[((c2 + c3)^2 - c1^2)/(4 c2 c3)];
     cond1 = (c1 Cos[3 y/2] + (c2 + c3) Cos[Sqrt[3] x/2] == 0);
     cond2 = (-c1 Sin[3 y/2] + (-c2 + c3) Sin[Sqrt[3] x/2] == 0);
     cond3 = (Sin[Sqrt[3] x/2] == Phi);
     sols = Solve[cond1 && cond2 && cond3, {x, y}];
     {x, y} /. Refine[sols, {C[1] == 1, C[2] == 1}][[1]]
     , {$MaxMachineNumber, $MaxMachineNumber}
     ]
    ]
   ];

Why doesn't this work? I either get errors to do with incompatibility of rank sizes but I'm pretty confident the If statement is returning two objects of the same type and size. If I change {MaxM...,MaxM..} to just MaxM... I lose that error and get the error

"CompiledFunction::cfse: "Compiled expression y should be a !(\"machine-size real number\").""

If I then include x and y into my Module beginning declarations I get the error

Compile::initvar: The variable y has not been initialized or has been initialized to Null.

What is the proper way to initialise the variables x and y, or is my problem something else?

Here is how I apply it:

res = 4;
AngleArray = 
  Transpose[
   Table[theta + I phi, {theta, 0, Pi, Pi/res}, {phi, 0, 2 Pi, 
     Pi/res}]];
start = AbsoluteTime[];
array = Map[DiracPoint, AngleArray, {2}];
end = AbsoluteTime[] - start

Answer 1

In reply to @Wizard's comment. This is how I ended up solving/speeding up my code. I computed all the possible solutions to equation cond2 and cond3 and then picked whichever one first matched cond3. Probably an ugly way of doing it but seems to work.

DiracPoint[{theta_, phi_}] :=

  Module[{c1, c2, c3, Phi, x1, x2, y1a, y1b, y2a, y2b, Dx, Dy},
   {c1, c2, c3} = 
    1 - 3 Sin[theta]^2 Cos[phi - 2 Pi (# - 1)/3]^2 & /@ {1, 2, 3};
   {Dx, Dy} = If[0 <= ((c2 + c3)^2 - c1^2)/(4 c2 c3) <= 1,
     Phi = 
      Sqrt[((c2 + c3)^2 - c1^2)/(4 c2 c3)];(*Minus Phi gives K' point*)

          asx = ArcSin[Phi];
     asy1 = ArcSin[((c2 - c3)/c1) Sin[Sqrt[3] x1/2]];
     asy2 = ArcSin[((c2 - c3)/c1) Sin[Sqrt[3] x2/2]];
     x1 = (2/Sqrt[3]) asx;
     y1a = (2/3) asy1;
     y1b = (2/3) Sign[asy1] (Pi - Abs[asy1]);
     x2 = (2/Sqrt[3]) Sign[asx] (Pi - Abs[asx]);
     y2a = (2/3) asy2;
     y2b = (2/3) Sign[asy2] (Pi - Abs[asy2]);
     tol = 1*^-10;

     Which[
      Abs[(c2 + c3) Cos[Sqrt[3] x1/2] + c1 Cos[3 y1a/2]] < tol,
      {x1, y1a},
      Abs[(c2 + c3) Cos[Sqrt[3] x1/2] + c1 Cos[3 y1b/2]] < tol,
      {x1, y1b},
      Abs[(c2 + c3) Cos[Sqrt[3] x2/2] + c1 Cos[3 y2a/2]] < tol,
      {x2, y2a},
      Abs[(c2 + c3) Cos[Sqrt[3] x2/2] + c1 Cos[3 y2b/2]] < tol,
      {x2, y2b},
      1 == 1,
      {$MaxMachineNumber, $MaxMachineNumber}
      ]
     , {$MaxMachineNumber, $MaxMachineNumber}];
   {theta, phi, Dx, Dy}
   ];