how to compile the function correctly?

Question

My calculations are running terribly slow (for a larger number of variables than in the given code). I can't compile the function correctly. Can anyone help me?

De[e_,h_,v_]:=((e h^3)/(12 (1-v^2)))
Alf[a_]:=((m Pi)/a)
Bet[b_]:=((n Pi)/b)
q[p1_,p2_,a_,b_,x0_,y0_]:=
 (
  ((4 p1)/(a b)) Sin[Alf[a] x0] Sin[Bet[b] y0]+
    ((4 p2)/(a b)) Sin[Alf[a] (x0+2.021)] Sin[Bet[b] (y0+0.065)]+
        ((4 p1)/(a b)) Sin[Alf[a] x0] Sin[Bet[b] (y0+1.050)]+
            ((4 p2)/(a b)) Sin[Alf[a] (x0+2.021)] Sin[Bet[b] (y0+0.985)]

  )
Delt[e_,h_,v_,a_,b_,k_]:=(De[e,h,v] (Alf[a]^2+Bet[b]^2)^2+k)
w[p1_,p2_,a_,b_,x0_,y0_,e_,h_,v_,k_]:=q[p1,p2,a,b,x0,y0]/Delt[e,h,v,a,b,k]
Mx[e_,h_,v_,a_,b_,p1_,p2_,k_,x0_,y0_,x_/;0<x<6,y_/;0<y<6]:=De[e,h,v]*Sum[Sum[((Alf[a]^2+v Bet[b]^2) w[p1,p2,a,b,x0,y0,e,h,v,k]) Sin[Alf[a] x] Sin[Bet[b] y],{n,1,20}],{m,1,20}]

k={4.846825833322111`*^6,5.238931521109236`*^6,5.077350158653542`*^6,4.536267905643828`*^6,4.375764694611544`*^6,4.564669230813377`*^6,5.839164643396563`*^6,5.2185564799462985`*^6,4.3317022672116915`*^6,4.595977212066057`*^6,4.394030477319662`*^6,4.662831240535733`*^6}

pairs={{5.200301369200046`,3.164827201205645`},{4.163004893697675`,1.7409758270465447`},{5.695791313032204`,1.0023134075279154`},{0.8082870663144348`,4.173725169069691`},{3.6129799469236428`,3.6510371020047483`},{5.050196700105754`,5.042860219651104`}}

Mx[27000000000,0.2,0.2,6,6,45597,5793,#1,Sequence@@#2,Sequence@@#2]&@@@Tuples[{k,pairs}]

Answer 1

As Henrik noted in the comments, your code is a low-key disaster, with m and n never being explicitly called, random parameters floating around but never passed into a function, tons of other parameters, etc. But whatever. Without much work it's possible to compile the specific case you present.

Here's a way to turn all your code into a single inert expression, then we can Compile that:

expr =
  Block[
   {Sum = sum},
   Mx[27000000000, 0.2, 0.2, 6, 6, 45597, 5793, #1, Sequence @@ #2, 
      Sequence @@ #2] & @@@ Tuples[{k, pairs}
     ]
   ];

Then we can compile this like so:

compVersion =
  With[{expr = expr},
   Hold[
      Compile[
       {},
       expr
       ]
      ] /. sum -> Sum // ReleaseHold
   ];

And this seems fast enough:

compVersion[] // RepeatedTiming

{1.03, {15086.2, 16534.2, 9884.55, 15243.3, 16329.8, 8927.37, 15028., 16385.9,
   9879.67, 15187.4, 16150.7, 8912.45, 15051.7, 16446., 9881.67, 15210.2, 
  16223.1, 8918.54, 15134.1, 16657.8, 9888.47, 15289.1, 16480.1, 8939.51, 
  15159.4, 16723.9, 9890.51, 15313.4, 16560.9, 8945.91, 15129.6, 16646.2, 
  9888.1, 15284.9, 16466., 8938.39, 14943.3, 16173.7, 9872.37, 15105.9, 
  15897., 8890.41, 15031., 16393.4, 9879.92, 15190.3, 16159.7, 8913.21, 
  15166.5, 16742.4, 9891.08, 15320.2, 16583.5, 8947.68, 15124.7, 16633.6, 
  9887.71, 15280.2, 16450.6, 8937.15, 15156.5, 16716.3, 9890.28, 15310.7, 
  16551.6, 8945.18, 15114.4, 16606.7, 9886.86, 15270.3, 16417.9, 8934.53}}