matrix - Faster trace of product of two matrices

I noticed that for the computation of the trace of a product of two matrices, using Tr[Dot[A,B]] is a little inefficient. Dot is computing all the elements of the matrix product, while Tr only needs the diagonal.

Is there a low-level, or fast implementation of trace-dot in Mathematica? (It needs to be able to work on matrices of mixed datatypes)

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Look, I made a top-level implementation of trace-dot that is faster than Trace[Dot[...]]:

myTrDot[m1_,m2_]:=Total[MapThread[Dot, {m1, Transpose[m2]}]];

exMat1 = RandomVariate[GaussianOrthogonalMatrixDistribution[1000]];

exMat2 = RandomVariate[GaussianOrthogonalMatrixDistribution[1000]];

Tr[Dot[exMat1, exMat2]]; // AbsoluteTiming
(* 0.020229 *)

myTrDot[exMat1, exMat2]; // AbsoluteTiming
(* 0.015503 *)

Answer

myTrDot2 = 
 Compile[{{m1, _Real, 2}, {m2, _Real, 2}}, 

  Flatten[m1].Flatten[Transpose[m2]]]

exMat1 = RandomVariate[GaussianOrthogonalMatrixDistribution[10000]];

exMat2 = RandomVariate[GaussianOrthogonalMatrixDistribution[10000]];

Tr[Dot[exMat1, exMat2]]; // AbsoluteTiming

myTrDot[exMat1, exMat2]; // AbsoluteTiming


myTrDot2[exMat1, exMat2]; // AbsoluteTiming

{18.0692, Null}

{1.64879, Null}

{1.42026, Null}