How do I calculate the determinant of a 6x6 matrix in C++ really fast?

In C++ I need to calculate the determinant of a 6x6 matrix really fast. This is how I would do this for a 2x2 matrix: ``` double det2(double A[2][2]) { return A[0][0]*A[1][1] - A[0][1]*A[1][0]; } ``` I want a similar function for the determinant of a 6x6 matrix but I do not want to write it by hand since it contains 6! = 720 terms where each term is the product of 6 elements in the matrix. Therefore I want to use Leibniz formula: ``` static int perms6[720][6]; static int signs6[720]; double det6(double A[6][6]) { double sum = 0.0; for(int i = 0; i < 720; i++) { int j0 = perms6[i][0]; int j1 = perms6[i][1]; int j2 = perms6[i][2]; int j3 = perms6[i][3]; int j4 = perms6[i][4]; int j5 = perms6[i][5]; sum += signs6[i]*A[0]*A[j0]*A[1]*A[j1]*A[2]*A[j2]*A[3]*A[j3]*A[4]*A[j4]*A[5]*A[j5]; } return sum; } ``` How do I find the permutations and the signs? Is there some way I could get the compiler to do more of the work (e.g. C macros or template metaprogramming) so that the function would be even faster? EDIT: I just timed the following code (Eigen): ``` Matrix<double,6,6> A; // ... fill A for(long i = 0; i < 1e6; i++) { PartialPivLU< Matrix<double,6,6> > LU(A); double d = LU.determinant(); } ``` to 1.25 s. So using LU or Gauss decomposition is definitely fast enough for my use!