#ifndef INVERSE_H #define INVERSE_H #include "Fastor/config/config.h" #include "Fastor/meta/meta.h" #include "Fastor/simd_vector/extintrin.h" namespace Fastor { template, bool> = false> FASTOR_INLINE void _inverse(const T *FASTOR_RESTRICT src, T *FASTOR_RESTRICT dst); template, bool> = false> FASTOR_INLINE void _inverse(const T *FASTOR_RESTRICT src, T *FASTOR_RESTRICT dst) { *dst = T(1) / (*src); } #ifdef FASTOR_SSE2_IMPL template && !is_same_v_ && !is_same_v_, bool> = false> #else template, bool> = false> #endif FASTOR_INLINE void _inverse(const T *FASTOR_RESTRICT src, T *FASTOR_RESTRICT dst) { T det; T src0 = src[0]; T src1 = src[1]; T src2 = src[2]; T src3 = src[3]; /* Compute adjoint: */ dst[0] = + src3; dst[1] = - src1; dst[2] = - src2; dst[3] = + src0; /* Compute determinant: */ det = src0 * dst[0] + src1 * dst[2]; /* Multiply adjoint with reciprocal of determinant: */ det = T(1.0) / det; dst[0] *= det; dst[1] *= det; dst[2] *= det; dst[3] *= det; } #ifdef FASTOR_SSE2_IMPL template && is_same_v_, bool> = false> FASTOR_INLINE void _inverse(const T *FASTOR_RESTRICT src, T *FASTOR_RESTRICT dst) { // This is much superior to the scalar code as // gcc/clang can't auto-vectorise the scalar code // 6 shuffles + 1 add + 1 mul + 1 div // Sky 6 + 4 + 4 + 11 = 25 __m128 mat = _mm_loadu_ps(src); // xor to swap off-diagonals sings __m128 nmat = _mm_neg_ps(mat); // two shuffles to get adjoint __m128 adj = _mm_shuffle_ps(mat, nmat, 0x009C ); adj = _mm_shuffle_ps(adj, adj , 0x39 ); // compute determinat __m128 tmp0 = _mm_shuffle_ps(mat , mat , 0x00D8); tmp0 = _mm_mul_ps (adj , tmp0 ); __m128 tmp1 = _mm_shuffle_ps(tmp0, tmp0, 0x1 ); __m128 det = _mm_div_ss (ONEPS, _mm_add_ss(tmp0,tmp1)); // broadcast det to all elements of __m128 det = _mm_shuffle_ps(det, det, 0x0 ); // divide adjoint by determinant __m128 inv = _mm_mul_ps (adj, det); _mm_storeu_ps(dst, inv); } template && is_same_v_, bool> = false> FASTOR_INLINE void _inverse(const T *FASTOR_RESTRICT src, T *FASTOR_RESTRICT dst) { // This is much superior to the scalar code as // gcc/clang can't auto-vectorise the scalar code // 8 shuffles + 1 add + 3 mul + 1 div // Sky 8 + 4 + 12 + 14 = 38 __m128d row0 = _mm_loadu_pd(src); __m128d row1 = _mm_loadu_pd(src+2); __m128d tmp = row0; row0 = _mm_shuffle_pd(row0,_mm_neg_pd(row0),0x2); row1 = _mm_shuffle_pd(_mm_neg_pd(row1),row1,0x2); // these two registers hold the adjoint __m128d irow0 = _mm_shuffle_pd(row1,row0,0x3); __m128d irow1 = _mm_shuffle_pd(row1,row0,0x0); // dot product to compute determinant __m128d det = _mm_mul_pd(tmp,_mm_reverse_pd(row1)); det = _mm_add_pd(det,_mm_reverse_pd(det)); // one by determinant __m128d invdet = _mm_div_pd(_mm_set1_pd(1.0),det); // scale irow0 = _mm_mul_pd(irow0,invdet); irow1 = _mm_mul_pd(irow1,invdet); _mm_storeu_pd(dst ,irow0); _mm_storeu_pd(dst+2,irow1); } #endif template, bool> = false> FASTOR_INLINE void _inverse(const T *FASTOR_RESTRICT src, T *FASTOR_RESTRICT dst) { T det; T src0 = src[0]; T src1 = src[1]; T src2 = src[2]; T src3 = src[3]; T src4 = src[4]; T src5 = src[5]; T src6 = src[6]; T src7 = src[7]; T src8 = src[8]; /* Compute adjoint: */ dst[0] = + src4 * src8 - src5 * src7; dst[1] = - src1 * src8 + src2 * src7; dst[2] = + src1 * src5 - src2 * src4; dst[3] = - src3 * src8 + src5 * src6; dst[4] = + src0 * src8 - src2 * src6; dst[5] = - src0 * src5 + src2 * src3; dst[6] = + src3 * src7 - src4 * src6; dst[7] = - src0 * src7 + src1 * src6; dst[8] = + src0 * src4 - src1 * src3; /* Compute determinant: */ det = src0 * dst[0] + src1 * dst[3] + src2 * dst[6]; /* Multiply adjoint with reciprocal of determinant: */ det = T(1.0) / det; dst[0] *= det; dst[1] *= det; dst[2] *= det; dst[3] *= det; dst[4] *= det; dst[5] *= det; dst[6] *= det; dst[7] *= det; dst[8] *= det; } #ifdef FASTOR_SSE2_IMPL template && !is_same_v_ && !is_same_v_, bool> = false> #else template, bool> = false> #endif FASTOR_INLINE void _inverse(const T *FASTOR_RESTRICT src, T *FASTOR_RESTRICT dst) { T t1 = src[2*4+2]*src[3*4+3] - src[2*4+3]*src[3*4+2]; T t2 = src[2*4+1]*src[3*4+3] - src[2*4+3]*src[3*4+1]; T t3 = src[2*4+1]*src[3*4+2] - src[2*4+2]*src[3*4+1]; dst[0] = src[1*4+1]*t1 - src[1*4+2]*t2 + src[1*4+3]*t3; dst[1] = src[0*4+2]*t2 - src[0*4+1]*t1 - src[0*4+3]*t3; T t4 = src[2*4+0]*src[3*4+3] - src[2*4+3]*src[3*4+0]; T t5 = src[2*4+0]*src[3*4+2] - src[2*4+2]*src[3*4+0]; dst[4] = src[1*4+2]*t4 - src[1*4+0]*t1 - src[1*4+3]*t5; dst[5] = src[0*4+0]*t1 - src[0*4+2]*t4 + src[0*4+3]*t5; t1 = src[2*4+0]*src[3*4+1] - src[2*4+1]*src[3*4+0]; dst[8] = src[1*4+0]*t2 - src[1*4+1]*t4 + src[1*4+3]*t1; dst[9] = src[0*4+1]*t4 - src[0*4+0]*t2 - src[0*4+3]*t1; dst[12] = src[1*4+1]*t5 - src[1*4+0]*t3 - src[1*4+2]*t1; dst[13] = src[0*4+0]*t3 - src[0*4+1]*t5 + src[0*4+2]*t1; t1 = src[0*4+2]*src[1*4+3] - src[0*4+3]*src[1*4+2]; t2 = src[0*4+1]*src[1*4+3] - src[0*4+3]*src[1*4+1]; t3 = src[0*4+1]*src[1*4+2] - src[0*4+2]*src[1*4+1]; dst[2] = src[3*4+1]*t1 - src[3*4+2]*t2 + src[3*4+3]*t3; dst[3] = src[2*4+2]*t2 - src[2*4+1]*t1 - src[2*4+3]*t3; t4 = src[0*4+0]*src[1*4+3] - src[0*4+3]*src[1*4+0]; t5 = src[0*4+0]*src[1*4+2] - src[0*4+2]*src[1*4+0]; dst[6] = src[3*4+2]*t4 - src[3*4+0]*t1 - src[3*4+3]*t5; dst[7] = src[2*4+0]*t1 - src[2*4+2]*t4 + src[2*4+3]*t5; t1 = src[0*4+0]*src[1*4+1] - src[0*4+1]*src[1*4+0]; dst[10] = src[3*4+0]*t2 - src[3*4+1]*t4 + src[3*4+3]*t1; dst[11] = src[2*4+1]*t4 - src[2*4+0]*t2 - src[2*4+3]*t1; dst[14] = src[3*4+1]*t5 - src[3*4+0]*t3 - src[3*4+2]*t1; dst[15] = src[2*4+0]*t3 - src[2*4+1]*t5 + src[2*4+2]*t1; const T __det = src[0]*dst[0] + src[1]*dst[4] + src[2]*dst[8] + src[3]*dst[12]; const T __invdet = T(1)/__det; for (int i=0; i<16; ++i) dst[i] *= __invdet; } #ifdef FASTOR_SSE2_IMPL template && is_same_v_, bool> = false> FASTOR_INLINE void _inverse(const T *FASTOR_RESTRICT src, T *FASTOR_RESTRICT dst) { // From Intel's SSE matrix library // The inverse is calculated using "Divide and Conquer" technique. The // original matrix is divide into four 2x2 sub-matrices. Since each // register of the matrix holds two elements, the smaller matrices are // consisted of two registers. Hence we get a better locality of the // calculations. const __m128 p4f_sign_PNNP = _mm_castsi128_ps(_mm_set_epi32(0x00000000, 0x80000000, 0x80000000, 0x00000000)); // Load the full matrix into registers __m128 _L1 = _mm_loadu_ps(src + 0); __m128 _L2 = _mm_loadu_ps(src + 4); __m128 _L3 = _mm_loadu_ps(src + 8); __m128 _L4 = _mm_loadu_ps(src + 12); __m128 A, B, C, D; // the four sub-matrices A = _mm_movelh_ps(_L1, _L2); B = _mm_movehl_ps(_L2, _L1); C = _mm_movelh_ps(_L3, _L4); D = _mm_movehl_ps(_L4, _L3); // partial inverse of the sub-matrices __m128 iA, iB, iC, iD, DC, AB; __m128 dA, dB, dC, dD; // determinant of the sub-matrices __m128 det, d, d1, d2; __m128 rd; // reciprocal of the determinant // AB = A# * B AB = _mm_mul_ps(_mm_shuffle_ps(A,A,0x0F), B); AB = _mm_sub_ps(AB,_mm_mul_ps(_mm_shuffle_ps(A,A,0xA5), _mm_shuffle_ps(B,B,0x4E))); // DC = D# * C DC = _mm_mul_ps(_mm_shuffle_ps(D,D,0x0F), C); DC = _mm_sub_ps(DC,_mm_mul_ps(_mm_shuffle_ps(D,D,0xA5), _mm_shuffle_ps(C,C,0x4E))); // dA = |A| dA = _mm_mul_ps(_mm_shuffle_ps(A, A, 0x5F),A); dA = _mm_sub_ss(dA, _mm_movehl_ps(dA,dA)); // dB = |B| dB = _mm_mul_ps(_mm_shuffle_ps(B, B, 0x5F),B); dB = _mm_sub_ss(dB, _mm_movehl_ps(dB,dB)); // dC = |C| dC = _mm_mul_ps(_mm_shuffle_ps(C, C, 0x5F),C); dC = _mm_sub_ss(dC, _mm_movehl_ps(dC,dC)); // dD = |D| dD = _mm_mul_ps(_mm_shuffle_ps(D, D, 0x5F),D); dD = _mm_sub_ss(dD, _mm_movehl_ps(dD,dD)); // d = trace(AB*DC) = trace(A#*B*D#*C) d = _mm_mul_ps(_mm_shuffle_ps(DC,DC,0xD8),AB); // iD = C*A#*B iD = _mm_mul_ps(_mm_shuffle_ps(C,C,0xA0), _mm_movelh_ps(AB,AB)); iD = _mm_add_ps(iD,_mm_mul_ps(_mm_shuffle_ps(C,C,0xF5), _mm_movehl_ps(AB,AB))); // iA = B*D#*C iA = _mm_mul_ps(_mm_shuffle_ps(B,B,0xA0), _mm_movelh_ps(DC,DC)); iA = _mm_add_ps(iA,_mm_mul_ps(_mm_shuffle_ps(B,B,0xF5), _mm_movehl_ps(DC,DC))); // d = trace(AB*DC) = trace(A#*B*D#*C) [continue] d = _mm_add_ps(d, _mm_movehl_ps(d, d)); d = _mm_add_ss(d, _mm_shuffle_ps(d, d, 1)); d1 = _mm_mul_ss(dA,dD); d2 = _mm_mul_ss(dB,dC); // iD = D*|A| - C*A#*B iD = _mm_sub_ps(_mm_mul_ps(D,_mm_shuffle_ps(dA,dA,0)), iD); // iA = A*|D| - B*D#*C; iA = _mm_sub_ps(_mm_mul_ps(A,_mm_shuffle_ps(dD,dD,0)), iA); // det = |A|*|D| + |B|*|C| - trace(A#*B*D#*C) det = _mm_sub_ss(_mm_add_ss(d1,d2),d); rd = _mm_div_ss(_mm_set_ss(1.0f), det); // iB = D * (A#B)# = D*B#*A iB = _mm_mul_ps(D, _mm_shuffle_ps(AB,AB,0x33)); iB = _mm_sub_ps(iB, _mm_mul_ps(_mm_shuffle_ps(D,D,0xB1), _mm_shuffle_ps(AB,AB,0x66))); // iC = A * (D#C)# = A*C#*D iC = _mm_mul_ps(A, _mm_shuffle_ps(DC,DC,0x33)); iC = _mm_sub_ps(iC, _mm_mul_ps(_mm_shuffle_ps(A,A,0xB1), _mm_shuffle_ps(DC,DC,0x66))); rd = _mm_shuffle_ps(rd,rd,0); rd = _mm_xor_ps(rd, p4f_sign_PNNP); // iB = C*|B| - D*B#*A iB = _mm_sub_ps(_mm_mul_ps(C,_mm_shuffle_ps(dB,dB,0)), iB); // iC = B*|C| - A*C#*D; iC = _mm_sub_ps(_mm_mul_ps(B,_mm_shuffle_ps(dC,dC,0)), iC); // iX = iX / det iA = _mm_mul_ps(rd,iA); iB = _mm_mul_ps(rd,iB); iC = _mm_mul_ps(rd,iC); iD = _mm_mul_ps(rd,iD); _mm_storeu_ps(dst+0, _mm_shuffle_ps(iA,iB,0x77)); _mm_storeu_ps(dst+4, _mm_shuffle_ps(iA,iB,0x22)); _mm_storeu_ps(dst+8, _mm_shuffle_ps(iC,iD,0x77)); _mm_storeu_ps(dst+12, _mm_shuffle_ps(iC,iD,0x22)); } template && is_same_v_, bool> = false> FASTOR_INLINE void _inverse(const T *FASTOR_RESTRICT src, T *FASTOR_RESTRICT dst) { // From Intel's SSE matrix library // The inverse is calculated using "Divide and Conquer" technique. The // original matrix is divide into four 2x2 sub-matrices. Since each // register of the matrix holds two elements, the smaller matrices are // consisted of two registers. Hence we get a better locality of the // calculations. const __m128d _Sign_NP = _mm_castsi128_pd(_mm_set_epi32(0x0,0x0,0x80000000,0x0)); const __m128d _Sign_PN = _mm_castsi128_pd(_mm_set_epi32(0x80000000,0x0,0x0,0x0)); // the four sub-matrices __m128d A1, A2, B1, B2, C1, C2, D1, D2; A1 = _mm_loadu_pd(src + 0); B1 = _mm_loadu_pd(src + 2); A2 = _mm_loadu_pd(src + 4); B2 = _mm_loadu_pd(src + 6); C1 = _mm_loadu_pd(src + 8); D1 = _mm_loadu_pd(src +10); C2 = _mm_loadu_pd(src +12); D2 = _mm_loadu_pd(src +14); // partial inverse of the sub-matrices __m128d iA1, iA2, iB1, iB2, iC1, iC2, iD1, iD2, DC1, DC2, AB1, AB2; __m128d dA, dB, dC, dD; // determinant of the sub-matrices __m128d det, d1, d2, rd; // dA = |A| dA = _mm_shuffle_pd(A2, A2, 1); dA = _mm_mul_pd(A1, dA); dA = _mm_sub_sd(dA, _mm_shuffle_pd(dA,dA,3)); // dB = |B| dB = _mm_shuffle_pd(B2, B2, 1); dB = _mm_mul_pd(B1, dB); dB = _mm_sub_sd(dB, _mm_shuffle_pd(dB,dB,3)); // AB = A# * B AB1 = _mm_mul_pd(B1, _mm_shuffle_pd(A2,A2,3)); AB2 = _mm_mul_pd(B2, _mm_shuffle_pd(A1,A1,0)); AB1 = _mm_sub_pd(AB1, _mm_mul_pd(B2, _mm_shuffle_pd(A1,A1,3))); AB2 = _mm_sub_pd(AB2, _mm_mul_pd(B1, _mm_shuffle_pd(A2,A2,0))); // dC = |C| dC = _mm_shuffle_pd(C2, C2, 1); dC = _mm_mul_pd(C1, dC); dC = _mm_sub_sd(dC, _mm_shuffle_pd(dC,dC,3)); // dD = |D| dD = _mm_shuffle_pd(D2, D2, 1); dD = _mm_mul_pd(D1, dD); dD = _mm_sub_sd(dD, _mm_shuffle_pd(dD,dD,3)); // DC = D# * C DC1 = _mm_mul_pd(C1, _mm_shuffle_pd(D2,D2,3)); DC2 = _mm_mul_pd(C2, _mm_shuffle_pd(D1,D1,0)); DC1 = _mm_sub_pd(DC1, _mm_mul_pd(C2, _mm_shuffle_pd(D1,D1,3))); DC2 = _mm_sub_pd(DC2, _mm_mul_pd(C1, _mm_shuffle_pd(D2,D2,0))); // rd = trace(AB*DC) = trace(A#*B*D#*C) d1 = _mm_mul_pd(AB1, _mm_shuffle_pd(DC1, DC2, 0)); d2 = _mm_mul_pd(AB2, _mm_shuffle_pd(DC1, DC2, 3)); rd = _mm_add_pd(d1, d2); rd = _mm_add_sd(rd, _mm_shuffle_pd(rd, rd,3)); // iD = C*A#*B iD1 = _mm_mul_pd(AB1, _mm_shuffle_pd(C1,C1,0)); iD2 = _mm_mul_pd(AB1, _mm_shuffle_pd(C2,C2,0)); iD1 = _mm_add_pd(iD1, _mm_mul_pd(AB2, _mm_shuffle_pd(C1,C1,3))); iD2 = _mm_add_pd(iD2, _mm_mul_pd(AB2, _mm_shuffle_pd(C2,C2,3))); // iA = B*D#*C iA1 = _mm_mul_pd(DC1, _mm_shuffle_pd(B1,B1,0)); iA2 = _mm_mul_pd(DC1, _mm_shuffle_pd(B2,B2,0)); iA1 = _mm_add_pd(iA1, _mm_mul_pd(DC2, _mm_shuffle_pd(B1,B1,3))); iA2 = _mm_add_pd(iA2, _mm_mul_pd(DC2, _mm_shuffle_pd(B2,B2,3))); // iD = D*|A| - C*A#*B dA = _mm_shuffle_pd(dA,dA,0); iD1 = _mm_sub_pd(_mm_mul_pd(D1, dA), iD1); iD2 = _mm_sub_pd(_mm_mul_pd(D2, dA), iD2); // iA = A*|D| - B*D#*C; dD = _mm_shuffle_pd(dD,dD,0); iA1 = _mm_sub_pd(_mm_mul_pd(A1, dD), iA1); iA2 = _mm_sub_pd(_mm_mul_pd(A2, dD), iA2); d1 = _mm_mul_sd(dA, dD); d2 = _mm_mul_sd(dB, dC); // iB = D * (A#B)# = D*B#*A iB1 = _mm_mul_pd(D1, _mm_shuffle_pd(AB2,AB1,1)); iB2 = _mm_mul_pd(D2, _mm_shuffle_pd(AB2,AB1,1)); iB1 = _mm_sub_pd(iB1, _mm_mul_pd(_mm_shuffle_pd(D1,D1,1), _mm_shuffle_pd(AB2,AB1,2))); iB2 = _mm_sub_pd(iB2, _mm_mul_pd(_mm_shuffle_pd(D2,D2,1), _mm_shuffle_pd(AB2,AB1,2))); // det = |A|*|D| + |B|*|C| - trace(A#*B*D#*C) det = _mm_add_sd(d1, d2); det = _mm_sub_sd(det, rd); // iC = A * (D#C)# = A*C#*D iC1 = _mm_mul_pd(A1, _mm_shuffle_pd(DC2,DC1,1)); iC2 = _mm_mul_pd(A2, _mm_shuffle_pd(DC2,DC1,1)); iC1 = _mm_sub_pd(iC1, _mm_mul_pd(_mm_shuffle_pd(A1,A1,1), _mm_shuffle_pd(DC2,DC1,2))); iC2 = _mm_sub_pd(iC2, _mm_mul_pd(_mm_shuffle_pd(A2,A2,1), _mm_shuffle_pd(DC2,DC1,2))); rd = _mm_div_sd(_mm_set_sd(1.0), det); rd = _mm_shuffle_pd(rd,rd,0); // iB = C*|B| - D*B#*A dB = _mm_shuffle_pd(dB,dB,0); iB1 = _mm_sub_pd(_mm_mul_pd(C1, dB), iB1); iB2 = _mm_sub_pd(_mm_mul_pd(C2, dB), iB2); d1 = _mm_xor_pd(rd, _Sign_PN); d2 = _mm_xor_pd(rd, _Sign_NP); // iC = B*|C| - A*C#*D; dC = _mm_shuffle_pd(dC,dC,0); iC1 = _mm_sub_pd(_mm_mul_pd(B1, dC), iC1); iC2 = _mm_sub_pd(_mm_mul_pd(B2, dC), iC2); _mm_storeu_pd(dst+0, _mm_mul_pd(_mm_shuffle_pd(iA2, iA1, 3), d1)); _mm_storeu_pd(dst+4, _mm_mul_pd(_mm_shuffle_pd(iA2, iA1, 0), d2)); _mm_storeu_pd(dst+2, _mm_mul_pd(_mm_shuffle_pd(iB2, iB1, 3), d1)); _mm_storeu_pd(dst+4+2, _mm_mul_pd(_mm_shuffle_pd(iB2, iB1, 0), d2)); _mm_storeu_pd(dst+2*4, _mm_mul_pd(_mm_shuffle_pd(iC2, iC1, 3), d1)); _mm_storeu_pd(dst+3*4, _mm_mul_pd(_mm_shuffle_pd(iC2, iC1, 0), d2)); _mm_storeu_pd(dst+2*4+2,_mm_mul_pd(_mm_shuffle_pd(iD2, iD1, 3), d1)); _mm_storeu_pd(dst+3*4+2,_mm_mul_pd(_mm_shuffle_pd(iD2, iD1, 0), d2)); } #endif } // end of namespace Fastor #endif // INVERSE_H