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CacheFriendlyProduct.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra. Eigen itself is part of the KDE project.
//
// Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr>
//
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 3 of the License, or (at your option) any later version.
//
// Alternatively, you can redistribute it and/or
// modify it under the terms of the GNU General Public License as
// published by the Free Software Foundation; either version 2 of
// the License, or (at your option) any later version.
//
// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public
// License and a copy of the GNU General Public License along with
// Eigen. If not, see <http://www.gnu.org/licenses/>.

#ifndef EIGEN_CACHE_FRIENDLY_PRODUCT_H
#define EIGEN_CACHE_FRIENDLY_PRODUCT_H

template <int L2MemorySize,typename Scalar>
struct ei_L2_block_traits {
  enum {width = 8 * ei_meta_sqrt<L2MemorySize/(64*sizeof(Scalar))>::ret };
};

#ifndef EIGEN_EXTERN_INSTANTIATIONS

template<typename Scalar>
static void ei_cache_friendly_product(
  int _rows, int _cols, int depth,
  bool _lhsRowMajor, const Scalar* _lhs, int _lhsStride,
  bool _rhsRowMajor, const Scalar* _rhs, int _rhsStride,
  bool resRowMajor, Scalar* res, int resStride)
{
  const Scalar* EIGEN_RESTRICT lhs;
  const Scalar* EIGEN_RESTRICT rhs;
  int lhsStride, rhsStride, rows, cols;
  bool lhsRowMajor;

  if (resRowMajor)
  {
    lhs = _rhs;
    rhs = _lhs;
    lhsStride = _rhsStride;
    rhsStride = _lhsStride;
    cols = _rows;
    rows = _cols;
    lhsRowMajor = !_rhsRowMajor;
    ei_assert(_lhsRowMajor);
  }
  else
  {
    lhs = _lhs;
    rhs = _rhs;
    lhsStride = _lhsStride;
    rhsStride = _rhsStride;
    rows = _rows;
    cols = _cols;
    lhsRowMajor = _lhsRowMajor;
    ei_assert(!_rhsRowMajor);
  }

  typedef typename ei_packet_traits<Scalar>::type PacketType;

  enum {
    PacketSize = sizeof(PacketType)/sizeof(Scalar),
    #if (defined __i386__)
    // i386 architecture provides only 8 xmm registers,
    // so let's reduce the max number of rows processed at once.
    MaxBlockRows = 4,
    MaxBlockRows_ClampingMask = 0xFFFFFC,
    #else
    MaxBlockRows = 8,
    MaxBlockRows_ClampingMask = 0xFFFFF8,
    #endif
    // maximal size of the blocks fitted in L2 cache
    MaxL2BlockSize = ei_L2_block_traits<EIGEN_TUNE_FOR_CPU_CACHE_SIZE,Scalar>::width
  };

  const bool resIsAligned = (PacketSize==1) || (((resStride%PacketSize) == 0) && (size_t(res)%16==0));

  const int remainingSize = depth % PacketSize;
  const int size = depth - remainingSize; // third dimension of the product clamped to packet boundaries
  const int l2BlockRows = MaxL2BlockSize > rows ? rows : MaxL2BlockSize;
  const int l2BlockCols = MaxL2BlockSize > cols ? cols : MaxL2BlockSize;
  const int l2BlockSize = MaxL2BlockSize > size ? size : MaxL2BlockSize;
  const int l2BlockSizeAligned = (1 + std::max(l2BlockSize,l2BlockCols)/PacketSize)*PacketSize;
  const bool needRhsCopy = (PacketSize>1) && ((rhsStride%PacketSize!=0) || (size_t(rhs)%16!=0));
  Scalar* EIGEN_RESTRICT block = 0;
  const int allocBlockSize = l2BlockRows*size;
  block = ei_aligned_stack_new(Scalar, allocBlockSize);
  Scalar* EIGEN_RESTRICT rhsCopy
    = ei_aligned_stack_new(Scalar, l2BlockSizeAligned*l2BlockSizeAligned);

  // loops on each L2 cache friendly blocks of the result
  for(int l2i=0; l2i<rows; l2i+=l2BlockRows)
  {
    const int l2blockRowEnd = std::min(l2i+l2BlockRows, rows);
    const int l2blockRowEndBW = l2blockRowEnd & MaxBlockRows_ClampingMask;    // end of the rows aligned to bw
    const int l2blockRemainingRows = l2blockRowEnd - l2blockRowEndBW;         // number of remaining rows
    //const int l2blockRowEndBWPlusOne = l2blockRowEndBW + (l2blockRemainingRows?0:MaxBlockRows);

    // build a cache friendly blocky matrix
    int count = 0;

    // copy l2blocksize rows of m_lhs to blocks of ps x bw
    for(int l2k=0; l2k<size; l2k+=l2BlockSize)
    {
      const int l2blockSizeEnd = std::min(l2k+l2BlockSize, size);

      for (int i = l2i; i<l2blockRowEndBW/*PlusOne*/; i+=MaxBlockRows)
      {
        // TODO merge the "if l2blockRemainingRows" using something like:
        // const int blockRows = std::min(i+MaxBlockRows, rows) - i;

        for (int k=l2k; k<l2blockSizeEnd; k+=PacketSize)
        {
          // TODO write these loops using meta unrolling
          // negligible for large matrices but useful for small ones
          if (lhsRowMajor)
          {
            for (int w=0; w<MaxBlockRows; ++w)
              for (int s=0; s<PacketSize; ++s)
                block[count++] = lhs[(i+w)*lhsStride + (k+s)];
          }
          else
          {
            for (int w=0; w<MaxBlockRows; ++w)
              for (int s=0; s<PacketSize; ++s)
                block[count++] = lhs[(i+w) + (k+s)*lhsStride];
          }
        }
      }
      if (l2blockRemainingRows>0)
      {
        for (int k=l2k; k<l2blockSizeEnd; k+=PacketSize)
        {
          if (lhsRowMajor)
          {
            for (int w=0; w<l2blockRemainingRows; ++w)
              for (int s=0; s<PacketSize; ++s)
                block[count++] = lhs[(l2blockRowEndBW+w)*lhsStride + (k+s)];
          }
          else
          {
            for (int w=0; w<l2blockRemainingRows; ++w)
              for (int s=0; s<PacketSize; ++s)
                block[count++] = lhs[(l2blockRowEndBW+w) + (k+s)*lhsStride];
          }
        }
      }
    }

    for(int l2j=0; l2j<cols; l2j+=l2BlockCols)
    {
      int l2blockColEnd = std::min(l2j+l2BlockCols, cols);

      for(int l2k=0; l2k<size; l2k+=l2BlockSize)
      {
        // acumulate bw rows of lhs time a single column of rhs to a bw x 1 block of res
        int l2blockSizeEnd = std::min(l2k+l2BlockSize, size);

        // if not aligned, copy the rhs block
        if (needRhsCopy)
          for(int l1j=l2j; l1j<l2blockColEnd; l1j+=1)
          {
            ei_internal_assert(l2BlockSizeAligned*(l1j-l2j)+(l2blockSizeEnd-l2k) < l2BlockSizeAligned*l2BlockSizeAligned);
            memcpy(rhsCopy+l2BlockSizeAligned*(l1j-l2j),&(rhs[l1j*rhsStride+l2k]),(l2blockSizeEnd-l2k)*sizeof(Scalar));
          }

        // for each bw x 1 result's block
        for(int l1i=l2i; l1i<l2blockRowEndBW; l1i+=MaxBlockRows)
        {
          int offsetblock = l2k * (l2blockRowEnd-l2i) + (l1i-l2i)*(l2blockSizeEnd-l2k) - l2k*MaxBlockRows;
          const Scalar* EIGEN_RESTRICT localB = &block[offsetblock];

          for(int l1j=l2j; l1j<l2blockColEnd; l1j+=1)
          {
            const Scalar* EIGEN_RESTRICT rhsColumn;
            if (needRhsCopy)
              rhsColumn = &(rhsCopy[l2BlockSizeAligned*(l1j-l2j)-l2k]);
            else
              rhsColumn = &(rhs[l1j*rhsStride]);

            PacketType dst[MaxBlockRows];
            dst[3] = dst[2] = dst[1] = dst[0] = ei_pset1(Scalar(0.));
            if (MaxBlockRows==8)
              dst[7] = dst[6] = dst[5] = dst[4] = dst[0];

            PacketType tmp;

            for(int k=l2k; k<l2blockSizeEnd; k+=PacketSize)
            {
              tmp = ei_ploadu(&rhsColumn[k]);
              PacketType A0, A1, A2, A3, A4, A5;
              A0 = ei_pload(localB + k*MaxBlockRows);
              A1 = ei_pload(localB + k*MaxBlockRows+1*PacketSize);
              A2 = ei_pload(localB + k*MaxBlockRows+2*PacketSize);
              A3 = ei_pload(localB + k*MaxBlockRows+3*PacketSize);
              if (MaxBlockRows==8) A4 = ei_pload(localB + k*MaxBlockRows+4*PacketSize);
              if (MaxBlockRows==8) A5 = ei_pload(localB + k*MaxBlockRows+5*PacketSize);
              dst[0] = ei_pmadd(tmp, A0, dst[0]);
              if (MaxBlockRows==8) A0 = ei_pload(localB + k*MaxBlockRows+6*PacketSize);
              dst[1] = ei_pmadd(tmp, A1, dst[1]);
              if (MaxBlockRows==8) A1 = ei_pload(localB + k*MaxBlockRows+7*PacketSize);
              dst[2] = ei_pmadd(tmp, A2, dst[2]);
              dst[3] = ei_pmadd(tmp, A3, dst[3]);
              if (MaxBlockRows==8)
              {
                dst[4] = ei_pmadd(tmp, A4, dst[4]);
                dst[5] = ei_pmadd(tmp, A5, dst[5]);
                dst[6] = ei_pmadd(tmp, A0, dst[6]);
                dst[7] = ei_pmadd(tmp, A1, dst[7]);
              }
            }

            Scalar* EIGEN_RESTRICT localRes = &(res[l1i + l1j*resStride]);

            if (PacketSize>1 && resIsAligned)
            {
              // the result is aligned: let's do packet reduction
              ei_pstore(&(localRes[0]), ei_padd(ei_pload(&(localRes[0])), ei_preduxp(&dst[0])));
              if (PacketSize==2)
                ei_pstore(&(localRes[2]), ei_padd(ei_pload(&(localRes[2])), ei_preduxp(&(dst[2]))));
              if (MaxBlockRows==8)
              {
                ei_pstore(&(localRes[4]), ei_padd(ei_pload(&(localRes[4])), ei_preduxp(&(dst[4]))));
                if (PacketSize==2)
                  ei_pstore(&(localRes[6]), ei_padd(ei_pload(&(localRes[6])), ei_preduxp(&(dst[6]))));
              }
            }
            else
            {
              // not aligned => per coeff packet reduction
              localRes[0] += ei_predux(dst[0]);
              localRes[1] += ei_predux(dst[1]);
              localRes[2] += ei_predux(dst[2]);
              localRes[3] += ei_predux(dst[3]);
              if (MaxBlockRows==8)
              {
                localRes[4] += ei_predux(dst[4]);
                localRes[5] += ei_predux(dst[5]);
                localRes[6] += ei_predux(dst[6]);
                localRes[7] += ei_predux(dst[7]);
              }
            }
          }
        }
        if (l2blockRemainingRows>0)
        {
          int offsetblock = l2k * (l2blockRowEnd-l2i) + (l2blockRowEndBW-l2i)*(l2blockSizeEnd-l2k) - l2k*l2blockRemainingRows;
          const Scalar* localB = &block[offsetblock];

          for(int l1j=l2j; l1j<l2blockColEnd; l1j+=1)
          {
            const Scalar* EIGEN_RESTRICT rhsColumn;
            if (needRhsCopy)
              rhsColumn = &(rhsCopy[l2BlockSizeAligned*(l1j-l2j)-l2k]);
            else
              rhsColumn = &(rhs[l1j*rhsStride]);

            PacketType dst[MaxBlockRows];
            dst[3] = dst[2] = dst[1] = dst[0] = ei_pset1(Scalar(0.));
            if (MaxBlockRows==8)
              dst[7] = dst[6] = dst[5] = dst[4] = dst[0];

            // let's declare a few other temporary registers
            PacketType tmp;

            for(int k=l2k; k<l2blockSizeEnd; k+=PacketSize)
            {
              tmp = ei_pload(&rhsColumn[k]);

                                           dst[0] = ei_pmadd(tmp, ei_pload(&(localB[k*l2blockRemainingRows             ])), dst[0]);
              if (l2blockRemainingRows>=2) dst[1] = ei_pmadd(tmp, ei_pload(&(localB[k*l2blockRemainingRows+  PacketSize])), dst[1]);
              if (l2blockRemainingRows>=3) dst[2] = ei_pmadd(tmp, ei_pload(&(localB[k*l2blockRemainingRows+2*PacketSize])), dst[2]);
              if (l2blockRemainingRows>=4) dst[3] = ei_pmadd(tmp, ei_pload(&(localB[k*l2blockRemainingRows+3*PacketSize])), dst[3]);
              if (MaxBlockRows==8)
              {
                if (l2blockRemainingRows>=5) dst[4] = ei_pmadd(tmp, ei_pload(&(localB[k*l2blockRemainingRows+4*PacketSize])), dst[4]);
                if (l2blockRemainingRows>=6) dst[5] = ei_pmadd(tmp, ei_pload(&(localB[k*l2blockRemainingRows+5*PacketSize])), dst[5]);
                if (l2blockRemainingRows>=7) dst[6] = ei_pmadd(tmp, ei_pload(&(localB[k*l2blockRemainingRows+6*PacketSize])), dst[6]);
                if (l2blockRemainingRows>=8) dst[7] = ei_pmadd(tmp, ei_pload(&(localB[k*l2blockRemainingRows+7*PacketSize])), dst[7]);
              }
            }

            Scalar* EIGEN_RESTRICT localRes = &(res[l2blockRowEndBW + l1j*resStride]);

            // process the remaining rows once at a time
                                         localRes[0] += ei_predux(dst[0]);
            if (l2blockRemainingRows>=2) localRes[1] += ei_predux(dst[1]);
            if (l2blockRemainingRows>=3) localRes[2] += ei_predux(dst[2]);
            if (l2blockRemainingRows>=4) localRes[3] += ei_predux(dst[3]);
            if (MaxBlockRows==8)
            {
              if (l2blockRemainingRows>=5) localRes[4] += ei_predux(dst[4]);
              if (l2blockRemainingRows>=6) localRes[5] += ei_predux(dst[5]);
              if (l2blockRemainingRows>=7) localRes[6] += ei_predux(dst[6]);
              if (l2blockRemainingRows>=8) localRes[7] += ei_predux(dst[7]);
            }

          }
        }
      }
    }
  }
  if (PacketSize>1 && remainingSize)
  {
    if (lhsRowMajor)
    {
      for (int j=0; j<cols; ++j)
        for (int i=0; i<rows; ++i)
        {
          Scalar tmp = lhs[i*lhsStride+size] * rhs[j*rhsStride+size];
          // FIXME this loop get vectorized by the compiler !
          for (int k=1; k<remainingSize; ++k)
            tmp += lhs[i*lhsStride+size+k] * rhs[j*rhsStride+size+k];
          res[i+j*resStride] += tmp;
        }
    }
    else
    {
      for (int j=0; j<cols; ++j)
        for (int i=0; i<rows; ++i)
        {
          Scalar tmp = lhs[i+size*lhsStride] * rhs[j*rhsStride+size];
          for (int k=1; k<remainingSize; ++k)
            tmp += lhs[i+(size+k)*lhsStride] * rhs[j*rhsStride+size+k];
          res[i+j*resStride] += tmp;
        }
    }
  }

  ei_aligned_stack_delete(Scalar, block, allocBlockSize);
  ei_aligned_stack_delete(Scalar, rhsCopy, l2BlockSizeAligned*l2BlockSizeAligned);
}

#endif // EIGEN_EXTERN_INSTANTIATIONS

/* Optimized col-major matrix * vector product:
 * This algorithm processes 4 columns at onces that allows to both reduce
 * the number of load/stores of the result by a factor 4 and to reduce
 * the instruction dependency. Moreover, we know that all bands have the
 * same alignment pattern.
 * TODO: since rhs gets evaluated only once, no need to evaluate it
 */
template<typename Scalar, typename RhsType>
static EIGEN_DONT_INLINE void ei_cache_friendly_product_colmajor_times_vector(
  int size,
  const Scalar* lhs, int lhsStride,
  const RhsType& rhs,
  Scalar* res)
{
  #ifdef _EIGEN_ACCUMULATE_PACKETS
  #error _EIGEN_ACCUMULATE_PACKETS has already been defined
  #endif
  #define _EIGEN_ACCUMULATE_PACKETS(A0,A13,A2) \
    ei_pstore(&res[j], \
      ei_padd(ei_pload(&res[j]), \
        ei_padd( \
          ei_padd(ei_pmul(ptmp0,EIGEN_CAT(ei_ploa , A0)(&lhs0[j])), \
                  ei_pmul(ptmp1,EIGEN_CAT(ei_ploa , A13)(&lhs1[j]))), \
          ei_padd(ei_pmul(ptmp2,EIGEN_CAT(ei_ploa , A2)(&lhs2[j])), \
                  ei_pmul(ptmp3,EIGEN_CAT(ei_ploa , A13)(&lhs3[j]))) )))

  typedef typename ei_packet_traits<Scalar>::type Packet;
  const int PacketSize = sizeof(Packet)/sizeof(Scalar);

  enum { AllAligned = 0, EvenAligned, FirstAligned, NoneAligned };
  const int columnsAtOnce = 4;
  const int peels = 2;
  const int PacketAlignedMask = PacketSize-1;
  const int PeelAlignedMask = PacketSize*peels-1;

  // How many coeffs of the result do we have to skip to be aligned.
  // Here we assume data are at least aligned on the base scalar type that is mandatory anyway.
  const int alignedStart = ei_alignmentOffset(res,size);
  const int alignedSize = PacketSize>1 ? alignedStart + ((size-alignedStart) & ~PacketAlignedMask) : 0;
  const int peeledSize  = peels>1 ? alignedStart + ((alignedSize-alignedStart) & ~PeelAlignedMask) : alignedStart;

  const int alignmentStep = PacketSize>1 ? (PacketSize - lhsStride % PacketSize) & PacketAlignedMask : 0;
  int alignmentPattern = alignmentStep==0 ? AllAligned
                       : alignmentStep==(PacketSize/2) ? EvenAligned
                       : FirstAligned;

  // we cannot assume the first element is aligned because of sub-matrices
  const int lhsAlignmentOffset = ei_alignmentOffset(lhs,size);

  // find how many columns do we have to skip to be aligned with the result (if possible)
  int skipColumns = 0;
  if (PacketSize>1)
  {
    ei_internal_assert(size_t(lhs+lhsAlignmentOffset)%sizeof(Packet)==0 || size<PacketSize);

    while (skipColumns<PacketSize &&
           alignedStart != ((lhsAlignmentOffset + alignmentStep*skipColumns)%PacketSize))
      ++skipColumns;
    if (skipColumns==PacketSize)
    {
      // nothing can be aligned, no need to skip any column
      alignmentPattern = NoneAligned;
      skipColumns = 0;
    }
    else
    {
      skipColumns = std::min(skipColumns,rhs.size());
      // note that the skiped columns are processed later.
    }

    ei_internal_assert((alignmentPattern==NoneAligned) || (size_t(lhs+alignedStart+lhsStride*skipColumns)%sizeof(Packet))==0);
  }

  int offset1 = (FirstAligned && alignmentStep==1?3:1);
  int offset3 = (FirstAligned && alignmentStep==1?1:3);

  int columnBound = ((rhs.size()-skipColumns)/columnsAtOnce)*columnsAtOnce + skipColumns;
  for (int i=skipColumns; i<columnBound; i+=columnsAtOnce)
  {
    Packet ptmp0 = ei_pset1(rhs[i]),   ptmp1 = ei_pset1(rhs[i+offset1]),
           ptmp2 = ei_pset1(rhs[i+2]), ptmp3 = ei_pset1(rhs[i+offset3]);

    // this helps a lot generating better binary code
    const Scalar *lhs0 = lhs + i*lhsStride, *lhs1 = lhs + (i+offset1)*lhsStride,
                 *lhs2 = lhs + (i+2)*lhsStride, *lhs3 = lhs + (i+offset3)*lhsStride;

    if (PacketSize>1)
    {
      /* explicit vectorization */
      // process initial unaligned coeffs
      for (int j=0; j<alignedStart; ++j)
        res[j] += ei_pfirst(ptmp0)*lhs0[j] + ei_pfirst(ptmp1)*lhs1[j] + ei_pfirst(ptmp2)*lhs2[j] + ei_pfirst(ptmp3)*lhs3[j];

      if (alignedSize>alignedStart)
      {
        switch(alignmentPattern)
        {
          case AllAligned:
            for (int j = alignedStart; j<alignedSize; j+=PacketSize)
              _EIGEN_ACCUMULATE_PACKETS(d,d,d);
            break;
          case EvenAligned:
            for (int j = alignedStart; j<alignedSize; j+=PacketSize)
              _EIGEN_ACCUMULATE_PACKETS(d,du,d);
            break;
          case FirstAligned:
            if(peels>1)
            {
              Packet A00, A01, A02, A03, A10, A11, A12, A13;

              A01 = ei_pload(&lhs1[alignedStart-1]);
              A02 = ei_pload(&lhs2[alignedStart-2]);
              A03 = ei_pload(&lhs3[alignedStart-3]);

              for (int j = alignedStart; j<peeledSize; j+=peels*PacketSize)
              {
                A11 = ei_pload(&lhs1[j-1+PacketSize]);  ei_palign<1>(A01,A11);
                A12 = ei_pload(&lhs2[j-2+PacketSize]);  ei_palign<2>(A02,A12);
                A13 = ei_pload(&lhs3[j-3+PacketSize]);  ei_palign<3>(A03,A13);

                A00 = ei_pload (&lhs0[j]);
                A10 = ei_pload (&lhs0[j+PacketSize]);
                A00 = ei_pmadd(ptmp0, A00, ei_pload(&res[j]));
                A10 = ei_pmadd(ptmp0, A10, ei_pload(&res[j+PacketSize]));

                A00 = ei_pmadd(ptmp1, A01, A00);
                A01 = ei_pload(&lhs1[j-1+2*PacketSize]);  ei_palign<1>(A11,A01);
                A00 = ei_pmadd(ptmp2, A02, A00);
                A02 = ei_pload(&lhs2[j-2+2*PacketSize]);  ei_palign<2>(A12,A02);
                A00 = ei_pmadd(ptmp3, A03, A00);
                ei_pstore(&res[j],A00);
                A03 = ei_pload(&lhs3[j-3+2*PacketSize]);  ei_palign<3>(A13,A03);
                A10 = ei_pmadd(ptmp1, A11, A10);
                A10 = ei_pmadd(ptmp2, A12, A10);
                A10 = ei_pmadd(ptmp3, A13, A10);
                ei_pstore(&res[j+PacketSize],A10);
              }
            }
            for (int j = peeledSize; j<alignedSize; j+=PacketSize)
              _EIGEN_ACCUMULATE_PACKETS(d,du,du);
            break;
          default:
            for (int j = alignedStart; j<alignedSize; j+=PacketSize)
              _EIGEN_ACCUMULATE_PACKETS(du,du,du);
            break;
        }
      }
    } // end explicit vectorization

    /* process remaining coeffs (or all if there is no explicit vectorization) */
    for (int j=alignedSize; j<size; ++j)
      res[j] += ei_pfirst(ptmp0)*lhs0[j] + ei_pfirst(ptmp1)*lhs1[j] + ei_pfirst(ptmp2)*lhs2[j] + ei_pfirst(ptmp3)*lhs3[j];
  }

  // process remaining first and last columns (at most columnsAtOnce-1)
  int end = rhs.size();
  int start = columnBound;
  do
  {
    for (int i=start; i<end; ++i)
    {
      Packet ptmp0 = ei_pset1(rhs[i]);
      const Scalar* lhs0 = lhs + i*lhsStride;

      if (PacketSize>1)
      {
        /* explicit vectorization */
        // process first unaligned result's coeffs
        for (int j=0; j<alignedStart; ++j)
          res[j] += ei_pfirst(ptmp0) * lhs0[j];

        // process aligned result's coeffs
        if ((size_t(lhs0+alignedStart)%sizeof(Packet))==0)
          for (int j = alignedStart;j<alignedSize;j+=PacketSize)
            ei_pstore(&res[j], ei_pmadd(ptmp0,ei_pload(&lhs0[j]),ei_pload(&res[j])));
        else
          for (int j = alignedStart;j<alignedSize;j+=PacketSize)
            ei_pstore(&res[j], ei_pmadd(ptmp0,ei_ploadu(&lhs0[j]),ei_pload(&res[j])));
      }

      // process remaining scalars (or all if no explicit vectorization)
      for (int j=alignedSize; j<size; ++j)
        res[j] += ei_pfirst(ptmp0) * lhs0[j];
    }
    if (skipColumns)
    {
      start = 0;
      end = skipColumns;
      skipColumns = 0;
    }
    else
      break;
  } while(PacketSize>1);
  #undef _EIGEN_ACCUMULATE_PACKETS
}

// TODO add peeling to mask unaligned load/stores
template<typename Scalar, typename ResType>
static EIGEN_DONT_INLINE void ei_cache_friendly_product_rowmajor_times_vector(
  const Scalar* lhs, int lhsStride,
  const Scalar* rhs, int rhsSize,
  ResType& res)
{
  #ifdef _EIGEN_ACCUMULATE_PACKETS
  #error _EIGEN_ACCUMULATE_PACKETS has already been defined
  #endif

  #define _EIGEN_ACCUMULATE_PACKETS(A0,A13,A2) {\
    Packet b = ei_pload(&rhs[j]); \
    ptmp0 = ei_pmadd(b, EIGEN_CAT(ei_ploa,A0) (&lhs0[j]), ptmp0); \
    ptmp1 = ei_pmadd(b, EIGEN_CAT(ei_ploa,A13)(&lhs1[j]), ptmp1); \
    ptmp2 = ei_pmadd(b, EIGEN_CAT(ei_ploa,A2) (&lhs2[j]), ptmp2); \
    ptmp3 = ei_pmadd(b, EIGEN_CAT(ei_ploa,A13)(&lhs3[j]), ptmp3); }

  typedef typename ei_packet_traits<Scalar>::type Packet;
  const int PacketSize = sizeof(Packet)/sizeof(Scalar);

  enum { AllAligned=0, EvenAligned=1, FirstAligned=2, NoneAligned=3 };
  const int rowsAtOnce = 4;
  const int peels = 2;
  const int PacketAlignedMask = PacketSize-1;
  const int PeelAlignedMask = PacketSize*peels-1;
  const int size = rhsSize;

  // How many coeffs of the result do we have to skip to be aligned.
  // Here we assume data are at least aligned on the base scalar type that is mandatory anyway.
  const int alignedStart = ei_alignmentOffset(rhs, size);
  const int alignedSize = PacketSize>1 ? alignedStart + ((size-alignedStart) & ~PacketAlignedMask) : 0;
  const int peeledSize  = peels>1 ? alignedStart + ((alignedSize-alignedStart) & ~PeelAlignedMask) : alignedStart;

  const int alignmentStep = PacketSize>1 ? (PacketSize - lhsStride % PacketSize) & PacketAlignedMask : 0;
  int alignmentPattern = alignmentStep==0 ? AllAligned
                       : alignmentStep==(PacketSize/2) ? EvenAligned
                       : FirstAligned;

  // we cannot assume the first element is aligned because of sub-matrices
  const int lhsAlignmentOffset = ei_alignmentOffset(lhs,size);

  // find how many rows do we have to skip to be aligned with rhs (if possible)
  int skipRows = 0;
  if (PacketSize>1)
  {
    ei_internal_assert(size_t(lhs+lhsAlignmentOffset)%sizeof(Packet)==0  || size<PacketSize);

    while (skipRows<PacketSize &&
           alignedStart != ((lhsAlignmentOffset + alignmentStep*skipRows)%PacketSize))
      ++skipRows;
    if (skipRows==PacketSize)
    {
      // nothing can be aligned, no need to skip any column
      alignmentPattern = NoneAligned;
      skipRows = 0;
    }
    else
    {
      skipRows = std::min(skipRows,res.size());
      // note that the skiped columns are processed later.
    }
    ei_internal_assert((alignmentPattern==NoneAligned) || PacketSize==1
      || (size_t(lhs+alignedStart+lhsStride*skipRows)%sizeof(Packet))==0);
  }

  int offset1 = (FirstAligned && alignmentStep==1?3:1);
  int offset3 = (FirstAligned && alignmentStep==1?1:3);

  int rowBound = ((res.size()-skipRows)/rowsAtOnce)*rowsAtOnce + skipRows;
  for (int i=skipRows; i<rowBound; i+=rowsAtOnce)
  {
    Scalar tmp0 = Scalar(0), tmp1 = Scalar(0), tmp2 = Scalar(0), tmp3 = Scalar(0);

    // this helps the compiler generating good binary code
    const Scalar *lhs0 = lhs + i*lhsStride,     *lhs1 = lhs + (i+offset1)*lhsStride,
                 *lhs2 = lhs + (i+2)*lhsStride, *lhs3 = lhs + (i+offset3)*lhsStride;

    if (PacketSize>1)
    {
      /* explicit vectorization */
      Packet ptmp0 = ei_pset1(Scalar(0)), ptmp1 = ei_pset1(Scalar(0)), ptmp2 = ei_pset1(Scalar(0)), ptmp3 = ei_pset1(Scalar(0));

      // process initial unaligned coeffs
      // FIXME this loop get vectorized by the compiler !
      for (int j=0; j<alignedStart; ++j)
      {
        Scalar b = rhs[j];
        tmp0 += b*lhs0[j]; tmp1 += b*lhs1[j]; tmp2 += b*lhs2[j]; tmp3 += b*lhs3[j];
      }

      if (alignedSize>alignedStart)
      {
        switch(alignmentPattern)
        {
          case AllAligned:
            for (int j = alignedStart; j<alignedSize; j+=PacketSize)
              _EIGEN_ACCUMULATE_PACKETS(d,d,d);
            break;
          case EvenAligned:
            for (int j = alignedStart; j<alignedSize; j+=PacketSize)
              _EIGEN_ACCUMULATE_PACKETS(d,du,d);
            break;
          case FirstAligned:
            if (peels>1)
            {
              /* Here we proccess 4 rows with with two peeled iterations to hide
               * tghe overhead of unaligned loads. Moreover unaligned loads are handled
               * using special shift/move operations between the two aligned packets
               * overlaping the desired unaligned packet. This is *much* more efficient
               * than basic unaligned loads.
               */
              Packet A01, A02, A03, b, A11, A12, A13;
              A01 = ei_pload(&lhs1[alignedStart-1]);
              A02 = ei_pload(&lhs2[alignedStart-2]);
              A03 = ei_pload(&lhs3[alignedStart-3]);

              for (int j = alignedStart; j<peeledSize; j+=peels*PacketSize)
              {
                b = ei_pload(&rhs[j]);
                A11 = ei_pload(&lhs1[j-1+PacketSize]);  ei_palign<1>(A01,A11);
                A12 = ei_pload(&lhs2[j-2+PacketSize]);  ei_palign<2>(A02,A12);
                A13 = ei_pload(&lhs3[j-3+PacketSize]);  ei_palign<3>(A03,A13);

                ptmp0 = ei_pmadd(b, ei_pload (&lhs0[j]), ptmp0);
                ptmp1 = ei_pmadd(b, A01, ptmp1);
                A01 = ei_pload(&lhs1[j-1+2*PacketSize]);  ei_palign<1>(A11,A01);
                ptmp2 = ei_pmadd(b, A02, ptmp2);
                A02 = ei_pload(&lhs2[j-2+2*PacketSize]);  ei_palign<2>(A12,A02);
                ptmp3 = ei_pmadd(b, A03, ptmp3);
                A03 = ei_pload(&lhs3[j-3+2*PacketSize]);  ei_palign<3>(A13,A03);

                b = ei_pload(&rhs[j+PacketSize]);
                ptmp0 = ei_pmadd(b, ei_pload (&lhs0[j+PacketSize]), ptmp0);
                ptmp1 = ei_pmadd(b, A11, ptmp1);
                ptmp2 = ei_pmadd(b, A12, ptmp2);
                ptmp3 = ei_pmadd(b, A13, ptmp3);
              }
            }
            for (int j = peeledSize; j<alignedSize; j+=PacketSize)
              _EIGEN_ACCUMULATE_PACKETS(d,du,du);
            break;
          default:
            for (int j = alignedStart; j<alignedSize; j+=PacketSize)
              _EIGEN_ACCUMULATE_PACKETS(du,du,du);
            break;
        }
        tmp0 += ei_predux(ptmp0);
        tmp1 += ei_predux(ptmp1);
        tmp2 += ei_predux(ptmp2);
        tmp3 += ei_predux(ptmp3);
      }
    } // end explicit vectorization

    // process remaining coeffs (or all if no explicit vectorization)
    // FIXME this loop get vectorized by the compiler !
    for (int j=alignedSize; j<size; ++j)
    {
      Scalar b = rhs[j];
      tmp0 += b*lhs0[j]; tmp1 += b*lhs1[j]; tmp2 += b*lhs2[j]; tmp3 += b*lhs3[j];
    }
    res[i] += tmp0; res[i+offset1] += tmp1; res[i+2] += tmp2; res[i+offset3] += tmp3;
  }

  // process remaining first and last rows (at most columnsAtOnce-1)
  int end = res.size();
  int start = rowBound;
  do
  {
    for (int i=start; i<end; ++i)
    {
      Scalar tmp0 = Scalar(0);
      Packet ptmp0 = ei_pset1(tmp0);
      const Scalar* lhs0 = lhs + i*lhsStride;
      // process first unaligned result's coeffs
      // FIXME this loop get vectorized by the compiler !
      for (int j=0; j<alignedStart; ++j)
        tmp0 += rhs[j] * lhs0[j];

      if (alignedSize>alignedStart)
      {
        // process aligned rhs coeffs
        if ((size_t(lhs0+alignedStart)%sizeof(Packet))==0)
          for (int j = alignedStart;j<alignedSize;j+=PacketSize)
            ptmp0 = ei_pmadd(ei_pload(&rhs[j]), ei_pload(&lhs0[j]), ptmp0);
        else
          for (int j = alignedStart;j<alignedSize;j+=PacketSize)
            ptmp0 = ei_pmadd(ei_pload(&rhs[j]), ei_ploadu(&lhs0[j]), ptmp0);
        tmp0 += ei_predux(ptmp0);
      }

      // process remaining scalars
      // FIXME this loop get vectorized by the compiler !
      for (int j=alignedSize; j<size; ++j)
        tmp0 += rhs[j] * lhs0[j];
      res[i] += tmp0;
    }
    if (skipRows)
    {
      start = 0;
      end = skipRows;
      skipRows = 0;
    }
    else
      break;
  } while(PacketSize>1);

  #undef _EIGEN_ACCUMULATE_PACKETS
}

#endif // EIGEN_CACHE_FRIENDLY_PRODUCT_H

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