ProductEvaluators.h

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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2006-2008 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2011 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
 
#ifndef EIGEN_PRODUCTEVALUATORS_H
#define EIGEN_PRODUCTEVALUATORS_H
 
// IWYU pragma: private
#include "./InternalHeaderCheck.h"
 
namespace Eigen {
 
namespace internal {
 
template <typename Lhs, typename Rhs, int Options>
struct evaluator<Product<Lhs, Rhs, Options>> : public product_evaluator<Product<Lhs, Rhs, Options>> {
  typedef Product<Lhs, Rhs, Options> XprType;
  typedef product_evaluator<XprType> Base;
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE explicit evaluator(const XprType& xpr) : Base(xpr) {}
};
 
// Catch "scalar * ( A * B )" and transform it to "(A*scalar) * B"
// TODO we should apply that rule only if that's really helpful
template <typename Lhs, typename Rhs, typename Scalar1, typename Scalar2, typename Plain1>
struct evaluator_assume_aliasing<CwiseBinaryOp<internal::scalar_product_op<Scalar1, Scalar2>,
                                               const CwiseNullaryOp<internal::scalar_constant_op<Scalar1>, Plain1>,
                                               const Product<Lhs, Rhs, DefaultProduct>>> {
  static const bool value = true;
};
template <typename Lhs, typename Rhs, typename Scalar1, typename Scalar2, typename Plain1>
struct evaluator<CwiseBinaryOp<internal::scalar_product_op<Scalar1, Scalar2>,
                               const CwiseNullaryOp<internal::scalar_constant_op<Scalar1>, Plain1>,
                               const Product<Lhs, Rhs, DefaultProduct>>>
    : public evaluator<Product<EIGEN_SCALAR_BINARYOP_EXPR_RETURN_TYPE(Scalar1, Lhs, product), Rhs, DefaultProduct>> {
  typedef CwiseBinaryOp<internal::scalar_product_op<Scalar1, Scalar2>,
                        const CwiseNullaryOp<internal::scalar_constant_op<Scalar1>, Plain1>,
                        const Product<Lhs, Rhs, DefaultProduct>>
      XprType;
  typedef evaluator<Product<EIGEN_SCALAR_BINARYOP_EXPR_RETURN_TYPE(Scalar1, Lhs, product), Rhs, DefaultProduct>> Base;
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE explicit evaluator(const XprType& xpr)
      : Base(xpr.lhs().functor().m_other * xpr.rhs().lhs() * xpr.rhs().rhs()) {}
};
 
template <typename Lhs, typename Rhs, int DiagIndex>
struct evaluator<Diagonal<const Product<Lhs, Rhs, DefaultProduct>, DiagIndex>>
    : public evaluator<Diagonal<const Product<Lhs, Rhs, LazyProduct>, DiagIndex>> {
  typedef Diagonal<const Product<Lhs, Rhs, DefaultProduct>, DiagIndex> XprType;
  typedef evaluator<Diagonal<const Product<Lhs, Rhs, LazyProduct>, DiagIndex>> Base;
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE explicit evaluator(const XprType& xpr)
      : Base(Diagonal<const Product<Lhs, Rhs, LazyProduct>, DiagIndex>(
            Product<Lhs, Rhs, LazyProduct>(xpr.nestedExpression().lhs(), xpr.nestedExpression().rhs()), xpr.index())) {}
};
 
// Helper class to perform a matrix product with the destination at hand.
// Depending on the sizes of the factors, there are different evaluation strategies
// as controlled by internal::product_type.
template <typename Lhs, typename Rhs, typename LhsShape = typename evaluator_traits<Lhs>::Shape,
          typename RhsShape = typename evaluator_traits<Rhs>::Shape,
          int ProductType = internal::product_type<Lhs, Rhs>::value>
struct generic_product_impl;
 
template <typename Lhs, typename Rhs>
struct evaluator_assume_aliasing<Product<Lhs, Rhs, DefaultProduct>> {
  static const bool value = true;
};
 
// This is the default evaluator implementation for products:
// It creates a temporary and call generic_product_impl
template <typename Lhs, typename Rhs, int Options, int ProductTag, typename LhsShape, typename RhsShape>
struct product_evaluator<Product<Lhs, Rhs, Options>, ProductTag, LhsShape, RhsShape>
    : public evaluator<typename Product<Lhs, Rhs, Options>::PlainObject> {
  typedef Product<Lhs, Rhs, Options> XprType;
  typedef typename XprType::PlainObject PlainObject;
  typedef evaluator<PlainObject> Base;
  enum { Flags = Base::Flags | EvalBeforeNestingBit };
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE explicit product_evaluator(const XprType& xpr)
      : m_result(xpr.rows(), xpr.cols()) {
    internal::construct_at<Base>(this, m_result);
 
    // FIXME shall we handle nested_eval here?,
    // if so, then we must take care at removing the call to nested_eval in the specializations (e.g., in
    // permutation_matrix_product, transposition_matrix_product, etc.)
    //     typedef typename internal::nested_eval<Lhs,Rhs::ColsAtCompileTime>::type LhsNested;
    //     typedef typename internal::nested_eval<Rhs,Lhs::RowsAtCompileTime>::type RhsNested;
    //     typedef internal::remove_all_t<LhsNested> LhsNestedCleaned;
    //     typedef internal::remove_all_t<RhsNested> RhsNestedCleaned;
    //
    //     const LhsNested lhs(xpr.lhs());
    //     const RhsNested rhs(xpr.rhs());
    //
    //     generic_product_impl<LhsNestedCleaned, RhsNestedCleaned>::evalTo(m_result, lhs, rhs);
 
    generic_product_impl<Lhs, Rhs, LhsShape, RhsShape, ProductTag>::evalTo(m_result, xpr.lhs(), xpr.rhs());
  }
 
 protected:
  PlainObject m_result;
};
 
// The following three shortcuts are enabled only if the scalar types match exactly.
// TODO: we could enable them for different scalar types when the product is not vectorized.
 
// Dense = Product
template <typename DstXprType, typename Lhs, typename Rhs, int Options, typename Scalar>
struct Assignment<DstXprType, Product<Lhs, Rhs, Options>, internal::assign_op<Scalar, Scalar>, Dense2Dense,
                  std::enable_if_t<(Options == DefaultProduct || Options == AliasFreeProduct)>> {
  typedef Product<Lhs, Rhs, Options> SrcXprType;
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(DstXprType& dst, const SrcXprType& src,
                                                        const internal::assign_op<Scalar, Scalar>&) {
    Index dstRows = src.rows();
    Index dstCols = src.cols();
    if ((dst.rows() != dstRows) || (dst.cols() != dstCols)) dst.resize(dstRows, dstCols);
    // FIXME shall we handle nested_eval here?
    generic_product_impl<Lhs, Rhs>::evalTo(dst, src.lhs(), src.rhs());
  }
};
 
// Dense += Product
template <typename DstXprType, typename Lhs, typename Rhs, int Options, typename Scalar>
struct Assignment<DstXprType, Product<Lhs, Rhs, Options>, internal::add_assign_op<Scalar, Scalar>, Dense2Dense,
                  std::enable_if_t<(Options == DefaultProduct || Options == AliasFreeProduct)>> {
  typedef Product<Lhs, Rhs, Options> SrcXprType;
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(DstXprType& dst, const SrcXprType& src,
                                                        const internal::add_assign_op<Scalar, Scalar>&) {
    eigen_assert(dst.rows() == src.rows() && dst.cols() == src.cols());
    // FIXME shall we handle nested_eval here?
    generic_product_impl<Lhs, Rhs>::addTo(dst, src.lhs(), src.rhs());
  }
};
 
// Dense -= Product
template <typename DstXprType, typename Lhs, typename Rhs, int Options, typename Scalar>
struct Assignment<DstXprType, Product<Lhs, Rhs, Options>, internal::sub_assign_op<Scalar, Scalar>, Dense2Dense,
                  std::enable_if_t<(Options == DefaultProduct || Options == AliasFreeProduct)>> {
  typedef Product<Lhs, Rhs, Options> SrcXprType;
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(DstXprType& dst, const SrcXprType& src,
                                                        const internal::sub_assign_op<Scalar, Scalar>&) {
    eigen_assert(dst.rows() == src.rows() && dst.cols() == src.cols());
    // FIXME shall we handle nested_eval here?
    generic_product_impl<Lhs, Rhs>::subTo(dst, src.lhs(), src.rhs());
  }
};
 
// Dense ?= scalar * Product
// TODO we should apply that rule if that's really helpful
// for instance, this is not good for inner products
template <typename DstXprType, typename Lhs, typename Rhs, typename AssignFunc, typename Scalar, typename ScalarBis,
          typename Plain>
struct Assignment<DstXprType,
                  CwiseBinaryOp<internal::scalar_product_op<ScalarBis, Scalar>,
                                const CwiseNullaryOp<internal::scalar_constant_op<ScalarBis>, Plain>,
                                const Product<Lhs, Rhs, DefaultProduct>>,
                  AssignFunc, Dense2Dense> {
  typedef CwiseBinaryOp<internal::scalar_product_op<ScalarBis, Scalar>,
                        const CwiseNullaryOp<internal::scalar_constant_op<ScalarBis>, Plain>,
                        const Product<Lhs, Rhs, DefaultProduct>>
      SrcXprType;
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(DstXprType& dst, const SrcXprType& src,
                                                        const AssignFunc& func) {
    call_assignment_no_alias(dst, (src.lhs().functor().m_other * src.rhs().lhs()) * src.rhs().rhs(), func);
  }
};
 
//----------------------------------------
// Catch "Dense ?= xpr + Product<>" expression to save one temporary
// FIXME we could probably enable these rules for any product, i.e., not only Dense and DefaultProduct
 
template <typename OtherXpr, typename Lhs, typename Rhs>
struct evaluator_assume_aliasing<
    CwiseBinaryOp<
        internal::scalar_sum_op<typename OtherXpr::Scalar, typename Product<Lhs, Rhs, DefaultProduct>::Scalar>,
        const OtherXpr, const Product<Lhs, Rhs, DefaultProduct>>,
    DenseShape> {
  static const bool value = true;
};
 
template <typename OtherXpr, typename Lhs, typename Rhs>
struct evaluator_assume_aliasing<
    CwiseBinaryOp<
        internal::scalar_difference_op<typename OtherXpr::Scalar, typename Product<Lhs, Rhs, DefaultProduct>::Scalar>,
        const OtherXpr, const Product<Lhs, Rhs, DefaultProduct>>,
    DenseShape> {
  static const bool value = true;
};
 
template <typename DstXprType, typename OtherXpr, typename ProductType, typename Func1, typename Func2>
struct assignment_from_xpr_op_product {
  template <typename SrcXprType, typename InitialFunc>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(DstXprType& dst, const SrcXprType& src,
                                                        const InitialFunc& /*func*/) {
    call_assignment_no_alias(dst, src.lhs(), Func1());
    call_assignment_no_alias(dst, src.rhs(), Func2());
  }
};
 
#define EIGEN_CATCH_ASSIGN_XPR_OP_PRODUCT(ASSIGN_OP, BINOP, ASSIGN_OP2)                             \
  template <typename DstXprType, typename OtherXpr, typename Lhs, typename Rhs, typename DstScalar, \
            typename SrcScalar, typename OtherScalar, typename ProdScalar>                          \
  struct Assignment<DstXprType,                                                                     \
                    CwiseBinaryOp<internal::BINOP<OtherScalar, ProdScalar>, const OtherXpr,         \
                                  const Product<Lhs, Rhs, DefaultProduct>>,                         \
                    internal::ASSIGN_OP<DstScalar, SrcScalar>, Dense2Dense>                         \
      : assignment_from_xpr_op_product<DstXprType, OtherXpr, Product<Lhs, Rhs, DefaultProduct>,     \
                                       internal::ASSIGN_OP<DstScalar, OtherScalar>,                 \
                                       internal::ASSIGN_OP2<DstScalar, ProdScalar>> {}
 
EIGEN_CATCH_ASSIGN_XPR_OP_PRODUCT(assign_op, scalar_sum_op, add_assign_op);
EIGEN_CATCH_ASSIGN_XPR_OP_PRODUCT(add_assign_op, scalar_sum_op, add_assign_op);
EIGEN_CATCH_ASSIGN_XPR_OP_PRODUCT(sub_assign_op, scalar_sum_op, sub_assign_op);
 
EIGEN_CATCH_ASSIGN_XPR_OP_PRODUCT(assign_op, scalar_difference_op, sub_assign_op);
EIGEN_CATCH_ASSIGN_XPR_OP_PRODUCT(add_assign_op, scalar_difference_op, sub_assign_op);
EIGEN_CATCH_ASSIGN_XPR_OP_PRODUCT(sub_assign_op, scalar_difference_op, add_assign_op);
 
//----------------------------------------
 
template <typename Lhs, typename Rhs>
struct generic_product_impl<Lhs, Rhs, DenseShape, DenseShape, InnerProduct> {
  using impl = default_inner_product_impl<Lhs, Rhs, false>;
  template <typename Dst>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dst& dst, const Lhs& lhs, const Rhs& rhs) {
    dst.coeffRef(0, 0) = impl::run(lhs, rhs);
  }
 
  template <typename Dst>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void addTo(Dst& dst, const Lhs& lhs, const Rhs& rhs) {
    dst.coeffRef(0, 0) += impl::run(lhs, rhs);
  }
 
  template <typename Dst>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void subTo(Dst& dst, const Lhs& lhs, const Rhs& rhs) {
    dst.coeffRef(0, 0) -= impl::run(lhs, rhs);
  }
};
 
/***********************************************************************
 *  Implementation of outer dense * dense vector product
 ***********************************************************************/
 
// Column major result
template <typename Dst, typename Lhs, typename Rhs, typename Func>
void EIGEN_DEVICE_FUNC outer_product_selector_run(Dst& dst, const Lhs& lhs, const Rhs& rhs, const Func& func,
                                                  const false_type&) {
  evaluator<Rhs> rhsEval(rhs);
  ei_declare_local_nested_eval(Lhs, lhs, Rhs::SizeAtCompileTime, actual_lhs);
  // FIXME if cols is large enough, then it might be useful to make sure that lhs is sequentially stored
  // FIXME not very good if rhs is real and lhs complex while alpha is real too
  const Index cols = dst.cols();
  for (Index j = 0; j < cols; ++j) func(dst.col(j), rhsEval.coeff(Index(0), j) * actual_lhs);
}
 
// Row major result
template <typename Dst, typename Lhs, typename Rhs, typename Func>
void EIGEN_DEVICE_FUNC outer_product_selector_run(Dst& dst, const Lhs& lhs, const Rhs& rhs, const Func& func,
                                                  const true_type&) {
  evaluator<Lhs> lhsEval(lhs);
  ei_declare_local_nested_eval(Rhs, rhs, Lhs::SizeAtCompileTime, actual_rhs);
  // FIXME if rows is large enough, then it might be useful to make sure that rhs is sequentially stored
  // FIXME not very good if lhs is real and rhs complex while alpha is real too
  const Index rows = dst.rows();
  for (Index i = 0; i < rows; ++i) func(dst.row(i), lhsEval.coeff(i, Index(0)) * actual_rhs);
}
 
template <typename Lhs, typename Rhs>
struct generic_product_impl<Lhs, Rhs, DenseShape, DenseShape, OuterProduct> {
  template <typename T>
  struct is_row_major : std::conditional_t<(int(T::Flags) & RowMajorBit), internal::true_type, internal::false_type> {};
  typedef typename Product<Lhs, Rhs>::Scalar Scalar;
 
  // TODO it would be nice to be able to exploit our *_assign_op functors for that purpose
  struct set {
    template <typename Dst, typename Src>
    EIGEN_DEVICE_FUNC void operator()(const Dst& dst, const Src& src) const {
      dst.const_cast_derived() = src;
    }
  };
  struct add {
    template <typename Dst, typename Src>
    EIGEN_DEVICE_FUNC void operator()(const Dst& dst, const Src& src) const {
      dst.const_cast_derived() += src;
    }
  };
  struct sub {
    template <typename Dst, typename Src>
    EIGEN_DEVICE_FUNC void operator()(const Dst& dst, const Src& src) const {
      dst.const_cast_derived() -= src;
    }
  };
  struct adds {
    Scalar m_scale;
    explicit adds(const Scalar& s) : m_scale(s) {}
    template <typename Dst, typename Src>
    void EIGEN_DEVICE_FUNC operator()(const Dst& dst, const Src& src) const {
      dst.const_cast_derived() += m_scale * src;
    }
  };
 
  template <typename Dst>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dst& dst, const Lhs& lhs, const Rhs& rhs) {
    internal::outer_product_selector_run(dst, lhs, rhs, set(), is_row_major<Dst>());
  }
 
  template <typename Dst>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void addTo(Dst& dst, const Lhs& lhs, const Rhs& rhs) {
    internal::outer_product_selector_run(dst, lhs, rhs, add(), is_row_major<Dst>());
  }
 
  template <typename Dst>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void subTo(Dst& dst, const Lhs& lhs, const Rhs& rhs) {
    internal::outer_product_selector_run(dst, lhs, rhs, sub(), is_row_major<Dst>());
  }
 
  template <typename Dst>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void scaleAndAddTo(Dst& dst, const Lhs& lhs, const Rhs& rhs,
                                                                  const Scalar& alpha) {
    internal::outer_product_selector_run(dst, lhs, rhs, adds(alpha), is_row_major<Dst>());
  }
};
 
// This base class provides default implementations for evalTo, addTo, subTo, in terms of scaleAndAddTo
template <typename Lhs, typename Rhs, typename Derived>
struct generic_product_impl_base {
  typedef typename Product<Lhs, Rhs>::Scalar Scalar;
 
  template <typename Dst>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dst& dst, const Lhs& lhs, const Rhs& rhs) {
    dst.setZero();
    scaleAndAddTo(dst, lhs, rhs, Scalar(1));
  }
 
  template <typename Dst>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void addTo(Dst& dst, const Lhs& lhs, const Rhs& rhs) {
    scaleAndAddTo(dst, lhs, rhs, Scalar(1));
  }
 
  template <typename Dst>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void subTo(Dst& dst, const Lhs& lhs, const Rhs& rhs) {
    scaleAndAddTo(dst, lhs, rhs, Scalar(-1));
  }
 
  template <typename Dst>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void scaleAndAddTo(Dst& dst, const Lhs& lhs, const Rhs& rhs,
                                                                  const Scalar& alpha) {
    Derived::scaleAndAddTo(dst, lhs, rhs, alpha);
  }
};
 
template <typename Lhs, typename Rhs>
struct generic_product_impl<Lhs, Rhs, DenseShape, DenseShape, GemvProduct>
    : generic_product_impl_base<Lhs, Rhs, generic_product_impl<Lhs, Rhs, DenseShape, DenseShape, GemvProduct>> {
  typedef typename nested_eval<Lhs, 1>::type LhsNested;
  typedef typename nested_eval<Rhs, 1>::type RhsNested;
  typedef typename Product<Lhs, Rhs>::Scalar Scalar;
  enum { Side = Lhs::IsVectorAtCompileTime ? OnTheLeft : OnTheRight };
  typedef internal::remove_all_t<std::conditional_t<int(Side) == OnTheRight, LhsNested, RhsNested>> MatrixType;
 
  template <typename Dest>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void scaleAndAddTo(Dest& dst, const Lhs& lhs, const Rhs& rhs,
                                                                  const Scalar& alpha) {
    // Fallback to inner product if both the lhs and rhs is a runtime vector.
    if (lhs.rows() == 1 && rhs.cols() == 1) {
      dst.coeffRef(0, 0) += alpha * lhs.row(0).conjugate().dot(rhs.col(0));
      return;
    }
    LhsNested actual_lhs(lhs);
    RhsNested actual_rhs(rhs);
    internal::gemv_dense_selector<Side, (int(MatrixType::Flags) & RowMajorBit) ? RowMajor : ColMajor,
                                  bool(internal::blas_traits<MatrixType>::HasUsableDirectAccess)>::run(actual_lhs,
                                                                                                       actual_rhs, dst,
                                                                                                       alpha);
  }
};
 
template <typename Lhs, typename Rhs>
struct generic_product_impl<Lhs, Rhs, DenseShape, DenseShape, CoeffBasedProductMode> {
  typedef typename Product<Lhs, Rhs>::Scalar Scalar;
 
  template <typename Dst>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dst& dst, const Lhs& lhs, const Rhs& rhs) {
    // Same as: dst.noalias() = lhs.lazyProduct(rhs);
    // but easier on the compiler side
    call_assignment_no_alias(dst, lhs.lazyProduct(rhs), internal::assign_op<typename Dst::Scalar, Scalar>());
  }
 
  template <typename Dst>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void addTo(Dst& dst, const Lhs& lhs, const Rhs& rhs) {
    // dst.noalias() += lhs.lazyProduct(rhs);
    call_assignment_no_alias(dst, lhs.lazyProduct(rhs), internal::add_assign_op<typename Dst::Scalar, Scalar>());
  }
 
  template <typename Dst>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void subTo(Dst& dst, const Lhs& lhs, const Rhs& rhs) {
    // dst.noalias() -= lhs.lazyProduct(rhs);
    call_assignment_no_alias(dst, lhs.lazyProduct(rhs), internal::sub_assign_op<typename Dst::Scalar, Scalar>());
  }
 
  // This is a special evaluation path called from generic_product_impl<...,GemmProduct> in file GeneralMatrixMatrix.h
  // This variant tries to extract scalar multiples from both the LHS and RHS and factor them out. For instance:
  //   dst {,+,-}= (s1*A)*(B*s2)
  // will be rewritten as:
  //   dst {,+,-}= (s1*s2) * (A.lazyProduct(B))
  // There are at least four benefits of doing so:
  //  1 - huge performance gain for heap-allocated matrix types as it save costly allocations.
  //  2 - it is faster than simply by-passing the heap allocation through stack allocation.
  //  3 - it makes this fallback consistent with the heavy GEMM routine.
  //  4 - it fully by-passes huge stack allocation attempts when multiplying huge fixed-size matrices.
  //      (see https://stackoverflow.com/questions/54738495)
  // For small fixed sizes matrices, however, the gains are less obvious, it is sometimes x2 faster, but sometimes x3
  // slower, and the behavior depends also a lot on the compiler... This is why this re-writing strategy is currently
  // enabled only when falling back from the main GEMM.
  template <typename Dst, typename Func>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void eval_dynamic(Dst& dst, const Lhs& lhs, const Rhs& rhs,
                                                                 const Func& func) {
    enum {
      HasScalarFactor = blas_traits<Lhs>::HasScalarFactor || blas_traits<Rhs>::HasScalarFactor,
      ConjLhs = blas_traits<Lhs>::NeedToConjugate,
      ConjRhs = blas_traits<Rhs>::NeedToConjugate
    };
    // FIXME: in c++11 this should be auto, and extractScalarFactor should also return auto
    //        this is important for real*complex_mat
    Scalar actualAlpha = combine_scalar_factors<Scalar>(lhs, rhs);
 
    eval_dynamic_impl(dst, blas_traits<Lhs>::extract(lhs).template conjugateIf<ConjLhs>(),
                      blas_traits<Rhs>::extract(rhs).template conjugateIf<ConjRhs>(), func, actualAlpha,
                      std::conditional_t<HasScalarFactor, true_type, false_type>());
  }
 
 protected:
  template <typename Dst, typename LhsT, typename RhsT, typename Func, typename Scalar>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void eval_dynamic_impl(Dst& dst, const LhsT& lhs, const RhsT& rhs,
                                                                      const Func& func, const Scalar& s /* == 1 */,
                                                                      false_type) {
    EIGEN_UNUSED_VARIABLE(s);
    eigen_internal_assert(numext::is_exactly_one(s));
    call_restricted_packet_assignment_no_alias(dst, lhs.lazyProduct(rhs), func);
  }
 
  template <typename Dst, typename LhsT, typename RhsT, typename Func, typename Scalar>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void eval_dynamic_impl(Dst& dst, const LhsT& lhs, const RhsT& rhs,
                                                                      const Func& func, const Scalar& s, true_type) {
    call_restricted_packet_assignment_no_alias(dst, s * lhs.lazyProduct(rhs), func);
  }
};
 
// This specialization enforces the use of a coefficient-based evaluation strategy
template <typename Lhs, typename Rhs>
struct generic_product_impl<Lhs, Rhs, DenseShape, DenseShape, LazyCoeffBasedProductMode>
    : generic_product_impl<Lhs, Rhs, DenseShape, DenseShape, CoeffBasedProductMode> {};
 
// Case 2: Evaluate coeff by coeff
//
// This is mostly taken from CoeffBasedProduct.h
// The main difference is that we add an extra argument to the etor_product_*_impl::run() function
// for the inner dimension of the product, because evaluator object do not know their size.
 
template <int Traversal, int UnrollingIndex, typename Lhs, typename Rhs, typename RetScalar>
struct etor_product_coeff_impl;
 
template <int StorageOrder, int UnrollingIndex, typename Lhs, typename Rhs, typename Packet, int LoadMode>
struct etor_product_packet_impl;
 
template <typename Lhs, typename Rhs, int ProductTag>
struct product_evaluator<Product<Lhs, Rhs, LazyProduct>, ProductTag, DenseShape, DenseShape>
    : evaluator_base<Product<Lhs, Rhs, LazyProduct>> {
  typedef Product<Lhs, Rhs, LazyProduct> XprType;
  typedef typename XprType::Scalar Scalar;
  typedef typename XprType::CoeffReturnType CoeffReturnType;
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE explicit product_evaluator(const XprType& xpr)
      : m_lhs(xpr.lhs()),
        m_rhs(xpr.rhs()),
        m_lhsImpl(m_lhs),  // FIXME the creation of the evaluator objects should result in a no-op, but check that!
        m_rhsImpl(m_rhs),  //       Moreover, they are only useful for the packet path, so we could completely disable
                           //       them when not needed, or perhaps declare them on the fly on the packet method... We
                           //       have experiment to check what's best.
        m_innerDim(xpr.lhs().cols()) {
    EIGEN_INTERNAL_CHECK_COST_VALUE(NumTraits<Scalar>::MulCost);
    EIGEN_INTERNAL_CHECK_COST_VALUE(NumTraits<Scalar>::AddCost);
    EIGEN_INTERNAL_CHECK_COST_VALUE(CoeffReadCost);
#if 0
    std::cerr << "LhsOuterStrideBytes=  " << LhsOuterStrideBytes << "\n";
    std::cerr << "RhsOuterStrideBytes=  " << RhsOuterStrideBytes << "\n";
    std::cerr << "LhsAlignment=         " << LhsAlignment << "\n";
    std::cerr << "RhsAlignment=         " << RhsAlignment << "\n";
    std::cerr << "CanVectorizeLhs=      " << CanVectorizeLhs << "\n";
    std::cerr << "CanVectorizeRhs=      " << CanVectorizeRhs << "\n";
    std::cerr << "CanVectorizeInner=    " << CanVectorizeInner << "\n";
    std::cerr << "EvalToRowMajor=       " << EvalToRowMajor << "\n";
    std::cerr << "Alignment=            " << Alignment << "\n";
    std::cerr << "Flags=                " << Flags << "\n";
#endif
  }
 
  // Everything below here is taken from CoeffBasedProduct.h
 
  typedef typename internal::nested_eval<Lhs, Rhs::ColsAtCompileTime>::type LhsNested;
  typedef typename internal::nested_eval<Rhs, Lhs::RowsAtCompileTime>::type RhsNested;
 
  typedef internal::remove_all_t<LhsNested> LhsNestedCleaned;
  typedef internal::remove_all_t<RhsNested> RhsNestedCleaned;
 
  typedef evaluator<LhsNestedCleaned> LhsEtorType;
  typedef evaluator<RhsNestedCleaned> RhsEtorType;
 
  enum {
    RowsAtCompileTime = LhsNestedCleaned::RowsAtCompileTime,
    ColsAtCompileTime = RhsNestedCleaned::ColsAtCompileTime,
    InnerSize = min_size_prefer_fixed(LhsNestedCleaned::ColsAtCompileTime, RhsNestedCleaned::RowsAtCompileTime),
    MaxRowsAtCompileTime = LhsNestedCleaned::MaxRowsAtCompileTime,
    MaxColsAtCompileTime = RhsNestedCleaned::MaxColsAtCompileTime
  };
 
  typedef typename find_best_packet<Scalar, RowsAtCompileTime>::type LhsVecPacketType;
  typedef typename find_best_packet<Scalar, ColsAtCompileTime>::type RhsVecPacketType;
 
  enum {
 
    LhsCoeffReadCost = LhsEtorType::CoeffReadCost,
    RhsCoeffReadCost = RhsEtorType::CoeffReadCost,
    CoeffReadCost = InnerSize == 0 ? NumTraits<Scalar>::ReadCost
                    : InnerSize == Dynamic
                        ? HugeCost
                        : InnerSize * (NumTraits<Scalar>::MulCost + int(LhsCoeffReadCost) + int(RhsCoeffReadCost)) +
                              (InnerSize - 1) * NumTraits<Scalar>::AddCost,
 
    Unroll = CoeffReadCost <= EIGEN_UNROLLING_LIMIT,
 
    LhsFlags = LhsEtorType::Flags,
    RhsFlags = RhsEtorType::Flags,
 
    LhsRowMajor = LhsFlags & RowMajorBit,
    RhsRowMajor = RhsFlags & RowMajorBit,
 
    LhsVecPacketSize = unpacket_traits<LhsVecPacketType>::size,
    RhsVecPacketSize = unpacket_traits<RhsVecPacketType>::size,
 
    // Here, we don't care about alignment larger than the usable packet size.
    LhsAlignment =
        plain_enum_min(LhsEtorType::Alignment, LhsVecPacketSize* int(sizeof(typename LhsNestedCleaned::Scalar))),
    RhsAlignment =
        plain_enum_min(RhsEtorType::Alignment, RhsVecPacketSize* int(sizeof(typename RhsNestedCleaned::Scalar))),
 
    SameType = is_same<typename LhsNestedCleaned::Scalar, typename RhsNestedCleaned::Scalar>::value,
 
    CanVectorizeRhs = bool(RhsRowMajor) && (RhsFlags & PacketAccessBit) && (ColsAtCompileTime != 1),
    CanVectorizeLhs = (!LhsRowMajor) && (LhsFlags & PacketAccessBit) && (RowsAtCompileTime != 1),
 
    EvalToRowMajor = (MaxRowsAtCompileTime == 1 && MaxColsAtCompileTime != 1) ? 1
                     : (MaxColsAtCompileTime == 1 && MaxRowsAtCompileTime != 1)
                         ? 0
                         : (bool(RhsRowMajor) && !CanVectorizeLhs),
 
    Flags = ((int(LhsFlags) | int(RhsFlags)) & HereditaryBits & ~RowMajorBit) |
            (EvalToRowMajor ? RowMajorBit : 0)
            // TODO enable vectorization for mixed types
            | (SameType && (CanVectorizeLhs || CanVectorizeRhs) ? PacketAccessBit : 0) |
            (XprType::IsVectorAtCompileTime ? LinearAccessBit : 0),
 
    LhsOuterStrideBytes =
        int(LhsNestedCleaned::OuterStrideAtCompileTime) * int(sizeof(typename LhsNestedCleaned::Scalar)),
    RhsOuterStrideBytes =
        int(RhsNestedCleaned::OuterStrideAtCompileTime) * int(sizeof(typename RhsNestedCleaned::Scalar)),
 
    Alignment = bool(CanVectorizeLhs)
                    ? (LhsOuterStrideBytes <= 0 || (int(LhsOuterStrideBytes) % plain_enum_max(1, LhsAlignment)) != 0
                           ? 0
                           : LhsAlignment)
                : bool(CanVectorizeRhs)
                    ? (RhsOuterStrideBytes <= 0 || (int(RhsOuterStrideBytes) % plain_enum_max(1, RhsAlignment)) != 0
                           ? 0
                           : RhsAlignment)
                    : 0,
 
    /* CanVectorizeInner deserves special explanation. It does not affect the product flags. It is not used outside
     * of Product. If the Product itself is not a packet-access expression, there is still a chance that the inner
     * loop of the product might be vectorized. This is the meaning of CanVectorizeInner. Since it doesn't affect
     * the Flags, it is safe to make this value depend on ActualPacketAccessBit, that doesn't affect the ABI.
     */
    CanVectorizeInner = SameType && LhsRowMajor && (!RhsRowMajor) &&
                        (int(LhsFlags) & int(RhsFlags) & ActualPacketAccessBit) &&
                        (int(InnerSize) % packet_traits<Scalar>::size == 0)
  };
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const CoeffReturnType coeff(Index row, Index col) const {
    return (m_lhs.row(row).transpose().cwiseProduct(m_rhs.col(col))).sum();
  }
 
  /* Allow index-based non-packet access. It is impossible though to allow index-based packed access,
   * which is why we don't set the LinearAccessBit.
   * TODO: this seems possible when the result is a vector
   */
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const CoeffReturnType coeff(Index index) const {
    const Index row = (RowsAtCompileTime == 1 || MaxRowsAtCompileTime == 1) ? 0 : index;
    const Index col = (RowsAtCompileTime == 1 || MaxRowsAtCompileTime == 1) ? index : 0;
    return (m_lhs.row(row).transpose().cwiseProduct(m_rhs.col(col))).sum();
  }
 
  template <int LoadMode, typename PacketType>
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const PacketType packet(Index row, Index col) const {
    PacketType res;
    typedef etor_product_packet_impl<bool(int(Flags) & RowMajorBit) ? RowMajor : ColMajor,
                                     Unroll ? int(InnerSize) : Dynamic, LhsEtorType, RhsEtorType, PacketType, LoadMode>
        PacketImpl;
    PacketImpl::run(row, col, m_lhsImpl, m_rhsImpl, m_innerDim, res);
    return res;
  }
 
  template <int LoadMode, typename PacketType>
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const PacketType packet(Index index) const {
    const Index row = (RowsAtCompileTime == 1 || MaxRowsAtCompileTime == 1) ? 0 : index;
    const Index col = (RowsAtCompileTime == 1 || MaxRowsAtCompileTime == 1) ? index : 0;
    return packet<LoadMode, PacketType>(row, col);
  }
 
 protected:
  add_const_on_value_type_t<LhsNested> m_lhs;
  add_const_on_value_type_t<RhsNested> m_rhs;
 
  LhsEtorType m_lhsImpl;
  RhsEtorType m_rhsImpl;
 
  // TODO: Get rid of m_innerDim if known at compile time
  Index m_innerDim;
};
 
template <typename Lhs, typename Rhs>
struct product_evaluator<Product<Lhs, Rhs, DefaultProduct>, LazyCoeffBasedProductMode, DenseShape, DenseShape>
    : product_evaluator<Product<Lhs, Rhs, LazyProduct>, CoeffBasedProductMode, DenseShape, DenseShape> {
  typedef Product<Lhs, Rhs, DefaultProduct> XprType;
  typedef Product<Lhs, Rhs, LazyProduct> BaseProduct;
  typedef product_evaluator<BaseProduct, CoeffBasedProductMode, DenseShape, DenseShape> Base;
  enum { Flags = Base::Flags | EvalBeforeNestingBit };
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE explicit product_evaluator(const XprType& xpr)
      : Base(BaseProduct(xpr.lhs(), xpr.rhs())) {}
};
 
/****************************************
*** Coeff based product, Packet path  ***
****************************************/
 
template <int UnrollingIndex, typename Lhs, typename Rhs, typename Packet, int LoadMode>
struct etor_product_packet_impl<RowMajor, UnrollingIndex, Lhs, Rhs, Packet, LoadMode> {
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Index row, Index col, const Lhs& lhs, const Rhs& rhs,
                                                        Index innerDim, Packet& res) {
    etor_product_packet_impl<RowMajor, UnrollingIndex - 1, Lhs, Rhs, Packet, LoadMode>::run(row, col, lhs, rhs,
                                                                                            innerDim, res);
    res = pmadd(pset1<Packet>(lhs.coeff(row, Index(UnrollingIndex - 1))),
                rhs.template packet<LoadMode, Packet>(Index(UnrollingIndex - 1), col), res);
  }
};
 
template <int UnrollingIndex, typename Lhs, typename Rhs, typename Packet, int LoadMode>
struct etor_product_packet_impl<ColMajor, UnrollingIndex, Lhs, Rhs, Packet, LoadMode> {
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Index row, Index col, const Lhs& lhs, const Rhs& rhs,
                                                        Index innerDim, Packet& res) {
    etor_product_packet_impl<ColMajor, UnrollingIndex - 1, Lhs, Rhs, Packet, LoadMode>::run(row, col, lhs, rhs,
                                                                                            innerDim, res);
    res = pmadd(lhs.template packet<LoadMode, Packet>(row, Index(UnrollingIndex - 1)),
                pset1<Packet>(rhs.coeff(Index(UnrollingIndex - 1), col)), res);
  }
};
 
template <typename Lhs, typename Rhs, typename Packet, int LoadMode>
struct etor_product_packet_impl<RowMajor, 1, Lhs, Rhs, Packet, LoadMode> {
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Index row, Index col, const Lhs& lhs, const Rhs& rhs,
                                                        Index /*innerDim*/, Packet& res) {
    res = pmul(pset1<Packet>(lhs.coeff(row, Index(0))), rhs.template packet<LoadMode, Packet>(Index(0), col));
  }
};
 
template <typename Lhs, typename Rhs, typename Packet, int LoadMode>
struct etor_product_packet_impl<ColMajor, 1, Lhs, Rhs, Packet, LoadMode> {
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Index row, Index col, const Lhs& lhs, const Rhs& rhs,
                                                        Index /*innerDim*/, Packet& res) {
    res = pmul(lhs.template packet<LoadMode, Packet>(row, Index(0)), pset1<Packet>(rhs.coeff(Index(0), col)));
  }
};
 
template <typename Lhs, typename Rhs, typename Packet, int LoadMode>
struct etor_product_packet_impl<RowMajor, 0, Lhs, Rhs, Packet, LoadMode> {
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Index /*row*/, Index /*col*/, const Lhs& /*lhs*/,
                                                        const Rhs& /*rhs*/, Index /*innerDim*/, Packet& res) {
    res = pset1<Packet>(typename unpacket_traits<Packet>::type(0));
  }
};
 
template <typename Lhs, typename Rhs, typename Packet, int LoadMode>
struct etor_product_packet_impl<ColMajor, 0, Lhs, Rhs, Packet, LoadMode> {
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Index /*row*/, Index /*col*/, const Lhs& /*lhs*/,
                                                        const Rhs& /*rhs*/, Index /*innerDim*/, Packet& res) {
    res = pset1<Packet>(typename unpacket_traits<Packet>::type(0));
  }
};
 
template <typename Lhs, typename Rhs, typename Packet, int LoadMode>
struct etor_product_packet_impl<RowMajor, Dynamic, Lhs, Rhs, Packet, LoadMode> {
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Index row, Index col, const Lhs& lhs, const Rhs& rhs,
                                                        Index innerDim, Packet& res) {
    res = pset1<Packet>(typename unpacket_traits<Packet>::type(0));
    for (Index i = 0; i < innerDim; ++i)
      res = pmadd(pset1<Packet>(lhs.coeff(row, i)), rhs.template packet<LoadMode, Packet>(i, col), res);
  }
};
 
template <typename Lhs, typename Rhs, typename Packet, int LoadMode>
struct etor_product_packet_impl<ColMajor, Dynamic, Lhs, Rhs, Packet, LoadMode> {
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Index row, Index col, const Lhs& lhs, const Rhs& rhs,
                                                        Index innerDim, Packet& res) {
    res = pset1<Packet>(typename unpacket_traits<Packet>::type(0));
    for (Index i = 0; i < innerDim; ++i)
      res = pmadd(lhs.template packet<LoadMode, Packet>(row, i), pset1<Packet>(rhs.coeff(i, col)), res);
  }
};
 
/***************************************************************************
 * Triangular products
 ***************************************************************************/
template <int Mode, bool LhsIsTriangular, typename Lhs, bool LhsIsVector, typename Rhs, bool RhsIsVector>
struct triangular_product_impl;
 
template <typename Lhs, typename Rhs, int ProductTag>
struct generic_product_impl<Lhs, Rhs, TriangularShape, DenseShape, ProductTag>
    : generic_product_impl_base<Lhs, Rhs, generic_product_impl<Lhs, Rhs, TriangularShape, DenseShape, ProductTag>> {
  typedef typename Product<Lhs, Rhs>::Scalar Scalar;
 
  template <typename Dest>
  static void scaleAndAddTo(Dest& dst, const Lhs& lhs, const Rhs& rhs, const Scalar& alpha) {
    triangular_product_impl<Lhs::Mode, true, typename Lhs::MatrixType, false, Rhs, Rhs::ColsAtCompileTime == 1>::run(
        dst, lhs.nestedExpression(), rhs, alpha);
  }
};
 
template <typename Lhs, typename Rhs, int ProductTag>
struct generic_product_impl<Lhs, Rhs, DenseShape, TriangularShape, ProductTag>
    : generic_product_impl_base<Lhs, Rhs, generic_product_impl<Lhs, Rhs, DenseShape, TriangularShape, ProductTag>> {
  typedef typename Product<Lhs, Rhs>::Scalar Scalar;
 
  template <typename Dest>
  static void scaleAndAddTo(Dest& dst, const Lhs& lhs, const Rhs& rhs, const Scalar& alpha) {
    triangular_product_impl<Rhs::Mode, false, Lhs, Lhs::RowsAtCompileTime == 1, typename Rhs::MatrixType, false>::run(
        dst, lhs, rhs.nestedExpression(), alpha);
  }
};
 
/***************************************************************************
 * SelfAdjoint products
 ***************************************************************************/
template <typename Lhs, int LhsMode, bool LhsIsVector, typename Rhs, int RhsMode, bool RhsIsVector>
struct selfadjoint_product_impl;
 
template <typename Lhs, typename Rhs, int ProductTag>
struct generic_product_impl<Lhs, Rhs, SelfAdjointShape, DenseShape, ProductTag>
    : generic_product_impl_base<Lhs, Rhs, generic_product_impl<Lhs, Rhs, SelfAdjointShape, DenseShape, ProductTag>> {
  typedef typename Product<Lhs, Rhs>::Scalar Scalar;
 
  template <typename Dest>
  static EIGEN_DEVICE_FUNC void scaleAndAddTo(Dest& dst, const Lhs& lhs, const Rhs& rhs, const Scalar& alpha) {
    selfadjoint_product_impl<typename Lhs::MatrixType, Lhs::Mode, false, Rhs, 0, Rhs::IsVectorAtCompileTime>::run(
        dst, lhs.nestedExpression(), rhs, alpha);
  }
};
 
template <typename Lhs, typename Rhs, int ProductTag>
struct generic_product_impl<Lhs, Rhs, DenseShape, SelfAdjointShape, ProductTag>
    : generic_product_impl_base<Lhs, Rhs, generic_product_impl<Lhs, Rhs, DenseShape, SelfAdjointShape, ProductTag>> {
  typedef typename Product<Lhs, Rhs>::Scalar Scalar;
 
  template <typename Dest>
  static void scaleAndAddTo(Dest& dst, const Lhs& lhs, const Rhs& rhs, const Scalar& alpha) {
    selfadjoint_product_impl<Lhs, 0, Lhs::IsVectorAtCompileTime, typename Rhs::MatrixType, Rhs::Mode, false>::run(
        dst, lhs, rhs.nestedExpression(), alpha);
  }
};
 
/***************************************************************************
 * Diagonal products
 ***************************************************************************/
 
template <typename MatrixType, typename DiagonalType, typename Derived, int ProductOrder>
struct diagonal_product_evaluator_base : evaluator_base<Derived> {
  typedef typename ScalarBinaryOpTraits<typename MatrixType::Scalar, typename DiagonalType::Scalar>::ReturnType Scalar;
 
 public:
  enum {
    CoeffReadCost = int(NumTraits<Scalar>::MulCost) + int(evaluator<MatrixType>::CoeffReadCost) +
                    int(evaluator<DiagonalType>::CoeffReadCost),
 
    MatrixFlags = evaluator<MatrixType>::Flags,
    DiagFlags = evaluator<DiagonalType>::Flags,
 
    StorageOrder_ = (Derived::MaxRowsAtCompileTime == 1 && Derived::MaxColsAtCompileTime != 1)   ? RowMajor
                    : (Derived::MaxColsAtCompileTime == 1 && Derived::MaxRowsAtCompileTime != 1) ? ColMajor
                    : MatrixFlags & RowMajorBit                                                  ? RowMajor
                                                                                                 : ColMajor,
    SameStorageOrder_ = int(StorageOrder_) == ((MatrixFlags & RowMajorBit) ? RowMajor : ColMajor),
 
    ScalarAccessOnDiag_ = !((int(StorageOrder_) == ColMajor && int(ProductOrder) == OnTheLeft) ||
                            (int(StorageOrder_) == RowMajor && int(ProductOrder) == OnTheRight)),
    SameTypes_ = is_same<typename MatrixType::Scalar, typename DiagonalType::Scalar>::value,
    // FIXME currently we need same types, but in the future the next rule should be the one
    // Vectorizable_ = bool(int(MatrixFlags)&PacketAccessBit) && ((!_PacketOnDiag) || (SameTypes_ &&
    // bool(int(DiagFlags)&PacketAccessBit))),
    Vectorizable_ = bool(int(MatrixFlags) & PacketAccessBit) && SameTypes_ &&
                    (SameStorageOrder_ || (MatrixFlags & LinearAccessBit) == LinearAccessBit) &&
                    (ScalarAccessOnDiag_ || (bool(int(DiagFlags) & PacketAccessBit))),
    LinearAccessMask_ =
        (MatrixType::RowsAtCompileTime == 1 || MatrixType::ColsAtCompileTime == 1) ? LinearAccessBit : 0,
    Flags =
        ((HereditaryBits | LinearAccessMask_) & (unsigned int)(MatrixFlags)) | (Vectorizable_ ? PacketAccessBit : 0),
    Alignment = evaluator<MatrixType>::Alignment,
 
    AsScalarProduct =
        (DiagonalType::SizeAtCompileTime == 1) ||
        (DiagonalType::SizeAtCompileTime == Dynamic && MatrixType::RowsAtCompileTime == 1 &&
         ProductOrder == OnTheLeft) ||
        (DiagonalType::SizeAtCompileTime == Dynamic && MatrixType::ColsAtCompileTime == 1 && ProductOrder == OnTheRight)
  };
 
  EIGEN_DEVICE_FUNC diagonal_product_evaluator_base(const MatrixType& mat, const DiagonalType& diag)
      : m_diagImpl(diag), m_matImpl(mat) {
    EIGEN_INTERNAL_CHECK_COST_VALUE(NumTraits<Scalar>::MulCost);
    EIGEN_INTERNAL_CHECK_COST_VALUE(CoeffReadCost);
  }
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const Scalar coeff(Index idx) const {
    if (AsScalarProduct)
      return m_diagImpl.coeff(0) * m_matImpl.coeff(idx);
    else
      return m_diagImpl.coeff(idx) * m_matImpl.coeff(idx);
  }
 
 protected:
  template <int LoadMode, typename PacketType>
  EIGEN_STRONG_INLINE PacketType packet_impl(Index row, Index col, Index id, internal::true_type) const {
    return internal::pmul(m_matImpl.template packet<LoadMode, PacketType>(row, col),
                          internal::pset1<PacketType>(m_diagImpl.coeff(id)));
  }
 
  template <int LoadMode, typename PacketType>
  EIGEN_STRONG_INLINE PacketType packet_impl(Index row, Index col, Index id, internal::false_type) const {
    enum {
      InnerSize = (MatrixType::Flags & RowMajorBit) ? MatrixType::ColsAtCompileTime : MatrixType::RowsAtCompileTime,
      DiagonalPacketLoadMode = plain_enum_min(
          LoadMode,
          ((InnerSize % 16) == 0) ? int(Aligned16) : int(evaluator<DiagonalType>::Alignment))  // FIXME hardcoded 16!!
    };
    return internal::pmul(m_matImpl.template packet<LoadMode, PacketType>(row, col),
                          m_diagImpl.template packet<DiagonalPacketLoadMode, PacketType>(id));
  }
 
  evaluator<DiagonalType> m_diagImpl;
  evaluator<MatrixType> m_matImpl;
};
 
// diagonal * dense
template <typename Lhs, typename Rhs, int ProductKind, int ProductTag>
struct product_evaluator<Product<Lhs, Rhs, ProductKind>, ProductTag, DiagonalShape, DenseShape>
    : diagonal_product_evaluator_base<Rhs, typename Lhs::DiagonalVectorType, Product<Lhs, Rhs, LazyProduct>,
                                      OnTheLeft> {
  typedef diagonal_product_evaluator_base<Rhs, typename Lhs::DiagonalVectorType, Product<Lhs, Rhs, LazyProduct>,
                                          OnTheLeft>
      Base;
  using Base::coeff;
  using Base::m_diagImpl;
  using Base::m_matImpl;
  typedef typename Base::Scalar Scalar;
 
  typedef Product<Lhs, Rhs, ProductKind> XprType;
  typedef typename XprType::PlainObject PlainObject;
  typedef typename Lhs::DiagonalVectorType DiagonalType;
 
  enum { StorageOrder = Base::StorageOrder_ };
 
  EIGEN_DEVICE_FUNC explicit product_evaluator(const XprType& xpr) : Base(xpr.rhs(), xpr.lhs().diagonal()) {}
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const Scalar coeff(Index row, Index col) const {
    return m_diagImpl.coeff(row) * m_matImpl.coeff(row, col);
  }
 
#ifndef EIGEN_GPUCC
  template <int LoadMode, typename PacketType>
  EIGEN_STRONG_INLINE PacketType packet(Index row, Index col) const {
    // FIXME: NVCC used to complain about the template keyword, but we have to check whether this is still the case.
    // See also similar calls below.
    return this->template packet_impl<LoadMode, PacketType>(
        row, col, row, std::conditional_t<int(StorageOrder) == RowMajor, internal::true_type, internal::false_type>());
  }
 
  template <int LoadMode, typename PacketType>
  EIGEN_STRONG_INLINE PacketType packet(Index idx) const {
    return packet<LoadMode, PacketType>(int(StorageOrder) == ColMajor ? idx : 0,
                                        int(StorageOrder) == ColMajor ? 0 : idx);
  }
#endif
};
 
// dense * diagonal
template <typename Lhs, typename Rhs, int ProductKind, int ProductTag>
struct product_evaluator<Product<Lhs, Rhs, ProductKind>, ProductTag, DenseShape, DiagonalShape>
    : diagonal_product_evaluator_base<Lhs, typename Rhs::DiagonalVectorType, Product<Lhs, Rhs, LazyProduct>,
                                      OnTheRight> {
  typedef diagonal_product_evaluator_base<Lhs, typename Rhs::DiagonalVectorType, Product<Lhs, Rhs, LazyProduct>,
                                          OnTheRight>
      Base;
  using Base::coeff;
  using Base::m_diagImpl;
  using Base::m_matImpl;
  typedef typename Base::Scalar Scalar;
 
  typedef Product<Lhs, Rhs, ProductKind> XprType;
  typedef typename XprType::PlainObject PlainObject;
 
  enum { StorageOrder = Base::StorageOrder_ };
 
  EIGEN_DEVICE_FUNC explicit product_evaluator(const XprType& xpr) : Base(xpr.lhs(), xpr.rhs().diagonal()) {}
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const Scalar coeff(Index row, Index col) const {
    return m_matImpl.coeff(row, col) * m_diagImpl.coeff(col);
  }
 
#ifndef EIGEN_GPUCC
  template <int LoadMode, typename PacketType>
  EIGEN_STRONG_INLINE PacketType packet(Index row, Index col) const {
    return this->template packet_impl<LoadMode, PacketType>(
        row, col, col, std::conditional_t<int(StorageOrder) == ColMajor, internal::true_type, internal::false_type>());
  }
 
  template <int LoadMode, typename PacketType>
  EIGEN_STRONG_INLINE PacketType packet(Index idx) const {
    return packet<LoadMode, PacketType>(int(StorageOrder) == ColMajor ? idx : 0,
                                        int(StorageOrder) == ColMajor ? 0 : idx);
  }
#endif
};
 
/***************************************************************************
 * Products with permutation matrices
 ***************************************************************************/
 
template <typename ExpressionType, int Side, bool Transposed, typename ExpressionShape>
struct permutation_matrix_product;
 
template <typename ExpressionType, int Side, bool Transposed>
struct permutation_matrix_product<ExpressionType, Side, Transposed, DenseShape> {
  typedef typename nested_eval<ExpressionType, 1>::type MatrixType;
  typedef remove_all_t<MatrixType> MatrixTypeCleaned;
 
  template <typename Dest, typename PermutationType>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Dest& dst, const PermutationType& perm,
                                                        const ExpressionType& xpr) {
    MatrixType mat(xpr);
    const Index n = Side == OnTheLeft ? mat.rows() : mat.cols();
    // FIXME we need an is_same for expression that is not sensitive to constness. For instance
    // is_same_xpr<Block<const Matrix>, Block<Matrix> >::value should be true.
    // if(is_same<MatrixTypeCleaned,Dest>::value && extract_data(dst) == extract_data(mat))
    if (is_same_dense(dst, mat)) {
      // apply the permutation inplace
      Matrix<bool, PermutationType::RowsAtCompileTime, 1, 0, PermutationType::MaxRowsAtCompileTime> mask(perm.size());
      mask.fill(false);
      Index r = 0;
      while (r < perm.size()) {
        // search for the next seed
        while (r < perm.size() && mask[r]) r++;
        if (r >= perm.size()) break;
        // we got one, let's follow it until we are back to the seed
        Index k0 = r++;
        Index kPrev = k0;
        mask.coeffRef(k0) = true;
        for (Index k = perm.indices().coeff(k0); k != k0; k = perm.indices().coeff(k)) {
          Block<Dest, Side == OnTheLeft ? 1 : Dest::RowsAtCompileTime,
                Side == OnTheRight ? 1 : Dest::ColsAtCompileTime>(dst, k)
              .swap(Block < Dest, Side == OnTheLeft ? 1 : Dest::RowsAtCompileTime,
                    Side == OnTheRight
                        ? 1
                        : Dest::ColsAtCompileTime > (dst, ((Side == OnTheLeft) ^ Transposed) ? k0 : kPrev));
 
          mask.coeffRef(k) = true;
          kPrev = k;
        }
      }
    } else {
      for (Index i = 0; i < n; ++i) {
        Block<Dest, Side == OnTheLeft ? 1 : Dest::RowsAtCompileTime, Side == OnTheRight ? 1 : Dest::ColsAtCompileTime>(
            dst, ((Side == OnTheLeft) ^ Transposed) ? perm.indices().coeff(i) : i)
 
            =
 
                Block < const MatrixTypeCleaned,
            Side == OnTheLeft ? 1 : MatrixTypeCleaned::RowsAtCompileTime,
            Side == OnTheRight ? 1
                               : MatrixTypeCleaned::ColsAtCompileTime >
                                     (mat, ((Side == OnTheRight) ^ Transposed) ? perm.indices().coeff(i) : i);
      }
    }
  }
};
 
template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape>
struct generic_product_impl<Lhs, Rhs, PermutationShape, MatrixShape, ProductTag> {
  template <typename Dest>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Lhs& lhs, const Rhs& rhs) {
    permutation_matrix_product<Rhs, OnTheLeft, false, MatrixShape>::run(dst, lhs, rhs);
  }
};
 
template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape>
struct generic_product_impl<Lhs, Rhs, MatrixShape, PermutationShape, ProductTag> {
  template <typename Dest>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Lhs& lhs, const Rhs& rhs) {
    permutation_matrix_product<Lhs, OnTheRight, false, MatrixShape>::run(dst, rhs, lhs);
  }
};
 
template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape>
struct generic_product_impl<Inverse<Lhs>, Rhs, PermutationShape, MatrixShape, ProductTag> {
  template <typename Dest>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Inverse<Lhs>& lhs, const Rhs& rhs) {
    permutation_matrix_product<Rhs, OnTheLeft, true, MatrixShape>::run(dst, lhs.nestedExpression(), rhs);
  }
};
 
template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape>
struct generic_product_impl<Lhs, Inverse<Rhs>, MatrixShape, PermutationShape, ProductTag> {
  template <typename Dest>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Lhs& lhs, const Inverse<Rhs>& rhs) {
    permutation_matrix_product<Lhs, OnTheRight, true, MatrixShape>::run(dst, rhs.nestedExpression(), lhs);
  }
};
 
/***************************************************************************
 * Products with transpositions matrices
 ***************************************************************************/
 
// FIXME could we unify Transpositions and Permutation into a single "shape"??
 
template <typename ExpressionType, int Side, bool Transposed, typename ExpressionShape>
struct transposition_matrix_product {
  typedef typename nested_eval<ExpressionType, 1>::type MatrixType;
  typedef remove_all_t<MatrixType> MatrixTypeCleaned;
 
  template <typename Dest, typename TranspositionType>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Dest& dst, const TranspositionType& tr,
                                                        const ExpressionType& xpr) {
    MatrixType mat(xpr);
    typedef typename TranspositionType::StorageIndex StorageIndex;
    const Index size = tr.size();
    StorageIndex j = 0;
 
    if (!is_same_dense(dst, mat)) dst = mat;
 
    for (Index k = (Transposed ? size - 1 : 0); Transposed ? k >= 0 : k < size; Transposed ? --k : ++k)
      if (Index(j = tr.coeff(k)) != k) {
        if (Side == OnTheLeft)
          dst.row(k).swap(dst.row(j));
        else if (Side == OnTheRight)
          dst.col(k).swap(dst.col(j));
      }
  }
};
 
template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape>
struct generic_product_impl<Lhs, Rhs, TranspositionsShape, MatrixShape, ProductTag> {
  template <typename Dest>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Lhs& lhs, const Rhs& rhs) {
    transposition_matrix_product<Rhs, OnTheLeft, false, MatrixShape>::run(dst, lhs, rhs);
  }
};
 
template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape>
struct generic_product_impl<Lhs, Rhs, MatrixShape, TranspositionsShape, ProductTag> {
  template <typename Dest>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Lhs& lhs, const Rhs& rhs) {
    transposition_matrix_product<Lhs, OnTheRight, false, MatrixShape>::run(dst, rhs, lhs);
  }
};
 
template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape>
struct generic_product_impl<Transpose<Lhs>, Rhs, TranspositionsShape, MatrixShape, ProductTag> {
  template <typename Dest>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Transpose<Lhs>& lhs, const Rhs& rhs) {
    transposition_matrix_product<Rhs, OnTheLeft, true, MatrixShape>::run(dst, lhs.nestedExpression(), rhs);
  }
};
 
template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape>
struct generic_product_impl<Lhs, Transpose<Rhs>, MatrixShape, TranspositionsShape, ProductTag> {
  template <typename Dest>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Lhs& lhs, const Transpose<Rhs>& rhs) {
    transposition_matrix_product<Lhs, OnTheRight, true, MatrixShape>::run(dst, rhs.nestedExpression(), lhs);
  }
};
 
/***************************************************************************
 * skew symmetric products
 * for now we just call the generic implementation
 ***************************************************************************/
template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape>
struct generic_product_impl<Lhs, Rhs, SkewSymmetricShape, MatrixShape, ProductTag> {
  template <typename Dest>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Lhs& lhs, const Rhs& rhs) {
    generic_product_impl<typename Lhs::DenseMatrixType, Rhs, DenseShape, MatrixShape, ProductTag>::evalTo(dst, lhs,
                                                                                                          rhs);
  }
};
 
template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape>
struct generic_product_impl<Lhs, Rhs, MatrixShape, SkewSymmetricShape, ProductTag> {
  template <typename Dest>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Lhs& lhs, const Rhs& rhs) {
    generic_product_impl<Lhs, typename Rhs::DenseMatrixType, MatrixShape, DenseShape, ProductTag>::evalTo(dst, lhs,
                                                                                                          rhs);
  }
};
 
template <typename Lhs, typename Rhs, int ProductTag>
struct generic_product_impl<Lhs, Rhs, SkewSymmetricShape, SkewSymmetricShape, ProductTag> {
  template <typename Dest>
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Lhs& lhs, const Rhs& rhs) {
    generic_product_impl<typename Lhs::DenseMatrixType, typename Rhs::DenseMatrixType, DenseShape, DenseShape,
                         ProductTag>::evalTo(dst, lhs, rhs);
  }
};
 
}  // end namespace internal
 
}  // end namespace Eigen
 
#endif  // EIGEN_PRODUCT_EVALUATORS_H