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 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
 // 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_CXX11_TENSOR_TENSOR_BLOCK_H
 #define EIGEN_CXX11_TENSOR_TENSOR_BLOCK_H
 
 namespace Eigen {
 namespace internal {
 
 // -------------------------------------------------------------------------- //
 // Forward declarations for templates defined below.
 template <typename Scalar, typename IndexType, int NumDims, int Layout>
 class TensorBlockIO;
 
 // -------------------------------------------------------------------------- //
 // Helper function to compute strides for densely stored buffer of given
 // dimensions.
 
 // TODO(ezhulenev): We compute strides 1000 times in different evaluators, use
 // this function instead everywhere.
 template <int Layout, typename IndexType, int NumDims>
 EIGEN_ALWAYS_INLINE DSizes<IndexType, NumDims> strides(
     const DSizes<IndexType, NumDims>& dimensions) {
   DSizes<IndexType, NumDims> strides;
   if (NumDims == 0) return strides;
 
   // TODO(ezhulenev): Use templates to unroll this loop (similar to
   // h_array_reduce in CXX11meta.h)? Benchmark it.
   if (static_cast<int>(Layout) == static_cast<int>(ColMajor)) {
     strides[0] = 1;
     for (int i = 1; i < NumDims; ++i) {
       strides[i] = strides[i - 1] * dimensions[i - 1];
     }
   } else {
     strides[NumDims - 1] = 1;
     for (int i = NumDims - 2; i >= 0; --i) {
       strides[i] = strides[i + 1] * dimensions[i + 1];
     }
   }
 
   return strides;
 }
 
 template <int Layout, typename IndexType, size_t NumDims>
 EIGEN_ALWAYS_INLINE DSizes<IndexType, NumDims> strides(
     const Eigen::array<IndexType, NumDims>& dimensions) {
   return strides<Layout>(DSizes<IndexType, NumDims>(dimensions));
 }
 
 template <int Layout, std::ptrdiff_t... Indices>
 EIGEN_STRONG_INLINE DSizes<std::ptrdiff_t, sizeof...(Indices)> strides(
     const Sizes<Indices...>& sizes) {
   return strides<Layout>(DSizes<std::ptrdiff_t, sizeof...(Indices)>(sizes));
 }
 
 // -------------------------------------------------------------------------- //
 
 // Tensor block shape type defines what are the shape preference for the blocks
 // extracted from the larger tensor.
 //
 // Example: blocks of 100 elements from the large 100x100 tensor:
 // - tensor: 100x100
 // - target_block_size: 100
 //
 // TensorBlockShapeType:
 //  - kUniformAllDims: 100 blocks of size 10x10
 //  - kSkewedInnerDims: 100 blocks of size 100x1 (or 1x100 depending on a column
 //                      or row major layout)
 enum class TensorBlockShapeType { kUniformAllDims, kSkewedInnerDims };
 
 struct TensorBlockResourceRequirements {
   TensorBlockShapeType shape_type;  // target block shape
   size_t size;                      // target block size
   TensorOpCost cost_per_coeff;      // cost of computing a single block element
 
 #ifdef EIGEN_HIPCC
   // For HIPCC, we need to explicitly declare as a "device fun", the constructor
   // which is implicitly invoked in the "merge" / "any" routines. else HIPCC
   // errors out complaining about the lack of a matching constructor
   EIGEN_DEVICE_FUNC
   TensorBlockResourceRequirements(TensorBlockShapeType shape_type_, size_t size_,
                   TensorOpCost cost_)
     : shape_type(shape_type_), size(size_), cost_per_coeff(cost_)
   {}
 #endif
 
   template <typename Scalar>
   EIGEN_DEVICE_FUNC static TensorBlockResourceRequirements withShapeAndSize(
       TensorBlockShapeType shape_type, size_t size_in_bytes,
       TensorOpCost cost) {
     const size_t size = numext::maxi(size_t(1), size_in_bytes / sizeof(Scalar));
     return {shape_type, size, cost};
   }
 
   template <typename Scalar>
   EIGEN_DEVICE_FUNC static TensorBlockResourceRequirements withShapeAndSize(
       TensorBlockShapeType shape_type, size_t size_in_bytes) {
     // This default cost per coefficient is valid for most materialized tensor
     // block evaluation implementations, because they typically just read
     // coefficients from the underlying tensor storage, and write to the tensor
     // block buffer (scratch or destination memory, reads and writes have linear
     // access pattern). We ignore the fixed cost of block evaluation, because in
     // practice it should negligible.
     //
     // Lazy block evaluation adds the cost of calling a functor for each
     // coefficient.
     //
     // All non-trivial block evaluation implementations must provide their own
     // cost approximation (e.g. shuffling inner dimension has a much higher cost
     // because it reads memory randomly, although the total number of moved
     // bytes is the same).
     return withShapeAndSize<Scalar>(shape_type, size_in_bytes,
                                     {/*bytes_loaded=*/sizeof(Scalar),
                                      /*bytes_stored=*/sizeof(Scalar),
                                      /*compute_cycles=*/0});
   }
 
   template <typename Scalar>
   EIGEN_DEVICE_FUNC static TensorBlockResourceRequirements skewed(
       size_t size_in_bytes) {
     return withShapeAndSize<Scalar>(TensorBlockShapeType::kSkewedInnerDims,
                                     size_in_bytes);
   }
 
   template <typename Scalar>
   EIGEN_DEVICE_FUNC static TensorBlockResourceRequirements uniform(
       size_t size_in_bytes) {
     return withShapeAndSize<Scalar>(TensorBlockShapeType::kUniformAllDims,
                                     size_in_bytes);
   }
 
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE TensorBlockResourceRequirements
   merge(const TensorBlockResourceRequirements& lhs,
         const TensorBlockResourceRequirements& rhs) {
     return {merge(lhs.shape_type, rhs.shape_type),           // shape_type
             merge(lhs.size, rhs.size),                       // size
             merge(lhs.cost_per_coeff, rhs.cost_per_coeff)};  // cost_per_coeff
   }
 
   EIGEN_DEVICE_FUNC TensorBlockResourceRequirements& addCostPerCoeff(
       TensorOpCost cost) {
     cost_per_coeff += cost;
     return *this;
   }
 
   // This is a resource requirement that should be returned from expressions
   // that do not have any block evaluation preference (e.g. default tensor
   // expression with raw buffer access).
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE TensorBlockResourceRequirements any() {
     return {TensorBlockShapeType::kUniformAllDims, 1, {0, 0, 0}};
   }
 
  private:
   using Requirements = TensorBlockResourceRequirements;
 
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE size_t merge(size_t lhs_size, size_t rhs_size) {
     return numext::maxi(lhs_size, rhs_size);
   }
 
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE TensorBlockShapeType
   merge(TensorBlockShapeType lhs, TensorBlockShapeType rhs) {
     return (lhs == TensorBlockShapeType::kSkewedInnerDims ||
             rhs == TensorBlockShapeType::kSkewedInnerDims)
                ? TensorBlockShapeType::kSkewedInnerDims
                : TensorBlockShapeType::kUniformAllDims;
   }
 
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE TensorOpCost merge(TensorOpCost lhs_cost,
                                                 TensorOpCost rhs_cost) {
     return lhs_cost + rhs_cost;
   }
 };
 
 // -------------------------------------------------------------------------- //
 // TensorBlockDescriptor specifies a block offset within a tensor and the block
 // sizes along each of the tensor dimensions.
 
 template <int NumDims, typename IndexType = Eigen::Index>
 class TensorBlockDescriptor {
  public:
   typedef DSizes<IndexType, NumDims> Dimensions;
 
   // If we evaluate a Tensor assignment, and expression on the left, already has
   // a memory buffer, then we might do performance optimization, and evaluate
   // the root expression directly into the final output memory. Some time it's
   // possible to reuse it for materializing subexpressions inside an expression
   // tree, to to avoid dynamic memory allocation.
   //
   // The pointer type of the underlying storage is erased, because passing
   // Scalar type through all the expression evaluation layers is way too many
   // templates. In practice destination buffer type should always match the
   // evaluated expression scalar type.
   class DestinationBuffer {
    public:
     enum DestinationBufferKind : int {
       // The above explicit specification of "int" as the enum basetype is
       // needed to get around a HIPCC link error ("the field type is not
       // amp-compatible")
       // which is issued for class members with the enum type.
       // TODO(rocm):
       // remove the "int" basetype once HIPCC has been fixed to not error out
       // in the above scenario.
 
       // Destination buffer is not defined (`m_data` == nullptr).
       kEmpty,
 
       // Tensor block defined by an owning tensor block descriptor can fit
       // contiguously into the destination buffer. In this case it's safe to
       // materialize tensor block in the destination buffer, wrap it in a
       // TensorMap, and use to build Eigen expression on top of it.
       kContiguous,
 
       // Destination buffer strides do not match strides of the contiguously
       // stored block, and it's impossible to define a TensorMap over this
       // buffer. However if we are evaluating a root of an expression tree, we
       // still can materialize an output into this destination, because we can
       // guarantee that no one will ever access it through block API.
       //
       // In theory it is possible to build valid TensorStriding<TensorMap>
       // expression on top of this destination buffer, however it has
       // inefficient coeff/packet access, and defeats the purpose of fast block
       // evaluation API.
       kStrided
     };
 
     template <typename Scalar>
     Scalar* data() const {
       eigen_assert(m_data_type_size == sizeof(Scalar));
       return static_cast<Scalar*>(m_data);
     }
 
     const Dimensions& strides() const { return m_strides; }
     const DestinationBufferKind& kind() const { return m_kind; }
 
    private:
     friend class TensorBlockDescriptor;
 
     DestinationBuffer() : m_data(NULL), m_data_type_size(0), m_kind(kEmpty) {}
 
     template <typename Scalar>
     DestinationBuffer(Scalar* data, const Dimensions& strides,
                       DestinationBufferKind kind)
         : m_data(static_cast<void*>(data)),
           m_data_type_size(sizeof(Scalar)),
           m_strides(strides),
           m_kind(kind) {}
 
     template <int Layout, typename Scalar>
     static DestinationBuffer make(const TensorBlockDescriptor& desc,
                                   Scalar* data, const Dimensions& strides) {
       return DestinationBuffer(data, strides, kind<Layout>(desc, strides));
     }
 
     template <int Layout>
     static DestinationBufferKind kind(const TensorBlockDescriptor& desc,
                                       const Dimensions& strides) {
       const Dimensions& desc_dims = desc.dimensions();
       const Dimensions& desc_strides = internal::strides<Layout>(desc_dims);
       for (int i = 0; i < NumDims; ++i) {
         if (desc_dims[i] == 1) continue;
         if (desc_strides[i] != strides[i]) return kStrided;
       }
       return kContiguous;
     }
 
     // Storage pointer is type erased, to reduce template bloat, but we still
     // keep the size of the underlying element type for error checking.
     void* m_data;
     size_t m_data_type_size;
 
     // Destination buffer dimensions always match the dimensions of a tensor
     // block descriptor it belongs to, however strides might be different.
     Dimensions m_strides;
 
     DestinationBufferKind m_kind;
   };
 
   TensorBlockDescriptor(const IndexType offset, const Dimensions& dimensions,
                         const DestinationBuffer& destination)
       : m_offset(offset),
         m_dimensions(dimensions),
         m_destination(destination) {}
 
   TensorBlockDescriptor(const IndexType offset, const Dimensions& dimensions)
       : m_offset(offset),
         m_dimensions(dimensions),
         m_destination(DestinationBuffer()) {}
 
   IndexType offset() const { return m_offset; }
   const Dimensions& dimensions() const { return m_dimensions; }
   IndexType dimension(int index) const { return m_dimensions[index]; }
   IndexType size() const { return array_prod<IndexType>(m_dimensions); }
 
   const DestinationBuffer& destination() const { return m_destination; }
 
   template <int Layout, typename Scalar>
   void AddDestinationBuffer(Scalar* dst_base, const Dimensions& dst_strides) {
     eigen_assert(dst_base != NULL);
     m_destination =
         DestinationBuffer::template make<Layout>(*this, dst_base, dst_strides);
   }
 
   template <int Layout, typename Scalar, typename DstStridesIndexType>
   void AddDestinationBuffer(
       Scalar* dst_base,
       const DSizes<DstStridesIndexType, NumDims>& dst_strides) {
     // DSizes constructor will do index type promotion if it's safe.
     AddDestinationBuffer<Layout>(dst_base, Dimensions(dst_strides));
   }
 
   TensorBlockDescriptor& DropDestinationBuffer() {
     m_destination.m_data = NULL;
     m_destination.m_kind = DestinationBuffer::kEmpty;
     return *this;
   }
 
   bool HasDestinationBuffer() const {
     return m_destination.kind() != DestinationBuffer::kEmpty;
   }
 
   // Returns a copy of `*this` with updated offset.
   TensorBlockDescriptor WithOffset(IndexType offset) const {
     return TensorBlockDescriptor(offset, m_dimensions, m_destination);
   }
 
  private:
   // Offset and dimensions are immutable after construction. Block descriptor
   // can only be mutated by adding or dropping destination.
   const IndexType m_offset;
   const Dimensions m_dimensions;
   DestinationBuffer m_destination;
 };
 
 // -------------------------------------------------------------------------- //
 // TensorBlockMapper is responsible for iterating over the blocks of a tensor.
 
 template <int NumDims, int Layout, typename IndexType = Eigen::Index>
 class TensorBlockMapper {
   typedef TensorBlockDescriptor<NumDims, IndexType> BlockDescriptor;
 
  public:
   typedef DSizes<IndexType, NumDims> Dimensions;
 
   TensorBlockMapper() = default;
   TensorBlockMapper(const DSizes<IndexType, NumDims>& dimensions,
                     const TensorBlockResourceRequirements& requirements)
       : m_tensor_dimensions(dimensions), m_requirements(requirements) {
     // Compute block dimensions and the total number of blocks.
     InitializeBlockDimensions();
   }
 
   EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE IndexType blockCount() const {
     return m_total_block_count;
   }
 
   EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE IndexType blockTotalSize() const {
     return m_block_dimensions.TotalSize();
   }
 
   EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const DSizes<IndexType, NumDims>&
   blockDimensions() const {
     return m_block_dimensions;
   }
 
   EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE BlockDescriptor
   blockDescriptor(IndexType block_index) const {
     static const bool isColMajor = Layout == static_cast<int>(ColMajor);
 
     IndexType offset = 0;
     DSizes<IndexType, NumDims> dimensions;
 
     if (NumDims == 0) return BlockDescriptor(offset, dimensions);
 
     // Iterate outer -> inner dimensions.
     for (int i = NumDims - 1; i >= 0; --i) {
       const int dim = isColMajor ? i : NumDims - i - 1;
 
       const IndexType idx = block_index / m_block_strides[dim];
       block_index -= idx * m_block_strides[dim];
 
       const IndexType coord = idx * m_block_dimensions[dim];
       dimensions[dim] = numext::mini(m_tensor_dimensions[dim] - coord,
                                      m_block_dimensions[dim]);
       offset += coord * m_tensor_strides[dim];
     }
 
     return {offset, dimensions};
   }
 
  private:
   void InitializeBlockDimensions() {
     // Requested block shape and size.
     const TensorBlockShapeType shape_type = m_requirements.shape_type;
     IndexType target_block_size =
         numext::maxi<IndexType>(1, static_cast<IndexType>(m_requirements.size));
 
     IndexType tensor_size = m_tensor_dimensions.TotalSize();
 
     // Corner case: one of the dimensions is zero. Logic below is too complex
     // to handle this case on a general basis, just use unit block size.
     // Note: we must not yield blocks with zero dimensions (recipe for
     // overflows/underflows, divisions by zero and NaNs later).
     if (tensor_size == 0) {
       for (int i = 0; i < NumDims; ++i) {
         m_block_dimensions[i] = 1;
       }
       m_total_block_count = 0;
       return;
     }
 
     // If tensor fits into a target block size, evaluate it as a single block.
     if (tensor_size <= target_block_size) {
       m_block_dimensions = m_tensor_dimensions;
       m_total_block_count = 1;
       // The only valid block index is `0`, and in this case we do not need
       // to compute real strides for tensor or blocks (see blockDescriptor).
       for (int i = 0; i < NumDims; ++i) {
         m_tensor_strides[i] = 0;
         m_block_strides[i] = 1;
       }
       return;
     }
 
     static const bool isColMajor = Layout == static_cast<int>(ColMajor);
 
     // Block shape skewed towards inner dimension.
     if (shape_type == TensorBlockShapeType::kSkewedInnerDims) {
       IndexType coeff_to_allocate = target_block_size;
 
       for (int i = 0; i < NumDims; ++i) {
         const int dim = isColMajor ? i : NumDims - i - 1;
         m_block_dimensions[dim] =
             numext::mini(coeff_to_allocate, m_tensor_dimensions[dim]);
         coeff_to_allocate = divup(
             coeff_to_allocate,
             numext::maxi(static_cast<IndexType>(1), m_block_dimensions[dim]));
       }
       eigen_assert(coeff_to_allocate == 1);
 
     } else if (shape_type == TensorBlockShapeType::kUniformAllDims) {
       // Tensor will not fit within 'target_block_size' budget: calculate tensor
       // block dimension sizes based on "square" dimension size target.
       const IndexType dim_size_target = convert_index<IndexType>(
           std::pow(static_cast<float>(target_block_size),
                    1.0f / static_cast<float>(m_block_dimensions.rank())));
 
       for (int i = 0; i < NumDims; ++i) {
         // TODO(andydavis) Adjust the inner most 'block_dim_size' to make it
         // a multiple of the packet size. Note that reducing
         // 'block_dim_size' in this manner can increase the number of
         // blocks, and so will amplify any per-block overhead.
         m_block_dimensions[i] =
             numext::mini(dim_size_target, m_tensor_dimensions[i]);
       }
 
       // Add any un-allocated coefficients to inner dimension(s).
       IndexType total_size = m_block_dimensions.TotalSize();
       for (int i = 0; i < NumDims; ++i) {
         const int dim = isColMajor ? i : NumDims - i - 1;
 
         if (m_block_dimensions[dim] < m_tensor_dimensions[dim]) {
           const IndexType total_size_other_dims =
               total_size / m_block_dimensions[dim];
           const IndexType alloc_avail =
               divup<IndexType>(target_block_size, total_size_other_dims);
           if (alloc_avail == m_block_dimensions[dim]) {
             // Insufficient excess coefficients to allocate.
             break;
           }
           m_block_dimensions[dim] =
               numext::mini(m_tensor_dimensions[dim], alloc_avail);
           total_size = total_size_other_dims * m_block_dimensions[dim];
         }
       }
 
     } else {
       eigen_assert(false);  // unknown block shape
     }
 
     eigen_assert(m_block_dimensions.TotalSize() >=
                  numext::mini<IndexType>(target_block_size,
                                          m_tensor_dimensions.TotalSize()));
 
     // Calculate block counts by dimension and total block count.
     DSizes<IndexType, NumDims> block_count;
     for (int i = 0; i < NumDims; ++i) {
       block_count[i] = divup(m_tensor_dimensions[i], m_block_dimensions[i]);
     }
     m_total_block_count = array_prod(block_count);
 
     // Calculate block strides (used for enumerating blocks).
     m_tensor_strides = strides<Layout>(m_tensor_dimensions);
     m_block_strides = strides<Layout>(block_count);
   }
 
   DSizes<IndexType, NumDims> m_tensor_dimensions;
   TensorBlockResourceRequirements m_requirements;
 
   DSizes<IndexType, NumDims> m_block_dimensions;
   IndexType m_total_block_count;
 
   DSizes<IndexType, NumDims> m_tensor_strides;
   DSizes<IndexType, NumDims> m_block_strides;
 };
 
 // -------------------------------------------------------------------------- //
 // TensorBlockScratchAllocator is responsible for allocating temporary buffers
 // for block evaluation (output or input block materialization). Given that
 // Eigen expression traversal order is deterministic, all temporary allocations
 // are happening in the same order, and usually have exactly the same size.
 // Scratch allocator keeps a trace of all dynamic allocations, and after the
 // first block evaluation is completed, we should be able to reuse all the
 // temporary buffers for the next block evaluation.
 
 template <typename Device>
 class TensorBlockScratchAllocator {
  public:
   explicit TensorBlockScratchAllocator(const Device& device)
       : m_device(device), m_allocation_index(0) {}
 
   ~TensorBlockScratchAllocator() {
     for (size_t i = 0; i < m_allocations.size(); ++i) {
       m_device.deallocate(m_allocations[i].ptr);
     }
   }
 
   void* allocate(size_t size) {
     // TODO(ezhulenev): Remove when replaced with inlined vector.
     if (m_allocations.capacity() == 0) m_allocations.reserve(8);
 
     // Check if we already have an existing allocation att current index.
     const int num_allocations = static_cast<int>(m_allocations.size());
     const bool has_allocation = m_allocation_index < num_allocations;
 
     // Allocation index can't be larger than the number of allocations.
     eigen_assert(m_allocation_index <= num_allocations);
 
     // If we have existing allocation, and its size is larger or equal to
     // requested size, we do nothing.
 
     // If current allocation can't fit requested size, we deallocate it, and
     // replace with a larger allocation.
     if (has_allocation && m_allocations[m_allocation_index].size < size) {
       m_device.deallocate(m_allocations[m_allocation_index].ptr);
       m_allocations[m_allocation_index].ptr = m_device.allocate(size);
       m_allocations[m_allocation_index].size = size;
     }
 
     // Make a new allocation if we don't have and existing one.
     if (!has_allocation) {
       Allocation allocation;
       allocation.ptr = m_device.allocate(size);
       allocation.size = size;
       m_allocations.push_back(allocation);
     }
 
     eigen_assert(m_allocations[m_allocation_index].ptr != NULL);
     eigen_assert(m_allocations[m_allocation_index].size >= size);
 
     return m_allocations[m_allocation_index++].ptr;
   }
 
   void reset() { m_allocation_index = 0; }
 
  private:
   struct Allocation {
     void* ptr;
     size_t size;
   };
 
   const Device& m_device;
   int m_allocation_index;
   // TODO(ezhulenev): This should be an inlined vector.
   std::vector<Allocation> m_allocations;
 };
 
 // -------------------------------------------------------------------------- //
 // TensorBlockKind represents all possible block kinds, that can be produced by
 // TensorEvaluator::evalBlock function.
 enum TensorBlockKind {
   // Tensor block that is a lazy expression that must be assigned to a
   // destination using TensorBlockAssign.
   kExpr,
 
   // Tensor block that is a view into a memory buffer owned by an underlying
   // Tensor expression (e.g. it can be a view into a Tensor buffer).
   kView,
 
   // Tensor block that was materialized in a scratch memory buffer, allocated
   // with TensorBlockScratchAllocator. This block must be copied to a
   // destination, similar to a block of `kExpr` type.
   kMaterializedInScratch,
 
   // Tensor block that was materialized directly into the final output memory
   // buffer. For example if the left side of an assignment is a Tensor, we can
   // directly materialize the block in the destination memory.
   //
   // If strides in the output buffer do not match tensor block strides, the
   // Tensor expression will be invalid, and should not be used by
   // TensorBlockAssign or for constructing another block expression.
   kMaterializedInOutput
 };
 
 // -------------------------------------------------------------------------- //
 // TensorBlockNotImplemented should be used to defined TensorBlock typedef in
 // TensorEvaluators that do not support block evaluation.
 
 class TensorBlockNotImplemented {
  public:
   typedef void XprType;
 };
 
 // -------------------------------------------------------------------------- //
 // XprScalar extracts Scalar type from the Eigen expressions (if expression type
 // is not void). It's required to be able to define lazy block expression for
 // argument types, that do not support block evaluation.
 
 template <typename XprType>
 struct XprScalar {
   typedef typename XprType::Scalar type;
 };
 template <>
 struct XprScalar<void> {
   typedef void type;
 };
 
 // -------------------------------------------------------------------------- //
 // TensorMaterializedBlock is a fully evaluated block of the original tensor,
 // and XprType is just a TensorMap over the data. This block type is typically
 // used to materialize blocks of tensor expressions, that can't be efficiently
 // represented as lazy Tensor expressions with fast coeff/packet operations,
 // e.g. we materialize all broadcasts into evaluated blocks.
 //
 // TensorMaterializedBlock does not own its memory buffer, it's either a memory
 // buffer that backs the original expression (e.g. block is just a view into a
 // Tensor), or a memory buffer allocated with scratch allocator, and in this
 // case the scratch allocator will deallocate it at the end of block based
 // expression execution.
 //
 // If the block was evaluated directly into the output buffer, and strides in
 // the output buffer do not match block strides, the TensorMap expression will
 // be invalid, and should never be used in block assignment or any other tensor
 // expression.
 
 template <typename Scalar, int NumDims, int Layout,
           typename IndexType = Eigen::Index>
 class TensorMaterializedBlock {
  public:
   typedef DSizes<IndexType, NumDims> Dimensions;
   typedef TensorMap<const Tensor<Scalar, NumDims, Layout> > XprType;
 
   TensorMaterializedBlock(TensorBlockKind kind, const Scalar* data,
                           const Dimensions& dimensions, bool valid_expr = true)
       : m_kind(kind),
         m_data(data),
         m_dimensions(dimensions),
         m_expr(m_data, m_dimensions),
         m_valid_expr(valid_expr) {
     eigen_assert(m_kind == internal::TensorBlockKind::kView ||
                  m_kind == internal::TensorBlockKind::kMaterializedInScratch ||
                  m_kind == internal::TensorBlockKind::kMaterializedInOutput);
   }
 
   TensorBlockKind kind() const { return m_kind; }
   // NOTE(ezhulenev): Returning XprType by value like in other block types
   // causes asan failures. The theory is that XprType::Nested doesn't work
   // properly for TensorMap.
   const XprType& expr() const {
     eigen_assert(m_valid_expr);
     return m_expr;
   }
   const Scalar* data() const { return m_data; }
   void cleanup() {}
 
   typedef internal::TensorBlockDescriptor<NumDims, IndexType> TensorBlockDesc;
 
   // TensorMaterializedBlock can be backed by different types of storage:
   //
   //   (1) Contiguous block of memory allocated with scratch allocator.
   //   (2) Contiguous block of memory reused from tensor block descriptor
   //       destination buffer.
   //   (3) Strided block of memory reused from tensor block descriptor
   //       destination buffer.
   //
   class Storage {
    public:
     Scalar* data() const { return m_data; }
     const Dimensions& dimensions() const { return m_dimensions; }
     const Dimensions& strides() const { return m_strides; }
 
     TensorMaterializedBlock AsTensorMaterializedBlock() const {
       return TensorMaterializedBlock(
           m_materialized_in_output
               ? internal::TensorBlockKind::kMaterializedInOutput
               : internal::TensorBlockKind::kMaterializedInScratch,
           m_data, m_dimensions, !m_strided_storage);
     }
 
    private:
     friend class TensorMaterializedBlock;
 
     Storage(Scalar* data, const Dimensions& dimensions,
             const Dimensions& strides, bool materialized_in_output,
             bool strided_storage)
         : m_data(data),
           m_dimensions(dimensions),
           m_strides(strides),
           m_materialized_in_output(materialized_in_output),
           m_strided_storage(strided_storage) {}
 
     Scalar* m_data;
     Dimensions m_dimensions;
     Dimensions m_strides;
     bool m_materialized_in_output;
     bool m_strided_storage;
   };
 
   // Creates a storage for materialized block either from the block descriptor
   // destination buffer, or allocates a new buffer with scratch allocator.
   template <typename TensorBlockScratch>
   EIGEN_STRONG_INLINE static Storage prepareStorage(
       TensorBlockDesc& desc, TensorBlockScratch& scratch,
       bool allow_strided_storage = false) {
     // Try to reuse destination as an output block buffer.
     typedef typename TensorBlockDesc::DestinationBuffer DestinationBuffer;
 
     if (desc.destination().kind() == DestinationBuffer::kContiguous) {
       Scalar* buffer = desc.destination().template data<Scalar>();
       desc.DropDestinationBuffer();
       return Storage(buffer, desc.dimensions(),
                      internal::strides<Layout>(desc.dimensions()),
                      /*materialized_in_output=*/true,
                      /*strided_storage=*/false);
 
     } else if (desc.destination().kind() == DestinationBuffer::kStrided &&
                allow_strided_storage) {
       Scalar* buffer = desc.destination().template data<Scalar>();
       desc.DropDestinationBuffer();
       return Storage(buffer, desc.dimensions(), desc.destination().strides(),
                      /*materialized_in_output=*/true, /*strided_storage=*/true);
 
     } else {
       void* mem = scratch.allocate(desc.size() * sizeof(Scalar));
       return Storage(static_cast<Scalar*>(mem), desc.dimensions(),
                      internal::strides<Layout>(desc.dimensions()),
                      /*materialized_in_output=*/false,
                      /*strided_storage=*/false);
     }
   }
 
   // Creates a materialized block for the given descriptor from a memory buffer.
   template <typename DataDimensions, typename TensorBlockScratch>
   EIGEN_STRONG_INLINE static TensorMaterializedBlock materialize(
       const Scalar* data, const DataDimensions& data_dims,
       TensorBlockDesc& desc, TensorBlockScratch& scratch) {
     eigen_assert(array_size<DataDimensions>::value == desc.dimensions().size());
 
     // If a tensor block dimensions covers a contiguous block of the underlying
     // memory, we can skip block buffer memory allocation, and construct a block
     // from existing `data` memory buffer.
     //
     // Example: (RowMajor layout)
     //   data_dims:          [11, 12, 13, 14]
     //   desc.dimensions():  [1,   1,  3, 14]
     //
     // In this case we can construct a TensorBlock starting at
     // `data + desc.offset()`, with a `desc.dimensions()` block sizes.
     static const bool is_col_major = Layout == ColMajor;
 
     // Find out how many inner dimensions have a matching size.
     int num_matching_inner_dims = 0;
     for (int i = 0; i < NumDims; ++i) {
       int dim = is_col_major ? i : NumDims - i - 1;
       if (data_dims[dim] != desc.dimensions()[dim]) break;
       ++num_matching_inner_dims;
     }
 
     // All the outer dimensions must be of size `1`, except a single dimension
     // before the matching inner dimension (`3` in the example above).
     bool can_use_direct_access = true;
     for (int i = num_matching_inner_dims + 1; i < NumDims; ++i) {
       int dim = is_col_major ? i : NumDims - i - 1;
       if (desc.dimension(dim) != 1) {
         can_use_direct_access = false;
         break;
       }
     }
 
     if (can_use_direct_access) {
       const Scalar* block_start = data + desc.offset();
       return TensorMaterializedBlock(internal::TensorBlockKind::kView,
                                      block_start, desc.dimensions());
 
     } else {
       // Reuse destination buffer or allocate new buffer with scratch allocator.
       const Storage storage = prepareStorage(desc, scratch);
 
       typedef internal::TensorBlockIO<Scalar, IndexType, NumDims, Layout>
           TensorBlockIO;
       typedef typename TensorBlockIO::Dst TensorBlockIODst;
       typedef typename TensorBlockIO::Src TensorBlockIOSrc;
 
       TensorBlockIOSrc src(internal::strides<Layout>(Dimensions(data_dims)),
                            data, desc.offset());
       TensorBlockIODst dst(storage.dimensions(), storage.strides(),
                            storage.data());
 
       TensorBlockIO::Copy(dst, src);
       return storage.AsTensorMaterializedBlock();
     }
   }
 
  private:
   TensorBlockKind m_kind;
   const Scalar* m_data;
   Dimensions m_dimensions;
   XprType m_expr;
   bool m_valid_expr;
 };
 
 // -------------------------------------------------------------------------- //
 // TensorCwiseUnaryBlock is a lazy tensor expression block that applies UnaryOp
 // functor to the blocks produced by the underlying Tensor expression.
 
 template <typename UnaryOp, typename ArgTensorBlock>
 class TensorCwiseUnaryBlock {
   static const bool NoArgBlockAccess =
       internal::is_void<typename ArgTensorBlock::XprType>::value;
 
  public:
   typedef typename conditional<
       NoArgBlockAccess, void,
       TensorCwiseUnaryOp<UnaryOp, const typename ArgTensorBlock::XprType> >::
       type XprType;
 
   typedef typename XprScalar<XprType>::type Scalar;
 
   TensorCwiseUnaryBlock(const ArgTensorBlock& arg_block, const UnaryOp& functor)
       : m_arg_block(arg_block), m_functor(functor) {}
 
   TensorBlockKind kind() const { return internal::TensorBlockKind::kExpr; }
 
   XprType expr() const { return XprType(m_arg_block.expr(), m_functor); }
   const Scalar* data() const { return NULL; }
   void cleanup() { m_arg_block.cleanup(); }
 
  private:
   ArgTensorBlock m_arg_block;
   UnaryOp m_functor;
 };
 
 // -------------------------------------------------------------------------- //
 // TensorCwiseUnaryBlock is a lazy tensor expression block that applies BinaryOp
 // functor to the blocks produced by the underlying Tensor expression.
 
 template <typename BinaryOp, typename LhsTensorBlock, typename RhsTensorBlock>
 class TensorCwiseBinaryBlock {
   static const bool NoArgBlockAccess =
       internal::is_void<typename LhsTensorBlock::XprType>::value ||
       internal::is_void<typename RhsTensorBlock::XprType>::value;
 
  public:
   typedef typename conditional<
       NoArgBlockAccess, void,
       TensorCwiseBinaryOp<BinaryOp, const typename LhsTensorBlock::XprType,
                           const typename RhsTensorBlock::XprType> >::type
       XprType;
 
   typedef typename XprScalar<XprType>::type Scalar;
 
   TensorCwiseBinaryBlock(const LhsTensorBlock& left_block,
                          const RhsTensorBlock& right_block,
                          const BinaryOp& functor)
       : m_left_block(left_block),
         m_right_block(right_block),
         m_functor(functor) {}
 
   TensorBlockKind kind() const { return internal::TensorBlockKind::kExpr; }
 
   XprType expr() const {
     return XprType(m_left_block.expr(), m_right_block.expr(), m_functor);
   }
 
   const Scalar* data() const { return NULL; }
 
   void cleanup() {
     m_left_block.cleanup();
     m_right_block.cleanup();
   }
 
  private:
   LhsTensorBlock m_left_block;
   RhsTensorBlock m_right_block;
   BinaryOp m_functor;
 };
 
 // -------------------------------------------------------------------------- //
 // TensorUnaryExprBlock is a lazy tensor expression block that can construct
 // an arbitrary tensor expression from a block of the underlying type (this is a
 // generalization of the TensorCwiseUnaryBlock for arbitrary expressions).
 
 template <typename BlockFactory, typename ArgTensorBlock>
 class TensorUnaryExprBlock {
   typedef typename ArgTensorBlock::XprType ArgXprType;
   static const bool NoArgBlockAccess = internal::is_void<ArgXprType>::value;
 
  public:
   typedef typename conditional<
       NoArgBlockAccess, void,
       typename BlockFactory::template XprType<ArgXprType>::type>::type XprType;
 
   typedef typename XprScalar<XprType>::type Scalar;
 
   TensorUnaryExprBlock(const ArgTensorBlock& arg_block,
                        const BlockFactory& factory)
       : m_arg_block(arg_block), m_factory(factory) {}
 
   TensorBlockKind kind() const { return internal::TensorBlockKind::kExpr; }
   XprType expr() const { return m_factory.expr(m_arg_block.expr()); }
   const Scalar* data() const { return NULL; }
   void cleanup() { m_arg_block.cleanup(); }
 
  private:
   ArgTensorBlock m_arg_block;
   BlockFactory m_factory;
 };
 
 // -------------------------------------------------------------------------- //
 // TensorTernaryExprBlock is a lazy tensor expression block that can construct
 // an arbitrary tensor expression from three blocks of the underlying type.
 
 template <typename BlockFactory, typename Arg1TensorBlock,
           typename Arg2TensorBlock, typename Arg3TensorBlock>
 class TensorTernaryExprBlock {
   typedef typename Arg1TensorBlock::XprType Arg1XprType;
   typedef typename Arg2TensorBlock::XprType Arg2XprType;
   typedef typename Arg3TensorBlock::XprType Arg3XprType;
 
   static const bool NoArgBlockAccess = internal::is_void<Arg1XprType>::value ||
                                        internal::is_void<Arg2XprType>::value ||
                                        internal::is_void<Arg3XprType>::value;
 
  public:
   typedef typename conditional<
       NoArgBlockAccess, void,
       typename BlockFactory::template XprType<Arg1XprType, Arg2XprType,
                                               Arg3XprType>::type>::type XprType;
 
   typedef typename XprScalar<XprType>::type Scalar;
 
   TensorTernaryExprBlock(const Arg1TensorBlock& arg1_block,
                          const Arg2TensorBlock& arg2_block,
                          const Arg3TensorBlock& arg3_block,
                          const BlockFactory& factory)
       : m_arg1_block(arg1_block),
         m_arg2_block(arg2_block),
         m_arg3_block(arg3_block),
         m_factory(factory) {}
 
   TensorBlockKind kind() const { return internal::TensorBlockKind::kExpr; }
   XprType expr() const {
     return m_factory.expr(m_arg1_block.expr(), m_arg2_block.expr(),
                           m_arg3_block.expr());
   }
   const Scalar* data() const { return NULL; }
   void cleanup() {
     m_arg1_block.cleanup();
     m_arg2_block.cleanup();
     m_arg3_block.cleanup();
   }
 
  private:
   Arg1TensorBlock m_arg1_block;
   Arg2TensorBlock m_arg2_block;
   Arg3TensorBlock m_arg3_block;
   BlockFactory m_factory;
 };
 
 // -------------------------------------------------------------------------- //
 // StridedLinearBufferCopy provides a method to copy data between two linear
 // buffers with different strides, with optimized paths for scatter/gather.
 
 template <typename Scalar, typename IndexType>
 class StridedLinearBufferCopy {
   typedef typename packet_traits<Scalar>::type Packet;
   enum {
     Vectorizable = packet_traits<Scalar>::Vectorizable,
     PacketSize = packet_traits<Scalar>::size
   };
 
  public:
   // Specifying linear copy kind statically gives ~30% speedup for small sizes.
   enum class Kind {
     Linear = 0,       // src_stride == 1 && dst_stride == 1
     Scatter = 1,      // src_stride == 1 && dst_stride != 1
     FillLinear = 2,   // src_stride == 0 && dst_stride == 1
     FillScatter = 3,  // src_stride == 0 && dst_stride != 1
     Gather = 4,       // dst_stride == 1
     Random = 5        // everything else
   };
 
   struct Dst {
     Dst(IndexType o, IndexType s, Scalar* d) : offset(o), stride(s), data(d) {}
 
     IndexType offset;
     IndexType stride;
     Scalar* data;
   };
 
   struct Src {
     Src(IndexType o, IndexType s, const Scalar* d)
         : offset(o), stride(s), data(d) {}
 
     IndexType offset;
     IndexType stride;
     const Scalar* data;
   };
 
   template <typename StridedLinearBufferCopy::Kind kind>
   static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void Run(const Dst& dst,
                                                         const Src& src,
                                                         const size_t count) {
     Run<kind>(count, dst.offset, dst.stride, dst.data, src.offset, src.stride,
               src.data);
   }
 
  private:
   template <typename StridedLinearBufferCopy::Kind kind>
   static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void Run(
       const IndexType count, const IndexType dst_offset,
       const IndexType dst_stride, Scalar* EIGEN_RESTRICT dst_data,
       const IndexType src_offset, const IndexType src_stride,
       const Scalar* EIGEN_RESTRICT src_data) {
     const Scalar* src = &src_data[src_offset];
     Scalar* dst = &dst_data[dst_offset];
 
     if (!Vectorizable) {
       for (Index i = 0; i < count; ++i) {
         dst[i * dst_stride] = src[i * src_stride];
       }
       return;
     }
 
     const IndexType vectorized_size = count - PacketSize;
     IndexType i = 0;
 
     if (kind == StridedLinearBufferCopy::Kind::Linear) {
       // ******************************************************************** //
       // Linear copy from `src` to `dst`.
       const IndexType unrolled_size = count - 4 * PacketSize;
       eigen_assert(src_stride == 1 && dst_stride == 1);
       for (; i <= unrolled_size; i += 4 * PacketSize) {
         for (int j = 0; j < 4; ++j) {
           Packet p = ploadu<Packet>(src + i + j * PacketSize);
           pstoreu<Scalar, Packet>(dst + i + j * PacketSize, p);
         }
       }
       for (; i <= vectorized_size; i += PacketSize) {
         Packet p = ploadu<Packet>(src + i);
         pstoreu<Scalar, Packet>(dst + i, p);
       }
       for (; i < count; ++i) {
         dst[i] = src[i];
       }
       // ******************************************************************** //
     } else if (kind == StridedLinearBufferCopy::Kind::Scatter) {
       // Scatter from `src` to `dst`.
       eigen_assert(src_stride == 1 && dst_stride != 1);
       for (; i <= vectorized_size; i += PacketSize) {
         Packet p = ploadu<Packet>(src + i);
         pscatter<Scalar, Packet>(dst + i * dst_stride, p, dst_stride);
       }
       for (; i < count; ++i) {
         dst[i * dst_stride] = src[i];
       }
       // ******************************************************************** //
     } else if (kind == StridedLinearBufferCopy::Kind::FillLinear) {
       // Fill `dst` with value at `*src`.
       eigen_assert(src_stride == 0 && dst_stride == 1);
       const IndexType unrolled_size = count - 4 * PacketSize;
       Packet p = pload1<Packet>(src);
       for (; i <= unrolled_size; i += 4 * PacketSize) {
         for (int j = 0; j < 4; ++j) {
           pstoreu<Scalar, Packet>(dst + i + j * PacketSize, p);
         }
       }
       for (; i <= vectorized_size; i += PacketSize) {
         pstoreu<Scalar, Packet>(dst + i, p);
       }
       for (; i < count; ++i) {
         dst[i] = *src;
       }
       // ******************************************************************** //
     } else if (kind == StridedLinearBufferCopy::Kind::FillScatter) {
       // Scatter `*src` into `dst`.
       eigen_assert(src_stride == 0 && dst_stride != 1);
       Packet p = pload1<Packet>(src);
       for (; i <= vectorized_size; i += PacketSize) {
         pscatter<Scalar, Packet>(dst + i * dst_stride, p, dst_stride);
       }
       for (; i < count; ++i) {
         dst[i * dst_stride] = *src;
       }
       // ******************************************************************** //
     } else if (kind == StridedLinearBufferCopy::Kind::Gather) {
       // Gather from `src` into `dst`.
       eigen_assert(dst_stride == 1);
       for (; i <= vectorized_size; i += PacketSize) {
         Packet p = pgather<Scalar, Packet>(src + i * src_stride, src_stride);
         pstoreu<Scalar, Packet>(dst + i, p);
       }
       for (; i < count; ++i) {
         dst[i] = src[i * src_stride];
       }
       // ******************************************************************** //
     } else if (kind == StridedLinearBufferCopy::Kind::Random) {
       // Random.
       for (; i < count; ++i) {
         dst[i * dst_stride] = src[i * src_stride];
       }
     } else {
       eigen_assert(false);
     }
   }
 };
 
 // -------------------------------------------------------------------------- //
 // TensorBlockIO copies data from `src` tensor block, to the `dst` tensor block.
 // It's possible to specify src->dst dimension mapping for the copy operation.
 // Dimensions of `dst` specify how many elements have to be copied, for the
 // `src` we need to know only stride to navigate through source memory buffer.
 
 template <typename Scalar, typename IndexType, int NumDims, int Layout>
 class TensorBlockIO {
   static const bool IsColMajor = (Layout == ColMajor);
 
   typedef StridedLinearBufferCopy<Scalar, IndexType> LinCopy;
 
  public:
   typedef DSizes<IndexType, NumDims> Dimensions;
   typedef DSizes<int, NumDims> DimensionsMap;
 
   struct Dst {
     Dst(const Dimensions& dst_dims, const Dimensions& dst_strides, Scalar* dst,
         IndexType dst_offset = 0)
         : dims(dst_dims), strides(dst_strides), data(dst), offset(dst_offset) {}
 
     Dimensions dims;
     Dimensions strides;
     Scalar* data;
     IndexType offset;
   };
 
   struct Src {
     Src(const Dimensions& src_strides, const Scalar* src,
         IndexType src_offset = 0)
         : strides(src_strides), data(src), offset(src_offset) {}
 
     Dimensions strides;
     const Scalar* data;
     IndexType offset;
   };
 
   // Copies data to `dst` from `src`, using provided dimensions mapping:
   //
   //   src_dimension_index = dst_to_src_dim_map[dst_dimension_index]
   //
   // Returns the number of copied elements.
   static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE IndexType Copy(
       const Dst& dst, const Src& src, const DimensionsMap& dst_to_src_dim_map) {
     // Copy single scalar value from `src` to `dst`.
     if (NumDims == 0) {
       *(dst.data + dst.offset) = *(src.data + src.offset);
       return 1;
     }
 
     // Both `dst` and `src` must have contiguous innermost dimension. We also
     // accept the special case with stride '0', because it's used as a trick to
     // implement broadcasting.
     {
       int inner_dim = IsColMajor ? 0 : NumDims - 1;
       EIGEN_UNUSED_VARIABLE(inner_dim);
       eigen_assert(dst.strides[inner_dim] == 1 || dst.strides[inner_dim] == 0);
       eigen_assert(src.strides[inner_dim] == 1 || src.strides[inner_dim] == 0);
     }
 
     // Give a shorter name to `dst_to_src_dim_map`.
     const DimensionsMap& dim_map = dst_to_src_dim_map;
 
     // Do not squeeze reordered inner dimensions.
     int num_squeezable_dims = NumSqueezableInnerDims(dim_map);
 
     // NOTE: We find the innermost dimension (contiguous in memory) in the dst
     // block, and we write data linearly into that dimension, reading it from
     // the src. If dimensions are reordered, we might end up reading data from
     // the src with `stride != 1`.
     //
     // NOTE: Random-Read/Linear-Write can be up to ~2X faster than
     // Linear-Read/Random-Write: https://stackoverflow.com/a/54935680
 
     // Find the innermost dimension in the dst whose size is not 1. This is the
     // effective inner dim.
     int num_size_one_inner_dims = 0;
     for (int i = 0; i < num_squeezable_dims; ++i) {
       const int dst_dim = IsColMajor ? i : NumDims - i - 1;
       if (dst.dims[dst_dim] != 1) break;
       num_size_one_inner_dims++;
     }
 
     // If all dimensions are of size 1, just copy a scalar from `src` to `dst`.
     if (num_size_one_inner_dims == NumDims) {
       *(dst.data + dst.offset) = *(src.data + src.offset);
       return 1;
     }
 
     // Outermost dimension in the dst with `stride == 1` (contiguous in memory).
     const int dst_stride1_dim = IsColMajor
                                     ? num_size_one_inner_dims
                                     : NumDims - num_size_one_inner_dims - 1;
 
     // Dimension in the src that corresponds to the dst innermost dimension.
     const int src_dim_for_dst_stride1_dim =
         NumDims == 0 ? 1 : dim_map[dst_stride1_dim];
 
     // Size of the innermost dimension (length of contiguous blocks of memory).
     IndexType dst_inner_dim_size = NumDims == 0 ? 1 : dst.dims[dst_stride1_dim];
 
     // Squeeze multiple inner dims into one if they are contiguous in `dst` and
     // `src` memory, so we can do less linear copy calls.
     for (int i = num_size_one_inner_dims + 1; i < num_squeezable_dims; ++i) {
       const int dst_dim = IsColMajor ? i : NumDims - i - 1;
       const IndexType dst_stride = dst.strides[dst_dim];
       const IndexType src_stride = src.strides[dim_map[dst_dim]];
       if (dst_inner_dim_size == dst_stride && dst_stride == src_stride) {
         dst_inner_dim_size *= dst.dims[dst_dim];
         ++num_size_one_inner_dims;
       } else {
         break;
       }
     }
 
     // Setup strides to read data from `src` and write to `dst`.
     IndexType input_offset = src.offset;
     IndexType output_offset = dst.offset;
     IndexType input_stride =
         NumDims == 0 ? 1 : src.strides[src_dim_for_dst_stride1_dim];
     IndexType output_stride = NumDims == 0 ? 1 : dst.strides[dst_stride1_dim];
 
     const int at_least_1_dim = NumDims <= 1 ? 1 : NumDims - 1;
     array<BlockIteratorState, at_least_1_dim> it;
 
     // Initialize block iterator state. Squeeze away any dimension of size 1.
     int idx = 0;  // currently initialized iterator state index
     for (int i = num_size_one_inner_dims; i < NumDims - 1; ++i) {
       const int dst_dim = IsColMajor ? i + 1 : NumDims - i - 2;
       if (dst.dims[dst_dim] == 1) continue;
 
       it[idx].size = dst.dims[dst_dim];
       it[idx].input_stride = src.strides[dim_map[dst_dim]];
       it[idx].output_stride = dst.strides[dst_dim];
 
       it[idx].input_span = it[idx].input_stride * (it[idx].size - 1);
       it[idx].output_span = it[idx].output_stride * (it[idx].size - 1);
 
       idx++;
     }
 
     // Iterate copying data from src to dst.
     const IndexType block_total_size = NumDims == 0 ? 1 : dst.dims.TotalSize();
 
 #define COPY_INNER_DIM(KIND)                                           \
   IndexType num_copied = 0;                                            \
   for (num_copied = 0; num_copied < block_total_size;                  \
        num_copied += dst_inner_dim_size) {                             \
     LinCopy::template Run<KIND>(                                       \
         typename LinCopy::Dst(output_offset, output_stride, dst.data), \
         typename LinCopy::Src(input_offset, input_stride, src.data),   \
         dst_inner_dim_size);                                           \
                                                                        \
     for (int j = 0; j < idx; ++j) {                                    \
       if (++it[j].count < it[j].size) {                                \
         input_offset += it[j].input_stride;                            \
         output_offset += it[j].output_stride;                          \
         break;                                                         \
       }                                                                \
       it[j].count = 0;                                                 \
       input_offset -= it[j].input_span;                                \
       output_offset -= it[j].output_span;                              \
     }                                                                  \
   }                                                                    \
   return num_copied;
 
     if (input_stride == 1 && output_stride == 1) {
       COPY_INNER_DIM(LinCopy::Kind::Linear);
     } else if (input_stride == 1 && output_stride != 1) {
       COPY_INNER_DIM(LinCopy::Kind::Scatter);
     } else if (input_stride == 0 && output_stride == 1) {
       COPY_INNER_DIM(LinCopy::Kind::FillLinear);
     } else if (input_stride == 0 && output_stride != 1) {
       COPY_INNER_DIM(LinCopy::Kind::FillScatter);
     } else if (output_stride == 1) {
       COPY_INNER_DIM(LinCopy::Kind::Gather);
     } else {
       COPY_INNER_DIM(LinCopy::Kind::Random);
     }
 
 #undef COPY_INNER_DIM
   }
 
   // Copy from `src` to `dst` with an identity src->dst dimension map. Returns
   // the number of copied elements.
   static EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE IndexType Copy(const Dst& dst,
                                                               const Src& src) {
     DimensionsMap dst_to_src_map;
     for (int i = 0; i < NumDims; ++i) dst_to_src_map[i] = i;
     return Copy(dst, src, dst_to_src_map);
   }
 
  private:
   struct BlockIteratorState {
     BlockIteratorState()
         : size(0),
           count(0),
           input_stride(0),
           output_stride(0),
           input_span(0),
           output_span(0) {}
 
     IndexType size;
     IndexType count;
     IndexType input_stride;
     IndexType output_stride;
     IndexType input_span;
     IndexType output_span;
   };
 
   // Compute how many inner dimensions it's allowed to squeeze when doing IO
   // between two tensor blocks. It's safe to squeeze inner dimensions, only
   // if they are not reordered.
   static int NumSqueezableInnerDims(const DimensionsMap& dim_map) {
     int num_squeezable_dims = 0;
     for (int i = 0; i < NumDims; ++i) {
       const int dim = IsColMajor ? i : NumDims - i - 1;
       if (dim_map[dim] != dim) break;
       num_squeezable_dims++;
     }
     return num_squeezable_dims;
   }
 };
 
 // -------------------------------------------------------------------------- //
 // TensorBlockAssignment assigns a block expression of type `TensorBlockExpr` to
 // a Tensor block defined by `desc`, backed by a memory buffer at `target`.
 //
 // Currently there is no way to write from a Tensor expression to a block of
 // memory, if dimensions are reordered. If you need to do that, you should
 // materialize a Tensor block expression into a memory buffer, and then use
 // TensorBlockIO to copy data between two memory buffers with a custom
 // `target->src` dimension map (see definition above).
 //
 // Also currently the innermost dimension of `target` must have a stride '1'
 // (contiguous in memory). This restriction could be lifted with a `pscatter`,
 // but in practice it's never needed, and there is a similar TensorBlockIO
 // workaround for that.
 //
 // TODO(ezhulenev): TensorBlockAssignment is a special case of TensorBlockIO
 // where `src` is a tensor expression. Explore if it is possible to rewrite IO
 // to use expressions instead of pointers, and after that TensorBlockAssignment
 // will become an alias to IO.
 template <typename Scalar, int NumDims, typename TensorBlockExpr,
           typename IndexType = Eigen::Index>
 class TensorBlockAssignment {
   // We will use coeff/packet path to evaluate block expressions.
   typedef TensorEvaluator<const TensorBlockExpr, DefaultDevice>
       TensorBlockEvaluator;
 
   typedef DSizes<IndexType, NumDims> Dimensions;
 
   enum {
     Vectorizable = packet_traits<Scalar>::Vectorizable,
     PacketSize = packet_traits<Scalar>::size
   };
 
   template <bool Vectorizable, typename Evaluator>
   struct InnerDimAssign {
     EIGEN_ALWAYS_INLINE static void Run(Scalar* target, IndexType count,
                                         const Evaluator& eval,
                                         IndexType eval_offset) {
       for (IndexType i = 0; i < count; ++i) {
         target[i] = eval.coeff(eval_offset + i);
       }
     }
   };
 
   template <typename Evaluator>
   struct InnerDimAssign<true, Evaluator> {
     EIGEN_ALWAYS_INLINE static void Run(Scalar* target, IndexType count,
                                         const Evaluator& eval,
                                         IndexType eval_offset) {
       typedef typename packet_traits<Scalar>::type Packet;
 
       const IndexType unrolled_size = count - 4 * PacketSize;
       const IndexType vectorized_size = count - PacketSize;
       IndexType i = 0;
 
       for (; i <= unrolled_size; i += 4 * PacketSize) {
         for (int j = 0; j < 4; ++j) {
           const IndexType idx = eval_offset + i + j * PacketSize;
           Packet p = eval.template packet<Unaligned>(idx);
           pstoreu<Scalar>(target + i + j * PacketSize, p);
         }
       }
 
       for (; i <= vectorized_size; i += PacketSize) {
         Packet p = eval.template packet<Unaligned>(eval_offset + i);
         pstoreu<Scalar>(target + i, p);
       }
 
       for (; i < count; ++i) {
         target[i] = eval.coeff(eval_offset + i);
       }
     }
   };
 
  public:
   struct Target {
     Target(const Dimensions& target_dims, const Dimensions& target_strides,
            Scalar* target_data, IndexType target_offset = 0)
         : dims(target_dims),
           strides(target_strides),
           data(target_data),
           offset(target_offset) {}
 
     Dimensions dims;
     Dimensions strides;
     Scalar* data;
     IndexType offset;
   };
 
   static Target target(const Dimensions& target_dims,
                        const Dimensions& target_strides, Scalar* target_data,
                        IndexType target_offset = 0) {
     return Target(target_dims, target_strides, target_data, target_offset);
   }
 
   template <typename TargetDimsIndexType, typename TargetStridesIndexType>
   static Target target(
       const DSizes<TargetDimsIndexType, NumDims>& target_dims,
       const DSizes<TargetStridesIndexType, NumDims>& target_strides,
       Scalar* target_data, IndexType target_offset = 0) {
     // DSizes constructor will do index type promotion if it's safe.
     return Target(Dimensions(target_dims), Dimensions(target_strides),
                   target_data, target_offset);
   }
 
   static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void Run(
       const Target& target, const TensorBlockExpr& expr) {
     // Prepare evaluator for block expression.
     DefaultDevice default_device;
     TensorBlockEvaluator eval(expr, default_device);
 
     // Tensor block expression dimension should match destination dimensions.
     eigen_assert(dimensions_match(target.dims, eval.dimensions()));
 
     static const int Layout = TensorBlockEvaluator::Layout;
     static const bool is_col_major = Layout == ColMajor;
 
     // Initialize output inner dimension size based on a layout.
     const IndexType output_size = NumDims == 0 ? 1 : target.dims.TotalSize();
     const int inner_dim_idx = is_col_major ? 0 : NumDims - 1;
     IndexType output_inner_dim_size = target.dims[inner_dim_idx];
 
     // Target inner dimension stride must be '1'.
     eigen_assert(target.strides[inner_dim_idx] == 1);
 
     // Squeeze multiple inner dims into one if they are contiguous in `target`.
     IndexType num_squeezed_dims = 0;
     for (Index i = 1; i < NumDims; ++i) {
       const Index dim = is_col_major ? i : NumDims - i - 1;
       const IndexType target_stride = target.strides[dim];
 
       if (output_inner_dim_size == target_stride) {
         output_inner_dim_size *= target.dims[dim];
         num_squeezed_dims++;
       } else {
         break;
       }
     }
 
     // Initialize output block iterator state. Dimension in this array are
     // always in inner_most -> outer_most order (col major layout).
     array<BlockIteratorState, NumDims> it;
 
     int idx = 0;  // currently initialized iterator state index
     for (Index i = num_squeezed_dims; i < NumDims - 1; ++i) {
       const Index dim = is_col_major ? i + 1 : NumDims - i - 2;
 
       it[idx].count = 0;
       it[idx].size = target.dims[dim];
       it[idx].output_stride = target.strides[dim];
       it[idx].output_span = it[idx].output_stride * (it[idx].size - 1);
       idx++;
     }
 
     // We read block expression from the beginning, and start writing data to
     // `target` at given offset.
     IndexType input_offset = 0;
     IndexType output_offset = target.offset;
 
     // Iterate copying data from `eval` to `target`.
     for (IndexType i = 0; i < output_size; i += output_inner_dim_size) {
       // Assign to `target` at current offset.
       InnerDimAssign<Vectorizable && TensorBlockEvaluator::PacketAccess,
                      TensorBlockEvaluator>::Run(target.data + output_offset,
                                                 output_inner_dim_size, eval,
                                                 input_offset);
 
       // Move input offset forward by the number of assigned coefficients.
       input_offset += output_inner_dim_size;
 
       // Update index.
       for (int j = 0; j < idx; ++j) {
         if (++it[j].count < it[j].size) {
           output_offset += it[j].output_stride;
           break;
         }
         it[j].count = 0;
         output_offset -= it[j].output_span;
       }
     }
   }
 
  private:
   struct BlockIteratorState {
     BlockIteratorState()
         : count(0), size(0), output_stride(0), output_span(0) {}
 
     IndexType count;
     IndexType size;
     IndexType output_stride;
     IndexType output_span;
   };
 };
 
 // -------------------------------------------------------------------------- //
 
 }  // namespace internal
 }  // namespace Eigen
 
 #endif  // EIGEN_CXX11_TENSOR_TENSOR_BLOCK_H

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