TensorBroadcasting.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2014 Benoit Steiner <benoit.steiner.goog@gmail.com>
//
// 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_BROADCASTING_H
#define EIGEN_CXX11_TENSOR_TENSOR_BROADCASTING_H
 
// IWYU pragma: private
#include "./InternalHeaderCheck.h"
 
namespace Eigen {
 
namespace internal {
template <typename Broadcast, typename XprType>
struct traits<TensorBroadcastingOp<Broadcast, XprType>> : public traits<XprType> {
  typedef typename XprType::Scalar Scalar;
  typedef traits<XprType> XprTraits;
  typedef typename XprTraits::StorageKind StorageKind;
  typedef typename XprTraits::Index Index;
  typedef typename XprType::Nested Nested;
  typedef std::remove_reference_t<Nested> Nested_;
  static constexpr int NumDimensions = XprTraits::NumDimensions;
  static constexpr int Layout = XprTraits::Layout;
  typedef typename XprTraits::PointerType PointerType;
  enum {
    // Broadcast is read-only.
    Flags = traits<XprType>::Flags & ~LvalueBit
  };
};
 
template <typename Broadcast, typename XprType>
struct eval<TensorBroadcastingOp<Broadcast, XprType>, Eigen::Dense> {
  typedef const TensorBroadcastingOp<Broadcast, XprType> EIGEN_DEVICE_REF type;
};
 
template <typename Broadcast, typename XprType>
struct nested<TensorBroadcastingOp<Broadcast, XprType>, 1,
              typename eval<TensorBroadcastingOp<Broadcast, XprType>>::type> {
  typedef TensorBroadcastingOp<Broadcast, XprType> type;
};
 
template <typename Dims>
struct is_input_scalar {
  static const bool value = false;
};
template <>
struct is_input_scalar<Sizes<>> {
  static const bool value = true;
};
template <typename std::ptrdiff_t... Indices>
struct is_input_scalar<Sizes<Indices...>> {
  static constexpr bool value = (Sizes<Indices...>::total_size == 1);
};
 
}  // end namespace internal

template <typename Broadcast, typename XprType>
class TensorBroadcastingOp : public TensorBase<TensorBroadcastingOp<Broadcast, XprType>, ReadOnlyAccessors> {
 public:
  typedef typename Eigen::internal::traits<TensorBroadcastingOp>::Scalar Scalar;
  typedef typename Eigen::NumTraits<Scalar>::Real RealScalar;
  typedef typename XprType::CoeffReturnType CoeffReturnType;
  typedef typename Eigen::internal::nested<TensorBroadcastingOp>::type Nested;
  typedef typename Eigen::internal::traits<TensorBroadcastingOp>::StorageKind StorageKind;
  typedef typename Eigen::internal::traits<TensorBroadcastingOp>::Index Index;
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE TensorBroadcastingOp(const XprType& expr, const Broadcast& broadcast)
      : m_xpr(expr), m_broadcast(broadcast) {}
 
  EIGEN_DEVICE_FUNC const Broadcast& broadcast() const { return m_broadcast; }
 
  EIGEN_DEVICE_FUNC const internal::remove_all_t<typename XprType::Nested>& expression() const { return m_xpr; }
 
 protected:
  typename XprType::Nested m_xpr;
  const Broadcast m_broadcast;
};
 
// Eval as rvalue
template <typename Broadcast, typename ArgType, typename Device>
struct TensorEvaluator<const TensorBroadcastingOp<Broadcast, ArgType>, Device> {
  typedef TensorBroadcastingOp<Broadcast, ArgType> XprType;
  typedef typename XprType::Index Index;
  static constexpr int NumDims = internal::array_size<typename TensorEvaluator<ArgType, Device>::Dimensions>::value;
  typedef DSizes<Index, NumDims> Dimensions;
  typedef typename XprType::Scalar Scalar;
  typedef typename TensorEvaluator<ArgType, Device>::Dimensions InputDimensions;
  typedef typename XprType::CoeffReturnType CoeffReturnType;
  typedef typename PacketType<CoeffReturnType, Device>::type PacketReturnType;
  static constexpr int PacketSize = PacketType<CoeffReturnType, Device>::size;
 
 protected:  //  all the non-static fields must have the same access control, otherwise the TensorEvaluator won't be
             //  standard layout;
  bool isCopy, nByOne, oneByN;
 
 public:
  typedef StorageMemory<CoeffReturnType, Device> Storage;
  typedef typename Storage::Type EvaluatorPointerType;
 
  enum {
    IsAligned = TensorEvaluator<ArgType, Device>::IsAligned,
    PacketAccess = TensorEvaluator<ArgType, Device>::PacketAccess,
    BlockAccess = TensorEvaluator<ArgType, Device>::BlockAccess,
    PreferBlockAccess = true,
    RawAccess = false
  };
  static constexpr int Layout = TensorEvaluator<ArgType, Device>::Layout;
 
  typedef std::remove_const_t<Scalar> ScalarNoConst;
 
  // We do block based broadcasting using a trick with 2x tensor rank and 0
  // strides. See block method implementation for details.
  typedef DSizes<Index, 2 * NumDims> BroadcastDimensions;
 
  //===- Tensor block evaluation strategy (see TensorBlock.h) -------------===//
  typedef internal::TensorBlockDescriptor<NumDims, Index> TensorBlockDesc;
  typedef internal::TensorBlockScratchAllocator<Device> TensorBlockScratch;
 
  typedef typename TensorEvaluator<const ArgType, Device>::TensorBlock ArgTensorBlock;
 
  typedef typename internal::TensorMaterializedBlock<ScalarNoConst, NumDims, Layout, Index> TensorBlock;
  //===--------------------------------------------------------------------===//
 
  EIGEN_STRONG_INLINE TensorEvaluator(const XprType& op, const Device& device)
      : isCopy(false),
        nByOne(false),
        oneByN(false),
        m_device(device),
        m_broadcast(op.broadcast()),
        m_impl(op.expression(), device) {
    // The broadcasting op doesn't change the rank of the tensor. One can't broadcast a scalar
    // and store the result in a scalar. Instead one should reshape the scalar into a N-D
    // tensor with N >= 1 of 1 element first and then broadcast.
    EIGEN_STATIC_ASSERT((NumDims > 0), YOU_MADE_A_PROGRAMMING_MISTAKE);
    const InputDimensions& input_dims = m_impl.dimensions();
    isCopy = true;
    for (int i = 0; i < NumDims; ++i) {
      eigen_assert(input_dims[i] > 0);
      m_dimensions[i] = input_dims[i] * m_broadcast[i];
      if (m_broadcast[i] != 1) {
        isCopy = false;
      }
    }
 
    if (static_cast<int>(Layout) == static_cast<int>(ColMajor)) {
      m_inputStrides[0] = 1;
      m_outputStrides[0] = 1;
      for (int i = 1; i < NumDims; ++i) {
        m_inputStrides[i] = m_inputStrides[i - 1] * input_dims[i - 1];
        m_outputStrides[i] = m_outputStrides[i - 1] * m_dimensions[i - 1];
      }
    } else {
      m_inputStrides[NumDims - 1] = 1;
      m_outputStrides[NumDims - 1] = 1;
      for (int i = NumDims - 2; i >= 0; --i) {
        m_inputStrides[i] = m_inputStrides[i + 1] * input_dims[i + 1];
        m_outputStrides[i] = m_outputStrides[i + 1] * m_dimensions[i + 1];
      }
    }
 
    if (input_dims[0] == 1) {
      oneByN = true;
      for (int i = 1; i < NumDims; ++i) {
        if (m_broadcast[i] != 1) {
          oneByN = false;
          break;
        }
      }
    } else if (input_dims[NumDims - 1] == 1) {
      nByOne = true;
      for (int i = 0; i < NumDims - 1; ++i) {
        if (m_broadcast[i] != 1) {
          nByOne = false;
          break;
        }
      }
    }
 
    // Handle special format like NCHW, its input shape is '[1, N..., 1]' and
    // broadcast shape is '[N, 1..., N]'
    if (!oneByN && !nByOne) {
      if (input_dims[0] == 1 && input_dims[NumDims - 1] == 1 && NumDims > 2) {
        nByOne = true;
        oneByN = true;
        for (int i = 1; i < NumDims - 1; ++i) {
          if (m_broadcast[i] != 1) {
            nByOne = false;
            oneByN = false;
            break;
          }
        }
      }
    }
  }
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const Dimensions& dimensions() const { return m_dimensions; }
 
  EIGEN_STRONG_INLINE bool evalSubExprsIfNeeded(EvaluatorPointerType) {
    m_impl.evalSubExprsIfNeeded(NULL);
    return true;
  }
 
#ifdef EIGEN_USE_THREADS
  template <typename EvalSubExprsCallback>
  EIGEN_STRONG_INLINE void evalSubExprsIfNeededAsync(EvaluatorPointerType, EvalSubExprsCallback done) {
    m_impl.evalSubExprsIfNeededAsync(nullptr, [done](bool) { done(true); });
  }
#endif  // EIGEN_USE_THREADS
 
  EIGEN_STRONG_INLINE void cleanup() { m_impl.cleanup(); }
 
  EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE CoeffReturnType coeff(Index index) const {
    if (internal::is_input_scalar<internal::remove_all_t<InputDimensions>>::value) {
      return m_impl.coeff(0);
    }
 
    if (static_cast<int>(Layout) == static_cast<int>(ColMajor)) {
      if (isCopy) {
        return m_impl.coeff(index);
      } else {
        return coeffColMajor(index);
      }
    } else {
      if (isCopy) {
        return m_impl.coeff(index);
      } else {
        return coeffRowMajor(index);
      }
    }
  }
 
  // TODO: attempt to speed this up. The integer divisions and modulo are slow
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Index indexColMajor(Index index) const {
    Index inputIndex = 0;
    EIGEN_UNROLL_LOOP
    for (int i = NumDims - 1; i > 0; --i) {
      const Index idx = index / m_outputStrides[i];
      if (internal::index_statically_eq<Broadcast>(i, 1)) {
        eigen_assert(idx < m_impl.dimensions()[i]);
        inputIndex += idx * m_inputStrides[i];
      } else {
        if (internal::index_statically_eq<InputDimensions>(i, 1)) {
          eigen_assert(idx % m_impl.dimensions()[i] == 0);
        } else {
          inputIndex += (idx % m_impl.dimensions()[i]) * m_inputStrides[i];
        }
      }
      index -= idx * m_outputStrides[i];
    }
    if (internal::index_statically_eq<Broadcast>(0, 1)) {
      eigen_assert(index < m_impl.dimensions()[0]);
      inputIndex += index;
    } else {
      if (internal::index_statically_eq<InputDimensions>(0, 1)) {
        eigen_assert(index % m_impl.dimensions()[0] == 0);
      } else {
        inputIndex += (index % m_impl.dimensions()[0]);
      }
    }
    return inputIndex;
  }
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE CoeffReturnType coeffColMajor(Index index) const {
    return m_impl.coeff(indexColMajor(index));
  }
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Index indexRowMajor(Index index) const {
    Index inputIndex = 0;
    EIGEN_UNROLL_LOOP
    for (int i = 0; i < NumDims - 1; ++i) {
      const Index idx = index / m_outputStrides[i];
      if (internal::index_statically_eq<Broadcast>(i, 1)) {
        eigen_assert(idx < m_impl.dimensions()[i]);
        inputIndex += idx * m_inputStrides[i];
      } else {
        if (internal::index_statically_eq<InputDimensions>(i, 1)) {
          eigen_assert(idx % m_impl.dimensions()[i] == 0);
        } else {
          inputIndex += (idx % m_impl.dimensions()[i]) * m_inputStrides[i];
        }
      }
      index -= idx * m_outputStrides[i];
    }
    if (internal::index_statically_eq<Broadcast>(NumDims - 1, 1)) {
      eigen_assert(index < m_impl.dimensions()[NumDims - 1]);
      inputIndex += index;
    } else {
      if (internal::index_statically_eq<InputDimensions>(NumDims - 1, 1)) {
        eigen_assert(index % m_impl.dimensions()[NumDims - 1] == 0);
      } else {
        inputIndex += (index % m_impl.dimensions()[NumDims - 1]);
      }
    }
    return inputIndex;
  }
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE CoeffReturnType coeffRowMajor(Index index) const {
    return m_impl.coeff(indexRowMajor(index));
  }
 
  template <int LoadMode>
  EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE PacketReturnType packet(Index index) const {
    if (internal::is_input_scalar<internal::remove_all_t<InputDimensions>>::value) {
      return internal::pset1<PacketReturnType>(m_impl.coeff(0));
    }
 
    if (static_cast<int>(Layout) == static_cast<int>(ColMajor)) {
      if (isCopy) {
#ifdef EIGEN_GPU_COMPILE_PHASE
        // See PR 437: on NVIDIA P100 and K20m we observed a x3-4 speed up by enforcing
        // unaligned loads here. The reason is unclear though.
        return m_impl.template packet<Unaligned>(index);
#else
        return m_impl.template packet<LoadMode>(index);
#endif
      } else if (oneByN && !nByOne) {
        return packetNByOne<LoadMode>(index);
      } else if (!oneByN && nByOne) {
        return packetOneByN<LoadMode>(index);
      } else if (oneByN && nByOne) {
        return packetOneByNByOne<LoadMode>(index);
      } else {
        return packetColMajor<LoadMode>(index);
      }
    } else {
      if (isCopy) {
#ifdef EIGEN_GPU_COMPILE_PHASE
        // See above.
        return m_impl.template packet<Unaligned>(index);
#else
        return m_impl.template packet<LoadMode>(index);
#endif
      } else if (oneByN && !nByOne) {
        return packetOneByN<LoadMode>(index);
      } else if (!oneByN && nByOne) {
        return packetNByOne<LoadMode>(index);
      } else if (oneByN && nByOne) {
        return packetOneByNByOne<LoadMode>(index);
      } else {
        return packetRowMajor<LoadMode>(index);
      }
    }
  }
 
  template <int LoadMode>
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE PacketReturnType packetOneByNByOne(Index index) const {
    eigen_assert(index + PacketSize - 1 < dimensions().TotalSize());
 
    EIGEN_ALIGN_MAX std::remove_const_t<CoeffReturnType> values[PacketSize];
    Index startDim, endDim;
    Index inputIndex, outputOffset, batchedIndex;
 
    if (static_cast<int>(Layout) == static_cast<int>(ColMajor)) {
      startDim = NumDims - 1;
      endDim = 1;
    } else {
      startDim = 0;
      endDim = NumDims - 2;
    }
 
    batchedIndex = index % m_outputStrides[startDim];
    inputIndex = batchedIndex / m_outputStrides[endDim];
    outputOffset = batchedIndex % m_outputStrides[endDim];
 
    if (outputOffset + PacketSize <= m_outputStrides[endDim]) {
      values[0] = m_impl.coeff(inputIndex);
      return internal::pload1<PacketReturnType>(values);
    } else {
      EIGEN_UNROLL_LOOP
      for (int i = 0, cur = 0; i < PacketSize; ++i, ++cur) {
        if (outputOffset + cur < m_outputStrides[endDim]) {
          values[i] = m_impl.coeff(inputIndex);
        } else {
          ++inputIndex;
          inputIndex = (inputIndex == m_inputStrides[startDim] ? 0 : inputIndex);
          values[i] = m_impl.coeff(inputIndex);
          outputOffset = 0;
          cur = 0;
        }
      }
      return internal::pload<PacketReturnType>(values);
    }
  }
 
  template <int LoadMode>
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE PacketReturnType packetOneByN(Index index) const {
    // Consider the flattened tensor [v0, ..., vN],
    // Concatenates m_broadcast[dim] copies,
    //    [v0, ..., vN, v0, ..., vN, ... ]
    // with dim == NumDims - 1 for col-major, dim == 0 for row-major.
    eigen_assert(index + PacketSize - 1 < dimensions().TotalSize());
 
    // Size of flattened tensor.
    const Index M =
        (static_cast<int>(Layout) == static_cast<int>(ColMajor)) ? m_inputStrides[NumDims - 1] : m_inputStrides[0];
    Index inputIndex = index % M;
    if (inputIndex + PacketSize <= M) {
      return m_impl.template packet<Unaligned>(inputIndex);
    } else {
      EIGEN_ALIGN_MAX std::remove_const_t<CoeffReturnType> values[PacketSize];
      EIGEN_UNROLL_LOOP
      for (int i = 0; i < PacketSize; ++i) {
        if (inputIndex > M - 1) {
          inputIndex = 0;
        }
        values[i] = m_impl.coeff(inputIndex++);
      }
      return internal::pload<PacketReturnType>(values);
    }
  }
 
  template <int LoadMode>
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE PacketReturnType packetNByOne(Index index) const {
    // Consider the flattened tensor [v0, ..., vN],
    // Interleaves m_broadcast[dim] copies,
    //    [v0, v0, ..., v1, v1, ..., vN, vN, ... ]
    // with dim == 0 for col-major, dim == NumDims - 1 for row-major.
    eigen_assert(index + PacketSize - 1 < dimensions().TotalSize());
 
    const Index M =
        (static_cast<int>(Layout) == static_cast<int>(ColMajor)) ? m_broadcast[0] : m_broadcast[NumDims - 1];
 
    Index inputIndex = index / M;
    Index outputOffset = index % M;
    if (outputOffset + PacketSize <= M) {
      return internal::pset1<PacketReturnType>(m_impl.coeff(inputIndex));
    } else {
      EIGEN_ALIGN_MAX std::remove_const_t<CoeffReturnType> values[PacketSize];
      EIGEN_UNROLL_LOOP
      for (int i = 0; i < PacketSize; ++i) {
        if (outputOffset < M) {
          values[i] = m_impl.coeff(inputIndex);
          ++outputOffset;
        } else {
          values[i] = m_impl.coeff(++inputIndex);
          outputOffset = 1;  // Next offset.
        }
      }
      return internal::pload<PacketReturnType>(values);
    }
  }
 
  // Ignore the LoadMode and always use unaligned loads since we can't guarantee
  // the alignment at compile time.
  template <int LoadMode>
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE PacketReturnType packetColMajor(Index index) const {
    eigen_assert(index + PacketSize - 1 < dimensions().TotalSize());
 
    const Index originalIndex = index;
 
    Index inputIndex = 0;
    EIGEN_UNROLL_LOOP
    for (int i = NumDims - 1; i > 0; --i) {
      const Index idx = index / m_outputStrides[i];
      if (internal::index_statically_eq<Broadcast>(i, 1)) {
        eigen_assert(idx < m_impl.dimensions()[i]);
        inputIndex += idx * m_inputStrides[i];
      } else {
        if (internal::index_statically_eq<InputDimensions>(i, 1)) {
          eigen_assert(idx % m_impl.dimensions()[i] == 0);
        } else {
          inputIndex += (idx % m_impl.dimensions()[i]) * m_inputStrides[i];
        }
      }
      index -= idx * m_outputStrides[i];
    }
    Index innermostLoc;
    if (internal::index_statically_eq<Broadcast>(0, 1)) {
      eigen_assert(index < m_impl.dimensions()[0]);
      innermostLoc = index;
    } else {
      if (internal::index_statically_eq<InputDimensions>(0, 1)) {
        eigen_assert(index % m_impl.dimensions()[0] == 0);
        innermostLoc = 0;
      } else {
        innermostLoc = index % m_impl.dimensions()[0];
      }
    }
    inputIndex += innermostLoc;
 
    // Todo: this could be extended to the second dimension if we're not
    // broadcasting alongside the first dimension, and so on.
    if (innermostLoc + PacketSize <= m_impl.dimensions()[0]) {
      return m_impl.template packet<Unaligned>(inputIndex);
    } else {
      EIGEN_ALIGN_MAX std::remove_const_t<CoeffReturnType> values[PacketSize];
      values[0] = m_impl.coeff(inputIndex);
      EIGEN_UNROLL_LOOP
      for (int i = 1; i < PacketSize; ++i) {
        if (innermostLoc + i < m_impl.dimensions()[0]) {
          values[i] = m_impl.coeff(inputIndex + i);
        } else {
          values[i] = coeffColMajor(originalIndex + i);
        }
      }
      PacketReturnType rslt = internal::pload<PacketReturnType>(values);
      return rslt;
    }
  }
 
  template <int LoadMode>
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE PacketReturnType packetRowMajor(Index index) const {
    eigen_assert(index + PacketSize - 1 < dimensions().TotalSize());
 
    const Index originalIndex = index;
 
    Index inputIndex = 0;
    EIGEN_UNROLL_LOOP
    for (int i = 0; i < NumDims - 1; ++i) {
      const Index idx = index / m_outputStrides[i];
      if (internal::index_statically_eq<Broadcast>(i, 1)) {
        eigen_assert(idx < m_impl.dimensions()[i]);
        inputIndex += idx * m_inputStrides[i];
      } else {
        if (internal::index_statically_eq<InputDimensions>(i, 1)) {
          eigen_assert(idx % m_impl.dimensions()[i] == 0);
        } else {
          inputIndex += (idx % m_impl.dimensions()[i]) * m_inputStrides[i];
        }
      }
      index -= idx * m_outputStrides[i];
    }
    Index innermostLoc;
    if (internal::index_statically_eq<Broadcast>(NumDims - 1, 1)) {
      eigen_assert(index < m_impl.dimensions()[NumDims - 1]);
      innermostLoc = index;
    } else {
      if (internal::index_statically_eq<InputDimensions>(NumDims - 1, 1)) {
        eigen_assert(index % m_impl.dimensions()[NumDims - 1] == 0);
        innermostLoc = 0;
      } else {
        innermostLoc = index % m_impl.dimensions()[NumDims - 1];
      }
    }
    inputIndex += innermostLoc;
 
    // Todo: this could be extended to the second dimension if we're not
    // broadcasting alongside the first dimension, and so on.
    if (innermostLoc + PacketSize <= m_impl.dimensions()[NumDims - 1]) {
      return m_impl.template packet<Unaligned>(inputIndex);
    } else {
      EIGEN_ALIGN_MAX std::remove_const_t<CoeffReturnType> values[PacketSize];
      values[0] = m_impl.coeff(inputIndex);
      EIGEN_UNROLL_LOOP
      for (int i = 1; i < PacketSize; ++i) {
        if (innermostLoc + i < m_impl.dimensions()[NumDims - 1]) {
          values[i] = m_impl.coeff(inputIndex + i);
        } else {
          values[i] = coeffRowMajor(originalIndex + i);
        }
      }
      PacketReturnType rslt = internal::pload<PacketReturnType>(values);
      return rslt;
    }
  }
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE TensorOpCost costPerCoeff(bool vectorized) const {
    double compute_cost = TensorOpCost::AddCost<Index>();
    if (!isCopy && NumDims > 0) {
      EIGEN_UNROLL_LOOP
      for (int i = NumDims - 1; i > 0; --i) {
        compute_cost += TensorOpCost::DivCost<Index>();
        if (internal::index_statically_eq<Broadcast>(i, 1)) {
          compute_cost += TensorOpCost::MulCost<Index>() + TensorOpCost::AddCost<Index>();
        } else {
          if (!internal::index_statically_eq<InputDimensions>(i, 1)) {
            compute_cost +=
                TensorOpCost::MulCost<Index>() + TensorOpCost::ModCost<Index>() + TensorOpCost::AddCost<Index>();
          }
        }
        compute_cost += TensorOpCost::MulCost<Index>() + TensorOpCost::AddCost<Index>();
      }
    }
    return m_impl.costPerCoeff(vectorized) + TensorOpCost(0, 0, compute_cost, vectorized, PacketSize);
  }
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE internal::TensorBlockResourceRequirements getResourceRequirements() const {
    // TODO(wuke): Targeting L1 size is 30% faster than targeting L{-1} on large
    // tensors. But this might need further tuning.
    const size_t target_size = m_device.firstLevelCacheSize();
    return internal::TensorBlockResourceRequirements::merge(
        m_impl.getResourceRequirements(), internal::TensorBlockResourceRequirements::skewed<Scalar>(target_size));
  }
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE TensorBlock block(TensorBlockDesc& desc, TensorBlockScratch& scratch,
                                                          bool /*root_of_expr_ast*/ = false) const {
    BlockBroadcastingParams params = blockBroadcastingParams(desc);
 
    if (params.inner_dim_size == 0 || params.bcast_dim_size == 0) {
      return emptyBlock();
    }
 
    // Prepare storage for the materialized broadcasting result.
    const typename TensorBlock::Storage block_storage = TensorBlock::prepareStorage(desc, scratch);
    ScalarNoConst* materialized_output = block_storage.data();
 
    // We potentially will need to materialize input blocks.
    size_t materialized_input_size = 0;
    ScalarNoConst* materialized_input = NULL;
 
    // Initialize block broadcating iterator state for outer dimensions (outer
    // with regard to bcast dimension). Dimension in this array are always in
    // inner_most -> outer_most order (col major layout).
    array<BlockBroadcastingIteratorState, NumDims> it;
    int idx = 0;
 
    for (int i = params.inner_dim_count + 1; i < NumDims; ++i) {
      const Index dim = IsColMajor ? i : NumDims - 1 - i;
      it[idx].size = params.output_dims[dim];
      it[idx].count = 0;
      it[idx].output_stride = m_outputStrides[dim];
      it[idx].output_span = it[idx].output_stride * (it[idx].size - 1);
      idx++;
    }
 
    // Write output into the beginning of `materialized_output`.
    Index output_offset = 0;
 
    // We will fill output block by broadcasting along the bcast dim, and
    // iterating over outer dimension.
    const Index output_size = NumDims == 0 ? 1 : params.output_dims.TotalSize();
 
    for (Index num_output_coeffs = 0; num_output_coeffs < output_size;) {
      ScalarNoConst* bcast_output = materialized_output + num_output_coeffs;
      Index bcast_offset = desc.offset() + output_offset;
 
      // Broadcast along the bcast dimension.
      num_output_coeffs += BroadcastBlockAlongBcastDim(params, bcast_offset, scratch, bcast_output, &materialized_input,
                                                       &materialized_input_size);
 
      // Switch to the next outer dimension.
      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;
      }
    }
 
    return block_storage.AsTensorMaterializedBlock();
  }
 
  EIGEN_DEVICE_FUNC EvaluatorPointerType data() const { return NULL; }
 
  const TensorEvaluator<ArgType, Device>& impl() const { return m_impl; }
 
  Broadcast functor() const { return m_broadcast; }
 
 private:
  static constexpr bool IsColMajor = static_cast<int>(Layout) == static_cast<int>(ColMajor);
 
  // We will build a general case block broadcasting on top of broadcasting
  // primitive that will do broadcasting only for the inner dimension(s) along
  // the first dimension smaller than the input size (it's called `bcast_dim`).
  //
  // Example:
  //           dim:  0  1  2   (ColMajor)
  //    input size: [9, 3, 6]
  //    block size: [9, 2, 6]
  //
  // We will compute broadcasted block by iterating over the outer dimensions
  // before `bcast_dim` (only dimension `2` in this example) and computing
  // broadcasts along the `bcast_dim` (dimension `1` in this example).
 
  // BlockBroadcastingParams holds precomputed parameters for broadcasting a
  // single block along the broadcasting dimension. Sizes and strides along the
  // `bcast_dim` might be invalid, they will be adjusted later in
  // `BroadcastBlockAlongBcastDim`.
  struct BlockBroadcastingParams {
    Dimensions input_dims;      // input expression dimensions
    Dimensions output_dims;     // output block sizes
    Dimensions output_strides;  // output block strides
 
    int inner_dim_count;   // count inner dimensions matching in size
    int bcast_dim;         // broadcasting dimension index
    Index bcast_dim_size;  // broadcasting dimension size
    Index inner_dim_size;  // inner dimensions size
 
    // Block sizes and strides for the input block where all dimensions before
    // `bcast_dim` are equal to `1`.
    Dimensions input_block_sizes;
    Dimensions input_block_strides;
 
    // Block sizes and strides for blocks with extra dimensions and strides `0`.
    BroadcastDimensions bcast_block_sizes;
    BroadcastDimensions bcast_block_strides;
    BroadcastDimensions bcast_input_strides;
  };
 
  struct BlockBroadcastingIteratorState {
    Index size;
    Index count;
    Index output_stride;
    Index output_span;
  };
 
  EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE BlockBroadcastingParams blockBroadcastingParams(TensorBlockDesc& desc) const {
    BlockBroadcastingParams params;
 
    params.input_dims = Dimensions(m_impl.dimensions());
 
    // Output block sizes and strides.
    params.output_dims = desc.dimensions();
    params.output_strides = internal::strides<Layout>(params.output_dims);
 
    // Find the broadcasting dimension (first dimension with output size smaller
    // that the input size).
    params.bcast_dim = 0;
    params.bcast_dim_size = 1;
    params.inner_dim_size = 1;
 
    // Count the number of inner dimensions that have the same size in the block
    // and in the broadcast expression.
    params.inner_dim_count = 0;
 
    for (int i = 0; i < NumDims; ++i) {
      const int dim = IsColMajor ? i : NumDims - i - 1;
 
      if (params.output_dims[dim] == m_dimensions[dim]) {
        params.inner_dim_size *= params.output_dims[dim];
        ++params.inner_dim_count;
        continue;
      }
 
      // First non-matching dimension is the broadcasting dimension.
      eigen_assert(params.output_dims[dim] < m_dimensions[dim]);
      params.bcast_dim = dim;
      params.bcast_dim_size = params.output_dims[dim];
      break;
    }
 
    // Calculate the input block size for looking into the input.
    for (int i = 0; i < params.inner_dim_count; ++i) {
      const int dim = IsColMajor ? i : NumDims - i - 1;
      params.input_block_sizes[dim] = params.input_dims[dim];
    }
    for (int i = params.inner_dim_count; i < NumDims; ++i) {
      const int dim = IsColMajor ? i : NumDims - i - 1;
      params.input_block_sizes[dim] = 1;
    }
    params.input_block_strides = internal::strides<Layout>(params.input_block_sizes);
 
    // Broadcast with the 0-stride trick: Create 1 extra dim for each
    // broadcast, set the input stride to 0.
    //
    // When ColMajor:
    //
    // - bcast_block_sizes:
    //   [d_0, b_0, d_1, b_1, ...]
    //
    // - bcast_block_strides:
    //   [output_block_strides[0], output_block_strides[0] * d_0,
    //    output_block_strides[1], output_block_strides[1] * d_1,
    //   ...]
    //
    // - bcast_input_strides:
    //   [input_block_strides[0], 0,
    //    input_block_strides[1], 0,
    //   ...].
    //
    for (int i = 0; i < params.inner_dim_count; ++i) {
      const int dim = IsColMajor ? i : NumDims - i - 1;
 
      const int copy_dim = IsColMajor ? 2 * i : 2 * NumDims - 2 * i - 1;
      const int broadcast_dim = IsColMajor ? copy_dim + 1 : copy_dim - 1;
 
      params.bcast_block_sizes[copy_dim] = params.input_dims[dim];
      params.bcast_block_sizes[broadcast_dim] = m_broadcast[dim];
      params.bcast_block_strides[copy_dim] = params.output_strides[dim];
      params.bcast_block_strides[broadcast_dim] = params.output_strides[dim] * params.input_dims[dim];
      params.bcast_input_strides[copy_dim] = params.input_block_strides[dim];
      params.bcast_input_strides[broadcast_dim] = 0;
    }
 
    for (int i = 2 * params.inner_dim_count; i < 2 * NumDims; ++i) {
      const int dim = IsColMajor ? i : 2 * NumDims - i - 1;
      params.bcast_block_sizes[dim] = 1;
      params.bcast_block_strides[dim] = 0;
      params.bcast_input_strides[dim] = 0;
    }
 
    return params;
  }
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE TensorBlock emptyBlock() const {
    DSizes<Index, NumDims> dimensions;
    for (int i = 0; i < NumDims; ++i) dimensions[i] = 0;
    return TensorBlock(internal::TensorBlockKind::kView, NULL, dimensions);
  }
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Index BroadcastBlockAlongBcastDim(
      BlockBroadcastingParams params, Index bcast_offset, TensorBlockScratch& scratch,
      ScalarNoConst* materialized_output, ScalarNoConst** materialized_input, size_t* materialized_input_size) const {
    if (params.bcast_dim_size == 1) {
      // We just need one block read using the ready-set values above.
      return BroadcastBlock(params.input_block_sizes, params.input_block_strides, params.bcast_block_sizes,
                            params.bcast_block_strides, params.bcast_input_strides, bcast_offset, 0, scratch,
                            materialized_output, materialized_input, materialized_input_size);
 
    } else if (params.input_dims[params.bcast_dim] == 1) {
      // Broadcast bcast dimension (< NumDims) by bcast_dim_size.
      const int broadcast_bcast_dim =
          IsColMajor ? 2 * params.inner_dim_count + 1 : 2 * NumDims - 2 * params.inner_dim_count - 2;
 
      params.bcast_block_sizes[broadcast_bcast_dim] = params.bcast_dim_size;
      params.bcast_input_strides[broadcast_bcast_dim] = 0;
      params.bcast_block_strides[broadcast_bcast_dim] = params.output_strides[params.bcast_dim];
 
      return BroadcastBlock(params.input_block_sizes, params.input_block_strides, params.bcast_block_sizes,
                            params.bcast_block_strides, params.bcast_input_strides, bcast_offset, 0, scratch,
                            materialized_output, materialized_input, materialized_input_size);
 
    } else {
      // Keep track of the total number of the coefficients written to the
      // output block.
      Index num_output_coeffs = 0;
 
      // The general case. Let's denote the output block as
      //
      //   x[..., a:a+bcast_dim_size, :, ..., :]
      //
      // where a:a+bcast_dim_size is a slice on the bcast_dim dimension
      // (< NumDims). We need to split the a:a+bcast_dim_size into possibly 3
      // sub-blocks:
      //
      // (1) a:b, where b is the smallest multiple of
      //     input_dims[bcast_dim_start] in [a, a+bcast_dim_size].
      //
      // (2) b:c, where c is the largest multiple of input_dims[bcast_dim_start]
      //     in [a, a+bcast_dim_size].
      //
      // (3) c:a+bcast_dim_size .
      //
      // Or, when b and c do not exist, we just need to process the whole block
      // together.
 
      // Find a.
      const Index bcast_dim_left_index = bcast_offset / m_outputStrides[params.bcast_dim];
 
      // Find b and c.
      const Index input_bcast_dim_size = params.input_dims[params.bcast_dim];
 
      // First multiple after a. This is b when <= bcast_dim_left_index +
      // bcast_dim_size.
      const Index first_multiple =
          numext::div_ceil<Index>(bcast_dim_left_index, input_bcast_dim_size) * input_bcast_dim_size;
 
      if (first_multiple <= bcast_dim_left_index + params.bcast_dim_size) {
        // b exists, so does c. Find it.
        const Index last_multiple =
            (bcast_dim_left_index + params.bcast_dim_size) / input_bcast_dim_size * input_bcast_dim_size;
        const int copy_bcast_dim =
            IsColMajor ? 2 * params.inner_dim_count : 2 * NumDims - 2 * params.inner_dim_count - 1;
        const int broadcast_bcast_dim =
            IsColMajor ? 2 * params.inner_dim_count + 1 : 2 * NumDims - 2 * params.inner_dim_count - 2;
 
        if (first_multiple > bcast_dim_left_index) {
          const Index head_size = first_multiple - bcast_dim_left_index;
          params.input_block_sizes[params.bcast_dim] = head_size;
          params.bcast_block_sizes[copy_bcast_dim] = head_size;
          params.bcast_input_strides[copy_bcast_dim] = params.input_block_strides[params.bcast_dim];
          params.bcast_block_strides[copy_bcast_dim] = params.output_strides[params.bcast_dim];
          params.bcast_block_sizes[broadcast_bcast_dim] = 1;
          params.bcast_input_strides[broadcast_bcast_dim] = 0;
          params.bcast_block_strides[broadcast_bcast_dim] =
              params.output_strides[params.bcast_dim] * params.input_dims[params.bcast_dim];
 
          num_output_coeffs +=
              BroadcastBlock(params.input_block_sizes, params.input_block_strides, params.bcast_block_sizes,
                             params.bcast_block_strides, params.bcast_input_strides, bcast_offset, 0, scratch,
                             materialized_output, materialized_input, materialized_input_size);
        }
        if (first_multiple < last_multiple) {
          params.input_block_sizes[params.bcast_dim] = input_bcast_dim_size;
          params.bcast_block_sizes[copy_bcast_dim] = input_bcast_dim_size;
          params.bcast_input_strides[copy_bcast_dim] = params.input_block_strides[params.bcast_dim];
          params.bcast_block_strides[copy_bcast_dim] = params.output_strides[params.bcast_dim];
          params.bcast_block_sizes[broadcast_bcast_dim] = (last_multiple - first_multiple) / input_bcast_dim_size;
          params.bcast_input_strides[broadcast_bcast_dim] = 0;
          params.bcast_block_strides[broadcast_bcast_dim] =
              params.output_strides[params.bcast_dim] * params.input_dims[params.bcast_dim];
          const Index offset = (first_multiple - bcast_dim_left_index) * m_outputStrides[params.bcast_dim];
 
          num_output_coeffs +=
              BroadcastBlock(params.input_block_sizes, params.input_block_strides, params.bcast_block_sizes,
                             params.bcast_block_strides, params.bcast_input_strides, bcast_offset, offset, scratch,
                             materialized_output, materialized_input, materialized_input_size);
        }
        if (last_multiple < bcast_dim_left_index + params.bcast_dim_size) {
          const Index tail_size = bcast_dim_left_index + params.bcast_dim_size - last_multiple;
          params.input_block_sizes[params.bcast_dim] = tail_size;
          params.bcast_block_sizes[copy_bcast_dim] = tail_size;
          params.bcast_input_strides[copy_bcast_dim] = params.input_block_strides[params.bcast_dim];
          params.bcast_block_strides[copy_bcast_dim] = params.output_strides[params.bcast_dim];
          params.bcast_block_sizes[broadcast_bcast_dim] = 1;
          params.bcast_input_strides[broadcast_bcast_dim] = 0;
          params.bcast_block_strides[broadcast_bcast_dim] =
              params.output_strides[params.bcast_dim] * params.input_dims[params.bcast_dim];
          const Index offset = (last_multiple - bcast_dim_left_index) * m_outputStrides[params.bcast_dim];
 
          num_output_coeffs +=
              BroadcastBlock(params.input_block_sizes, params.input_block_strides, params.bcast_block_sizes,
                             params.bcast_block_strides, params.bcast_input_strides, bcast_offset, offset, scratch,
                             materialized_output, materialized_input, materialized_input_size);
        }
      } else {
        // b and c do not exist.
        const int copy_bcast_dim =
            IsColMajor ? 2 * params.inner_dim_count : 2 * NumDims - 2 * params.inner_dim_count - 1;
        params.input_block_sizes[params.bcast_dim] = params.bcast_dim_size;
        params.bcast_block_sizes[copy_bcast_dim] = params.bcast_dim_size;
        params.bcast_input_strides[copy_bcast_dim] = params.input_block_strides[params.bcast_dim];
        params.bcast_block_strides[copy_bcast_dim] = params.output_strides[params.bcast_dim];
 
        num_output_coeffs +=
            BroadcastBlock(params.input_block_sizes, params.input_block_strides, params.bcast_block_sizes,
                           params.bcast_block_strides, params.bcast_input_strides, bcast_offset, 0, scratch,
                           materialized_output, materialized_input, materialized_input_size);
      }
 
      return num_output_coeffs;
    }
  }
 
  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Index BroadcastBlock(
      const Dimensions& input_block_sizes, const Dimensions& input_block_strides,
      const BroadcastDimensions& bcast_block_sizes, const BroadcastDimensions& bcast_block_strides,
      const BroadcastDimensions& bcast_input_strides, Index bcast_offset, Index offset, TensorBlockScratch& scratch,
      ScalarNoConst* materialized_output, ScalarNoConst** materialized_input, size_t* materialized_input_size) const {
    // ---------------------------------------------------------------------- //
    // Tensor block descriptor for reading block from the input.
    const Index input_offset = bcast_offset + offset;
    TensorBlockDesc input_desc(IsColMajor ? indexColMajor(input_offset) : indexRowMajor(input_offset),
                               input_block_sizes);
 
    ArgTensorBlock input_block = m_impl.block(input_desc, scratch);
 
    // ---------------------------------------------------------------------- //
    // Materialize input block into a temporary memory buffer only if it's not
    // already available in the arg block.
    const ScalarNoConst* input_buffer = NULL;
 
    if (input_block.data() != NULL) {
      // Input block already has raw data, there is no need to materialize it.
      input_buffer = input_block.data();
 
    } else {
      // Otherwise we have to do block assignment into a temporary buffer.
 
      // Maybe reuse previously allocated buffer, or allocate a new one with a
      // scratch allocator.
      const size_t input_total_size = input_block_sizes.TotalSize();
      if (*materialized_input == NULL || *materialized_input_size < input_total_size) {
        *materialized_input_size = input_total_size;
        void* mem = scratch.allocate(*materialized_input_size * sizeof(Scalar));
        *materialized_input = static_cast<ScalarNoConst*>(mem);
      }
 
      typedef internal::TensorBlockAssignment<ScalarNoConst, NumDims, typename ArgTensorBlock::XprType, Index>
          TensorBlockAssignment;
 
      TensorBlockAssignment::Run(
          TensorBlockAssignment::target(input_block_sizes, input_block_strides, *materialized_input),
          input_block.expr());
 
      input_buffer = *materialized_input;
    }
 
    // ---------------------------------------------------------------------- //
    // Copy data from materialized input block to the materialized output, using
    // given broadcast strides (strides with zeroes).
    typedef internal::TensorBlockIO<ScalarNoConst, Index, 2 * NumDims, Layout> TensorBlockIO;
 
    typename TensorBlockIO::Src src(bcast_input_strides, input_buffer);
    typename TensorBlockIO::Dst dst(bcast_block_sizes, bcast_block_strides, materialized_output + offset);
 
    return TensorBlockIO::Copy(dst, src);
  }
 
 protected:
  const Device EIGEN_DEVICE_REF m_device;
  const std::remove_reference_t<Broadcast> m_broadcast;
  Dimensions m_dimensions;
  array<Index, NumDims> m_outputStrides;
  array<Index, NumDims> m_inputStrides;
  TensorEvaluator<ArgType, Device> m_impl;
};
 
}  // end namespace Eigen
 
#endif  // EIGEN_CXX11_TENSOR_TENSOR_BROADCASTING_H