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Composable Kernel: /home/docs/checkouts/readthedocs.org/user_builds/advanced-micro-devices-composable-kernel/checkouts/develop/include/ck_tile/ops/flatmm/kernel/flatmm_kernel.hpp Source File
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 // SPDX-License-Identifier: MIT
 // Copyright (c) 2018-2025, Advanced Micro Devices, Inc. All rights reserved.
  
 #pragma once
  
 #include <iostream>
 #include <string>
  
 #include "ck_tile/core.hpp"
 #include "ck_tile/ops/common.hpp"
 #include "ck_tile/ops/gemm/pipeline/gemm_pipeline_ag_bg_cr_scheduler.hpp"
  
 namespace ck_tile {
  
 template <index_t NumDTensor = 0>
 struct FlatmmHostArgs
 {
     CK_TILE_HOST FlatmmHostArgs() = default;
     CK_TILE_HOST FlatmmHostArgs(const void* a_ptr_,
                                 const void* b_ptr_,
                                 const std::array<const void*, NumDTensor>& ds_ptr_,
                                 void* e_ptr_,
                                 index_t k_batch_,
                                 index_t M_,
                                 index_t N_,
                                 index_t K_,
                                 index_t stride_A_,
                                 index_t stride_B_,
                                 const std::array<index_t, NumDTensor>& stride_Ds_,
                                 index_t stride_E_)
         : a_ptr(a_ptr_),
           b_ptr(b_ptr_),
           ds_ptr(ds_ptr_),
           e_ptr(e_ptr_),
           M(M_),
           N(N_),
           K(K_),
           stride_A(stride_A_),
           stride_B(stride_B_),
           stride_Ds(stride_Ds_),
           stride_E(stride_E_),
           k_batch(k_batch_)
     {
     }
  
     const void* a_ptr;
     const void* b_ptr;
     const std::array<const void*, NumDTensor> ds_ptr;
     union
     {
         void* e_ptr;
         void* c_ptr;
     };
     index_t M;
     index_t N;
     index_t K;
     index_t stride_A;
     index_t stride_B;
     const std::array<index_t, NumDTensor> stride_Ds;
     union
     {
         index_t stride_E;
         index_t stride_C;
     };
  
     index_t k_batch;
 };
  
 template <index_t NumDTensor = 0>
 struct FlatmmKernelArgs
 {
     const void* a_ptr;
     // const void* b_shuffle_ptr;
     const void* b_ptr;
     const std::array<const void*, NumDTensor> ds_ptr;
     void* e_ptr;
     index_t M;
     index_t N;
     index_t K;
     index_t stride_A;
     index_t stride_B;
     std::array<index_t, NumDTensor> stride_Ds;
     index_t stride_E;
     index_t k_batch;
 };
  
 template <typename TilePartitioner_, typename FlatmmPipeline_, typename EpiloguePipeline_>
 struct FlatmmKernel
 {
     using TilePartitioner = remove_cvref_t<TilePartitioner_>;
     using FlatmmPipeline  = remove_cvref_t<FlatmmPipeline_>;
     using BlockGemmShape =
         remove_cvref_t<typename FlatmmPipeline::BlockGemmShape>; // TileFlatmmShape
     using EpiloguePipeline              = remove_cvref_t<EpiloguePipeline_>;
     using ALayout                       = remove_cvref_t<typename FlatmmPipeline::ALayout>;
     using BLayout                       = remove_cvref_t<typename FlatmmPipeline::BLayout>;
     using ELayout                       = remove_cvref_t<typename FlatmmPipeline::CLayout>;
     using DsLayout                      = remove_cvref_t<typename EpiloguePipeline::DsLayout>;
     using DsDataType                    = remove_cvref_t<typename EpiloguePipeline::DsDataType>;
     static constexpr index_t kBlockSize = FlatmmPipeline::BlockSize;
  
     using ADataType = remove_cvref_t<typename FlatmmPipeline::ADataType>;
     using BDataType = remove_cvref_t<typename FlatmmPipeline::BDataType>;
     // Below type is actually accumulation data type - the output of block GEMM.
     using EDataType = remove_cvref_t<typename EpiloguePipeline::ODataType>;
  
     static constexpr index_t NumDTensor = DsDataType::size();
  
     static constexpr auto I0 = number<0>();
     static constexpr auto I1 = number<1>();
     static constexpr auto I2 = number<2>();
     static constexpr auto I3 = number<3>();
  
     static_assert(DsLayout::size() == DsDataType::size(),
                   "The size of DsLayout and DsDataType should be the same");
     using KernelArgs = FlatmmKernelArgs<DsLayout::size()>;
  
     [[nodiscard]] CK_TILE_HOST static const std::string GetName()
     {
         // clang-format off
         return concat('_', "gemm", gemm_prec_str<ADataType, BDataType>, FlatmmPipeline::GetName());
         // clang-format on
     }
  
     CK_TILE_HOST static constexpr auto GridSize(index_t M, index_t N, index_t KBatch)
     {
         return dim3(TilePartitioner::GridSize(M, N), 1, KBatch);
     }
  
     CK_TILE_HOST static constexpr auto BlockSize()
     {
         return is_wave32() ? dim3(kBlockSize / 2) : dim3(kBlockSize);
     }
  
     CK_TILE_HOST static constexpr KernelArgs
     MakeKernelArgs(const FlatmmHostArgs<NumDTensor>& hostArgs)
     {
         return KernelArgs{hostArgs.a_ptr,
                           hostArgs.b_ptr,
                           hostArgs.ds_ptr,
                           hostArgs.e_ptr,
                           hostArgs.M,
                           hostArgs.N,
                           hostArgs.K,
                           hostArgs.stride_A,
                           hostArgs.stride_B,
                           hostArgs.stride_Ds,
                           hostArgs.stride_E,
                           hostArgs.k_batch};
     }
  
     CK_TILE_HOST_DEVICE static constexpr index_t GetSmemSize()
     {
         return max(FlatmmPipeline::GetSmemSize(), EpiloguePipeline::GetSmemSize());
     }
  
     struct SplitKBatchOffset
     {
         __device__ SplitKBatchOffset(const KernelArgs& kargs, const std::size_t k_id = blockIdx.z)
         {
             constexpr auto K1   = TilePartitioner::BlockGemmShape::WarpTile::at(number<2>{});
             const index_t K_t   = kargs.k_batch * K1;
             const index_t KRead = (kargs.K + K_t - 1) / K_t * K1;
  
             if constexpr(std::is_same_v<tensor_layout::gemm::RowMajor, ALayout>)
             {
                 a_k_split_offset = k_id * KRead;
             }
             else if constexpr(std::is_same_v<tensor_layout::gemm::ColumnMajor, ALayout>)
             {
                 a_k_split_offset = k_id * KRead * kargs.stride_A;
             }
  
             if constexpr(std::is_same_v<tensor_layout::gemm::RowMajor, BLayout>)
             {
                 b_k_split_offset = k_id * KRead * kargs.stride_B;
             }
             else if constexpr(std::is_same_v<tensor_layout::gemm::ColumnMajor, BLayout>)
             {
                 b_k_split_offset = k_id * KRead;
             }
  
             if(k_id < static_cast<uint32_t>(kargs.k_batch - 1))
             {
                 splitted_k = KRead;
             }
             else
             {
                 splitted_k = kargs.K - KRead * (kargs.k_batch - 1);
             }
         }
  
         index_t a_k_split_offset;
         index_t b_k_split_offset;
         index_t splitted_k;
     };
  
     CK_TILE_HOST static bool IsSupportedArgument(const KernelArgs& kargs)
     {
         if constexpr(EpiloguePipeline::GetVectorSizeC() % 2 != 0 &&
                      is_any_of<EDataType, fp16_t, bf16_t>::value)
         {
             if(kargs.k_batch != 1)
             {
                 std::cerr << "Conditions not met for Kbatch >1 !" << std::endl;
                 return false;
             }
         }
  
         if constexpr(std::is_same_v<ALayout, tensor_layout::gemm::RowMajor>)
         {
             if(kargs.K % TilePartitioner::KPerBlock != 0 && FlatmmPipeline::kPadK == false)
             {
                 std::cerr << "Can't support K that is not a multiple of KPerBlock"
                              " without padding!"
                           << std::endl;
                 return false;
             }
             if(kargs.K % FlatmmPipeline::GetVectorSizeA() != 0)
             {
                 std::cerr << "K is not a multiple of vector load size for A tensor!" << std::endl;
                 return false;
             }
         }
         else
         {
             if(kargs.M % TilePartitioner::MPerBlock != 0 && FlatmmPipeline::kPadM == false)
             {
                 std::cerr << "Can't support M that is not a multiple of MPerBlock"
                              " without padding!"
                           << std::endl;
                 return false;
             }
             if(kargs.M % FlatmmPipeline::GetVectorSizeA() != 0)
             {
                 std::cerr << "M is not a multiple of vector load size for A tensor!" << std::endl;
                 return false;
             }
         }
  
         if constexpr(std::is_same_v<BLayout, tensor_layout::gemm::RowMajor>)
         {
             if(kargs.N % TilePartitioner::NPerBlock != 0 && FlatmmPipeline::kPadN == false)
             {
                 std::cerr << "Can't support N that is not a multiple of NPerBlock"
                              " without padding!"
                           << std::endl;
                 return false;
             }
             if(kargs.N % FlatmmPipeline::GetVectorSizeB() != 0)
             {
                 std::cerr << "N is not a multiple of vector load size for B tensor!" << std::endl;
                 return false;
             }
         }
         else
         {
             if(kargs.K % TilePartitioner::KPerBlock != 0 && FlatmmPipeline::kPadK == false)
             {
                 std::cerr << "Can't support K that is not a multiple of KPerBlock"
                              " without padding!"
                           << std::endl;
                 return false;
             }
             if(kargs.K % FlatmmPipeline::GetVectorSizeB() != 0)
             {
                 std::cerr << "K is not a multiple of vector load size for B tensor!" << std::endl;
                 return false;
             }
         }
  
         bool DTesnorIsValid = {true};
         static_for<0, NumDTensor, 1>{}([&](auto index) {
             using DiLayout = remove_cvref_t<std::tuple_element_t<index.value, DsLayout>>;
             if(std::is_same_v<DiLayout, ELayout> == false)
             {
                 DTesnorIsValid = false;
             }
             if constexpr(std::is_same_v<DiLayout, tensor_layout::gemm::RowMajor>)
             {
                 if(kargs.N % TilePartitioner::NPerBlock != 0 && FlatmmPipeline::kPadN == false)
                 {
                     CK_TILE_ERROR("Can't support N for tensor D that is not a multiple of "
                                   "NPerBlock without padding!");
                     DTesnorIsValid = false;
                 }
                 if(kargs.N % EpiloguePipeline::GetVectorSizeD(index) != 0)
                 {
                     CK_TILE_ERROR("N is not a multiple of vector load size for D tensor!");
                     DTesnorIsValid = false;
                 }
             }
             else
             {
                 if(kargs.M % TilePartitioner::MPerBlock != 0 && FlatmmPipeline::kPadM == false)
                 {
                     CK_TILE_ERROR("Can't support M for tensor D that is not a multiple of "
                                   "MPerBlock without padding!");
  
                     DTesnorIsValid = false;
                 }
                 if(kargs.M % EpiloguePipeline::GetVectorSizeD(index) != 0)
                 {
                     CK_TILE_ERROR("M is not a multiple of vector load size for D tensor!");
                     DTesnorIsValid = false;
                 }
             }
         });
  
         if constexpr(std::is_same_v<ELayout, tensor_layout::gemm::RowMajor>)
         {
             if(kargs.N % TilePartitioner::NPerBlock != 0 && FlatmmPipeline::kPadN == false)
             {
                 std::cerr << "Can't support N that is not a multiple of NPerBlock"
                              " without padding!"
                           << std::endl;
                 return false;
             }
             if(kargs.N % EpiloguePipeline::GetVectorSizeC() != 0)
             {
                 std::cerr << "N is not a multiple of vector load size for C tensor!" << std::endl;
                 return false;
             }
         }
         else
         {
             if(kargs.M % TilePartitioner::MPerBlock != 0 && FlatmmPipeline::kPadM == false)
             {
                 std::cerr << "Can't support M that is not a multiple of MPerBlock"
                              " without padding!"
                           << std::endl;
                 return false;
             }
             if(kargs.M % EpiloguePipeline::GetVectorSizeC() != 0)
             {
                 std::cerr << "M is not a multiple of vector load size for C tensor!" << std::endl;
                 return false;
             }
         }
         return DTesnorIsValid;
     }
  
     template <memory_operation_enum DstInMemOp = memory_operation_enum::set>
     CK_TILE_DEVICE static auto
     MakeGemmTensorViews(const ADataType* a_ptr,
                         const BDataType* b_flat_ptr,
                         const std::array<const void*, NumDTensor>& ds_ptr,
                         EDataType* e_ptr,
                         const KernelArgs& kargs,
                         const SplitKBatchOffset& splitk_batch_offset)
     {
         const auto& a_tensor_view = [&]() {
             if constexpr(std::is_same_v<ALayout, tensor_layout::gemm::RowMajor>)
             {
                 return make_naive_tensor_view<address_space_enum::global>(
                     a_ptr,
                     make_tuple(kargs.M, splitk_batch_offset.splitted_k),
                     make_tuple(kargs.stride_A, 1),
                     number<FlatmmPipeline::GetVectorSizeA()>{},
                     number<1>{});
             }
             else
             {
                 return make_naive_tensor_view<address_space_enum::global>(
                     a_ptr,
                     make_tuple(splitk_batch_offset.splitted_k, kargs.M),
                     make_tuple(kargs.stride_A, 1),
                     number<FlatmmPipeline::GetVectorSizeA()>{},
                     number<1>{});
             }
         }();
  
         index_t kFlatK = FlatmmPipeline::flatKPerWarp * (splitk_batch_offset.splitted_k /
                                                          BlockGemmShape::WarpTile::at(number<2>{}));
         index_t kFlatN = kargs.N * kargs.K / kFlatK;
         const auto& b_flat_tensor_view = [&]() {
             return make_naive_tensor_view<address_space_enum::global>(
                 b_flat_ptr,
                 make_tuple(kFlatN, kFlatK),
                 make_tuple(kFlatK, 1),
                 number<FlatmmPipeline::GetVectorSizeB()>{},
                 number<1>{});
         }();
  
         const auto& ds_tensor_view = generate_tuple(
             [&](auto i) {
                 using DiLayout   = remove_cvref_t<std::tuple_element_t<i.value, DsLayout>>;
                 using DDataType_ = remove_cvref_t<std::tuple_element_t<i.value, DsDataType>>;
                 if constexpr(std::is_same_v<DiLayout, tensor_layout::gemm::RowMajor>)
                 {
                     return make_naive_tensor_view<address_space_enum::global>(
                         static_cast<const DDataType_*>(ds_ptr[i]),
                         make_tuple(kargs.M, kargs.N),
                         make_tuple(kargs.stride_Ds[i], 1),
                         number<EpiloguePipeline::GetVectorSizeD(i)>{},
                         number<1>{});
                 }
                 else
                 {
                     return make_naive_tensor_view<address_space_enum::global>(
                         static_cast<const DDataType_*>(ds_ptr[i]),
                         make_tuple(kargs.N, kargs.M),
                         make_tuple(kargs.stride_Ds[i], 1),
                         number<EpiloguePipeline::GetVectorSizeD(i)>{},
                         number<1>{});
                 }
             },
             number<NumDTensor>{});
  
         // TODO: enable vector write for C in ColMajor
         const auto& e_tensor_view = [&]() {
             if constexpr(std::is_same_v<ELayout, tensor_layout::gemm::RowMajor>)
             {
                 return make_naive_tensor_view<address_space_enum::global>(
                     e_ptr,
                     make_tuple(kargs.M, kargs.N),
                     make_tuple(kargs.stride_E, 1),
                     number<EpiloguePipeline::GetVectorSizeC()>{},
                     number<1>{});
             }
             else
             {
                 return make_naive_tensor_view<address_space_enum::global>(
                     e_ptr,
                     make_tuple(kargs.N, kargs.M),
                     make_tuple(kargs.stride_E, 1),
                     number<1>{},
                     number<1>{});
             }
         }();
  
         return make_tuple(a_tensor_view, b_flat_tensor_view, ds_tensor_view, e_tensor_view);
     }
  
     template <typename TensorView>
     CK_TILE_DEVICE static auto MakeGemmPadViews(const TensorView& views)
     {
         const auto& a_pad_view = [&]() {
             const auto& a_tensor_view = views.at(I0);
             if constexpr(std::is_same_v<ALayout, tensor_layout::gemm::RowMajor>)
             {
                 return pad_tensor_view(a_tensor_view,
                                        make_tuple(number<TilePartitioner::MPerBlock>{},
                                                   number<TilePartitioner::KPerBlock>{}),
                                        sequence<false, FlatmmPipeline::kPadK>{});
             }
             else
             {
                 return pad_tensor_view(a_tensor_view,
                                        make_tuple(number<TilePartitioner::KPerBlock>{},
                                                   number<TilePartitioner::MPerBlock>{}),
                                        sequence<false, FlatmmPipeline::kPadM>{});
             }
         }();
  
         const auto& b_flat_tensor_view = views.at(I1);
  
         const auto& ds_pad_view = generate_tuple(
             [&](auto i) {
                 const auto& d_tensor_view = views.at(I2);
                 using DiLayout            = remove_cvref_t<std::tuple_element_t<i.value, DsLayout>>;
                 if constexpr(std::is_same_v<DiLayout, tensor_layout::gemm::RowMajor>)
                 {
                     return pad_tensor_view(d_tensor_view[i],
                                            make_tuple(number<TilePartitioner::MPerBlock>{},
                                                       number<TilePartitioner::NPerBlock>{}),
                                            sequence<false, FlatmmPipeline::kPadN>{});
                 }
                 else
                 {
                     return pad_tensor_view(d_tensor_view[i],
                                            make_tuple(number<TilePartitioner::NPerBlock>{},
                                                       number<TilePartitioner::MPerBlock>{}),
                                            sequence<false, FlatmmPipeline::kPadM>{});
                 }
             },
             number<NumDTensor>{});
  
         // TODO vector write in for C in ColMajor
         const auto& e_pad_view = [&]() {
             const auto& e_tensor_view = views.at(I3);
             if constexpr(std::is_same_v<ELayout, tensor_layout::gemm::RowMajor>)
             {
                 return pad_tensor_view(e_tensor_view,
                                        make_tuple(number<TilePartitioner::MPerBlock>{},
                                                   number<TilePartitioner::NPerBlock>{}),
                                        sequence<false, FlatmmPipeline::kPadN>{});
             }
             else
             {
                 return pad_tensor_view(e_tensor_view,
                                        make_tuple(number<TilePartitioner::MPerBlock>{},
                                                   number<TilePartitioner::NPerBlock>{}),
                                        sequence<FlatmmPipeline::kPadM, false>{});
             }
         }();
  
         return make_tuple(a_pad_view, b_flat_tensor_view, ds_pad_view, e_pad_view);
     }
  
     template <typename PadView>
     CK_TILE_DEVICE static auto
     MakeGemmTileWindows(const PadView& views, const index_t i_m, const index_t i_n)
     {
         const auto& a_pad_view      = views.at(I0);
         const auto& b_flat_pad_view = views.at(I1);
         const auto& ds_pad_view     = views.at(I2);
         const auto& e_pad_view      = views.at(I3);
  
         const auto& a_block_window = [&]() {
             if constexpr(std::is_same_v<ALayout, tensor_layout::gemm::RowMajor>)
             {
                 return make_tile_window(a_pad_view,
                                         make_tuple(number<TilePartitioner::MPerBlock>{},
                                                    number<TilePartitioner::KPerBlock>{}),
                                         {i_m, 0});
             }
             else
             {
                 return make_tile_window(a_pad_view,
                                         make_tuple(number<TilePartitioner::KPerBlock>{},
                                                    number<TilePartitioner::MPerBlock>{}),
                                         {0, i_m});
             }
         }();
  
         const auto& b_flat_block_window =
             make_tile_window(b_flat_pad_view,
                              make_tuple(number<FlatmmPipeline::flatNPerWarp>{},
                                         number<FlatmmPipeline::flatKPerWarp>{}),
                              {static_cast<int>(i_n / BlockGemmShape::WarpTile::at(I1)), 0});
  
         const auto ds_block_window = generate_tuple(
             [&](auto i) {
                 using DiLayout = remove_cvref_t<std::tuple_element_t<i.value, DsLayout>>;
                 if constexpr(std::is_same_v<DiLayout, tensor_layout::gemm::RowMajor>)
                 {
                     return make_tile_window(ds_pad_view[i],
                                             make_tuple(number<TilePartitioner::MPerBlock>{},
                                                        number<TilePartitioner::NPerBlock>{}),
                                             {i_m, i_n});
                 }
                 else
                 {
                     return make_tile_window(ds_pad_view[i],
                                             make_tuple(number<TilePartitioner::NPerBlock>{},
                                                        number<TilePartitioner::MPerBlock>{}),
                                             {i_n, i_m});
                 }
             },
             number<NumDTensor>{});
  
         auto e_block_window = make_tile_window(
             e_pad_view,
             make_tuple(number<TilePartitioner::MPerBlock>{}, number<TilePartitioner::NPerBlock>{}),
             {i_m, i_n});
  
         return make_tuple(a_block_window, b_flat_block_window, ds_block_window, e_block_window);
     }
  
     template <bool UseDefaultScheduler = true>
     CK_TILE_DEVICE static void RunFlatmm(const ADataType* a_ptr,
                                          const BDataType* b_flat_ptr,
                                          const std::array<const void*, NumDTensor>& ds_ptr,
                                          EDataType* e_ptr,
                                          void* smem_ptr,
                                          const KernelArgs& kargs,
                                          const SplitKBatchOffset& splitk_batch_offset,
                                          const index_t block_idx_m,
                                          const index_t block_idx_n)
     {
         // Create Gemm tensor views, pad views and tile windows
         const auto& gemm_tensor_views_tuple =
             MakeGemmTensorViews<EpiloguePipeline::MemoryOperation>(
                 a_ptr, b_flat_ptr, ds_ptr, e_ptr, kargs, splitk_batch_offset);
         const auto& gemm_pad_views = MakeGemmPadViews(gemm_tensor_views_tuple);
         auto gemm_tile_windows     = MakeGemmTileWindows(gemm_pad_views, block_idx_m, block_idx_n);
  
         const index_t num_loop = TilePartitioner::GetLoopNum(splitk_batch_offset.splitted_k);
  
         // Run GEMM cooperatively by whole workgroup.
         const auto& a_block_window      = gemm_tile_windows.at(I0);
         const auto& b_flat_block_window = gemm_tile_windows.at(I1);
         const auto& d_block_window      = gemm_tile_windows.at(I2);
         const auto& c_block_tile        = FlatmmPipeline{}.template operator()(
             a_block_window, b_flat_block_window, num_loop, smem_ptr);
         if(UseDefaultScheduler || (get_warp_id() == 0))
         {
             // Run Epilogue Pipeline
             auto& c_block_window = gemm_tile_windows.at(I3);
  
             EpiloguePipeline{}.template
             operator()<decltype(c_block_window), decltype(c_block_tile), decltype(d_block_window)>(
                 c_block_window, c_block_tile, d_block_window, smem_ptr);
         }
     }
  
     CK_TILE_DEVICE void operator()(KernelArgs kargs) const
     {
         const auto [iM, iN] = TilePartitioner{kargs.M, kargs.N}.GetOutputTileIndex(blockIdx.x);
         const index_t i_m   = amd_wave_read_first_lane(iM * TilePartitioner::MPerBlock);
         const index_t i_n   = amd_wave_read_first_lane(iN * TilePartitioner::NPerBlock);
  
         const SplitKBatchOffset splitk_batch_offset(kargs);
         // options
         const ADataType* a_ptr =
             static_cast<const ADataType*>(kargs.a_ptr) + splitk_batch_offset.a_k_split_offset;
         const BDataType* b_flat_ptr =
             static_cast<const BDataType*>(kargs.b_ptr) + splitk_batch_offset.b_k_split_offset;
         EDataType* e_ptr = static_cast<EDataType*>(kargs.e_ptr);
  
         // allocate LDS
         __shared__ char smem_ptr[GetSmemSize()];
  
         if constexpr(!(EpiloguePipeline::MemoryOperation == memory_operation_enum::atomic_add &&
                        EpiloguePipeline::GetVectorSizeC() % 2 != 0 &&
                        is_any_of<EDataType, fp16_t, bf16_t>::value))
         {
             constexpr auto scheduler_type = (FlatmmPipeline::NumWaveGroups == 1);
             RunFlatmm<scheduler_type>(a_ptr,
                                       b_flat_ptr,
                                       kargs.ds_ptr,
                                       e_ptr,
                                       smem_ptr,
                                       kargs,
                                       splitk_batch_offset,
                                       i_m,
                                       i_n);
         }
     }
 };
  
 } // namespace ck_tile