/home/docs/checkouts/readthedocs.org/user_builds/advanced-micro-devices-composable-kernel/checkouts/docs-6.4.3/include/ck_tile/core/algorithm/coordinate_transform.hpp Source File

/home/docs/checkouts/readthedocs.org/user_builds/advanced-micro-devices-composable-kernel/checkouts/docs-6.4.3/include/ck_tile/core/algorithm/coordinate_transform.hpp Source File#

Composable Kernel: /home/docs/checkouts/readthedocs.org/user_builds/advanced-micro-devices-composable-kernel/checkouts/docs-6.4.3/include/ck_tile/core/algorithm/coordinate_transform.hpp Source File
Go to the documentation of this file.
 // SPDX-License-Identifier: MIT
 // Copyright (c) 2018-2024, Advanced Micro Devices, Inc. All rights reserved.
  
 #pragma once
  
 #include "ck_tile/core/config.hpp"
 #include "ck_tile/core/container/multi_index.hpp"
 #include "ck_tile/core/container/container_helper.hpp"
 #include "ck_tile/core/utility/functional.hpp"
 #include "ck_tile/core/utility/type_traits.hpp"
 #include "ck_tile/core/utility/magic_div.hpp"
  
 namespace ck_tile {
  
 enum struct coord_transform_enum
 {
     undefined,
     pass_through,
     pad,
     embed,
     merge,
     unmerge,
     replicate,
     xor_t,
     offset,
     indexing,
 };
  
 template <index_t NDimLow, index_t NDimUp>
 struct base_transform
 {
     CK_TILE_HOST_DEVICE static constexpr auto get_type_enum()
     {
         return coord_transform_enum::undefined;
     }
  
     CK_TILE_HOST_DEVICE static constexpr index_t get_num_of_lower_dimension() { return NDimLow; }
  
     CK_TILE_HOST_DEVICE static constexpr index_t get_num_of_upper_dimension() { return NDimUp; }
  
     // return safe value for vector length/stride, based on compile-time known only
     // variables
     // MUST be static function
     template <typename LowVectorLengths, typename LowVectorStrides>
     CK_TILE_HOST_DEVICE static constexpr auto
     calculate_upper_dimension_safe_vector_length_strides(const LowVectorLengths&,
                                                          const LowVectorStrides&)
     {
         if constexpr(NDimUp > 0)
         {
             array<index_t, NDimUp> up_vector_lengths{-1};
             array<index_t, NDimUp> up_vector_strides{-1};
  
             return make_tuple(up_vector_lengths, up_vector_strides);
         }
         else
         {
             return make_tuple(array<index_t, 0>{}, array<index_t, 0>{});
         }
     }
 };
  
 template <typename LowLength>
 struct pass_through : public base_transform<1, 1>
 {
     static constexpr auto type_enum = coord_transform_enum::pass_through;
  
     using LowerIndex = multi_index<1>;
     using UpperIndex = multi_index<1>;
  
     using UpLengths = decltype(make_tuple(LowLength{}));
  
     UpLengths up_lengths_;
  
     CK_TILE_HOST_DEVICE constexpr pass_through() = default;
  
     CK_TILE_HOST_DEVICE constexpr pass_through(const LowLength& low_length)
         : up_lengths_{make_tuple(low_length)}
     {
     }
  
     CK_TILE_HOST_DEVICE static constexpr auto get_type_enum()
     {
         return coord_transform_enum::pass_through;
     }
  
     CK_TILE_HOST_DEVICE constexpr const auto& get_upper_lengths() const { return up_lengths_; }
  
     template <typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE static constexpr void calculate_lower_index(LowIdx& idx_low,
                                                                     const UpIdx& idx_up)
     {
         static_assert(LowIdx::size() == 1 && UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         idx_low(number<0>{}) = idx_up[number<0>{}];
     }
  
     template <typename LowIdxDiff, typename UpIdxDiff, typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE static void update_lower_index(LowIdxDiff& idx_diff_low,
                                                        const UpIdxDiff& idx_diff_up,
                                                        LowIdx& idx_low,
                                                        const UpIdx&)
     {
         static_assert(LowIdxDiff::size() == 1 && UpIdxDiff::size() == 1 && LowIdx::size() == 1 &&
                           UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         constexpr auto I0 = number<0>{};
  
         idx_diff_low[I0] = idx_diff_up[I0];
  
         idx_low += idx_diff_low;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_always_mapped_to_valid_lower_index()
     {
         return true;
     }
  
     template <typename UpIdx>
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_mapped_to_valid_lower_index(const UpIdx& /* idx_up */)
     {
         return true;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool is_known_at_compile_time()
     {
         return ck_tile::is_known_at_compile_time<UpLengths>::value;
     }
  
     // MUST be static function
     template <typename LowVectorLengths, typename LowVectorStrides>
     CK_TILE_HOST_DEVICE static constexpr auto
     calculate_upper_dimension_safe_vector_length_strides(const LowVectorLengths& low_vector_lengths,
                                                          const LowVectorStrides& low_vector_strides)
     {
         return make_tuple(low_vector_lengths, low_vector_strides);
     }
  
     CK_TILE_HOST_DEVICE void print() const
     {
         printf("pass_through{");
  
         //
         printf("up_lengths_:");
         print(up_lengths_);
  
         //
         printf("}");
     }
 };
  
 template <typename LowLength,
           typename LeftPadLength,
           typename RightPadLength,
           bool SkipIsValidCheck = false>
 struct pad : public base_transform<1, 1>
 {
     using LowerIndex = multi_index<1>;
     using UpperIndex = multi_index<1>;
  
     using UpLengths = decltype(make_tuple(LowLength{} + LeftPadLength{} + RightPadLength{}));
  
     UpLengths up_lengths_;
     LeftPadLength left_pad_length_;
     RightPadLength right_pad_length_;
  
     CK_TILE_HOST_DEVICE constexpr pad() : up_lengths_{}, left_pad_length_{}, right_pad_length_{} {}
  
     CK_TILE_HOST_DEVICE constexpr pad(const LowLength& low_length,
                                       const LeftPadLength& left_pad_length,
                                       const RightPadLength& right_pad_length)
         : up_lengths_{make_tuple(low_length + left_pad_length + right_pad_length)},
           left_pad_length_{left_pad_length},
           right_pad_length_{right_pad_length}
     {
     }
  
     CK_TILE_HOST_DEVICE constexpr const auto& get_upper_lengths() const { return up_lengths_; }
  
     template <typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr void calculate_lower_index(LowIdx& idx_low,
                                                              const UpIdx& idx_up) const
     {
         static_assert(LowIdx::size() == 1 && UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         idx_low(number<0>{}) = idx_up[number<0>{}] - left_pad_length_;
     }
  
     template <typename LowIdxDiff, typename UpIdxDiff, typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE static void update_lower_index(LowIdxDiff& idx_diff_low,
                                                        const UpIdxDiff& idx_diff_up,
                                                        LowIdx& idx_low,
                                                        const UpIdx&)
     {
         static_assert(LowIdxDiff::size() == 1 && UpIdxDiff::size() == 1 && LowIdx::size() == 1 &&
                           UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         constexpr auto I0 = number<0>{};
  
         idx_diff_low[I0] = idx_diff_up[I0];
  
         idx_low += idx_diff_low;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_always_mapped_to_valid_lower_index()
     {
         return SkipIsValidCheck;
     }
  
     template <typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr bool
     is_valid_upper_index_mapped_to_valid_lower_index(const UpIdx& idx_up) const
     {
         return SkipIsValidCheck ||
                ((idx_up[number<0>{}] >= left_pad_length_) &&
                 (idx_up[number<0>{}] < up_lengths_[number<0>{}] - right_pad_length_));
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool is_known_at_compile_time()
     {
         return ck_tile::is_known_at_compile_time<UpLengths>::value &&
                ck_tile::is_known_at_compile_time<LeftPadLength>::value &&
                ck_tile::is_known_at_compile_time<RightPadLength>::value;
     }
  
     CK_TILE_HOST_DEVICE void print() const
     {
         printf("pad{");
  
         //
         printf("up_lengths_: ");
         print(up_lengths_);
         printf(", ");
  
         //
         printf("left_pad_length_: ");
         print(left_pad_length_);
         printf(", ");
  
         //
         printf("right_pad_length_: ");
         print(right_pad_length_);
  
         printf("}");
     }
 };
  
 template <typename LowLength, typename LeftPadLength, bool SkipIsValidCheck = false>
 struct left_pad
 {
     using LowerIndex = multi_index<1>;
     using UpperIndex = multi_index<1>;
  
     using UpLengths = decltype(make_tuple(LowLength{} + LeftPadLength{}));
  
     UpLengths up_lengths_;
     LeftPadLength left_pad_length_;
  
     CK_TILE_HOST_DEVICE constexpr left_pad() = default;
  
     CK_TILE_HOST_DEVICE constexpr left_pad(const LowLength& low_length,
                                            const LeftPadLength& left_pad_length)
         : up_lengths_{make_tuple(low_length + left_pad_length)}, left_pad_length_{left_pad_length}
     {
     }
  
     CK_TILE_HOST_DEVICE constexpr const auto& get_upper_lengths() const { return up_lengths_; }
  
     template <typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr void calculate_lower_index(LowIdx& idx_low,
                                                              const UpIdx& idx_up) const
     {
         static_assert(LowIdx::size() == 1 && UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         idx_low(number<0>{}) = idx_up[number<0>{}] - left_pad_length_;
     }
  
     template <typename LowIdxDiff, typename UpIdxDiff, typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE static void update_lower_index(LowIdxDiff& idx_diff_low,
                                                        const UpIdxDiff& idx_diff_up,
                                                        LowIdx& idx_low,
                                                        const UpIdx&)
     {
         static_assert(LowIdxDiff::size() == 1 && UpIdxDiff::size() == 1 && LowIdx::size() == 1 &&
                           UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         constexpr auto I0 = number<0>{};
  
         idx_diff_low[I0] = idx_diff_up[I0];
  
         idx_low += idx_diff_low;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_always_mapped_to_valid_lower_index()
     {
         return SkipIsValidCheck;
     }
  
     template <typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr bool
     is_valid_upper_index_mapped_to_valid_lower_index(const UpIdx& idx_up) const
     {
         return SkipIsValidCheck || (idx_up[number<0>{}] >= left_pad_length_);
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool is_known_at_compile_time()
     {
         return ck_tile::is_known_at_compile_time<UpLengths>::value &&
                ck_tile::is_known_at_compile_time<LeftPadLength>::value;
     }
  
     // MUST be static function
     template <typename LowVectorLengths, typename LowVectorStrides>
     CK_TILE_HOST_DEVICE static constexpr auto
     calculate_upper_dimension_safe_vector_length_strides(const LowVectorLengths& low_vector_lengths,
                                                          const LowVectorStrides& low_vector_strides)
     {
         // TODO: we allow pass through this vector length. If one need per-pixel check,
         //       should change the guaranteed vector length while creating the tensor view.
         //       It's up to runtime to check the padding length should be multiple of vector length
         return make_tuple(low_vector_lengths, low_vector_strides);
     }
  
     CK_TILE_HOST_DEVICE void print() const
     {
         printf("left_pad{");
  
         //
         printf("up_lengths_: ");
         print(up_lengths_);
         printf(", ");
  
         //
         printf("left_pad_length_: ");
         print(left_pad_length_);
  
         printf("}");
     }
 };
  
 template <typename LowLength, typename RightPadLength, bool SkipIsValidCheck = false>
 struct right_pad : public base_transform<1, 1>
 {
     using LowerIndex = multi_index<1>;
     using UpperIndex = multi_index<1>;
  
     using UpLengths = decltype(make_tuple(LowLength{} + RightPadLength{}));
  
     UpLengths up_lengths_;
     LowLength low_length_;
     RightPadLength right_pad_length_;
  
     CK_TILE_HOST_DEVICE constexpr right_pad() = default;
  
     CK_TILE_HOST_DEVICE constexpr right_pad(const LowLength& low_length,
                                             const RightPadLength& right_pad_length)
         : up_lengths_{make_tuple(low_length + right_pad_length)},
           low_length_{low_length},
           right_pad_length_{right_pad_length}
     {
     }
  
     CK_TILE_HOST_DEVICE constexpr const auto& get_upper_lengths() const { return up_lengths_; }
  
     template <typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE static constexpr void calculate_lower_index(LowIdx& idx_low,
                                                                     const UpIdx& idx_up)
     {
         static_assert(LowIdx::size() == 1 && UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         idx_low(number<0>{}) = idx_up[number<0>{}];
     }
  
     template <typename LowIdxDiff, typename UpIdxDiff, typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE static void update_lower_index(LowIdxDiff& idx_diff_low,
                                                        const UpIdxDiff& idx_diff_up,
                                                        LowIdx& idx_low,
                                                        const UpIdx&)
     {
         static_assert(LowIdxDiff::size() == 1 && UpIdxDiff::size() == 1 && LowIdx::size() == 1 &&
                           UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         constexpr auto I0 = number<0>{};
  
         idx_diff_low[I0] = idx_diff_up[I0];
  
         idx_low += idx_diff_low;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_always_mapped_to_valid_lower_index()
     {
         return SkipIsValidCheck;
     }
  
     template <typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr bool
     is_valid_upper_index_mapped_to_valid_lower_index(const UpIdx& idx_up) const
     {
         return SkipIsValidCheck || (idx_up[number<0>{}] < low_length_);
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool is_known_at_compile_time()
     {
         return ck_tile::is_known_at_compile_time<UpLengths>::value &&
                ck_tile::is_known_at_compile_time<LowLength>::value &&
                ck_tile::is_known_at_compile_time<RightPadLength>::value;
     }
  
     // MUST be static function
     template <typename LowVectorLengths, typename LowVectorStrides>
     CK_TILE_HOST_DEVICE static constexpr auto
     calculate_upper_dimension_safe_vector_length_strides(const LowVectorLengths& low_vector_lengths,
                                                          const LowVectorStrides& low_vector_strides)
     {
         // TODO: we allow pass through this vector length. If one need per-pixel check,
         //       should change the guaranteed vector length while creating the tensor view.
         //       It's up to runtime to check the padding length should be multiple of vector length
         return make_tuple(low_vector_lengths, low_vector_strides);
     }
  
     CK_TILE_HOST_DEVICE void print() const
     {
         printf("right_pad{");
  
         //
         printf("up_lengths_: ");
         print(up_lengths_);
         printf(", ");
  
         //
         printf("right_pad_length_: ");
         print(right_pad_length_);
  
         printf("}");
     }
 };
  
 // idx_low = coefficients[0, ...nDimUp-1] * idx_up[0, ...nDimUp-1]
 // UpLengths and Coefficients can be either of the followings:
 //   1) Tuple of index_t, which is known at run-time, or
 //   2) Tuple of number, which is known at compile-time, or
 //   3) Tuple of mixture of index_t and number, which is known partially at run-time and partially
 //   at compile-time
 template <typename UpLengths,
           typename Coefficients,
           typename std::enable_if<UpLengths::size() == Coefficients::size(), bool>::type = false>
 struct embed : public base_transform<1, UpLengths::size()>
 {
     static constexpr index_t NDimUp = UpLengths::size();
  
     using LowerIndex = multi_index<1>;
     using UpperIndex = multi_index<NDimUp>;
  
     UpLengths up_lengths_;
     Coefficients coefficients_;
  
     CK_TILE_HOST_DEVICE constexpr embed() = default;
  
     CK_TILE_HOST_DEVICE constexpr embed(const UpLengths& up_lengths,
                                         const Coefficients& coefficients)
         : up_lengths_{up_lengths}, coefficients_{coefficients}
     {
     }
  
     CK_TILE_HOST_DEVICE static constexpr auto get_type_enum()
     {
         return coord_transform_enum::embed;
     }
  
     CK_TILE_HOST_DEVICE constexpr const auto& get_upper_lengths() const { return up_lengths_; }
  
     template <typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr void calculate_lower_index(LowIdx& idx_low,
                                                              const UpIdx& idx_up) const
     {
         static_assert(LowIdx::size() == 1 && UpIdx::size() == NDimUp,
                       "wrong! inconsistent # of dimension");
  
         idx_low(number<0>{}) = 0;
  
         static_for<0, NDimUp, 1>{}([&idx_low, &idx_up, this](auto i) {
             idx_low(number<0>{}) += idx_up[i] * this->coefficients_[i];
         });
     }
  
     template <typename LowIdxDiff, typename UpIdxDiff, typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE void update_lower_index(LowIdxDiff& idx_diff_low,
                                                 const UpIdxDiff& idx_diff_up,
                                                 LowIdx& idx_low,
                                                 const UpIdx&) const
     {
         static_assert(LowIdxDiff::size() == 1 && UpIdxDiff::size() == NDimUp &&
                           LowIdx::size() == 1 && UpIdx::size() == NDimUp,
                       "wrong! inconsistent # of dimension");
  
         idx_diff_low(number<0>{}) = 0;
  
         static_for<0, NDimUp, 1>{}(
             [&](auto i) { idx_diff_low(number<0>{}) += idx_diff_up[i] * coefficients_[i]; });
  
         idx_low += idx_diff_low;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_always_mapped_to_valid_lower_index()
     {
         return true;
     }
  
     template <typename UpIdx>
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_mapped_to_valid_lower_index(const UpIdx& /* idx_up */)
     {
         return true;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool is_known_at_compile_time()
     {
         return ck_tile::is_known_at_compile_time<UpLengths>::value &&
                ck_tile::is_known_at_compile_time<Coefficients>::value;
     }
  
     CK_TILE_HOST_DEVICE void print() const
     {
         printf("embed{");
  
         //
         printf("up_lengths_: ");
         print(up_lengths_);
         printf(", ");
  
         //
         printf("coefficients_: ");
         print(coefficients_);
  
         printf("}");
     }
 };
  
 template <typename LowLengths>
 struct lambda_merge_generate_MagicDivision_calculate_magic_divisor
 {
     template <index_t I>
     CK_TILE_HOST_DEVICE constexpr auto operator()(number<I> i) const
     {
         return magic_division::calculate_magic_numbers(LowLengths{}[i]);
     }
 };
  
 // Implementation of "merge" transformation primitive that uses magic-number-division to do lowering
 // of both multi-index and delta of multi-index
 // Caution:
 //   1. The magic number division implementation being used would produce correct result if the
 //   dividended is uint32_t and its value is with in 31-bit value range of uint32_t.
 //   2. The magic number division for int32_t dividened has not been implemented, the int32_t
 //   dividend would be bit-wise interpreted as uint32_t and magic number division implementation for
 //   uint32_t is then used.
 //   3. For merge primitive, upper-index is the dividend.
 //   4. When upper-index is uint32_t, its value need to be within 31-bit range.
 //   5. When upper-index is int32_t type (when index_t is int32_t), its value need to be
 //   non-negative.
 template <typename LowLengths>
 struct merge_v2_magic_division : public base_transform<LowLengths::size(), 1>
 {
     static constexpr index_t NDimLow = LowLengths::size();
  
     using LowerIndex = multi_index<NDimLow>;
     using UpperIndex = multi_index<1>;
  
     using UpLengths =
         decltype(make_tuple(container_reduce(LowLengths{}, multiplies{}, number<1>{})));
  
     using LowLengthsMagicDivisor = decltype(generate_tuple(
         lambda_merge_generate_MagicDivision_calculate_magic_divisor<LowLengths>{},
         number<NDimLow>{}));
  
     LowLengths low_lengths_;
     LowLengthsMagicDivisor low_lengths_magic_divisor_;
     UpLengths up_lengths_;
  
     static constexpr auto I0 = number<0>{};
     static constexpr auto I1 = number<1>{};
  
     CK_TILE_HOST_DEVICE constexpr merge_v2_magic_division() = default;
  
     CK_TILE_HOST_DEVICE constexpr merge_v2_magic_division(const LowLengths& low_lengths)
         : low_lengths_{low_lengths},
           low_lengths_magic_divisor_{generate_tuple(
               [&](auto i) { return magic_division::calculate_magic_numbers(low_lengths[i]); },
               number<NDimLow>{})},
           up_lengths_{make_tuple(container_reduce(low_lengths, multiplies{}, I1))}
     {
         static_assert(LowerIndex::size() == NDimLow, "wrong!");
     }
  
     CK_TILE_HOST_DEVICE static constexpr auto get_type_enum()
     {
         return coord_transform_enum::merge;
     }
  
     CK_TILE_HOST_DEVICE constexpr const auto& get_upper_lengths() const { return up_lengths_; }
  
     template <typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr void calculate_lower_index(LowIdx& idx_low,
                                                              const UpIdx& idx_up) const
     {
         static_assert(LowIdx::size() == NDimLow && UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         index_t tmp = idx_up[I0];
  
         static_for<NDimLow - 1, 0, -1>{}([&, this](auto i) {
             index_t tmp2 =
                 magic_division::do_magic_division(tmp,
                                                   this->low_lengths_magic_divisor_[i][I0],
                                                   this->low_lengths_magic_divisor_[i][I1]);
             idx_low(i) = tmp - tmp2 * this->low_lengths_[i];
             tmp        = tmp2;
         });
  
         idx_low(number<0>{}) = tmp;
     }
  
     template <typename LowIdxDiff, typename UpIdxDiff, typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE void update_lower_index(LowIdxDiff& idx_diff_low,
                                                 const UpIdxDiff&,
                                                 LowIdx& idx_low,
                                                 const UpIdx& idx_up_new) const
     {
         static_assert(LowIdxDiff::size() == NDimLow && UpIdxDiff::size() == 1 &&
                           LowIdx::size() == NDimLow && UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         index_t tmp = idx_up_new[number<0>{}];
  
         static_for<NDimLow - 1, 0, -1>{}([&, this](auto i) {
             index_t tmp2 =
                 magic_division::do_magic_division(tmp,
                                                   this->low_lengths_magic_divisor_[i][I0],
                                                   this->low_lengths_magic_divisor_[i][I1]);
  
             index_t idx_low_old = idx_low[i];
  
             idx_low(i) = tmp - tmp2 * this->low_lengths_[i];
             tmp        = tmp2;
  
             idx_diff_low(i) = idx_low[i] - idx_low_old;
         });
  
         idx_diff_low(number<0>{}) = tmp - idx_low(number<0>{});
  
         idx_low(number<0>{}) = tmp;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_always_mapped_to_valid_lower_index()
     {
         return true;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool is_known_at_compile_time()
     {
         return ck_tile::is_known_at_compile_time<LowLengths>::value &&
                ck_tile::is_known_at_compile_time<LowLengthsMagicDivisor>::value &&
                ck_tile::is_known_at_compile_time<UpLengths>::value;
     }
  
     template <typename UpIdx>
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_mapped_to_valid_lower_index(const UpIdx& /* idx_up */)
     {
         return true;
     }
  
     // MUST be static function
     template <typename LowVectorLengths, typename LowVectorStrides>
     CK_TILE_HOST_DEVICE static constexpr auto
     calculate_upper_dimension_safe_vector_length_strides(const LowVectorLengths& low_vector_lengths,
                                                          const LowVectorStrides& low_vector_strides)
     {
         array<index_t, 1> up_vector_lengths{-1};
         array<index_t, 1> up_vector_strides{-1};
  
         up_vector_lengths[0] = low_vector_lengths[number<NDimLow - 1>{}];
         up_vector_strides[0] = low_vector_strides[number<NDimLow - 1>{}];
  
         return make_tuple(up_vector_lengths, up_vector_strides);
     }
  
     CK_TILE_HOST_DEVICE void print() const
     {
         printf("merge_v2_magic_division{");
  
         //
         printf("low_lengths_ ");
         print(low_lengths_);
         printf(", ");
  
         //
         printf("up_lengths_ ");
         print(up_lengths_);
  
         printf("}");
     }
 };
  
 // Implementation of "merge" transformation primitive that uses division and mod. It is supposed to
 // be used for low_lengths that are known at compile time and are power of 2, otherwise performance
 // will be very bad
 template <typename LowLengths>
 struct merge_v3_division_mod : public base_transform<LowLengths::size(), 1>
 {
     static constexpr index_t NDimLow = LowLengths::size();
  
     using LowerIndex = multi_index<NDimLow>;
     using UpperIndex = multi_index<1>;
  
     using LowLengthsScan =
         decltype(container_reverse_exclusive_scan(LowLengths{}, multiplies{}, number<1>{}));
  
     using UpLengths =
         decltype(make_tuple(container_reduce(LowLengths{}, multiplies{}, number<1>{})));
  
     LowLengths low_lengths_;
     LowLengthsScan low_lengths_scan_;
     UpLengths up_lengths_;
  
     CK_TILE_HOST_DEVICE constexpr merge_v3_division_mod() = default;
  
     CK_TILE_HOST_DEVICE constexpr merge_v3_division_mod(const LowLengths& low_lengths)
         : low_lengths_{low_lengths},
           low_lengths_scan_{
               container_reverse_exclusive_scan(low_lengths, multiplies{}, number<1>{})},
           up_lengths_{make_tuple(container_reduce(low_lengths, multiplies{}, number<1>{}))}
     {
         static_assert(LowerIndex::size() == NDimLow, "wrong!");
     }
  
     CK_TILE_HOST_DEVICE constexpr const auto& get_upper_lengths() const { return up_lengths_; }
  
     template <typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr void calculate_lower_index(LowIdx& idx_low,
                                                              const UpIdx& idx_up) const
     {
         static_assert(LowIdx::size() == NDimLow && UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         index_t tmp = idx_up[number<0>{}];
  
         // division and mod
         static_for<0, NDimLow - 1, 1>{}([&](auto i) {
             idx_low(i) = tmp / this->low_lengths_scan_[i];
             tmp %= this->low_lengths_scan_[i];
         });
  
         idx_low(number<NDimLow - 1>{}) = tmp;
     }
  
     template <typename LowIdxDiff, typename UpIdxDiff, typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE void update_lower_index(LowIdxDiff& idx_diff_low,
                                                 const UpIdxDiff&,
                                                 LowIdx& idx_low,
                                                 const UpIdx& idx_up_new) const
     {
         static_assert(LowIdxDiff::size() == NDimLow && UpIdxDiff::size() == 1 &&
                           LowIdx::size() == NDimLow && UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         constexpr auto I0   = number<0>{};
         constexpr auto INm1 = number<NDimLow - 1>{};
  
         index_t tmp = idx_up_new[I0];
  
         static_for<0, NDimLow - 1, 1>{}([&](auto i) {
             const index_t tmp2 = idx_low[i];
             idx_low(i)         = tmp / this->low_lengths_scan_[i];
             idx_diff_low(i)    = idx_low[i] - tmp2;
             tmp %= this->low_lengths_scan_[i];
         });
  
         const index_t tmp2 = idx_low[INm1];
         idx_low(INm1)      = tmp;
         idx_diff_low(INm1) = idx_low[INm1] - tmp2;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_always_mapped_to_valid_lower_index()
     {
         return true;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool is_known_at_compile_time()
     {
         return ck_tile::is_known_at_compile_time<LowLengths>::value &&
                ck_tile::is_known_at_compile_time<LowLengthsScan>::value &&
                ck_tile::is_known_at_compile_time<UpLengths>::value;
     }
  
     template <typename UpIdx>
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_mapped_to_valid_lower_index(const UpIdx& /* idx_up */)
     {
         return true;
     }
  
     // MUST be static function
     template <typename LowVectorLengths, typename LowVectorStrides>
     CK_TILE_HOST_DEVICE static constexpr auto
     calculate_upper_dimension_safe_vector_length_strides(const LowVectorLengths& low_vector_lengths,
                                                          const LowVectorStrides& low_vector_strides)
     {
         array<index_t, 1> up_vector_lengths{-1};
         array<index_t, 1> up_vector_strides{-1};
  
         up_vector_lengths[0] = low_vector_lengths[number<NDimLow - 1>{}];
         up_vector_strides[0] = low_vector_strides[number<NDimLow - 1>{}];
  
         return make_tuple(up_vector_lengths, up_vector_strides);
     }
  
     CK_TILE_HOST_DEVICE void print() const
     {
         printf("Merge_v3_direct_division_mod{");
  
         //
         printf("low_lengths_ ");
         print(low_lengths_);
         printf(", ");
  
         //
         printf("low_lengths_scan_ ");
         print(low_lengths_scan_);
         printf(", ");
  
         //
         printf("up_lengths_ ");
         print(up_lengths_);
  
         printf("}");
     }
 };
  
 template <typename UpLengths, bool Use24BitIntegerCalculation>
 struct unmerge : public base_transform<1, UpLengths::size()>
 {
     static constexpr index_t NDimUp = UpLengths::size();
  
     using LowerIndex = multi_index<1>;
     using UpperIndex = multi_index<NDimUp>;
  
     using UpLengthsScan =
         decltype(container_reverse_exclusive_scan(UpLengths{}, multiplies{}, number<1>{}));
  
     UpLengths up_lengths_;
     UpLengthsScan up_lengths_scan_;
  
     CK_TILE_HOST_DEVICE constexpr unmerge() = default;
  
     CK_TILE_HOST_DEVICE constexpr unmerge(const UpLengths& up_lengths)
         : up_lengths_{up_lengths},
           up_lengths_scan_{container_reverse_exclusive_scan(up_lengths, multiplies{}, number<1>{})}
     {
     }
  
     CK_TILE_HOST_DEVICE static constexpr auto get_type_enum()
     {
         return coord_transform_enum::unmerge;
     }
  
     CK_TILE_HOST_DEVICE constexpr const auto& get_upper_lengths() const { return up_lengths_; }
  
     template <typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr void calculate_lower_index(LowIdx& idx_low,
                                                              const UpIdx& idx_up) const
     {
         if constexpr(!Use24BitIntegerCalculation)
         {
             idx_low(number<0>{}) = idx_up[number<NDimUp - 1>{}];
  
             static_for<0, NDimUp - 1, 1>{}(
                 [&](auto i) { idx_low(number<0>{}) += idx_up[i] * up_lengths_scan_[i]; });
         }
         else
         {
             idx_low(number<0>{}) = idx_up[number<NDimUp - 1>{}];
  
             static_for<0, NDimUp - 1, 1>{}([&](auto i) {
                 idx_low(number<0>{}) =
                     (0x00ffffff & idx_low[number<0>{}]) +
                     (0x00ffffff & idx_up[i]) * (0x00ffffff & up_lengths_scan_[i]);
             });
         }
     }
  
     template <typename LowIdxDiff, typename UpIdxDiff, typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE void update_lower_index(LowIdxDiff& idx_diff_low,
                                                 const UpIdxDiff& idx_diff_up,
                                                 LowIdx& idx_low,
                                                 const UpIdx&) const
     {
         calculate_lower_index(idx_diff_low, idx_diff_up);
  
         idx_low += idx_diff_low;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_always_mapped_to_valid_lower_index()
     {
         return true;
     }
  
     template <typename UpIdx>
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_mapped_to_valid_lower_index(const UpIdx& /* idx_up */)
     {
         return true;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool is_known_at_compile_time()
     {
         return ck_tile::is_known_at_compile_time<UpLengths>::value &&
                ck_tile::is_known_at_compile_time<UpLengthsScan>::value;
     }
  
     // MUST be static function
     template <typename LowVectorLengths, typename LowVectorStrides>
     CK_TILE_HOST_DEVICE static constexpr auto
     calculate_upper_dimension_safe_vector_length_strides(const LowVectorLengths& low_vector_lengths,
                                                          const LowVectorStrides& low_vector_strides)
     {
         array<index_t, NDimUp> up_vector_lengths{-1};
         array<index_t, NDimUp> up_vector_strides{-1};
  
         constexpr auto up_length_last = UpLengths{}[number<NDimUp - 1>{}];
  
         if constexpr(ck_tile::is_known_at_compile_time<decltype(up_length_last)>::value)
         {
             if(low_vector_lengths[0] != -1)
             {
                 up_vector_lengths(NDimUp - 1) = gcd(low_vector_lengths[0], up_length_last);
             }
         }
  
         up_vector_strides(NDimUp - 1) = low_vector_strides[0];
  
         return make_tuple(up_vector_lengths, up_vector_strides);
     }
  
     CK_TILE_HOST_DEVICE void print() const
     {
         printf("unmerge{");
  
         //
         printf("up_lengths_");
         print(up_lengths_);
         printf(", ");
  
         //
         printf("up_lengths_scan_");
         print(up_lengths_scan_);
  
         printf("}");
     }
 };
  
 template <typename LowerIndex>
 struct freeze : public base_transform<1, 0>
 {
     LowerIndex low_idx_;
  
     CK_TILE_HOST_DEVICE constexpr freeze() = default;
  
     CK_TILE_HOST_DEVICE constexpr freeze(const LowerIndex& low_idx) : low_idx_{low_idx} {}
  
     CK_TILE_HOST_DEVICE static constexpr auto get_upper_lengths() { return tuple<>{}; }
  
     template <typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr void calculate_lower_index(LowIdx& idx_low,
                                                              const UpIdx& /* idx_up */) const
     {
         static_assert(LowIdx::size() == 1 && UpIdx::size() == 0,
                       "wrong! inconsistent # of dimension");
  
         idx_low(number<0>{}) = low_idx_;
     }
  
     template <typename LowIdxDiff, typename UpIdxDiff, typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE static void update_lower_index(LowIdxDiff& idx_diff_low,
                                                        const UpIdxDiff& /* idx_diff_up */,
                                                        LowIdx& /* idx_low */,
                                                        const UpIdx& /* idx_up_new */)
     {
         idx_diff_low(number<0>{}) = 0;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_always_mapped_to_valid_lower_index()
     {
         return true;
     }
  
     template <typename UpIdx>
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_mapped_to_valid_lower_index(const UpIdx& /* idx_up */)
     {
         return true;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool is_known_at_compile_time()
     {
         return ck_tile::is_known_at_compile_time<LowerIndex>::value;
     }
  
     CK_TILE_HOST_DEVICE void print() const
     {
         printf("freeze{");
  
         //
         printf("low_idx_: ");
         print(low_idx_);
  
         printf("}");
     }
 };
  
 // insert a dangling upper dimension without lower dimension
 template <typename UpperLength>
 struct insert : public base_transform<0, 1>
 {
     using UpLengths = decltype(make_tuple(UpperLength{}));
  
     UpLengths up_lengths_;
  
     CK_TILE_HOST_DEVICE constexpr insert() = default;
  
     CK_TILE_HOST_DEVICE constexpr insert(const UpperLength& up_length)
         : up_lengths_{make_tuple(up_length)}
     {
     }
  
     CK_TILE_HOST_DEVICE static constexpr index_t get_num_of_lower_dimension() { return 0; }
  
     CK_TILE_HOST_DEVICE static constexpr index_t get_num_of_upper_dimension() { return 1; }
  
     CK_TILE_HOST_DEVICE constexpr auto get_upper_lengths() const { return up_lengths_; }
  
     template <typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr void calculate_lower_index(LowIdx&, const UpIdx&) const
     {
         static_assert(LowIdx::size() == 0 && UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
     }
  
     template <typename LowIdxDiff, typename UpIdxDiff, typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE static void
     update_lower_index(LowIdxDiff&, const UpIdxDiff&, LowIdx&, const UpIdx&)
     {
         static_assert(LowIdxDiff::size() == 0 && UpIdxDiff::size() == 1 && LowIdx::size() == 0 &&
                           UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool IsLinearTransform() { return true; }
  
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_always_mapped_to_valid_lower_index()
     {
         return true;
     }
  
     template <typename UpIdx>
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_mapped_to_valid_lower_index(const UpIdx& /* idx_up */)
     {
         return true;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool is_known_at_compile_time()
     {
         return ck_tile::is_known_at_compile_time<UpperLength>::value;
     }
  
     CK_TILE_HOST_DEVICE void print() const
     {
         printf("insert{");
  
         //
         print(up_lengths_);
  
         printf("}");
     }
 };
  
 // replicate the original tensor and create a higher dimensional tensor
 template <typename UpLengths>
 struct replicate : public base_transform<0, UpLengths::size()>
 {
     static constexpr index_t NDimUp = UpLengths::size();
  
     CK_TILE_HOST_DEVICE constexpr replicate() = default;
  
     CK_TILE_HOST_DEVICE constexpr replicate(const UpLengths& up_lengths) : up_lengths_{up_lengths}
     {
     }
  
     CK_TILE_HOST_DEVICE constexpr auto get_upper_lengths() const { return up_lengths_; }
  
     template <typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr void calculate_lower_index(LowIdx&, const UpIdx&) const
     {
         static_assert(LowIdx::size() == 0 && UpIdx::size() == NDimUp,
                       "wrong! inconsistent # of dimension");
     }
  
     template <typename LowIdxDiff, typename UpIdxDiff, typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE static void
     update_lower_index(LowIdxDiff&, const UpIdxDiff&, LowIdx&, const UpIdx&)
     {
         static_assert(LowIdxDiff::size() == 0 && UpIdxDiff::size() == NDimUp &&
                           LowIdx::size() == 0 && UpIdx::size() == NDimUp,
                       "wrong! inconsistent # of dimension");
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_always_mapped_to_valid_lower_index()
     {
         return true;
     }
  
     template <typename UpIdx>
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_mapped_to_valid_lower_index(const UpIdx& /* idx_up */)
     {
         return true;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool is_known_at_compile_time()
     {
         return ck_tile::is_known_at_compile_time<UpLengths>::value;
     }
  
     CK_TILE_HOST_DEVICE void print() const
     {
         printf("replicate{");
  
         //
         printf("up_lengths_: ");
         print(up_lengths_);
  
         printf("}");
     }
  
     //
     UpLengths up_lengths_;
 };
  
 template <typename LowLength, typename SliceBegin, typename SliceEnd>
 struct slice : public base_transform<1, 1>
 {
     using LowerIndex = multi_index<1>;
     using UpperIndex = multi_index<1>;
  
     using UpLengths = decltype(make_tuple(SliceEnd{} - SliceBegin{}));
  
     UpLengths up_lengths_;
     SliceBegin slice_begin_;
     SliceEnd slice_end_;
  
     CK_TILE_HOST_DEVICE constexpr slice() = default;
  
     CK_TILE_HOST_DEVICE constexpr slice(const LowLength&,
                                         const SliceBegin& slice_begin,
                                         const SliceEnd& slice_end)
         : up_lengths_{make_tuple(slice_end - slice_begin)},
           slice_begin_{slice_begin},
           slice_end_{slice_end}
     {
     }
  
     CK_TILE_HOST_DEVICE constexpr const auto& get_upper_lengths() const { return up_lengths_; }
  
     template <typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr void calculate_lower_index(LowIdx& idx_low,
                                                              const UpIdx& idx_up) const
     {
         static_assert(LowIdx::size() == 1 && UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         idx_low(number<0>{}) = idx_up[number<0>{}] + slice_begin_;
     }
  
     template <typename LowIdxDiff, typename UpIdxDiff, typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE static void update_lower_index(LowIdxDiff& idx_diff_low,
                                                        const UpIdxDiff& idx_diff_up,
                                                        LowIdx& idx_low,
                                                        const UpIdx&)
     {
         static_assert(LowIdxDiff::size() == 1 && UpIdxDiff::size() == 1 && LowIdx::size() == 1 &&
                           UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         constexpr auto I0 = number<0>{};
  
         idx_diff_low[I0] = idx_diff_up[I0];
  
         idx_low += idx_diff_low;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_always_mapped_to_valid_lower_index()
     {
         return true;
     }
  
     template <typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr bool
     is_valid_upper_index_mapped_to_valid_lower_index(const UpIdx&) const
     {
         return true;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool is_known_at_compile_time()
     {
         return ck_tile::is_known_at_compile_time<UpLengths>::value &&
                ck_tile::is_known_at_compile_time<SliceBegin>::value &&
                ck_tile::is_known_at_compile_time<SliceEnd>::value;
     }
  
     CK_TILE_HOST_DEVICE void print() const
     {
         printf("slice{");
  
         //
         printf("up_lengths_: ");
         print(up_lengths_);
         printf(", ");
  
         //
         printf("slice_begin_: ");
         print(slice_begin_);
         printf(", ");
  
         //
         printf("slice_end_: ");
         print(slice_end_);
  
         printf("}");
     } // namespace ck
 };    // namespace ck
  
 /*
  * \brief lower_idx = upper_idx % modulus.
  * TODO: Need an improved implementation since the modulo operation is expensive.
  */
 template <typename Modulus, typename UpLength>
 struct modulo : public base_transform<1, 1>
 {
     using LowerIndex = multi_index<1>;
     using UpperIndex = multi_index<1>;
     using UpLengths  = decltype(make_tuple(UpLength{}));
  
     Modulus modulus_;
     UpLengths up_lengths_;
  
     CK_TILE_HOST_DEVICE constexpr modulo() = default;
  
     CK_TILE_HOST_DEVICE constexpr modulo(const Modulus& modulus, const UpLength& up_length)
         : modulus_{modulus}, up_lengths_{make_tuple(up_length)}
     {
     }
  
     CK_TILE_HOST_DEVICE constexpr const auto& get_upper_lengths() const { return up_lengths_; }
  
     template <typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr void calculate_lower_index(LowIdx& idx_low,
                                                              const UpIdx& idx_up) const
     {
         static_assert(LowIdx::size() == 1 && UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         idx_low(number<0>{}) = idx_up[number<0>{}] % modulus_;
     }
  
     template <typename LowIdxDiff, typename UpIdxDiff, typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE void update_lower_index(LowIdxDiff& idx_diff_low,
                                                 const UpIdxDiff& idx_diff_up,
                                                 LowIdx& idx_low,
                                                 const UpIdx& up_idx) const
     {
         static_assert(LowIdxDiff::size() == 1 && UpIdxDiff::size() == 1 && LowIdx::size() == 1 &&
                           UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         constexpr auto I0 = number<0>{};
  
         const auto idx_low_old = idx_low;
         idx_low[I0]            = (up_idx[I0] + idx_diff_up[I0]) % modulus_;
         idx_diff_low[I0]       = idx_low - idx_low_old;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_always_mapped_to_valid_lower_index()
     {
         return true;
     }
  
     template <typename UpIdx>
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_mapped_to_valid_lower_index(const UpIdx& /* idx_up */)
     {
         return true;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool is_known_at_compile_time()
     {
         return ck_tile::is_known_at_compile_time<UpLengths>::value;
     }
  
     CK_TILE_HOST_DEVICE void print() const
     {
         printf("Modulus{");
  
         //
         printf("up_lengths_: ");
         print(up_lengths_);
  
         printf("}");
     }
 };
  
 // 2D XOR, NOTE: "xor" is a keyword
 template <typename LowLengths>
 struct xor_t : public base_transform<2, 2>
 {
     static constexpr auto type_enum = coord_transform_enum::xor_t;
  
     using LowerIndex = multi_index<2>;
     using UpperIndex = multi_index<2>;
  
     using UpLengths = LowLengths;
  
     UpLengths up_lengths_;
  
     CK_TILE_HOST_DEVICE constexpr xor_t() : up_lengths_{} {}
  
     CK_TILE_HOST_DEVICE constexpr xor_t(const LowLengths& low_lengths) : up_lengths_{low_lengths} {}
  
     CK_TILE_HOST_DEVICE static constexpr auto get_type_enum()
     {
         return coord_transform_enum::xor_t;
     }
  
     CK_TILE_HOST_DEVICE constexpr const auto& get_upper_lengths() const { return up_lengths_; }
  
     template <typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr void calculate_lower_index(LowIdx& idx_low,
                                                              const UpIdx& idx_up) const
     {
         static_assert(LowIdx::size() == 2 && UpIdx::size() == 2,
                       "wrong! inconsistent # of dimension");
  
         idx_low(number<0>{}) = idx_up[number<0>{}];
  
         idx_low(number<1>{}) =
             idx_up[number<1>{}] ^ (idx_up[number<0>{}] % up_lengths_[number<1>{}]);
     }
  
     template <typename LowIdxDiff, typename UpIdxDiff, typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE void update_lower_index(LowIdxDiff& idx_diff_low,
                                                 const UpIdxDiff&,
                                                 LowIdx& idx_low,
                                                 const UpIdx& idx_up) const
     {
         static_assert(LowIdxDiff::size() == 2 && UpIdxDiff::size() == 2 && LowIdx::size() == 2 &&
                           UpIdx::size() == 2,
                       "wrong! inconsistent # of dimension");
  
         const auto idx_low_old = idx_low;
  
         calculate_lower_index(idx_low, idx_up);
  
         idx_diff_low = idx_low - idx_low_old;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_always_mapped_to_valid_lower_index()
     {
         return true;
     }
  
     template <typename UpIdx>
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_mapped_to_valid_lower_index(const UpIdx& /* idx_up */)
     {
         return true;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool is_known_at_compile_time()
     {
         return ck_tile::is_known_at_compile_time<UpLengths>::value;
     }
  
     // MUST be static function
     template <typename LowVectorLengths, typename LowVectorStrides>
     CK_TILE_HOST_DEVICE constexpr auto calculate_upper_dimension_safe_vector_length_strides(
         const LowVectorLengths& low_vector_lengths,
         const LowVectorStrides& low_vector_strides) const
     {
         array<index_t, 2> up_vector_lengths = low_vector_lengths;
         array<index_t, 2> up_vector_strides = low_vector_strides;
  
         return make_tuple(up_vector_lengths, up_vector_strides);
     }
  
     CK_TILE_HOST_DEVICE void print() const
     {
         printf("xor_t{");
  
         //
         printf("up_lengths_: ");
         print(up_lengths_);
         printf(", ");
  
         printf("}");
     }
 };
  
 template <typename LowLength, typename OffsetLength>
 struct offset : public base_transform<1, 1>
 {
     using LowerIndex = multi_index<1>;
     using UpperIndex = multi_index<1>;
  
     using UpLengths = decltype(make_tuple(LowLength{}));
  
     UpLengths up_lengths_;
     OffsetLength offset_length_;
  
     CK_TILE_HOST_DEVICE constexpr offset() = default;
  
     CK_TILE_HOST_DEVICE constexpr offset(const LowLength& low_length,
                                          const OffsetLength& offset_length)
         : up_lengths_{make_tuple(low_length)}, offset_length_{offset_length}
     {
     }
  
     CK_TILE_HOST_DEVICE static constexpr auto get_type_enum()
     {
         return coord_transform_enum::offset;
     }
  
     CK_TILE_HOST_DEVICE constexpr const auto& get_upper_lengths() const { return up_lengths_; }
  
     template <typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr void calculate_lower_index(LowIdx& idx_low,
                                                              const UpIdx& idx_up) const
     {
         static_assert(LowIdx::size() == 1 && UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         idx_low(number<0>{}) = idx_up[number<0>{}] + offset_length_;
     }
  
     template <typename LowIdxDiff, typename UpIdxDiff, typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE static void update_lower_index(LowIdxDiff& idx_diff_low,
                                                        const UpIdxDiff& idx_diff_up,
                                                        LowIdx& idx_low,
                                                        const UpIdx&)
     {
         static_assert(LowIdxDiff::size() == 1 && UpIdxDiff::size() == 1 && LowIdx::size() == 1 &&
                           UpIdx::size() == 1,
                       "wrong! inconsistent # of dimension");
  
         constexpr auto I0 = number<0>{};
  
         idx_diff_low[I0] = idx_diff_up[I0];
  
         idx_low += idx_diff_low;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_always_mapped_to_valid_lower_index()
     {
         return true;
     }
  
     template <typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr bool
     is_valid_upper_index_mapped_to_valid_lower_index(const UpIdx&) const
     {
         return true;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool is_known_at_compile_time()
     {
         return ck_tile::is_known_at_compile_time<UpLengths>::value &&
                ck_tile::is_known_at_compile_time<OffsetLength>::value;
     }
  
     CK_TILE_HOST_DEVICE void print() const
     {
         printf("offset{");
  
         //
         printf("up_lengths_: ");
         print(up_lengths_);
         printf(", ");
  
         //
         printf("offset_length_: ");
         print(offset_length_);
  
         printf("}");
     }
 };
  
 template <typename UpLength, typename IndexingAdaptor>
 struct indexing : public base_transform<1, 1>
 {
     static constexpr index_t NDimUp = 1;
  
     using LowerIndex = multi_index<1>;
     using UpperIndex = multi_index<1>;
  
     using UpLengths = decltype(make_tuple(UpLength{}));
     UpLengths up_lengths_;
     IndexingAdaptor iadaptor_;
  
     CK_TILE_HOST_DEVICE constexpr indexing() = default;
  
     CK_TILE_HOST_DEVICE constexpr indexing(const UpLength& up_length,
                                            const IndexingAdaptor& iadaptor)
         : up_lengths_{make_tuple(up_length)}, iadaptor_{iadaptor}
     {
     }
  
     CK_TILE_HOST_DEVICE static constexpr auto get_type_enum()
     {
         return coord_transform_enum::indexing;
     }
  
     CK_TILE_HOST_DEVICE constexpr const auto& get_upper_lengths() const { return up_lengths_; }
  
     template <typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE constexpr void calculate_lower_index(LowIdx& idx_low,
                                                              const UpIdx& idx_up) const
     {
         static_assert(LowIdx::size() == 1 && UpIdx::size() == NDimUp,
                       "wrong! inconsistent # of dimension");
         iadaptor_.calculate_lower_index(idx_low, idx_up);
     }
  
     template <typename LowIdxDiff, typename UpIdxDiff, typename LowIdx, typename UpIdx>
     CK_TILE_HOST_DEVICE void update_lower_index(LowIdxDiff& idx_diff_low,
                                                 const UpIdxDiff& idx_diff_up,
                                                 LowIdx& idx_low,
                                                 const UpIdx& idx_up) const
     {
         // TODO: nonthing changed here
         static_assert(LowIdxDiff::size() == 1 && UpIdxDiff::size() == NDimUp &&
                           LowIdx::size() == 1 && UpIdx::size() == NDimUp,
                       "wrong! inconsistent # of dimension");
  
         iadaptor_.update_lower_index(idx_diff_low, idx_diff_up, idx_low, idx_up);
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_always_mapped_to_valid_lower_index()
     {
         return true;
     }
  
     template <typename UpIdx>
     CK_TILE_HOST_DEVICE static constexpr bool
     is_valid_upper_index_mapped_to_valid_lower_index(const UpIdx& /* idx_up */)
     {
         return true;
     }
  
     CK_TILE_HOST_DEVICE static constexpr bool is_known_at_compile_time()
     {
         return ck_tile::is_known_at_compile_time<UpLengths>::value &&
                IndexingAdaptor::is_known_at_compile_time();
     }
  
     CK_TILE_HOST_DEVICE void print() const
     {
         printf("embed{");
  
         //
         printf("up_lengths_: ");
         print(up_lengths_);
         printf(", ");
  
         printf("}");
     }
 };
  
 //*******************************************************************************************************
  
 template <typename LowLength>
 CK_TILE_HOST_DEVICE constexpr auto make_pass_through_transform(const LowLength& low_length)
 {
     return pass_through<LowLength>{low_length};
 }
  
 template <typename LowLength, typename LeftPad, typename RightPad, bool SkipIsValidCheck = false>
 CK_TILE_HOST_DEVICE constexpr auto
 make_pad_transform(const LowLength& low_length,
                    const LeftPad& left_pad,
                    const RightPad& right_pad,
                    bool_constant<SkipIsValidCheck> = bool_constant<false>{})
 {
     return pad<LowLength, LeftPad, RightPad, SkipIsValidCheck>{low_length, left_pad, right_pad};
 }
  
 template <typename LowLength, typename LeftPadLength, bool SkipIsValidCheck = false>
 CK_TILE_HOST_DEVICE constexpr auto
 make_left_pad_transform(const LowLength& low_length,
                         const LeftPadLength& left_pad_,
                         bool_constant<SkipIsValidCheck> = bool_constant<false>{})
 {
     return left_pad<LowLength, LeftPadLength, SkipIsValidCheck>{low_length, left_pad_};
 }
  
 template <typename LowLength, typename RightPadLength, bool SkipIsValidCheck = false>
 CK_TILE_HOST_DEVICE constexpr auto
 make_right_pad_transform(const LowLength& low_length,
                          const RightPadLength& right_pad_,
                          bool_constant<SkipIsValidCheck> = bool_constant<false>{})
 {
     return right_pad<LowLength, RightPadLength, SkipIsValidCheck>{low_length, right_pad_};
 }
  
 template <typename UpLengths,
           typename Coefficients,
           typename std::enable_if<UpLengths::size() == Coefficients::size(), bool>::type = false>
 CK_TILE_HOST_DEVICE constexpr auto make_embed_transform(const UpLengths& up_lengths,
                                                         const Coefficients& coefficients)
 {
     return embed<UpLengths, Coefficients>{up_lengths, coefficients};
 }
  
 template <typename LowLengths>
 CK_TILE_HOST_DEVICE constexpr auto
 make_merge_transform_v2_magic_division(const LowLengths& low_lengths)
 {
     return merge_v2_magic_division<LowLengths>{low_lengths};
 }
  
 template <typename LowLengths>
 CK_TILE_HOST_DEVICE constexpr auto
 make_merge_transform_v3_division_mod(const LowLengths& low_lengths)
 {
     return merge_v3_division_mod<LowLengths>{low_lengths};
 }
  
 template <typename LowLengths>
 CK_TILE_HOST_DEVICE constexpr auto make_merge_transform(const LowLengths& low_lengths)
 {
     return make_merge_transform_v2_magic_division(low_lengths);
 }
  
 template <typename UpLengths, bool Use24BitIntegerCalculation = false>
 CK_TILE_HOST_DEVICE constexpr auto
 make_unmerge_transform(const UpLengths& up_lengths,
                        bool_constant<Use24BitIntegerCalculation> = bool_constant<false>{})
 {
     return unmerge<UpLengths, Use24BitIntegerCalculation>{up_lengths};
 }
  
 template <typename LowerIndex>
 CK_TILE_HOST_DEVICE constexpr auto make_freeze_transform(const LowerIndex& low_idx)
 {
     return freeze<LowerIndex>{low_idx};
 }
  
 template <typename UpperIndex>
 CK_TILE_HOST_DEVICE constexpr auto make_insert_transform(const UpperIndex& up_idx)
 {
     return insert<UpperIndex>{up_idx};
 }
  
 template <typename UpLengths>
 CK_TILE_HOST_DEVICE constexpr auto make_replicate_transform(const UpLengths& up_lengths)
 {
     return replicate<UpLengths>{up_lengths};
 }
  
 template <typename LowLength, typename SliceBegin, typename SliceEnd>
 CK_TILE_HOST_DEVICE constexpr auto make_slice_transform(const LowLength& low_length,
                                                         const SliceBegin& slice_begin,
                                                         const SliceEnd& slice_end)
 {
     return slice<LowLength, SliceBegin, SliceEnd>{low_length, slice_begin, slice_end};
 }
  
 template <typename Modulus, typename UpLength>
 CK_TILE_HOST_DEVICE constexpr auto make_modulo_transform(const Modulus& modulus,
                                                          const UpLength& up_length)
 {
     return modulo<Modulus, UpLength>{modulus, up_length};
 }
  
 template <typename LowLengths>
 CK_TILE_HOST_DEVICE constexpr auto make_xor_transform(const LowLengths& low_lengths)
 {
     return xor_t<LowLengths>{low_lengths};
 }
  
 template <typename LowLength, typename OffsetLength>
 CK_TILE_HOST_DEVICE constexpr auto make_offset_transform(const LowLength& low_length,
                                                          const OffsetLength& offset_length)
 {
     return offset<LowLength, OffsetLength>{low_length, offset_length};
 }
  
 } // namespace ck_tile
  
 #include "ck_tile/core/algorithm/indexing_adaptor.hpp"
 namespace ck_tile {
  
 template <typename UpLength, typename Indices>
 CK_TILE_HOST_DEVICE constexpr auto make_indexing_transform(const UpLength& up_lengths,
                                                            const Indices& indices)
 {
     // by default we use the simplest one
     return indexing<UpLength, indexing_adaptor_onshot_cached<remove_cvref_t<Indices>>>{
         up_lengths, indexing_adaptor_onshot_cached<remove_cvref_t<Indices>>{indices}};
 }
  
 template <typename UpLength, typename IndexingAdaptor>
 CK_TILE_HOST_DEVICE constexpr auto
 make_indexing_transform_with_adaptor(const UpLength& up_lengths, const IndexingAdaptor& iadaptor)
 {
     return indexing<UpLength, IndexingAdaptor>{up_lengths, iadaptor};
 }
  
 } // namespace ck_tile