Merge remote-tracking branch 'origin/develop' into aosewski/ggemm_multi_d2

include/ck_tile/core/container/map.hpp

+// 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/array.hpp"
+#include "ck_tile/core/container/sequence.hpp"
+#include "ck_tile/core/container/tuple.hpp"
+
+namespace ck_tile {
+
+// naive map
+template <typename key, typename data, index_t max_size = 128>
+struct map
+{
+    using pair_type = tuple<key, data>;
+    using impl_type = array<pair_type, max_size>;
+
+    impl_type impl_;
+    index_t size_;
+
+    struct iterator
+    {
+        impl_type& impl_;
+        index_t pos_;
+
+        CK_TILE_HOST_DEVICE constexpr iterator(impl_type& impl, index_t pos)
+            : impl_{impl}, pos_{pos}
+        {
+        }
+
+        CK_TILE_HOST_DEVICE constexpr iterator& operator++()
+        {
+            pos_++;
+            return *this;
+        }
+
+        CK_TILE_HOST_DEVICE constexpr bool operator!=(const iterator& other) const
+        {
+            return other.pos_ != pos_;
+        }
+
+        CK_TILE_HOST_DEVICE constexpr pair_type& operator*() { return impl_.at(pos_); }
+    };
+
+    struct const_iterator
+    {
+        const impl_type& impl_;
+        index_t pos_;
+
+        CK_TILE_HOST_DEVICE constexpr const_iterator(const impl_type& impl, index_t pos)
+            : impl_{impl}, pos_{pos}
+        {
+        }
+
+        CK_TILE_HOST_DEVICE constexpr const_iterator& operator++()
+        {
+            pos_++;
+
+            return *this;
+        }
+
+        CK_TILE_HOST_DEVICE constexpr bool operator!=(const const_iterator& other) const
+        {
+            return other.pos_ != pos_;
+        }
+
+        CK_TILE_HOST_DEVICE constexpr const pair_type& operator*() const { return impl_.at(pos_); }
+    };
+
+    CK_TILE_HOST_DEVICE constexpr map() : impl_{}, size_{0} {}
+
+    CK_TILE_HOST_DEVICE constexpr index_t size() const { return size_; }
+
+    CK_TILE_HOST_DEVICE void clear() { size_ = 0; }
+
+    CK_TILE_HOST_DEVICE constexpr index_t find_position(const key& k) const
+    {
+        for(index_t i = 0; i < size(); i++)
+        {
+            if(impl_[i].template at<0>() == k)
+            {
+                return i;
+            }
+        }
+
+        return size_;
+    }
+
+    CK_TILE_HOST_DEVICE constexpr const_iterator find(const key& k) const
+    {
+        return const_iterator{impl_, find_position(k)};
+    }
+
+    CK_TILE_HOST_DEVICE constexpr iterator find(const key& k)
+    {
+        return iterator{impl_, find_position(k)};
+    }
+
+    CK_TILE_HOST_DEVICE constexpr const data& operator[](const key& k) const
+    {
+        const auto it = find(k);
+
+        // FIXME
+        // assert(it.pos_ < size());
+
+        return impl_[it.pos_].template at<1>();
+    }
+
+    CK_TILE_HOST_DEVICE constexpr data& operator()(const key& k)
+    {
+        auto it = find(k);
+
+        // if entry not found
+        if(it.pos_ == size())
+        {
+            impl_(it.pos_).template at<0>() = k;
+            size_++;
+        }
+
+        // FIXME
+        // assert(size_ <= max_size);
+
+        return impl_(it.pos_).template at<1>();
+    }
+
+    // WARNING: needed by compiler for C++ range-based for loop only, don't use this function!
+    CK_TILE_HOST_DEVICE constexpr const_iterator begin() const { return const_iterator{impl_, 0}; }
+
+    // WARNING: needed by compiler for C++ range-based for loop only, don't use this function!
+    CK_TILE_HOST_DEVICE constexpr const_iterator end() const
+    {
+        return const_iterator{impl_, size_};
+    }
+
+    // WARNING: needed by compiler for C++ range-based for loop only, don't use this function!
+    CK_TILE_HOST_DEVICE constexpr iterator begin() { return iterator{impl_, 0}; }
+
+    // WARNING: needed by compiler for C++ range-based for loop only, don't use this function!
+    CK_TILE_HOST_DEVICE constexpr iterator end() { return iterator{impl_, size_}; }
+
+    CK_TILE_HOST_DEVICE void print() const
+    {
+        printf("map{size_: %d, ", size_);
+        //
+        printf("impl_: [");
+        //
+        for(const auto& [k, d] : *this)
+        {
+            printf("{key: ");
+            print(k);
+            printf(", data: ");
+            print(d);
+            printf("}, ");
+        }
+        //
+        printf("]");
+        //
+        printf("}");
+    }
+};
+
+} // namespace ck_tile

include/ck_tile/core/container/meta_data_buffer.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2023, Advanced Micro Devices, Inc. All rights reserved.
+
+#pragma once
+
+#include "ck_tile/core/config.hpp"
+#include "ck_tile/core/container/array.hpp"
+#include "ck_tile/core/utility/bit_cast.hpp"
+#include <cstddef>
+
+namespace ck_tile {
+
+// TODO: this structure is not intented to be used by user
+template <index_t MaxSize>
+struct meta_data_buffer
+{
+    CK_TILE_HOST_DEVICE constexpr meta_data_buffer() : buffer_{}, size_{0} {}
+
+    template <typename X, typename... Xs>
+    CK_TILE_HOST_DEVICE constexpr meta_data_buffer(const X& x, const Xs&... xs)
+        : buffer_{}, size_{0}
+    {
+        push(x, xs...);
+    }
+
+    template <typename T>
+    CK_TILE_HOST_DEVICE constexpr void push(const T& data)
+    {
+        if constexpr(!std::is_empty_v<T>)
+        {
+            constexpr index_t size = sizeof(T);
+
+            auto tmp = bit_cast<array<std::byte, size>>(data);
+
+            for(int i = 0; i < size; i++)
+            {
+                buffer_(size_) = tmp[i];
+
+                size_++;
+            }
+        }
+    }
+
+    template <typename X, typename... Xs>
+    CK_TILE_HOST_DEVICE constexpr void push(const X& x, const Xs&... xs)
+    {
+        push(x);
+        push(xs...);
+    }
+
+    template <typename T>
+    CK_TILE_HOST_DEVICE constexpr T pop(index_t& pos) const
+    {
+        T data;
+
+        if constexpr(!std::is_empty_v<T>)
+        {
+            constexpr index_t size = sizeof(T);
+
+            array<std::byte, size> tmp;
+
+            for(int i = 0; i < size; i++)
+            {
+                tmp(i) = buffer_[pos];
+
+                pos++;
+            }
+
+            data = bit_cast<T>(tmp);
+        }
+
+        return data;
+    }
+
+    template <typename T>
+    CK_TILE_HOST_DEVICE constexpr T get(index_t pos) const
+    {
+        constexpr index_t size = sizeof(T);
+
+        array<std::byte, size> tmp;
+
+        for(int i = 0; i < size; i++)
+        {
+            tmp(i) = buffer_[pos];
+
+            pos++;
+        }
+
+        auto data = bit_cast<T>(tmp);
+
+        return data;
+    }
+
+    //
+    array<std::byte, MaxSize> buffer_;
+    index_t size_ = 0;
+};
+
+} // namespace ck_tile

include/ck_tile/core/container/multi_index.hpp

+// 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/array.hpp"
+#include "ck_tile/core/container/sequence.hpp"
+#include "ck_tile/core/container/tuple.hpp"
+#include "ck_tile/core/utility/functional.hpp"
+
+namespace ck_tile {
+
+// Don't use tihs directly. This is for old CK's internal usage,
+// in the future always use array instead
+template <index_t N>
+using multi_index = array<index_t, N>;
+
+template <typename... Xs>
+CK_TILE_HOST_DEVICE constexpr auto make_multi_index(Xs&&... xs)
+{
+    return make_array<index_t>(index_t{xs}...);
+}
+
+template <index_t NSize>
+CK_TILE_HOST_DEVICE constexpr auto make_zero_multi_index()
+{
+    return unpack([](auto... xs) { return make_multi_index(xs...); },
+                  typename uniform_sequence_gen<NSize, 0>::type{});
+}
+
+template <typename T>
+CK_TILE_HOST_DEVICE constexpr auto to_multi_index(const T& x)
+{
+    return unpack([](auto... ys) { return make_multi_index(ys...); }, x);
+}
+
+template <index_t NSize, typename X>
+CK_TILE_HOST_DEVICE constexpr auto operator+=(multi_index<NSize>& y, const X& x)
+{
+    static_assert(X::size() == NSize, "wrong! size not the same");
+    static_for<0, NSize, 1>{}([&](auto i) { y[i] += x[i]; });
+    return y;
+}
+
+template <index_t NSize, typename X>
+CK_TILE_HOST_DEVICE constexpr auto operator-=(multi_index<NSize>& y, const X& x)
+{
+    static_assert(X::size() == NSize, "wrong! size not the same");
+    static_for<0, NSize, 1>{}([&](auto i) { y[i] -= x[i]; });
+    return y;
+}
+
+template <index_t NSize, typename T>
+CK_TILE_HOST_DEVICE constexpr auto operator+(const multi_index<NSize>& a, const T& b)
+{
+    using type = multi_index<NSize>;
+    static_assert(T::size() == NSize, "wrong! size not the same");
+    type r;
+    static_for<0, NSize, 1>{}([&](auto i) { r[i] = a[i] + b[i]; });
+    return r;
+}
+
+template <index_t NSize, typename T>
+CK_TILE_HOST_DEVICE constexpr auto operator-(const multi_index<NSize>& a, const T& b)
+{
+    using type = multi_index<NSize>;
+    static_assert(T::size() == NSize, "wrong! size not the same");
+    type r;
+    static_for<0, NSize, 1>{}([&](auto i) { r[i] = a[i] - b[i]; });
+    return r;
+}
+
+template <index_t NSize, typename T>
+CK_TILE_HOST_DEVICE constexpr auto operator*(const multi_index<NSize>& a, const T& b)
+{
+    using type = multi_index<NSize>;
+    static_assert(T::size() == NSize, "wrong! size not the same");
+    type r;
+    static_for<0, NSize, 1>{}([&](auto i) { r[i] = a[i] * b[i]; });
+    return r;
+}
+
+// multi_index = index_t * multi_index
+template <index_t NSize>
+CK_TILE_HOST_DEVICE constexpr auto operator*(index_t a, const multi_index<NSize>& x)
+{
+    multi_index<NSize> r;
+    static_for<0, NSize, 1>{}([&](auto i) { r[i] = a * x[i]; });
+    return r;
+}
+
+// multi_index = multi_index * index_t
+template <index_t NSize>
+CK_TILE_HOST_DEVICE constexpr auto operator*(const multi_index<NSize>& x, index_t a)
+{
+    return a * x;
+}
+
+} // namespace ck_tile

include/ck_tile/core/container/sequence.hpp

+// 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/numeric/integer.hpp"
+#include "ck_tile/core/numeric/integral_constant.hpp"
+#include "ck_tile/core/numeric/math.hpp"
+#include "ck_tile/core/utility/to_sequence.hpp"
+#include "ck_tile/core/utility/type_traits.hpp"
+#include "ck_tile/core/utility/functional.hpp"
+
+namespace ck_tile {
+
+template <index_t, index_t, index_t>
+struct static_for;
+
+template <index_t...>
+struct sequence;
+
+template <typename Seq, index_t I>
+struct sequence_split;
+
+template <typename>
+struct sequence_reverse;
+
+template <typename>
+struct sequence_map_inverse;
+
+template <typename>
+struct is_valid_sequence_map;
+
+template <index_t I, index_t... Is>
+CK_TILE_HOST_DEVICE constexpr auto sequence_pop_front(sequence<I, Is...>);
+
+template <typename Seq>
+CK_TILE_HOST_DEVICE constexpr auto sequence_pop_back(Seq);
+
+namespace impl {
+// static_assert(__has_builtin(__type_pack_element), "can't find __type_pack_element");
+template <index_t I, typename... Ts>
+using at_index_t = __type_pack_element<I, Ts...>;
+} // namespace impl
+
+// we could implement as below, similiar to std. But let's reduce the symbol name...
+// template< class T, T... Ints >
+// class integer_sequence;
+
+template <index_t... Is>
+struct sequence
+{
+    using type       = sequence;
+    using value_type = index_t;
+
+    CK_TILE_HOST_DEVICE static constexpr index_t size() { return sizeof...(Is); }
+    CK_TILE_HOST_DEVICE static constexpr bool is_static() { return true; };
+
+    template <index_t I>
+    CK_TILE_HOST_DEVICE static constexpr auto get()
+    {
+        static_assert(I < size(), "wrong! I too large");
+        return number<impl::at_index_t<I, constant<Is>...>{}>{};
+    }
+
+    template <index_t I>
+    CK_TILE_HOST_DEVICE static constexpr auto get(number<I>)
+    {
+        static_assert(I < size(), "wrong! I too large");
+        return number<get<I>()>{};
+    }
+
+    CK_TILE_HOST_DEVICE static constexpr index_t at(index_t I)
+    {
+        // the last dummy element is to prevent compiler complain about empty array, when mSize = 0
+        const index_t mData[size() + 1] = {Is..., 0};
+        return mData[I];
+    }
+
+    template <index_t I>
+    CK_TILE_HOST_DEVICE static constexpr auto at()
+    {
+        static_assert(I < size(), "wrong! I too large");
+        return number<impl::at_index_t<I, constant<Is>...>{}>{};
+    }
+
+    template <index_t I>
+    CK_TILE_HOST_DEVICE static constexpr auto at(number<I>)
+    {
+        static_assert(I < size(), "wrong! I too large");
+        return number<get<I>()>{};
+    }
+
+    template <typename I>
+    CK_TILE_HOST_DEVICE constexpr auto operator[](I i) const
+    {
+        return at(i);
+    }
+
+    template <index_t... IRs>
+    CK_TILE_HOST_DEVICE static constexpr auto reorder_new_to_old(sequence<IRs...> /*new2old*/)
+    {
+        static_assert(sizeof...(Is) == sizeof...(IRs),
+                      "wrong! reorder map should have the same size as sequence to be rerodered");
+
+        static_assert(is_valid_sequence_map<sequence<IRs...>>::value, "wrong! invalid reorder map");
+
+        return sequence<type::get(number<IRs>{})...>{};
+    }
+
+    // MapOld2New is sequence<...>
+    template <typename MapOld2New>
+    CK_TILE_HOST_DEVICE static constexpr auto reorder_old_to_new(MapOld2New)
+    {
+        static_assert(MapOld2New::size() == size(),
+                      "wrong! reorder map should have the same size as sequence to be rerodered");
+
+        static_assert(is_valid_sequence_map<MapOld2New>::value, "wrong! invalid reorder map");
+
+        return reorder_new_to_old(typename sequence_map_inverse<MapOld2New>::type{});
+    }
+
+    CK_TILE_HOST_DEVICE static constexpr auto reverse()
+    {
+        return typename sequence_reverse<type>::type{};
+    }
+
+    CK_TILE_HOST_DEVICE static constexpr auto front()
+    {
+        static_assert(size() > 0, "wrong!");
+        return get(number<0>{});
+    }
+
+    CK_TILE_HOST_DEVICE static constexpr auto back()
+    {
+        static_assert(size() > 0, "wrong!");
+        return get(number<size() - 1>{});
+    }
+
+    CK_TILE_HOST_DEVICE static constexpr auto pop_front() { return sequence_pop_front(type{}); }
+
+    CK_TILE_HOST_DEVICE static constexpr auto pop_back() { return sequence_pop_back(type{}); }
+
+    template <index_t... Xs>
+    CK_TILE_HOST_DEVICE static constexpr auto push_front(sequence<Xs...>)
+    {
+        return sequence<Xs..., Is...>{};
+    }
+
+    template <index_t... Xs>
+    CK_TILE_HOST_DEVICE static constexpr auto push_front(number<Xs>...)
+    {
+        return sequence<Xs..., Is...>{};
+    }
+
+    template <index_t... Xs>
+    CK_TILE_HOST_DEVICE static constexpr auto push_back(sequence<Xs...>)
+    {
+        return sequence<Is..., Xs...>{};
+    }
+
+    template <index_t... Xs>
+    CK_TILE_HOST_DEVICE static constexpr auto push_back(number<Xs>...)
+    {
+        return sequence<Is..., Xs...>{};
+    }
+
+    // pickup element at index <Ids...>
+    template <index_t... Ids>
+    CK_TILE_HOST_DEVICE static constexpr auto extract(number<Ids>...)
+    {
+        return sequence<type::get(number<Ids>{})...>{};
+    }
+
+    template <index_t... Ids>
+    CK_TILE_HOST_DEVICE static constexpr auto extract(sequence<Ids...>)
+    {
+        return sequence<type::get(number<Ids>{})...>{};
+    }
+
+    // modify element at index "I" with value "X"
+    template <index_t I, index_t X>
+    CK_TILE_HOST_DEVICE static constexpr auto modify(number<I>, number<X>)
+    {
+        static_assert(I < size(), "wrong!");
+
+        using seq_split          = sequence_split<type, I>;
+        constexpr auto seq_left  = typename seq_split::left_type{};
+        constexpr auto seq_right = typename seq_split::right_type{}.pop_front();
+
+        return seq_left.push_back(number<X>{}).push_back(seq_right);
+    }
+
+    template <typename F>
+    CK_TILE_HOST_DEVICE static constexpr auto transform(F f)
+    {
+        return sequence<f(Is)...>{};
+    }
+
+    CK_TILE_HOST_DEVICE static void print()
+    {
+        printf("sequence{size: %d, data: [", size());
+        ((printf("%d ", Is)), ...);
+        printf("]}");
+    }
+};
+
+namespace impl {
+template <typename T, T... Ints>
+struct __integer_sequence;
+
+template <index_t... Ints>
+struct __integer_sequence<index_t, Ints...>
+{
+    using seq_type = sequence<Ints...>;
+};
+} // namespace impl
+
+// similiar
+template <index_t N>
+using make_index_sequence =
+    typename __make_integer_seq<impl::__integer_sequence, index_t, N>::seq_type;
+
+// merge sequence
+template <typename Seq, typename... Seqs>
+struct sequence_merge
+{
+    using type = typename sequence_merge<Seq, typename sequence_merge<Seqs...>::type>::type;
+};
+
+template <index_t... Xs, index_t... Ys>
+struct sequence_merge<sequence<Xs...>, sequence<Ys...>>
+{
+    using type = sequence<Xs..., Ys...>;
+};
+
+template <typename Seq>
+struct sequence_merge<Seq>
+{
+    using type = Seq;
+};
+
+// generate sequence
+template <index_t NSize, typename F>
+struct sequence_gen
+{
+    template <index_t IBegin, index_t NRemain, typename G>
+    struct sequence_gen_impl
+    {
+        static constexpr index_t NRemainLeft  = NRemain / 2;
+        static constexpr index_t NRemainRight = NRemain - NRemainLeft;
+        static constexpr index_t IMiddle      = IBegin + NRemainLeft;
+
+        using type = typename sequence_merge<
+            typename sequence_gen_impl<IBegin, NRemainLeft, G>::type,
+            typename sequence_gen_impl<IMiddle, NRemainRight, G>::type>::type;
+    };
+
+    template <index_t I, typename G>
+    struct sequence_gen_impl<I, 1, G>
+    {
+        static constexpr index_t Is = G{}(number<I>{});
+        using type                  = sequence<Is>;
+    };
+
+    template <index_t I, typename G>
+    struct sequence_gen_impl<I, 0, G>
+    {
+        using type = sequence<>;
+    };
+
+    using type = typename sequence_gen_impl<0, NSize, F>::type;
+};
+
+// arithmetic sequence
+template <index_t IBegin, index_t IEnd, index_t Increment>
+struct arithmetic_sequence_gen
+{
+    struct F
+    {
+        CK_TILE_HOST_DEVICE constexpr index_t operator()(index_t i) const
+        {
+            return i * Increment + IBegin;
+        }
+    };
+
+    using type0 = typename sequence_gen<(IEnd - IBegin) / Increment, F>::type;
+    using type1 = sequence<>;
+
+    static constexpr bool kHasContent =
+        (Increment > 0 && IBegin < IEnd) || (Increment < 0 && IBegin > IEnd);
+
+    using type = typename std::conditional<kHasContent, type0, type1>::type;
+};
+
+template <index_t IEnd>
+struct arithmetic_sequence_gen<0, IEnd, 1>
+{
+    using type = make_index_sequence<IEnd>;
+};
+
+// uniform sequence
+template <index_t NSize, index_t I>
+struct uniform_sequence_gen
+{
+    struct F
+    {
+        CK_TILE_HOST_DEVICE constexpr index_t operator()(index_t) const { return I; }
+    };
+
+    using type = typename sequence_gen<NSize, F>::type;
+};
+
+// reverse inclusive scan (with init) sequence
+template <typename, typename, index_t>
+struct sequence_reverse_inclusive_scan;
+
+template <index_t I, index_t... Is, typename Reduce, index_t Init>
+struct sequence_reverse_inclusive_scan<sequence<I, Is...>, Reduce, Init>
+{
+    using old_scan = typename sequence_reverse_inclusive_scan<sequence<Is...>, Reduce, Init>::type;
+
+    static constexpr index_t new_reduce = Reduce{}(I, old_scan{}.front());
+
+    using type = typename sequence_merge<sequence<new_reduce>, old_scan>::type;
+};
+
+template <index_t I, typename Reduce, index_t Init>
+struct sequence_reverse_inclusive_scan<sequence<I>, Reduce, Init>
+{
+    using type = sequence<Reduce{}(I, Init)>;
+};
+
+template <typename Reduce, index_t Init>
+struct sequence_reverse_inclusive_scan<sequence<>, Reduce, Init>
+{
+    using type = sequence<>;
+};
+
+// split sequence
+template <typename Seq, index_t I>
+struct sequence_split
+{
+    static constexpr index_t NSize = Seq{}.size();
+
+    using range0 = typename arithmetic_sequence_gen<0, I, 1>::type;
+    using range1 = typename arithmetic_sequence_gen<I, NSize, 1>::type;
+
+    using left_type  = decltype(Seq::extract(range0{}));
+    using right_type = decltype(Seq::extract(range1{}));
+};
+
+#if 0
+// reverse sequence
+template <typename Seq>
+struct sequence_reverse
+{
+    static constexpr index_t NSize = Seq{}.size();
+
+    using seq_split = sequence_split<Seq, NSize / 2>;
+    using type      = typename sequence_merge<
+        typename sequence_reverse<typename seq_split::right_type>::type,
+        typename sequence_reverse<typename seq_split::left_type>::type>::type;
+};
+
+template <index_t I>
+struct sequence_reverse<sequence<I>>
+{
+    using type = sequence<I>;
+};
+
+template <index_t I0, index_t I1>
+struct sequence_reverse<sequence<I0, I1>>
+{
+    using type = sequence<I1, I0>;
+};
+#endif
+
+namespace impl {
+template <typename Id, index_t... Ns>
+struct seq_reverse;
+
+template <index_t... Ids, index_t... Ns>
+struct seq_reverse<sequence<Ids...>, Ns...>
+{
+    template <index_t I>
+    using element = impl::at_index_t<I, constant<Ns>...>;
+    using type    = sequence<element<(sizeof...(Ns) - 1 - Ids)>::value...>;
+};
+} // namespace impl
+
+template <index_t... Ns>
+struct sequence_reverse<sequence<Ns...>>
+    : impl::seq_reverse<make_index_sequence<sizeof...(Ns)>, Ns...>
+{
+};
+
+// template <index_t... Ns>
+// using sequence_reverse_t = typename sequence_reverse<Ns...>::type;
+
+#if 1
+template <typename Reduce, typename Seq, typename... Seqs>
+struct sequence_reduce
+{
+    using type = typename sequence_reduce<Reduce,
+                                          Seq,
+                                          typename sequence_reduce<Reduce, Seqs...>::type>::type;
+};
+
+template <typename Reduce, index_t... Xs, index_t... Ys>
+struct sequence_reduce<Reduce, sequence<Xs...>, sequence<Ys...>>
+{
+    using type = sequence<Reduce{}(Xs, Ys)...>;
+};
+
+template <typename Reduce, typename Seq>
+struct sequence_reduce<Reduce, Seq>
+{
+    using type = Seq;
+};
+#endif
+
+template <typename Values, typename Ids, typename Compare>
+struct sequence_sort_impl
+{
+    template <typename LeftValues,
+              typename LeftIds,
+              typename RightValues,
+              typename RightIds,
+              typename MergedValues,
+              typename MergedIds,
+              typename Comp>
+    struct sorted_sequence_merge_impl
+    {
+        static constexpr bool choose_left = LeftValues::front() < RightValues::front();
+
+        static constexpr index_t chosen_value =
+            choose_left ? LeftValues::front() : RightValues::front();
+        static constexpr index_t chosen_id = choose_left ? LeftIds::front() : RightIds::front();
+
+        using new_merged_values = decltype(MergedValues::push_back(number<chosen_value>{}));
+        using new_merged_ids    = decltype(MergedIds::push_back(number<chosen_id>{}));
+
+        using new_left_values = typename std::
+            conditional<choose_left, decltype(LeftValues::pop_front()), LeftValues>::type;
+        using new_left_ids =
+            typename std::conditional<choose_left, decltype(LeftIds::pop_front()), LeftIds>::type;
+
+        using new_right_values = typename std::
+            conditional<choose_left, RightValues, decltype(RightValues::pop_front())>::type;
+        using new_right_ids =
+            typename std::conditional<choose_left, RightIds, decltype(RightIds::pop_front())>::type;
+
+        using merge = sorted_sequence_merge_impl<new_left_values,
+                                                 new_left_ids,
+                                                 new_right_values,
+                                                 new_right_ids,
+                                                 new_merged_values,
+                                                 new_merged_ids,
+                                                 Comp>;
+        // this is output
+        using merged_values = typename merge::merged_values;
+        using merged_ids    = typename merge::merged_ids;
+    };
+
+    template <typename LeftValues,
+              typename LeftIds,
+              typename MergedValues,
+              typename MergedIds,
+              typename Comp>
+    struct sorted_sequence_merge_impl<LeftValues,
+                                      LeftIds,
+                                      sequence<>,
+                                      sequence<>,
+                                      MergedValues,
+                                      MergedIds,
+                                      Comp>
+    {
+        using merged_values = typename sequence_merge<MergedValues, LeftValues>::type;
+        using merged_ids    = typename sequence_merge<MergedIds, LeftIds>::type;
+    };
+
+    template <typename RightValues,
+              typename RightIds,
+              typename MergedValues,
+              typename MergedIds,
+              typename Comp>
+    struct sorted_sequence_merge_impl<sequence<>,
+                                      sequence<>,
+                                      RightValues,
+                                      RightIds,
+                                      MergedValues,
+                                      MergedIds,
+                                      Comp>
+    {
+        using merged_values = typename sequence_merge<MergedValues, RightValues>::type;
+        using merged_ids    = typename sequence_merge<MergedIds, RightIds>::type;
+    };
+
+    template <typename LeftValues,
+              typename LeftIds,
+              typename RightValues,
+              typename RightIds,
+              typename Comp>
+    struct sorted_sequence_merge
+    {
+        using merge = sorted_sequence_merge_impl<LeftValues,
+                                                 LeftIds,
+                                                 RightValues,
+                                                 RightIds,
+                                                 sequence<>,
+                                                 sequence<>,
+                                                 Comp>;
+
+        using merged_values = typename merge::merged_values;
+        using merged_ids    = typename merge::merged_ids;
+    };
+
+    static constexpr index_t nsize = Values::size();
+
+    using split_unsorted_values = sequence_split<Values, nsize / 2>;
+    using split_unsorted_ids    = sequence_split<Ids, nsize / 2>;
+
+    using left_unsorted_values = typename split_unsorted_values::left_type;
+    using left_unsorted_ids    = typename split_unsorted_ids::left_type;
+    using left_sort          = sequence_sort_impl<left_unsorted_values, left_unsorted_ids, Compare>;
+    using left_sorted_values = typename left_sort::sorted_values;
+    using left_sorted_ids    = typename left_sort::sorted_ids;
+
+    using right_unsorted_values = typename split_unsorted_values::right_type;
+    using right_unsorted_ids    = typename split_unsorted_ids::right_type;
+    using right_sort = sequence_sort_impl<right_unsorted_values, right_unsorted_ids, Compare>;
+    using right_sorted_values = typename right_sort::sorted_values;
+    using right_sorted_ids    = typename right_sort::sorted_ids;
+
+    using merged_sorted = sorted_sequence_merge<left_sorted_values,
+                                                left_sorted_ids,
+                                                right_sorted_values,
+                                                right_sorted_ids,
+                                                Compare>;
+
+    using sorted_values = typename merged_sorted::merged_values;
+    using sorted_ids    = typename merged_sorted::merged_ids;
+};
+
+template <index_t ValueX, index_t ValueY, index_t IdX, index_t IdY, typename Compare>
+struct sequence_sort_impl<sequence<ValueX, ValueY>, sequence<IdX, IdY>, Compare>
+{
+    static constexpr bool choose_x = Compare{}(ValueX, ValueY);
+
+    using sorted_values = typename std::
+        conditional<choose_x, sequence<ValueX, ValueY>, sequence<ValueY, ValueX>>::type;
+    using sorted_ids =
+        typename std::conditional<choose_x, sequence<IdX, IdY>, sequence<IdY, IdX>>::type;
+};
+
+template <index_t Value, index_t Id, typename Compare>
+struct sequence_sort_impl<sequence<Value>, sequence<Id>, Compare>
+{
+    using sorted_values = sequence<Value>;
+    using sorted_ids    = sequence<Id>;
+};
+
+template <typename Compare>
+struct sequence_sort_impl<sequence<>, sequence<>, Compare>
+{
+    using sorted_values = sequence<>;
+    using sorted_ids    = sequence<>;
+};
+
+template <typename Values, typename Compare>
+struct sequence_sort
+{
+    using unsorted_ids = typename arithmetic_sequence_gen<0, Values::size(), 1>::type;
+    using sort         = sequence_sort_impl<Values, unsorted_ids, Compare>;
+
+    // this is output
+    using type                = typename sort::sorted_values;
+    using sorted2unsorted_map = typename sort::sorted_ids;
+};
+
+template <typename Values, typename Less, typename Equal>
+struct sequence_unique_sort
+{
+    template <typename RemainValues,
+              typename RemainIds,
+              typename UniquifiedValues,
+              typename UniquifiedIds,
+              typename Eq>
+    struct sorted_sequence_uniquify_impl
+    {
+        static constexpr index_t current_value = RemainValues::front();
+        static constexpr index_t current_id    = RemainIds::front();
+
+        static constexpr bool is_unique_value = (current_value != UniquifiedValues::back());
+
+        using new_remain_values = decltype(RemainValues::pop_front());
+        using new_remain_ids    = decltype(RemainIds::pop_front());
+
+        using new_uniquified_values =
+            typename std::conditional<is_unique_value,
+                                      decltype(UniquifiedValues::push_back(
+                                          number<current_value>{})),
+                                      UniquifiedValues>::type;
+
+        using new_uniquified_ids =
+            typename std::conditional<is_unique_value,
+                                      decltype(UniquifiedIds::push_back(number<current_id>{})),
+                                      UniquifiedIds>::type;
+
+        using uniquify = sorted_sequence_uniquify_impl<new_remain_values,
+                                                       new_remain_ids,
+                                                       new_uniquified_values,
+                                                       new_uniquified_ids,
+                                                       Eq>;
+
+        // this is output
+        using uniquified_values = typename uniquify::uniquified_values;
+        using uniquified_ids    = typename uniquify::uniquified_ids;
+    };
+
+    template <typename UniquifiedValues, typename UniquifiedIds, typename Eq>
+    struct sorted_sequence_uniquify_impl<sequence<>,
+                                         sequence<>,
+                                         UniquifiedValues,
+                                         UniquifiedIds,
+                                         Eq>
+    {
+        using uniquified_values = UniquifiedValues;
+        using uniquified_ids    = UniquifiedIds;
+    };
+
+    template <typename SortedValues, typename SortedIds, typename Eq>
+    struct sorted_sequence_uniquify
+    {
+        using uniquify = sorted_sequence_uniquify_impl<decltype(SortedValues::pop_front()),
+                                                       decltype(SortedIds::pop_front()),
+                                                       sequence<SortedValues::front()>,
+                                                       sequence<SortedIds::front()>,
+                                                       Eq>;
+
+        using uniquified_values = typename uniquify::uniquified_values;
+        using uniquified_ids    = typename uniquify::uniquified_ids;
+    };
+
+    using sort          = sequence_sort<Values, Less>;
+    using sorted_values = typename sort::type;
+    using sorted_ids    = typename sort::sorted2unsorted_map;
+
+    using uniquify = sorted_sequence_uniquify<sorted_values, sorted_ids, Equal>;
+
+    // this is output
+    using type                = typename uniquify::uniquified_values;
+    using sorted2unsorted_map = typename uniquify::uniquified_ids;
+};
+
+template <typename SeqMap>
+struct is_valid_sequence_map
+    : std::is_same<typename arithmetic_sequence_gen<0, SeqMap::size(), 1>::type,
+                   typename sequence_sort<SeqMap, less<index_t>>::type>
+{
+};
+
+template <typename SeqMap>
+struct sequence_map_inverse
+{
+    template <typename X2Y, typename WorkingY2X, index_t XBegin, index_t XRemain>
+    struct sequence_map_inverse_impl
+    {
+        static constexpr auto new_y2x =
+            WorkingY2X::modify(X2Y::get(number<XBegin>{}), number<XBegin>{});
+
+        using type =
+            typename sequence_map_inverse_impl<X2Y, decltype(new_y2x), XBegin + 1, XRemain - 1>::
+                type;
+    };
+
+    template <typename X2Y, typename WorkingY2X, index_t XBegin>
+    struct sequence_map_inverse_impl<X2Y, WorkingY2X, XBegin, 0>
+    {
+        using type = WorkingY2X;
+    };
+
+    using type =
+        typename sequence_map_inverse_impl<SeqMap,
+                                           typename uniform_sequence_gen<SeqMap::size(), 0>::type,
+                                           0,
+                                           SeqMap::size()>::type;
+};
+
+template <index_t... Xs, index_t... Ys>
+CK_TILE_HOST_DEVICE constexpr bool operator==(sequence<Xs...>, sequence<Ys...>)
+{
+    return ((Xs == Ys) && ...);
+}
+
+template <index_t... Xs, index_t... Ys>
+CK_TILE_HOST_DEVICE constexpr bool operator!=(sequence<Xs...> x, sequence<Ys...> y)
+{
+    return !(x == y);
+}
+
+template <index_t... Xs, index_t... Ys>
+CK_TILE_HOST_DEVICE constexpr auto operator+(sequence<Xs...>, sequence<Ys...>)
+{
+    static_assert(sizeof...(Xs) == sizeof...(Ys), "wrong! inconsistent size");
+
+    return sequence<(Xs + Ys)...>{};
+}
+
+template <index_t... Xs, index_t... Ys>
+CK_TILE_HOST_DEVICE constexpr auto operator-(sequence<Xs...>, sequence<Ys...>)
+{
+    static_assert(sizeof...(Xs) == sizeof...(Ys), "wrong! inconsistent size");
+
+    return sequence<(Xs - Ys)...>{};
+}
+
+template <index_t... Xs, index_t... Ys>
+CK_TILE_HOST_DEVICE constexpr auto operator*(sequence<Xs...>, sequence<Ys...>)
+{
+    static_assert(sizeof...(Xs) == sizeof...(Ys), "wrong! inconsistent size");
+
+    return sequence<(Xs * Ys)...>{};
+}
+
+template <index_t... Xs, index_t... Ys>
+CK_TILE_HOST_DEVICE constexpr auto operator/(sequence<Xs...>, sequence<Ys...>)
+{
+    static_assert(sizeof...(Xs) == sizeof...(Ys), "wrong! inconsistent size");
+
+    return sequence<(Xs / Ys)...>{};
+}
+
+template <index_t... Xs, index_t... Ys>
+CK_TILE_HOST_DEVICE constexpr auto operator%(sequence<Xs...>, sequence<Ys...>)
+{
+    static_assert(sizeof...(Xs) == sizeof...(Ys), "wrong! inconsistent size");
+
+    return sequence<(Xs % Ys)...>{};
+}
+
+template <index_t... Xs, index_t Y>
+CK_TILE_HOST_DEVICE constexpr auto operator+(sequence<Xs...>, number<Y>)
+{
+    return sequence<(Xs + Y)...>{};
+}
+
+template <index_t... Xs, index_t Y>
+CK_TILE_HOST_DEVICE constexpr auto operator-(sequence<Xs...>, number<Y>)
+{
+    return sequence<(Xs - Y)...>{};
+}
+
+template <index_t... Xs, index_t Y>
+CK_TILE_HOST_DEVICE constexpr auto operator*(sequence<Xs...>, number<Y>)
+{
+    return sequence<(Xs * Y)...>{};
+}
+
+template <index_t... Xs, index_t Y>
+CK_TILE_HOST_DEVICE constexpr auto operator/(sequence<Xs...>, number<Y>)
+{
+    return sequence<(Xs / Y)...>{};
+}
+
+template <index_t... Xs, index_t Y>
+CK_TILE_HOST_DEVICE constexpr auto operator%(sequence<Xs...>, number<Y>)
+{
+    return sequence<(Xs % Y)...>{};
+}
+
+template <index_t Y, index_t... Xs>
+CK_TILE_HOST_DEVICE constexpr auto operator+(number<Y>, sequence<Xs...>)
+{
+    return sequence<(Y + Xs)...>{};
+}
+
+template <index_t Y, index_t... Xs>
+CK_TILE_HOST_DEVICE constexpr auto operator-(number<Y>, sequence<Xs...>)
+{
+    return sequence<(Y - Xs)...>{};
+}
+
+template <index_t Y, index_t... Xs>
+CK_TILE_HOST_DEVICE constexpr auto operator*(number<Y>, sequence<Xs...>)
+{
+    return sequence<(Y * Xs)...>{};
+}
+
+template <index_t Y, index_t... Xs>
+CK_TILE_HOST_DEVICE constexpr auto operator/(number<Y>, sequence<Xs...>)
+{
+    return sequence<(Y / Xs)...>{};
+}
+
+template <index_t Y, index_t... Xs>
+CK_TILE_HOST_DEVICE constexpr auto operator%(number<Y>, sequence<Xs...>)
+{
+    return sequence<(Y % Xs)...>{};
+}
+
+template <index_t I, index_t... Is>
+CK_TILE_HOST_DEVICE constexpr auto sequence_pop_front(sequence<I, Is...>)
+{
+    return sequence<Is...>{};
+}
+
+template <typename Seq>
+CK_TILE_HOST_DEVICE constexpr auto sequence_pop_back(Seq)
+{
+    static_assert(Seq::size() > 0, "wrong! cannot pop an empty sequence!");
+    return sequence_pop_front(Seq::reverse()).reverse();
+}
+
+template <typename... Seqs>
+CK_TILE_HOST_DEVICE constexpr auto merge_sequences(Seqs...)
+{
+    return typename sequence_merge<Seqs...>::type{};
+}
+
+template <typename F, index_t... Xs>
+CK_TILE_HOST_DEVICE constexpr auto transform_sequences(F f, sequence<Xs...>)
+{
+    return sequence<f(Xs)...>{};
+}
+
+template <typename F, index_t... Xs, index_t... Ys>
+CK_TILE_HOST_DEVICE constexpr auto transform_sequences(F f, sequence<Xs...>, sequence<Ys...>)
+{
+    static_assert(sequence<Xs...>::size() == sequence<Ys...>::size(), "Dim not the same");
+
+    return sequence<f(Xs, Ys)...>{};
+}
+
+template <typename F, index_t... Xs, index_t... Ys, index_t... Zs>
+CK_TILE_HOST_DEVICE constexpr auto
+transform_sequences(F f, sequence<Xs...>, sequence<Ys...>, sequence<Zs...>)
+{
+    static_assert(sequence<Xs...>::size() == sequence<Ys...>::size() &&
+                      sequence<Xs...>::size() == sequence<Zs...>::size(),
+                  "Dim not the same");
+
+    return sequence<f(Xs, Ys, Zs)...>{};
+}
+
+template <typename Seq, typename Reduce, index_t Init>
+CK_TILE_HOST_DEVICE constexpr auto reverse_inclusive_scan_sequence(Seq, Reduce, number<Init>)
+{
+    return typename sequence_reverse_inclusive_scan<Seq, Reduce, Init>::type{};
+}
+
+template <typename Seq, typename Reduce, index_t Init>
+CK_TILE_HOST_DEVICE constexpr auto reverse_exclusive_scan_sequence(Seq, Reduce, number<Init>)
+{
+    return reverse_inclusive_scan_sequence(Seq::pop_front(), Reduce{}, number<Init>{})
+        .push_back(number<Init>{});
+}
+
+template <typename Seq, typename Reduce, index_t Init>
+CK_TILE_HOST_DEVICE constexpr auto inclusive_scan_sequence(Seq, Reduce, number<Init>)
+{
+    return reverse_inclusive_scan_sequence(Seq{}.reverse(), Reduce{}, number<Init>{}).reverse();
+}
+
+// e.g. Seq<2, 3, 4> --> Seq<0, 2, 5>, Init=0, Reduce=Add
+//      ResultSeq  TargetSeq  Reduce
+template <typename, typename, typename>
+struct sequence_exclusive_scan;
+
+template <index_t... Xs, index_t Y, index_t... Ys, typename Reduce>
+struct sequence_exclusive_scan<sequence<Xs...>, sequence<Y, Ys...>, Reduce>
+{
+    using old_scan = typename sequence_merge<sequence<Xs...>,
+                                             sequence<Reduce{}(Y, sequence<Xs...>{}.back())>>::type;
+    using type     = typename sequence_exclusive_scan<old_scan, sequence<Ys...>, Reduce>::type;
+};
+
+template <index_t... Xs, index_t Y, typename Reduce>
+struct sequence_exclusive_scan<sequence<Xs...>, sequence<Y>, Reduce>
+{
+    using type = sequence<Xs...>;
+};
+
+template <index_t... Xs, typename Reduce>
+struct sequence_exclusive_scan<sequence<Xs...>, sequence<>, Reduce>
+{
+    using type = sequence<Xs...>;
+};
+
+template <typename Seq, typename Reduce, index_t Init>
+constexpr auto exclusive_scan_sequence(Seq, Reduce, number<Init>)
+{
+    // TODO: c++20 and later can pass in Reduce with a lambda expression
+    return typename sequence_exclusive_scan<sequence<Init>, Seq, Reduce>::type{};
+}
+
+template <typename Seq>
+constexpr auto prefix_sum_sequence(Seq)
+{
+    return typename sequence_exclusive_scan<sequence<0>,
+                                            typename sequence_merge<Seq, sequence<0>>::type,
+                                            plus<index_t>>::type{};
+}
+
+template <typename Seq, index_t... Is>
+CK_TILE_HOST_DEVICE constexpr auto pick_sequence_elements_by_ids(Seq, sequence<Is...> /* ids */)
+{
+    return sequence<Seq::get(number<Is>{})...>{};
+}
+
+#if 1
+namespace detail {
+template <typename WorkSeq, typename RemainSeq, typename RemainMask>
+struct pick_sequence_elements_by_mask_impl
+{
+    using new_work_seq = typename std::conditional<RemainMask::front(),
+                                                   decltype(WorkSeq::push_back(RemainSeq::front())),
+                                                   WorkSeq>::type;
+
+    using type =
+        typename pick_sequence_elements_by_mask_impl<new_work_seq,
+                                                     decltype(RemainSeq::pop_front()),
+                                                     decltype(RemainMask::pop_front())>::type;
+};
+
+template <typename WorkSeq>
+struct pick_sequence_elements_by_mask_impl<WorkSeq, sequence<>, sequence<>>
+{
+    using type = WorkSeq;
+};
+
+} // namespace detail
+
+template <typename Seq, typename Mask>
+CK_TILE_HOST_DEVICE constexpr auto pick_sequence_elements_by_mask(Seq, Mask)
+{
+    static_assert(Seq::size() == Mask::size(), "wrong!");
+
+    return typename detail::pick_sequence_elements_by_mask_impl<sequence<>, Seq, Mask>::type{};
+}
+
+namespace detail {
+template <typename WorkSeq, typename RemainValues, typename RemainIds>
+struct modify_sequence_elements_by_ids_impl
+{
+    using new_work_seq = decltype(WorkSeq::modify(RemainIds::front(), RemainValues::front()));
+
+    using type =
+        typename modify_sequence_elements_by_ids_impl<new_work_seq,
+                                                      decltype(RemainValues::pop_front()),
+                                                      decltype(RemainIds::pop_front())>::type;
+};
+
+template <typename WorkSeq>
+struct modify_sequence_elements_by_ids_impl<WorkSeq, sequence<>, sequence<>>
+{
+    using type = WorkSeq;
+};
+} // namespace detail
+
+template <typename Seq, typename Values, typename Ids>
+CK_TILE_HOST_DEVICE constexpr auto modify_sequence_elements_by_ids(Seq, Values, Ids)
+{
+    static_assert(Values::size() == Ids::size() && Seq::size() >= Values::size(), "wrong!");
+
+    return typename detail::modify_sequence_elements_by_ids_impl<Seq, Values, Ids>::type{};
+}
+#endif
+
+template <typename Seq, typename Reduce, index_t Init>
+CK_TILE_HOST_DEVICE constexpr index_t
+reduce_on_sequence(Seq, Reduce f, number<Init> /*initial_value*/)
+{
+    index_t result = Init;
+
+    for(index_t i = 0; i < Seq::size(); ++i)
+    {
+        result = f(result, Seq::at(i));
+    }
+
+    return result;
+}
+
+// TODO: a generic any_of for any container
+template <typename Seq, typename F>
+CK_TILE_HOST_DEVICE constexpr bool sequence_any_of(Seq, F f)
+{
+    bool flag = false;
+
+    for(index_t i = 0; i < Seq::size(); ++i)
+    {
+        flag = flag || f(Seq::at(i));
+    }
+
+    return flag;
+}
+
+// TODO: a generic all_of for any container
+template <typename Seq, typename F>
+CK_TILE_HOST_DEVICE constexpr bool sequence_all_of(Seq, F f)
+{
+    bool flag = true;
+
+    for(index_t i = 0; i < Seq::size(); ++i)
+    {
+        flag = flag && f(Seq::at(i));
+    }
+
+    return flag;
+}
+
+template <typename... Seqs>
+using sequence_merge_t = typename sequence_merge<Seqs...>::type;
+
+template <index_t NSize, index_t I>
+using uniform_sequence_gen_t = typename uniform_sequence_gen<NSize, I>::type;
+
+template <index_t... Is>
+CK_TILE_HOST_DEVICE constexpr auto make_sequence(number<Is>...)
+{
+    return sequence<Is...>{};
+}
+
+// F() returns index_t
+// F use default constructor, so F cannot be lambda function
+template <typename F, index_t N>
+CK_TILE_HOST_DEVICE constexpr auto generate_sequence(F, number<N>)
+{
+    return typename sequence_gen<N, F>::type{};
+}
+
+// F() returns number<>
+// F could be lambda function
+template <typename F, index_t N>
+CK_TILE_HOST_DEVICE constexpr auto generate_sequence_v2(F&& f, number<N>)
+{
+    return unpack([&f](auto&&... xs) { return make_sequence(f(xs)...); },
+                  typename arithmetic_sequence_gen<0, N, 1>::type{});
+}
+
+template <class... T>
+struct tuple;
+
+template <index_t... Is>
+CK_TILE_HOST_DEVICE constexpr auto to_sequence(tuple<number<Is>...>)
+{
+    return sequence<Is...>{};
+}
+
+namespace detail {
+template <index_t h_idx, typename SeqSortedSamples, typename SeqRange>
+struct sorted_sequence_histogram;
+
+template <index_t h_idx, index_t x, index_t... xs, index_t r, index_t... rs>
+struct sorted_sequence_histogram<h_idx, sequence<x, xs...>, sequence<r, rs...>>
+{
+    template <typename Histogram>
+    constexpr auto operator()(Histogram& h)
+    {
+        if constexpr(x < r)
+        {
+            h.template at<h_idx>() += 1;
+            sorted_sequence_histogram<h_idx, sequence<xs...>, sequence<r, rs...>>{}(h);
+        }
+        else
+        {
+            h.template at<h_idx + 1>() = 1;
+            sorted_sequence_histogram<h_idx + 1, sequence<xs...>, sequence<rs...>>{}(h);
+        }
+    }
+};
+
+template <index_t h_idx, index_t x, index_t r, index_t... rs>
+struct sorted_sequence_histogram<h_idx, sequence<x>, sequence<r, rs...>>
+{
+    template <typename Histogram>
+    constexpr auto operator()(Histogram& h)
+    {
+        if constexpr(x < r)
+        {
+            h.template at<h_idx>() += 1;
+        }
+    }
+};
+} // namespace detail
+
+template <typename, index_t>
+struct array; // declare for later use (array->seq utility)
+
+// SeqSortedSamples: <0, 2, 3, 5, 7>, SeqRange: <0, 3, 6, 9> -> SeqHistogram : <2, 2, 1>
+template <typename SeqSortedSamples, index_t r, index_t... rs>
+CK_TILE_HOST_DEVICE constexpr auto histogram_sorted_sequence(SeqSortedSamples, sequence<r, rs...>)
+{
+    constexpr auto bins      = sizeof...(rs); // or categories
+    constexpr auto histogram = [&]() {
+        array<index_t, bins> h{0}; // make sure this can clear all element to zero
+        detail::sorted_sequence_histogram<0, SeqSortedSamples, sequence<rs...>>{}(h);
+        return h;
+    }();
+
+    return TO_SEQUENCE(histogram, bins);
+}
+
+template <typename F, index_t N>
+CK_TILE_HOST_DEVICE constexpr auto generate_array(F&& f, number<N>)
+{
+    using T = remove_cvref_t<decltype(f(number<0>{}))>;
+
+    return unpack([&f](auto&&... is) { return array<T, N>{f(is)...}; },
+                  typename arithmetic_sequence_gen<0, N, 1>::type{});
+}
+
+} // namespace ck_tile

include/ck_tile/core/container/span.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2023, Advanced Micro Devices, Inc. All rights reserved.
+
+#pragma once
+
+#include "ck_tile/core/config.hpp"
+#include <cstddef>
+#include <array>
+#include <type_traits>
+
+namespace ck_tile {
+
+// implement the c++20 std::span, lightweight, non-owning reference to a sequence
+// weather it is dynamic or static range. Or can be seen as a view of a contiguous sequence
+// TODO: do we need in device consider this is pointer?
+template <typename T>
+class span
+{
+    public:
+    using element_type    = T;
+    using value_type      = std::remove_cv_t<element_type>;
+    using size_type       = std::size_t;
+    using difference_type = std::ptrdiff_t;
+    using pointer         = element_type*;
+    using const_pointer   = const element_type*;
+    using reference       = element_type&;
+    using const_reference = const element_type&;
+    using iterator        = pointer;
+    using const_iterator  = pointer;
+
+    CK_TILE_HOST_DEVICE constexpr span() : span(nullptr, size_type{0}) {}
+
+    CK_TILE_HOST_DEVICE constexpr span(pointer first, size_type count) : ptr_(first), size_(count)
+    {
+    }
+
+    CK_TILE_HOST_DEVICE constexpr span(pointer first, pointer last) : span(first, last - first) {}
+
+    template <std::size_t N>
+    CK_TILE_HOST_DEVICE constexpr span(element_type (&arr)[N]) noexcept : span(arr, N)
+    {
+    }
+
+    template <std::size_t N>
+    CK_TILE_HOST_DEVICE constexpr span(std::array<value_type, N>& arr) noexcept
+        : span(arr.data(), N)
+    {
+    }
+
+    template <typename Container>
+    CK_TILE_HOST_DEVICE constexpr span(const Container& container)
+        : span(container.data(), container.size())
+    {
+    }
+
+    CK_TILE_HOST_DEVICE constexpr iterator begin() const noexcept { return ptr_; }
+    CK_TILE_HOST_DEVICE constexpr const_iterator cbegin() const noexcept { return begin(); }
+
+    CK_TILE_HOST_DEVICE constexpr iterator end() const noexcept { return begin() + size(); }
+    CK_TILE_HOST_DEVICE constexpr const_iterator cend() const noexcept { return end(); }
+
+    CK_TILE_HOST_DEVICE constexpr reference front() const { return *begin(); }
+    CK_TILE_HOST_DEVICE constexpr reference back() const { return *(--end()); }
+
+    CK_TILE_HOST_DEVICE constexpr reference operator[](size_type idx) const
+    {
+        return *(begin() + idx);
+    }
+    CK_TILE_HOST_DEVICE constexpr pointer data() const noexcept { return ptr_; }
+
+    CK_TILE_HOST_DEVICE constexpr size_type size() const noexcept { return size_; }
+
+    private:
+    pointer ptr_;
+    size_type size_;
+};
+
+} // namespace ck_tile

include/ck_tile/core/container/statically_indexed_array.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2023, Advanced Micro Devices, Inc. All rights reserved.
+
+#pragma once
+
+#include "ck_tile/core/config.hpp"
+#include "ck_tile/core/container/array.hpp"
+#include "ck_tile/core/container/tuple.hpp"
+#include "ck_tile/core/numeric/integer.hpp"
+
+namespace ck_tile {
+
+#if CK_TILE_STATICALLY_INDEXED_ARRAY_DEFAULT == CK_TILE_STATICALLY_INDEXED_ARRAY_USE_TUPLE
+
+template <typename T, index_t N>
+using statically_indexed_array = tuple_array<T, N>;
+
+#else
+
+// consider mark this struct as deprecated
+template <typename T, index_t N>
+using statically_indexed_array = array<T, N>;
+
+#endif
+
+// consider always use ck_tile::array for this purpose
+#if 0
+template <typename X, typename... Xs>
+CK_TILE_HOST_DEVICE constexpr auto make_statically_indexed_array(const X& x, const Xs&... xs)
+{
+    return statically_indexed_array<X, sizeof...(Xs) + 1>(x, static_cast<X>(xs)...);
+}
+
+// make empty statically_indexed_array
+template <typename X>
+CK_TILE_HOST_DEVICE constexpr auto make_statically_indexed_array()
+{
+    return statically_indexed_array<X, 0>();
+}
+#endif
+} // namespace ck_tile

include/ck_tile/core/container/thread_buffer.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2023, Advanced Micro Devices, Inc. All rights reserved.
+
+#pragma once
+
+#include "ck_tile/core/config.hpp"
+#include "ck_tile/core/container/array.hpp"
+#include "ck_tile/core/container/tuple.hpp"
+
+namespace ck_tile {
+
+#if CK_TILE_THREAD_BUFFER_DEFAULT == CK_TILE_THREAD_BUFFER_USE_TUPLE
+template <typename T, index_t N>
+using thread_buffer = tuple_array<T, N>;
+
+template <typename... Ts>
+CK_TILE_HOST_DEVICE constexpr auto make_thread_buffer(Ts&&... ts)
+{
+    return make_tuple(ts...);
+}
+#else
+
+#if 0
+template <typename T, index_t N>
+using thread_buffer = array<T, N>;
+
+template <typename... Ts>
+CK_TILE_HOST_DEVICE constexpr auto make_thread_buffer(Ts&&... ts)
+{
+    return make_array(ts...);
+}
+
+#endif
+
+// clang-format off
+template<typename T_, index_t N_>
+struct thread_buffer {
+    using value_type = remove_cvref_t<T_>;
+    static constexpr index_t N = N_;
+
+    value_type data[N];
+
+    // TODO: this ctor can't ignore
+    CK_TILE_HOST_DEVICE constexpr thread_buffer() : data{} {}
+    CK_TILE_HOST_DEVICE constexpr thread_buffer(const value_type & o) : data{o} {}
+
+    CK_TILE_HOST_DEVICE static constexpr auto size() { return N; }
+    CK_TILE_HOST_DEVICE auto & get() {return data; }
+    CK_TILE_HOST_DEVICE const auto & get() const {return data; }
+    CK_TILE_HOST_DEVICE auto & get(index_t i) {return data[i]; }
+    CK_TILE_HOST_DEVICE const auto & get(index_t i) const {return data[i]; }
+    CK_TILE_HOST_DEVICE constexpr const auto& operator[](index_t i) const { return get(i); }
+    CK_TILE_HOST_DEVICE constexpr auto& operator[](index_t i)             { return get(i); }
+    CK_TILE_HOST_DEVICE constexpr auto& operator()(index_t i)             { return get(i); }     // TODO: compatible
+    CK_TILE_HOST_DEVICE constexpr auto& at(index_t i)                                   { return get(i); }
+    CK_TILE_HOST_DEVICE constexpr const auto& at(index_t i) const                       { return get(i); }
+    template <index_t I> CK_TILE_HOST_DEVICE constexpr auto& at()                       { return get(I); }
+    template <index_t I> CK_TILE_HOST_DEVICE constexpr const auto& at() const           { return get(I); }
+    template <index_t I> CK_TILE_HOST_DEVICE constexpr auto& at(number<I>)              { return get(I); }
+    template <index_t I> CK_TILE_HOST_DEVICE constexpr const auto& at(number<I>) const  { return get(I); }
+
+    template <typename X_,
+              typename std::enable_if<has_same_scalar_type<value_type, X_>::value, bool>::type = false>
+    CK_TILE_HOST_DEVICE constexpr auto _get_as() const
+    {
+        using X = remove_cvref_t<X_>;
+
+        constexpr index_t kSPerX = vector_traits<X>::vector_size;
+        static_assert(N % kSPerX == 0);
+
+        union {
+            thread_buffer<X_, N / kSPerX> data {};
+            // tuple_array<value_type, kSPerX> sub_data;
+            value_type sub_data[N];
+        } vx;
+        static_for<0, N, 1>{}(
+            [&](auto j) { vx.sub_data[j] = data[j]; });
+        return vx.data;
+    }
+
+    template <typename X_,
+              index_t Is,
+              typename std::enable_if<has_same_scalar_type<value_type, X_>::value, bool>::type = false>
+    CK_TILE_HOST_DEVICE const constexpr remove_reference_t<X_> _get_as(number<Is> is) const
+    {
+        using X = remove_cvref_t<X_>;
+
+        constexpr index_t kSPerX = vector_traits<X>::vector_size;
+
+        union {
+            X_ data {};
+            tuple_array<value_type, kSPerX> sub_data;
+        } vx;
+        static_for<0, kSPerX, 1>{}(
+            [&](auto j) { vx.sub_data(j) = operator[]((is * number<sizeof(X_)/sizeof(value_type)>{}) + j); });
+        return vx.data;
+    }
+
+#if 0
+    template <typename X_,
+              index_t Is,
+              typename std::enable_if<has_same_scalar_type<value_type, X_>::value, bool>::type = false>
+    CK_TILE_HOST_DEVICE constexpr void _set_as(number<Is> is, X_ x)
+    {
+        using X = remove_cvref_t<X_>;
+
+        constexpr index_t kSPerX = vector_traits<X>::vector_size;
+
+        union {
+            X_ data;
+            tuple_array<value_type, kSPerX> sub_data;
+        } vx {x};
+
+        static_for<0, kSPerX, 1>{}(
+           [&](auto j) { operator()((is * number<sizeof(X_)/sizeof(value_type)>{}) + j) = vx.sub_data[j]; });
+    }
+#endif
+
+
+#define TB_COMMON_AS() \
+            static_assert(sizeof(value_type) * N % sizeof(Tx) == 0); \
+            constexpr int vx = sizeof(value_type) * N / sizeof(Tx)
+
+    template<typename Tx>
+    CK_TILE_HOST_DEVICE auto & get_as() {TB_COMMON_AS();
+            return reinterpret_cast<thread_buffer<Tx, vx>&>(data);}
+    template<typename Tx>
+    CK_TILE_HOST_DEVICE constexpr auto get_as() const {TB_COMMON_AS();
+            if constexpr(sizeof(value_type) <= 1 )
+            return _get_as<Tx>();   // TODO: current compiler for 8bit data need use union to get data back, should fix in the future
+            else
+            return reinterpret_cast<const thread_buffer<Tx, vx>&>(data);}
+    template<typename Tx, index_t I>
+    CK_TILE_HOST_DEVICE auto & get_as(number<I>) {TB_COMMON_AS();
+            return reinterpret_cast<thread_buffer<Tx, vx>&>(data).get(number<I>{});}
+    template<typename Tx, index_t I>
+    CK_TILE_HOST_DEVICE constexpr auto get_as(number<I>) const {TB_COMMON_AS();
+            if constexpr(sizeof(value_type) <= 1 )
+            return _get_as<Tx>(number<I>{});   // TODO: current compiler for 8bit data need use union to get data back, should fix in the future
+            else
+            return reinterpret_cast<const thread_buffer<Tx, vx>&>(data).get(number<I>{});}
+
+    template <typename Tx> CK_TILE_HOST_DEVICE constexpr void set_as(index_t i, const Tx & x)
+            { TB_COMMON_AS();    reinterpret_cast<thread_buffer<Tx, vx>&>(data).at(i) = x; }
+    template <typename Tx, index_t I> CK_TILE_HOST_DEVICE constexpr void set_as(number<I>, const Tx & x)
+            { TB_COMMON_AS();    reinterpret_cast<thread_buffer<Tx, vx>&>(data).at(number<I>{}) = x; }
+
+#undef TB_COMMON_AS
+};
+// clang-format on
+
+template <typename>
+struct vector_traits;
+
+// specialization for array
+template <typename T, index_t N>
+struct vector_traits<thread_buffer<T, N>>
+{
+    using scalar_type                    = T;
+    static constexpr index_t vector_size = N;
+};
+
+#endif
+
+} // namespace ck_tile

include/ck_tile/core/container/tuple.hpp

+// 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/sequence.hpp"
+#include "ck_tile/core/numeric/integer.hpp"
+#include "ck_tile/core/numeric/integral_constant.hpp"
+#include "ck_tile/core/numeric/math.hpp"
+#include "ck_tile/core/utility/functional.hpp"
+#include "ck_tile/core/utility/type_traits.hpp"
+#include <utility>
+#include <initializer_list>
+
+#ifndef CK_TILE_TUPLE_IMPL
+#define CK_TILE_TUPLE_IMPL 1
+#endif
+
+namespace ck_tile {
+
+namespace impl {
+template <typename T, index_t N>
+struct tuple_array_impl;
+}
+
+template <typename T, index_t N>
+using tuple_array = typename impl::tuple_array_impl<T, N>::type;
+
+namespace impl {
+
+// the place where content is stored
+template <index_t idx, typename T, bool is_empty = std::is_empty_v<T>>
+struct tuple_object
+{
+};
+
+template <index_t idx, typename T>
+struct tuple_object<idx, T, true>
+{
+    CK_TILE_HOST_DEVICE constexpr tuple_object() {}
+#if CK_TILE_TUPLE_IMPL == 0
+    template <typename U>
+    CK_TILE_HOST_DEVICE constexpr tuple_object(U&&)
+    {
+    }
+    template <typename U>
+    CK_TILE_HOST_DEVICE constexpr tuple_object(const U&)
+    {
+    }
+    template <typename U>
+    CK_TILE_HOST_DEVICE constexpr tuple_object(U&)
+    {
+    }
+#elif CK_TILE_TUPLE_IMPL == 1
+    template <typename U,
+              typename std::enable_if<!std::is_same<remove_cvref_t<U>, tuple_object>::value,
+                                      bool>::type = false>
+    CK_TILE_HOST_DEVICE constexpr tuple_object(U&&)
+    {
+    }
+#endif
+};
+
+template <index_t idx, typename T>
+struct tuple_object<idx, T, false>
+{
+    CK_TILE_HOST_DEVICE constexpr tuple_object() : element{} {}
+#if CK_TILE_TUPLE_IMPL == 0
+    template <typename U>
+    CK_TILE_HOST_DEVICE constexpr tuple_object(U&& e) : element(std::forward<U>(e))
+    {
+    }
+    template <typename U>
+    CK_TILE_HOST_DEVICE constexpr tuple_object(const U& e) : element(e)
+    {
+    }
+    template <typename U>
+    CK_TILE_HOST_DEVICE constexpr tuple_object(U& e) : element(e)
+    {
+    }
+#elif CK_TILE_TUPLE_IMPL == 1
+    template <typename U,
+              typename std::enable_if<!std::is_same<remove_cvref_t<U>, tuple_object>::value,
+                                      bool>::type = false>
+    CK_TILE_HOST_DEVICE constexpr tuple_object(U&& e) : element(std::forward<U>(e))
+    {
+    }
+#endif
+    T element;
+};
+
+// NOTE: we return a instance(not a reference) if content is empty
+template <index_t I, class T>
+CK_TILE_HOST_DEVICE constexpr T getv(const tuple_object<I, T, true>&)
+{
+    return {};
+}
+
+template <index_t I, class T>
+CK_TILE_HOST_DEVICE constexpr const T& getv(const tuple_object<I, T, false>& x)
+{
+    return x.element;
+}
+
+template <index_t I, class T>
+CK_TILE_HOST_DEVICE constexpr T& getv(tuple_object<I, T, false>& x)
+{
+    return x.element;
+}
+
+template <index_t I, class T>
+CK_TILE_HOST_DEVICE constexpr T&& getv(tuple_object<I, T, false>&& x)
+{
+    return static_cast<T&&>(x.element);
+}
+
+template <typename index_seq, typename... T>
+struct tuple_base;
+
+template <index_t... I, typename... T>
+struct tuple_base<sequence<I...>, T...> : tuple_object<I, T>...
+{
+    CK_TILE_HOST_DEVICE constexpr tuple_base() = default;
+
+#if CK_TILE_TUPLE_CTOR_WITH_INITIALIZER_LIST
+#define _ILE() (std::initializer_list<U>{}.size() - 1)
+    template <typename U>
+    CK_TILE_HOST_DEVICE constexpr tuple_base(std::initializer_list<U> us)
+        : tuple_object<I, T>(static_cast<T>(*(us.begin() + (I >= _ILE() ? _ILE() : I))))...
+    {
+    }
+#undef _ILE
+#endif
+
+#if CK_TILE_TUPLE_IMPL == 0
+    template <class... U>
+    CK_TILE_HOST_DEVICE constexpr explicit tuple_base(U&&... u)
+        : tuple_object<I, T>(std::forward<U>(u))...
+    {
+    }
+
+    template <class... U>
+    CK_TILE_HOST_DEVICE constexpr explicit tuple_base(const U&... u) : tuple_object<I, T>(u)...
+    {
+    }
+
+    template <class... U>
+    CK_TILE_HOST_DEVICE constexpr explicit tuple_base(U&... u) : tuple_object<I, T>(u)...
+    {
+    }
+
+    template <class... U>
+    CK_TILE_HOST_DEVICE constexpr tuple_base(tuple_base<sequence<I...>, U...>&& u)
+        : tuple_object<I, T>(getv(static_cast<tuple_object<I, U>&&>(u)))...
+    {
+    }
+
+    template <class... U>
+    CK_TILE_HOST_DEVICE constexpr tuple_base(const tuple_base<sequence<I...>, U...>& u)
+        : tuple_object<I, T>(getv(static_cast<const tuple_object<I, U>&>(u)))...
+    {
+    }
+
+    template <class... U>
+    CK_TILE_HOST_DEVICE constexpr tuple_base(tuple_base<sequence<I...>, U...>& u)
+        : tuple_object<I, T>(getv(static_cast<tuple_object<I, U>&>(u)))...
+    {
+    }
+#elif CK_TILE_TUPLE_IMPL == 1
+    template <class U,
+              typename std::enable_if<sizeof...(I) == 1 && sizeof...(T) == 1 &&
+                                          !std::is_same<remove_cvref_t<U>, tuple_base>::value,
+                                      bool>::type = false>
+    CK_TILE_HOST_DEVICE constexpr tuple_base(U&& u) : tuple_object<I, T>(std::forward<U>(u))...
+    {
+    }
+
+    template <typename... U, typename std::enable_if<sizeof...(U) >= 2, bool>::type = false>
+    CK_TILE_HOST_DEVICE constexpr tuple_base(U&&... u) : tuple_object<I, T>(std::forward<U>(u))...
+    {
+        static_assert(sizeof...(I) == sizeof...(T) && sizeof...(I) == sizeof...(U),
+                      "wrong! inconsistent size");
+    }
+
+#endif
+};
+} // namespace impl
+
+template <class... T>
+struct tuple : impl::tuple_base<make_index_sequence<sizeof...(T)>, T...>
+{
+    CK_TILE_HOST_DEVICE
+    static constexpr auto size() { return sizeof...(T); }
+    using base = impl::tuple_base<make_index_sequence<sizeof...(T)>, T...>;
+    CK_TILE_HOST_DEVICE constexpr tuple() = default;
+
+#if CK_TILE_TUPLE_CTOR_WITH_INITIALIZER_LIST
+    template <typename U>
+    CK_TILE_HOST_DEVICE constexpr tuple(std::initializer_list<U> us) : base(us)
+    {
+    }
+#endif
+
+#if CK_TILE_TUPLE_IMPL == 0
+    template <class... U>
+    CK_TILE_HOST_DEVICE constexpr tuple(U&&... u) : base(std::forward<U>(u)...)
+    {
+    }
+
+    template <class... U>
+    CK_TILE_HOST_DEVICE constexpr tuple(const U&... u) : base(u...)
+    {
+    }
+
+    template <class... U>
+    CK_TILE_HOST_DEVICE constexpr tuple(U&... u) : base(u...)
+    {
+    }
+
+    template <class... U>
+    CK_TILE_HOST_DEVICE constexpr tuple(tuple<U...>&& u)
+        : base(static_cast<impl::tuple_base<make_index_sequence<sizeof...(U)>, U...>&&>(u))
+    {
+    }
+
+    template <class... U>
+    CK_TILE_HOST_DEVICE constexpr tuple(const tuple<U...>& u)
+        : base(static_cast<const impl::tuple_base<make_index_sequence<sizeof...(U)>, U...>&>(u))
+    {
+    }
+
+    template <class... U>
+    CK_TILE_HOST_DEVICE constexpr tuple(tuple<U...>& u)
+        : base(static_cast<impl::tuple_base<make_index_sequence<sizeof...(U)>, U...>&>(u))
+    {
+    }
+#elif CK_TILE_TUPLE_IMPL == 1
+    template <
+        typename U,
+        typename std::enable_if<sizeof...(T) == 1 && !std::is_same<remove_cvref_t<U>, tuple>::value,
+                                bool>::type = false>
+    CK_TILE_HOST_DEVICE constexpr tuple(U&& u) : base(std::forward<U>(u))
+    {
+    }
+
+    template <typename... U,
+              typename std::enable_if<sizeof...(U) == sizeof...(T) && sizeof...(U) >= 2,
+                                      bool>::type = false>
+    CK_TILE_HOST_DEVICE constexpr tuple(U&&... u) : base(std::forward<U>(u)...)
+    {
+    }
+#endif
+    CK_TILE_HOST_DEVICE static constexpr bool is_static()
+    {
+        bool flag = true;
+
+        static_for<0, sizeof...(T), 1>{}([&flag](auto i) {
+            flag &= is_static_v<remove_cvref_t<__type_pack_element<i.value, T...>>>;
+        });
+
+        return flag;
+    }
+
+#define TP_COM_() static_assert(I < size(), "wrong! out of range")
+    // clang-format off
+    template<index_t I> CK_TILE_HOST_DEVICE constexpr decltype(auto) get() const          { TP_COM_(); return impl::getv<I>(*this); }
+    template<index_t I> CK_TILE_HOST_DEVICE constexpr decltype(auto) get(number<I>) const { TP_COM_(); return get<I>(); }
+    template<index_t I> CK_TILE_HOST_DEVICE constexpr decltype(auto) get()                      { TP_COM_(); return impl::getv<I>(*this); }
+    template<index_t I> CK_TILE_HOST_DEVICE constexpr decltype(auto) get(number<I>)             { TP_COM_(); return get<I>(); }
+
+    template<index_t I> CK_TILE_HOST_DEVICE constexpr decltype(auto) at() const          { TP_COM_(); return impl::getv<I>(*this); }
+    template<index_t I> CK_TILE_HOST_DEVICE constexpr decltype(auto) at(number<I>) const { TP_COM_(); return get<I>(); }
+    template<index_t I> CK_TILE_HOST_DEVICE constexpr decltype(auto) at()                      { TP_COM_(); return impl::getv<I>(*this); }
+    template<index_t I> CK_TILE_HOST_DEVICE constexpr decltype(auto) at(number<I>)             { TP_COM_(); return get<I>(); }
+
+    template<index_t I> CK_TILE_HOST_DEVICE constexpr decltype(auto) operator[](number<I>)             { TP_COM_(); return get<I>(); }
+    template<index_t I> CK_TILE_HOST_DEVICE constexpr decltype(auto) operator[](number<I>) const { TP_COM_(); return get<I>(); }
+    template<index_t I> CK_TILE_HOST_DEVICE constexpr decltype(auto) operator()(number<I>)             { TP_COM_(); return get<I>(); }  // TODO: compatible
+
+    // below function should be used under tuple_array<> type, no extra check will perform here
+    template <typename Tx> CK_TILE_HOST_DEVICE constexpr decltype(auto) get_as()                            { return reinterpret_cast<tuple_array<Tx, size()>&>(*this); }
+    template <typename Tx> CK_TILE_HOST_DEVICE constexpr decltype(auto) get_as() const                      { return reinterpret_cast<const tuple_array<Tx, size()>&>(*this); }
+    // below index is for index *AFTER* type convert, not before
+    //template <typename Tx> CK_TILE_HOST_DEVICE constexpr decltype(auto) get_as(index_t i)                   { TP_COM_(); return reinterpret_cast<tuple_array<Tx, size()>&>(*this).at(i); }
+    //template <typename Tx> CK_TILE_HOST_DEVICE constexpr decltype(auto) get_as(index_t i) const             { TP_COM_(); return reinterpret_cast<const tuple_array<Tx, size()>&>(*this).at(i); }
+    template <typename Tx, index_t I> CK_TILE_HOST_DEVICE constexpr decltype(auto) get_as(number<I>)        { TP_COM_(); return reinterpret_cast<tuple_array<Tx, size()>&>(*this).at(number<I>{}); }
+    template <typename Tx, index_t I> CK_TILE_HOST_DEVICE constexpr decltype(auto) get_as(number<I>) const  { TP_COM_(); return reinterpret_cast<const tuple_array<Tx, size()>&>(*this).at(number<I>{}); }
+    
+    // template <typename Tx> CK_TILE_HOST_DEVICE constexpr void set_as(index_t i, const Tx & x)               { TP_COM_(); reinterpret_cast<tuple_array<Tx, size()>&>(*this).at(i) = x; }
+    template <typename Tx, index_t I> CK_TILE_HOST_DEVICE constexpr void set_as(number<I>, const Tx & x)    { TP_COM_(); reinterpret_cast<tuple_array<Tx, size()>&>(*this).at(number<I>{}) = x; }
+
+    // clang-format on
+#undef TP_COM_
+};
+
+template <typename>
+struct vector_traits;
+
+// specialization for array
+template <typename... T>
+struct vector_traits<tuple<T...>>
+{
+    using scalar_type                    = __type_pack_element<0, T...>;
+    static constexpr index_t vector_size = sizeof...(T);
+};
+
+// template <class... T>
+// CK_TILE_HOST_DEVICE constexpr
+// tuple<T...>
+// make_tuple(T const&... t)
+// {
+//     return {t...};
+// }
+template <typename... Xs>
+CK_TILE_HOST_DEVICE constexpr bool operator==(const tuple<Xs...>& a, const tuple<Xs...>& b)
+{
+    bool same = true;
+
+    static_for<0, sizeof...(Xs), 1>{}([&](auto i) {
+        if(a[i] != b[i])
+        {
+            same = false;
+        }
+    });
+
+    return same;
+}
+
+template <typename... Xs>
+CK_TILE_HOST_DEVICE constexpr bool operator!=(const tuple<Xs...>& a, const tuple<Xs...>& b)
+{
+    return !(a == b);
+}
+
+template <typename... Xs>
+CK_TILE_HOST_DEVICE constexpr auto make_tuple(Xs&&... xs)
+{
+    // here xs is always a lvalue as function arg
+    // Xs may deduced as (e.g try to pass in a integer in following cases)
+    //  1). if pass in a rvalue (like function return or int{}) -> Xs is "int"
+    //  2). if pass in a const lvalue -> Xs is "const int &"
+    //  3). if pass in a non-const lvalue -> Xs is "int &"
+    // so the return type of std::forward will dependes on Xs
+    //  1). std::forward -> int&&
+    //  2). std::forward -> const int&
+    //  3). std::forward -> int&
+    return tuple<remove_cvref_t<Xs>...>(std::forward<Xs>(xs)...);
+}
+
+// https://en.cppreference.com/w/cpp/utility/tuple/tie
+template <typename... Args>
+constexpr tuple<Args&...> tie(Args&... args) noexcept
+{
+    return {args...};
+}
+
+template <typename X, typename Y>
+struct tuple_concat;
+
+template <typename... Xs, typename... Ys>
+struct tuple_concat<tuple<Xs...>, tuple<Ys...>>
+{
+    using type = tuple<Xs..., Ys...>;
+};
+
+namespace impl {
+// be very careful using this type (because we want the internal type)
+// template deduction will fail if infering the inner type
+// e.g.
+// template<typename T, index_t N> using some_wrapper = typename tuple_array_impl<T, N>::type;
+// template<typename T, index_t N> void foo(const some_wrapper<T, N>&) {}
+//   -> compiler will fail to deduce this type, because this is under non-deduced context
+//   (https://en.cppreference.com/w/cpp/language/template_argument_deduction, "Non-deduced
+//   contexts")
+//
+// -> use this instead
+// template<typename Tup> void foo(const Tup&) {}
+template <typename T, index_t N>
+struct tuple_array_impl
+{
+    using type = typename tuple_concat<typename tuple_array_impl<T, N / 2>::type,
+                                       typename tuple_array_impl<T, N - N / 2>::type>::type;
+};
+
+template <typename T>
+struct tuple_array_impl<T, 0>
+{
+    using type = tuple<>;
+};
+
+template <typename T>
+struct tuple_array_impl<T, 1>
+{
+    using type = tuple<T>;
+};
+} // namespace impl
+
+template <typename F, index_t N>
+CK_TILE_HOST_DEVICE constexpr auto generate_tuple(F&& f, number<N>)
+{
+    return unpack([&f](auto&&... is) { return make_tuple(f(is)...); },
+                  typename arithmetic_sequence_gen<0, N, 1>::type{});
+}
+
+template <typename F, index_t N>
+CK_TILE_HOST_DEVICE constexpr auto generate_tie(F&& f, number<N>)
+{
+    return unpack([&f](auto&&... is) { return tie(f(is)...); },
+                  typename arithmetic_sequence_gen<0, N, 1>::type{});
+}
+
+// tx and ty are tuple of references, return type of will tuple of referennce (not rvalue)
+template <typename... X, typename... Y>
+CK_TILE_HOST_DEVICE constexpr auto concat_tuple_of_reference(const tuple<X&...>& tx,
+                                                             const tuple<Y&...>& ty)
+{
+    return unpack2(
+        [&](auto&&... zs) { return tuple<decltype(zs)...>{std::forward<decltype(zs)>(zs)...}; },
+        tx,
+        ty);
+}
+
+template <typename... X, typename... Y>
+CK_TILE_HOST_DEVICE constexpr auto concat_tuple(const tuple<X...>& tx, const tuple<Y...>& ty)
+{
+    return unpack2(
+        [&](auto... zs) { return tuple<decltype(zs)...>{std::forward<decltype(zs)>(zs)...}; },
+        tx,
+        ty);
+}
+
+// Support any number of tuples to concat (also 1)
+template <typename... X>
+CK_TILE_HOST_DEVICE constexpr auto concat_tuple(const tuple<X...>& tx)
+{
+    return tx;
+}
+
+template <typename... X, typename... Tuples>
+CK_TILE_HOST_DEVICE constexpr auto concat_tuple(const tuple<X...>& tx, const Tuples&... tuples)
+{
+    return concat_tuple(tx, concat_tuple(tuples...));
+}
+
+namespace detail {
+
+template <typename F, typename X, index_t... Is>
+CK_TILE_HOST_DEVICE constexpr auto transform_tuples_impl(F f, const X& x, sequence<Is...>)
+{
+    return make_tuple(f(x.at(number<Is>{}))...);
+}
+
+template <typename F, typename X, typename Y, index_t... Is>
+CK_TILE_HOST_DEVICE constexpr auto
+transform_tuples_impl(F f, const X& x, const Y& y, sequence<Is...>)
+{
+    return make_tuple(f(x.at(number<Is>{}), y.at(number<Is>{}))...);
+}
+
+template <typename F, typename X, typename Y, typename Z, index_t... Is>
+CK_TILE_HOST_DEVICE constexpr auto
+transform_tuples_impl(F f, const X& x, const Y& y, const Z& z, sequence<Is...>)
+{
+    return make_tuple(f(x.at(number<Is>{}), y.at(number<Is>{}), z.at(number<Is>{}))...);
+}
+
+} // namespace detail
+
+template <typename F, typename X>
+CK_TILE_HOST_DEVICE constexpr auto transform_tuples(F f, const X& x)
+{
+    return detail::transform_tuples_impl(
+        f, x, typename arithmetic_sequence_gen<0, X::size(), 1>::type{});
+}
+
+template <typename F, typename X, typename Y>
+CK_TILE_HOST_DEVICE constexpr auto transform_tuples(F f, const X& x, const Y& y)
+{
+    return detail::transform_tuples_impl(
+        f, x, y, typename arithmetic_sequence_gen<0, X::size(), 1>::type{});
+}
+
+template <typename F, typename X, typename Y, typename Z>
+CK_TILE_HOST_DEVICE constexpr auto transform_tuples(F f, const X& x, const Y& y, const Z& z)
+{
+    return detail::transform_tuples_impl(
+        f, x, y, z, typename arithmetic_sequence_gen<0, X::size(), 1>::type{});
+}
+
+// By default unroll to the flatten
+template <index_t Depth = 0, index_t MaxDepth = -1>
+CK_TILE_HOST_DEVICE constexpr auto unroll_nested_tuple(const tuple<>& t)
+{
+    return t;
+}
+
+template <index_t Depth = 0, index_t MaxDepth = -1, typename T>
+CK_TILE_HOST_DEVICE constexpr auto unroll_nested_tuple(const T& t)
+{
+    return make_tuple(t);
+}
+
+template <index_t Depth = 0, index_t MaxDepth = -1, typename... Ts>
+CK_TILE_HOST_DEVICE constexpr auto unroll_nested_tuple(const tuple<Ts...>& t)
+{
+    if constexpr(Depth == MaxDepth)
+    {
+        return t;
+    }
+    else
+    {
+        return unpack(
+            [&](auto&&... ts) {
+                return concat_tuple(unroll_nested_tuple<Depth + 1, MaxDepth>(ts)...);
+            },
+            t);
+    }
+}
+
+template <typename... Ts>
+CK_TILE_HOST_DEVICE constexpr auto tuple_reverse(const tuple<Ts...>& t)
+{
+    return generate_tuple(
+        [&](auto i) {
+            using Idx = number<tuple<Ts...>::size() - i - 1>;
+            return t.at(Idx{});
+        },
+        number<tuple<Ts...>::size()()>{});
+}
+
+// Reduce tuple values in specific range using Function
+template <index_t Idx, index_t End, typename F, typename... Ts>
+CK_TILE_HOST_DEVICE constexpr auto tuple_reduce(F&& f, const tuple<Ts...>& t)
+{
+    static_assert(Idx < End, "Wrong parameters for tuple_reduce");
+    if constexpr(Idx + 1 == End)
+    {
+        return t.at(number<Idx>{});
+    }
+    else
+    {
+        return f(t.at(number<Idx>{}), tuple_reduce<Idx + 1, End>(f, t));
+    }
+}
+
+template <typename T>
+using is_tuple = decltype(std::declval<T&>().IsTuple());
+
+template <typename... Ts>
+CK_TILE_HOST_DEVICE constexpr auto is_nested_tuple(const tuple<Ts...>&)
+{
+    return (is_detected<is_tuple, Ts>::value || ...);
+}
+
+template <index_t depth = 0, typename T>
+CK_TILE_HOST_DEVICE constexpr auto tuple_depth(const T&)
+{
+    return depth;
+}
+
+template <index_t depth = 0, typename... Ts>
+CK_TILE_HOST_DEVICE constexpr auto tuple_depth(const tuple<Ts...>&)
+{
+    return max(tuple_depth<depth + 1>(Ts{})...);
+}
+
+template <typename... Seqs>
+CK_TILE_HOST_DEVICE constexpr auto to_array_of_array(tuple<Seqs...> t_of_s)
+{
+    constexpr index_t n0 = sizeof...(Seqs);
+
+    constexpr index_t max_n1 = [&] {
+        index_t max_n1_ = 0;
+
+        static_for<0, n0, 1>{}([&](auto i0) {
+            constexpr index_t n1 = t_of_s[i0].size();
+
+            max_n1_ = max_n1_ < n1 ? n1 : max_n1_;
+        });
+
+        return max_n1_;
+    }();
+
+    array<array<index_t, max_n1>, n0> a_of_a{{-1}};
+
+    static_for<0, n0, 1>{}([&](auto i0) {
+        constexpr index_t n1 = t_of_s[i0].size();
+
+        static_for<0, n1, 1>{}([&](auto i1) { a_of_a(i0)(i1) = t_of_s[i0][i1]; });
+    });
+
+    return a_of_a;
+}
+
+// Here should use MultiIndex<NSize>, instead of tuple<Ys...>, although the former
+// is the alias of the latter. This is because compiler cannot infer the NSize if
+// using MultiIndex<NSize>
+// TODO: how to fix this?
+template <typename... Ys,
+          typename X,
+          std::enable_if_t<!std::is_integral<X>::value && !std::is_floating_point<X>::value, bool> =
+              false>
+CK_TILE_HOST_DEVICE constexpr auto operator+=(tuple<Ys...>& y, const X& x)
+{
+    static_assert(X::Size() == sizeof...(Ys), "wrong! size not the same");
+    constexpr index_t NSize = sizeof...(Ys);
+    static_for<0, NSize, 1>{}([&](auto i) { y[i] += x[i]; });
+    return y;
+}
+
+template <typename... Ys,
+          typename X,
+          std::enable_if_t<!std::is_integral<X>::value && !std::is_floating_point<X>::value, bool> =
+              false>
+CK_TILE_HOST_DEVICE constexpr auto operator-=(tuple<Ys...>& y, const X& x)
+{
+    static_assert(X::Size() == sizeof...(Ys), "wrong! size not the same");
+    constexpr index_t NSize = sizeof...(Ys);
+    static_for<0, NSize, 1>{}([&](auto i) { y[i] -= x[i]; });
+    return y;
+}
+
+template <typename... Xs,
+          typename Y,
+          std::enable_if_t<!std::is_integral<Y>::value && !std::is_floating_point<Y>::value, bool> =
+              false>
+CK_TILE_HOST_DEVICE constexpr auto operator+(const tuple<Xs...>& x, const Y& y)
+{
+    static_assert(Y::Size() == sizeof...(Xs), "wrong! size not the same");
+    constexpr index_t NSize = sizeof...(Xs);
+
+    tuple<Xs...> r;
+    static_for<0, NSize, 1>{}([&](auto i) { r[i] = x[i] + y[i]; });
+    return r;
+}
+
+template <typename... Xs,
+          typename Y,
+          std::enable_if_t<!std::is_integral<Y>::value && !std::is_floating_point<Y>::value, bool> =
+              false>
+CK_TILE_HOST_DEVICE constexpr auto operator-(const tuple<Xs...>& x, const Y& y)
+{
+    static_assert(Y::Size() == sizeof...(Xs), "wrong! size not the same");
+    constexpr index_t NSize = sizeof...(Xs);
+
+    tuple<Xs...> r;
+    static_for<0, NSize, 1>{}([&](auto i) { r[i] = x[i] - y[i]; });
+    return r;
+}
+
+template <typename... Xs,
+          typename Y,
+          std::enable_if_t<!std::is_integral<Y>::value && !std::is_floating_point<Y>::value, bool> =
+              false>
+CK_TILE_HOST_DEVICE constexpr auto operator*(const tuple<Xs...>& x, const Y& y)
+{
+    static_assert(Y::Size() == sizeof...(Xs), "wrong! size not the same");
+    constexpr index_t NSize = sizeof...(Xs);
+
+    tuple<Xs...> r;
+    static_for<0, NSize, 1>{}([&](auto i) { r[i] = x[i] * y[i]; });
+    return r;
+}
+
+// MultiIndex = scalar * MultiIndex
+template <
+    typename... Xs,
+    typename Y,
+    std::enable_if_t<std::is_integral<Y>::value || std::is_floating_point<Y>::value, bool> = false>
+CK_TILE_HOST_DEVICE constexpr auto operator*(Y a, const tuple<Xs...>& x)
+{
+    constexpr index_t NSize = sizeof...(Xs);
+    tuple<Xs...> r;
+    static_for<0, NSize, 1>{}([&](auto i) { r[i] = a * x[i]; });
+    return r;
+}
+
+// MultiIndex = MultiIndex * scalar
+template <
+    typename... Xs,
+    typename Y,
+    std::enable_if_t<std::is_integral<Y>::value || std::is_floating_point<Y>::value, bool> = false>
+CK_TILE_HOST_DEVICE constexpr auto operator*(const tuple<Xs...>& x, Y a)
+{
+    return a * x;
+}
+
+template <typename... Xs, typename... Ys>
+CK_TILE_HOST_DEVICE constexpr auto operator/(const tuple<Xs...>& x, const tuple<Ys...>& y)
+{
+    static_assert(sizeof...(Xs) == sizeof...(Ys), "wrong!");
+    constexpr index_t NSize = sizeof...(Xs);
+    return generate_tuple([&](auto i) { return x[i] / y[i]; }, number<NSize>{});
+}
+
+} // namespace ck_tile
+
+#include <tuple>
+// WARNING: needed by compiler for C++ structured binding support only, don't use this
+namespace std {
+
+template <typename... Ts>
+struct tuple_size<ck_tile::tuple<Ts...>> : std::integral_constant<std::size_t, sizeof...(Ts)>
+{
+};
+
+template <std::size_t I, typename... Ts>
+struct tuple_element<I, ck_tile::tuple<Ts...>> : std::tuple_element<I, std::tuple<Ts...>>
+{
+};
+
+template <typename... Ts>
+struct tuple_size<const ck_tile::tuple<Ts...>> : std::integral_constant<std::size_t, sizeof...(Ts)>
+{
+};
+
+template <std::size_t I, typename... Ts>
+struct tuple_element<I, const ck_tile::tuple<Ts...>>
+    : std::tuple_element<I, const std::tuple<Ts...>>
+{
+};
+
+} // namespace std
+
+#if 1
+#define TO_TUPLE_OF_NUMBER(a, n)                                                             \
+    _Pragma("clang diagnostic push") _Pragma(                                                \
+        "clang diagnostic ignored \"-Wc++20-extensions\"")[a]<ck_tile::index_t... IDX_IDX_>( \
+        ck_tile::sequence<IDX_IDX_...>)                                                      \
+    {                                                                                        \
+        return ck_tile::tuple<ck_tile::number<a[ck_tile::number<IDX_IDX_>{}]>...>{};         \
+    }                                                                                        \
+    (ck_tile::make_index_sequence<n>{}) _Pragma("clang diagnostic pop")
+#else
+#define TO_TUPLE_OF_NUMBER(arr, n_)                                                      \
+    [&arr, n_] {                                                                         \
+        static_assert(arr.size() >= n_, "wrong! out of bound");                          \
+                                                                                         \
+        static_assert(n_ < 7, "not implemented");                                        \
+                                                                                         \
+        if constexpr(n_ == 0)                                                            \
+        {                                                                                \
+            return ck_tile::tuple<>{};                                                   \
+        }                                                                                \
+        else if constexpr(n_ == 1)                                                       \
+        {                                                                                \
+            return ck_tile::tuple<number<arr[0]>>{};                                     \
+        }                                                                                \
+        else if constexpr(n_ == 2)                                                       \
+        {                                                                                \
+            return ck_tile::tuple<number<arr[0]>, number<arr[1]>>{};                     \
+        }                                                                                \
+        else if constexpr(n_ == 3)                                                       \
+        {                                                                                \
+            return ck_tile::tuple<number<arr[0]>, number<arr[1]>, number<arr[2]>>{};     \
+        }                                                                                \
+        else if constexpr(n_ == 4)                                                       \
+        {                                                                                \
+            return ck_tile::                                                             \
+                tuple<number<arr[0]>, number<arr[1]>, number<arr[2]>, number<arr[3]>>{}; \
+        }                                                                                \
+        else if constexpr(n_ == 5)                                                       \
+        {                                                                                \
+            return ck_tile::tuple<number<arr[0]>,                                        \
+                                  number<arr[1]>,                                        \
+                                  number<arr[2]>,                                        \
+                                  number<arr[3]>,                                        \
+                                  number<arr[4]>>{};                                     \
+        }                                                                                \
+        else if constexpr(n_ == 6)                                                       \
+        {                                                                                \
+            return ck_tile::tuple<number<arr[0]>,                                        \
+                                  number<arr[1]>,                                        \
+                                  number<arr[2]>,                                        \
+                                  number<arr[3]>,                                        \
+                                  number<arr[4]>,                                        \
+                                  number<arr[5]>>{};                                     \
+        }                                                                                \
+    }()
+#endif

include/ck_tile/core/numeric/bfloat16.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2024, Advanced Micro Devices, Inc. All rights reserved.
+
+#include "ck_tile/core/config.hpp"
+#include "ck_tile/core/utility/bit_cast.hpp"
+#include "ck_tile/core/numeric/half.hpp"
+#include "ck_tile/core/numeric/integral_constant.hpp"
+#include "ck_tile/core/numeric/numeric.hpp"
+#include <stdint.h>
+
+#pragma once
+
+namespace ck_tile {
+
+enum class bf16_rounding_mode
+{
+    standard = 0, // rtn
+    truncate_with_nan,
+    truncate,
+};
+
+template <bf16_rounding_mode rounding =
+              static_cast<bf16_rounding_mode>(CK_TILE_FLOAT_TO_BFLOAT16_DEFAULT)>
+CK_TILE_HOST_DEVICE constexpr uint16_t float_to_bf16_raw(float f, constant<rounding> = {});
+
+template <bf16_rounding_mode rounding =
+              static_cast<bf16_rounding_mode>(CK_TILE_FLOAT_TO_BFLOAT16_DEFAULT)>
+CK_TILE_HOST_DEVICE constexpr uint16_t double_to_bf16_raw(double f, constant<rounding> = {});
+
+CK_TILE_HOST_DEVICE
+constexpr float bf16_to_float_raw(uint16_t x);
+
+CK_TILE_HOST_DEVICE
+constexpr double bf16_to_double_raw(uint16_t x);
+
+#if CK_TILE_USE_CUSTOM_DATA_TYPE
+// HIP use __hip_bfloat16 as struct
+struct alignas(2) bfloat16_t
+{
+    using raw_type = uint16_t;
+    raw_type data;
+
+    CK_TILE_HOST_DEVICE
+    static constexpr bfloat16_t bit_cast(raw_type x)
+    {
+        bfloat16_t y;
+        y.data = x;
+        return y;
+    }
+
+    // constructor
+    constexpr bfloat16_t() : data() {}
+
+    // construct from float
+    CK_TILE_HOST_DEVICE
+    explicit constexpr bfloat16_t(const float& x) : data(float_to_bf16_raw(x)) {}
+
+    // construct from double
+    CK_TILE_HOST_DEVICE
+    explicit constexpr bfloat16_t(const double& x) : data(double_to_bf16_raw(x)) {}
+
+    // construct from int
+    CK_TILE_HOST_DEVICE
+    explicit constexpr bfloat16_t(const int& x) : data(float_to_bf16_raw(static_cast<float>(x))) {}
+
+    // construct from unsigned int
+    CK_TILE_HOST_DEVICE
+    explicit constexpr bfloat16_t(const unsigned int& x)
+        : data(float_to_bf16_raw(static_cast<float>(x)))
+    {
+    }
+
+    // cast to float
+    CK_TILE_HOST_DEVICE
+    explicit constexpr operator float() const { return bf16_to_float_raw(data); }
+
+    // cast to float
+    CK_TILE_HOST_DEVICE
+    explicit constexpr operator double() const { return bf16_to_double_raw(data); }
+
+    // cast to int
+    CK_TILE_HOST_DEVICE
+    explicit constexpr operator int() const { return static_cast<int>(bf16_to_float_raw(data)); }
+
+    // internal access
+    CK_TILE_HOST_DEVICE
+    constexpr raw_type& get() { return data; }
+
+    CK_TILE_HOST_DEVICE
+    constexpr raw_type get() const { return data; }
+};
+template <typename>
+struct native_t;
+
+template <>
+struct native_t<bfloat16_t>
+{
+    using type = ushort;
+};
+using bf16_t     = bfloat16_t;
+using bf16_raw_t = typename bf16_t::raw_type;
+#else
+using bfloat16_t = ushort;
+using bf16_t     = bfloat16_t;
+using bf16_raw_t = uint16_t;
+#endif
+// round to nearest
+CK_TILE_HOST_DEVICE
+constexpr uint16_t float_to_bf16_rtn_raw(float f)
+{
+    union
+    {
+        float fp32;
+        uint32_t int32;
+    } u = {f};
+    if(~u.int32 & 0x7f800000)
+    {
+        // When the exponent bits are not all 1s, then the value is zero, normal,
+        // or subnormal. We round the bfloat16 mantissa up by adding 0x7FFF, plus
+        // 1 if the least significant bit of the bfloat16 mantissa is 1 (odd).
+        // This causes the bfloat16's mantissa to be incremented by 1 if the 16
+        // least significant bits of the float mantissa are greater than 0x8000,
+        // or if they are equal to 0x8000 and the least significant bit of the
+        // bfloat16 mantissa is 1 (odd). This causes it to be rounded to even when
+        // the lower 16 bits are exactly 0x8000. If the bfloat16 mantissa already
+        // has the value 0x7f, then incrementing it causes it to become 0x00 and
+        // the exponent is incremented by one, which is the next higher FP value
+        // to the unrounded bfloat16 value. When the bfloat16 value is subnormal
+        // with an exponent of 0x00 and a mantissa of 0x7F, it may be rounded up
+        // to a normal value with an exponent of 0x01 and a mantissa of 0x00.
+        // When the bfloat16 value has an exponent of 0xFE and a mantissa of 0x7F,
+        // incrementing it causes it to become an exponent of 0xFF and a mantissa
+        // of 0x00, which is Inf, the next higher value to the unrounded value.
+        u.int32 += 0x7fff + ((u.int32 >> 16) & 1); // Round to nearest, round to even
+    }
+    else if(u.int32 & 0xffff)
+    {
+        // When all of the exponent bits are 1, the value is Inf or NaN.
+        // Inf is indicated by a zero mantissa. NaN is indicated by any nonzero
+        // mantissa bit. Quiet NaN is indicated by the most significant mantissa
+        // bit being 1. Signaling NaN is indicated by the most significant
+        // mantissa bit being 0 but some other bit(s) being 1. If any of the
+        // lower 16 bits of the mantissa are 1, we set the least significant bit
+        // of the bfloat16 mantissa, in order to preserve signaling NaN in case
+        // the bloat16's mantissa bits are all 0.
+        u.int32 |= 0x10000; // Preserve signaling NaN
+    }
+    return uint16_t(u.int32 >> 16);
+}
+
+// Truncate instead of rounding, preserving SNaN
+CK_TILE_HOST_DEVICE
+constexpr uint16_t float_to_bf16_truc_nan_raw(float f)
+{
+    union
+    {
+        float fp32;
+        uint32_t int32;
+    } u = {f};
+    return uint16_t(u.int32 >> 16) | (!(~u.int32 & 0x7f800000) && (u.int32 & 0xffff));
+}
+
+// Fast truncate instead of rounding, RTZ
+CK_TILE_HOST_DEVICE
+constexpr uint16_t float_to_bf16_truc_raw(float f)
+{
+    union
+    {
+        float fp32;
+        uint32_t int32;
+    } u = {f};
+    return uint16_t(u.int32 >> 16);
+}
+
+template <bf16_rounding_mode rounding>
+CK_TILE_HOST_DEVICE constexpr uint16_t float_to_bf16_raw(float f, constant<rounding>)
+{
+    if constexpr(rounding == bf16_rounding_mode::standard)
+        return float_to_bf16_rtn_raw(f);
+    else if constexpr(rounding == bf16_rounding_mode::truncate_with_nan)
+        return float_to_bf16_truc_nan_raw(f);
+    else
+        return float_to_bf16_truc_raw(f);
+}
+
+template <bf16_rounding_mode rounding>
+CK_TILE_HOST_DEVICE constexpr uint16_t double_to_bf16_raw(double f, constant<rounding>)
+{
+    return float_to_bf16_raw(static_cast<float>(f), constant<rounding>{});
+}
+
+CK_TILE_HOST_DEVICE
+constexpr float bf16_to_float_raw(uint16_t x)
+{
+    union
+    {
+        uint32_t int32;
+        float fp32;
+    } u = {uint32_t(x) << 16};
+    return u.fp32;
+}
+
+CK_TILE_HOST_DEVICE
+constexpr double bf16_to_double_raw(uint16_t x)
+{
+    return static_cast<double>(bf16_to_float_raw(x));
+}
+
+template <bf16_rounding_mode rounding =
+              static_cast<bf16_rounding_mode>(CK_TILE_FLOAT_TO_BFLOAT16_DEFAULT)>
+CK_TILE_HOST_DEVICE constexpr bfloat16_t float_to_bf16(float f, constant<rounding> = {})
+{
+    return bit_cast<bfloat16_t>(float_to_bf16_raw(f, constant<rounding>{}));
+}
+
+template <bf16_rounding_mode rounding =
+              static_cast<bf16_rounding_mode>(CK_TILE_FLOAT_TO_BFLOAT16_DEFAULT)>
+CK_TILE_HOST_DEVICE constexpr bfloat16_t double_to_bf16(double f, constant<rounding> = {})
+{
+    return bit_cast<bfloat16_t>(double_to_bf16_raw(f, constant<rounding>{}));
+}
+
+CK_TILE_HOST_DEVICE
+constexpr float bf16_to_float(bfloat16_t x) { return bf16_to_float_raw(bit_cast<uint16_t>(x)); }
+
+CK_TILE_HOST_DEVICE
+constexpr double bf16_to_double(bfloat16_t x) { return static_cast<double>(bf16_to_float_raw(x)); }
+
+template <bf16_rounding_mode rounding =
+              static_cast<bf16_rounding_mode>(CK_TILE_FLOAT_TO_BFLOAT16_DEFAULT)>
+CK_TILE_HOST_DEVICE bfloat16_t constexpr fp16_to_bf16(half_t f, constant<rounding> = {})
+{
+    return bit_cast<bfloat16_t>(float_to_bf16_raw(static_cast<float>(f), constant<rounding>{}));
+}
+
+CK_TILE_HOST_DEVICE
+constexpr half_t bf16_to_fp16(bfloat16_t x) { return static_cast<fp16_t>(static_cast<float>(x)); }
+
+template <class T>
+struct numeric;
+
+template <>
+struct numeric<bfloat16_t>
+{
+    // minimum finite value, or minimum positive normalized value for float
+    CK_TILE_HOST_DEVICE static constexpr bfloat16_t min()
+    {
+        return bit_cast<bfloat16_t>(static_cast<bf16_raw_t>(0x0080));
+    }
+
+    // minumum finite value
+    CK_TILE_HOST_DEVICE static constexpr bfloat16_t lowest()
+    {
+        return bit_cast<bfloat16_t>(static_cast<bf16_raw_t>(0xff7f));
+    }
+
+    // maximum finite value
+    CK_TILE_HOST_DEVICE static constexpr bfloat16_t max()
+    {
+        return bit_cast<bfloat16_t>(static_cast<bf16_raw_t>(0x7f7f));
+    }
+
+    // difference between 1.0 and next value representable by float
+    CK_TILE_HOST_DEVICE static constexpr bfloat16_t epsilon()
+    {
+        return bit_cast<bfloat16_t>(static_cast<bf16_raw_t>(0x1000));
+    }
+
+    // maximum rounding error
+    // maximum rounding error
+    // bin :  f edcba 9876543210
+    // bits:  s eeeeeeee mmmmmmm
+    //        0 01111110 0000000 (0.5)
+    //
+    CK_TILE_HOST_DEVICE static constexpr bfloat16_t round_error()
+    {
+        return bit_cast<bfloat16_t>(static_cast<bf16_raw_t>(0x3f00));
+    }
+
+    // positive infinity value
+    CK_TILE_HOST_DEVICE static constexpr bfloat16_t infinity()
+    {
+        return bit_cast<bfloat16_t>(static_cast<bf16_raw_t>(0x7f80));
+    }
+
+    // quiet NaN
+    CK_TILE_HOST_DEVICE static constexpr bfloat16_t quiet_NaN()
+    {
+        return bit_cast<bfloat16_t>(static_cast<bf16_raw_t>(0x7FFF));
+    }
+
+    // signaling NaN
+    CK_TILE_HOST_DEVICE static constexpr bfloat16_t signaling_NaN()
+    {
+        return bit_cast<bfloat16_t>(static_cast<bf16_raw_t>(0x7FFF));
+    }
+
+    // smallest positive subnormal value
+    CK_TILE_HOST_DEVICE static constexpr bfloat16_t denorm_min()
+    {
+        return bit_cast<bfloat16_t>(static_cast<bf16_raw_t>(0x0001));
+    }
+    CK_TILE_HOST_DEVICE static constexpr bfloat16_t zero()
+    {
+        return bit_cast<bfloat16_t>(static_cast<bf16_raw_t>(0));
+    }
+};
+
+#if CK_TILE_USE_CUSTOM_DATA_TYPE
+CK_TILE_ARITHMETIC_USING_FLOAT(CK_TILE_HOST_DEVICE, bfloat16_t)
+#endif
+
+// math
+CK_TILE_HOST_DEVICE
+bfloat16_t abs(const bfloat16_t& x)
+{
+    return bit_cast<bfloat16_t>(static_cast<bf16_raw_t>(bit_cast<bf16_raw_t>(x) & 0x7fff));
+}
+
+CK_TILE_HOST_DEVICE
+bool isnan(const bfloat16_t& x)
+{
+    uint16_t xx = bit_cast<bf16_raw_t>(x);
+    return (xx & 0x7FFF) > 0x7C00;
+}
+
+CK_TILE_DEVICE
+bfloat16_t sqrt(bfloat16_t x)
+{
+    return static_cast<bfloat16_t>(__builtin_amdgcn_sqrtf(static_cast<float>(x)));
+};
+
+CK_TILE_DEVICE
+bfloat16_t exp(bfloat16_t x) { return static_cast<bfloat16_t>(__expf(static_cast<float>(x))); };
+
+CK_TILE_DEVICE
+bfloat16_t exp2(bfloat16_t x) { return static_cast<bfloat16_t>(exp2f(static_cast<float>(x))); };
+
+CK_TILE_DEVICE
+bfloat16_t log(bfloat16_t x) { return static_cast<bfloat16_t>(__logf(static_cast<float>(x))); };
+
+} // namespace ck_tile

include/ck_tile/core/numeric/float8.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2024, Advanced Micro Devices, Inc. All rights reserved.
+
+#include "ck_tile/core/config.hpp"
+#include "ck_tile/core/utility/bit_cast.hpp"
+#include "ck_tile/core/numeric/numeric.hpp"
+#include "ck_tile/core/utility/random.hpp"
+#include "ck_tile/core/numeric/half.hpp"
+#include "ck_tile/core/numeric/math.hpp"
+#include "ck_tile/core/numeric/integral_constant.hpp"
+#include "ck_tile/core/numeric/numeric.hpp"
+#include <stdint.h>
+#include <type_traits>
+
+#pragma once
+
+namespace ck_tile {
+
+// fp8 rounding modes
+// use standard for rounding to nearest, the faster one
+// use stochastic for stochastic rounding, helps to avoid error accumulation
+enum class fp8_rounding_mode
+{
+    standard = 0,
+    stochastic
+};
+
+/*
+ *                ______________NANOO_________________    |   ______________IEEE________________
+ *                   e4m3               e5m2              |    e4m3                e5m2
+ *      bias :        8                  16               |     7                   15
+ *      inf  :  1.0000.000           1.00000.00           |    N/A              s.11111.00
+ *      Nan  :  1.0000.000           1.00000.00           | s.1111.111          s.11111.{01, 10, 11}
+ *      zero :  0.0000.000           0.00000.00           | s.0000.000          s.00000.00
+ * Max(norm) :  s.1111.111 (240)     s.11111.11(57344)    | s.1111.110(448)     s.11110.11(57344)
+ * Max(snorm):  s.0000.111           s.00000.11           | s.0000.111(448)     s.00000.11(57344)
+ *                0.0068359375         2.288818e-05       |   0.013671875         4.57763671875e-05
+ * Min(norm) :  s.0001.000           s.00001.00           | s.0001.000          s.00001.00
+ *                2^-7(0.00078125)     2^-15(3.05176e-05) |   2^-6(0.015625)      2^-14(6.10352e-05)
+ * Min(snorm):  s.0000.001           s.00000.01           | s.0000.001          s.00000.01
+ *                2^-10(0.00097656)    2^-17(7.629395e-06)|   2^-9(0.001953125)   2^-16(1.52588e-05)
+ */
+
+template <fp8_rounding_mode rounding = static_cast<fp8_rounding_mode>(CK_TILE_FLOAT_TO_FP8_DEFAULT)>
+CK_TILE_HOST_DEVICE uint8_t float_to_fp8_raw(float, constant<rounding> = {});
+
+template <fp8_rounding_mode rounding = static_cast<fp8_rounding_mode>(CK_TILE_FLOAT_TO_FP8_DEFAULT)>
+CK_TILE_HOST_DEVICE uint8_t float_to_bf8_raw(float, constant<rounding> = {});
+
+CK_TILE_HOST_DEVICE float fp8_to_float_raw(uint8_t);
+CK_TILE_HOST_DEVICE float bf8_to_float_raw(uint8_t);
+
+#if CK_TILE_USE_CUSTOM_DATA_TYPE
+struct alignas(1) float8_e4m3_t
+{
+    static constexpr int exponent = 4;
+    static constexpr int mantissa = 3;
+#if defined(__gfx940__) || defined(__gfx941__) || defined(__gfx942__)
+    static constexpr int bias = 1 << (exponent - 1); // NANOO
+#else
+    static constexpr int bias = (1 << (exponent - 1)) - 1; // IEEE
+#endif
+    using raw_type = uint8_t;
+    raw_type data;
+
+    CK_TILE_HOST_DEVICE
+    static constexpr float8_e4m3_t bit_cast(raw_type x)
+    {
+        float8_e4m3_t y;
+        y.data = x;
+        return y;
+    }
+
+    // constructor
+    constexpr float8_e4m3_t() : data() {}
+
+    // construct from float
+    CK_TILE_HOST_DEVICE
+    explicit constexpr float8_e4m3_t(const float& x) : data(float_to_fp8_raw(x)) {}
+
+    // construct from int
+    CK_TILE_HOST_DEVICE
+    explicit constexpr float8_e4m3_t(const int& x) : data(float_to_fp8_raw(static_cast<float>(x)))
+    {
+    }
+
+    // construct from unsigned int
+    CK_TILE_HOST_DEVICE
+    explicit constexpr float8_e4m3_t(const unsigned int& x)
+        : data(float_to_fp8_raw(static_cast<float>(x)))
+    {
+    }
+
+    // cast to float
+    CK_TILE_HOST_DEVICE
+    explicit constexpr operator float() const { return fp8_to_float_raw(data); }
+
+    // cast to int
+    CK_TILE_HOST_DEVICE
+    explicit constexpr operator int() const { return static_cast<int>(fp8_to_float_raw(data)); }
+
+    // internal access
+    CK_TILE_HOST_DEVICE
+    constexpr raw_type& get() { return data; }
+
+    CK_TILE_HOST_DEVICE
+    constexpr raw_type get() const { return data; }
+};
+using fp8_t     = float8_e4m3_t;
+using fp8_raw_t = typename fp8_t::raw_type;
+
+struct alignas(1) float8_e5m2_t
+{
+    static constexpr int exponent = 5;
+    static constexpr int mantissa = 2;
+#if defined(__gfx940__) || defined(__gfx941__) || defined(__gfx942__)
+    static constexpr int bias = 1 << (exponent - 1); // NANOO
+#else
+    static constexpr int bias = (1 << (exponent - 1)) - 1; // IEEE
+#endif
+    using raw_type = uint8_t;
+    raw_type data;
+
+    CK_TILE_HOST_DEVICE
+    static constexpr float8_e5m2_t bit_cast(raw_type x)
+    {
+        float8_e5m2_t y;
+        y.data = x;
+        return y;
+    }
+
+    // constructor
+    constexpr float8_e5m2_t() : data() {}
+
+    // construct from float
+    CK_TILE_HOST_DEVICE
+    explicit constexpr float8_e5m2_t(const float& x) : data(float_to_bf8_raw(x)) {}
+
+    // construct from int
+    CK_TILE_HOST_DEVICE
+    explicit constexpr float8_e5m2_t(const int& x) : data(float_to_bf8_raw(static_cast<float>(x)))
+    {
+    }
+
+    // construct from unsigned int
+    CK_TILE_HOST_DEVICE
+    explicit constexpr float8_e5m2_t(const unsigned int& x)
+        : data(float_to_bf8_raw(static_cast<float>(x)))
+    {
+    }
+
+    // cast to float
+    CK_TILE_HOST_DEVICE
+    explicit constexpr operator float() const { return bf8_to_float_raw(data); }
+
+    // cast to int
+    CK_TILE_HOST_DEVICE
+    explicit constexpr operator int() const { return static_cast<int>(bf8_to_float_raw(data)); }
+
+    // internal access
+    CK_TILE_HOST_DEVICE
+    constexpr raw_type& get() { return data; }
+
+    CK_TILE_HOST_DEVICE
+    constexpr raw_type get() const { return data; }
+};
+using bf8_t     = float8_e5m2_t;
+using bf8_raw_t = typename bf8_t::raw_type;
+
+template <typename>
+struct native_t;
+
+template <>
+struct native_t<fp8_t>
+{
+    using type = _BitInt(8);
+};
+
+template <>
+struct native_t<bf8_t>
+{
+    using type = unsigned _BitInt(8);
+};
+
+#else
+using fp8_t     = _BitInt(8);
+using fp8_raw_t = uint8_t;
+using bf8_t     = unsigned _BitInt(8);
+using bf8_raw_t = uint8_t;
+#endif
+
+// below is sw fp8 conversion, not utilizing hw instruction
+namespace impl {
+
+template <typename X, typename Y, bool negative_zero_nan, bool clip, bool stoch>
+CK_TILE_HOST_DEVICE Y run_cast_to_f8(X x, uint32_t rng)
+{
+    // fp8/bf8 exponent/mantissa layout
+    constexpr int out_exp  = numeric_traits<Y>::exp;
+    constexpr int out_mant = numeric_traits<Y>::mant;
+
+    // original type exponent/mantissa layout
+    constexpr int in_exp  = numeric_traits<X>::exp;
+    constexpr int in_mant = numeric_traits<X>::mant;
+
+    int exponent, bias;
+    uint32_t head, mantissa, sign;
+    // nan code is same for float and half
+#if CK_TILE_USE_CUSTOM_DATA_TYPE
+    constexpr Y nan_code =
+        numeric<Y>::quiet_NaN(); // __builtin_bit_cast(Y, static_cast<uint8_t>(0x80));
+#else
+    constexpr Y nan_code             = 0x80;
+#endif
+
+    constexpr uint32_t nan_mask = numeric_traits<X>::nan_mask;
+
+    // convert to bitwise
+    using T_bitwise     = typename numeric_traits<X>::bitwise_type;
+    T_bitwise x_bitwise = *(reinterpret_cast<T_bitwise*>(&x));
+
+    // unpack the input, depends on datatype
+    head     = x_bitwise & numeric_traits<X>::head_mask;
+    mantissa = x_bitwise & numeric_traits<X>::mant_mask;
+    exponent = (head >> in_mant) & numeric_traits<X>::exp_mask;
+    sign     = head >> (in_exp + in_mant);
+    bias     = numeric_traits<X>::bias;
+
+    uint32_t signed_inf   = (sign << (in_exp + in_mant)) + (((1 << in_exp) - 1) << in_mant);
+    uint32_t drop_mask    = (1 << (in_mant - out_mant)) - 1;
+    constexpr int max_exp = (1 << out_exp) - (negative_zero_nan ? 1 : 2);
+
+    if constexpr(negative_zero_nan)
+    {
+        if((x_bitwise & nan_mask) == nan_mask)
+            return nan_code;
+    }
+    else
+    {
+        if((x_bitwise & nan_mask) == nan_mask)
+            return signed_inf + (mantissa != 0 ? 1 : 0);
+    }
+
+    // check if x is 0.0
+    if(x_bitwise == 0)
+        return __builtin_bit_cast(Y, static_cast<uint8_t>(0));
+
+    // First need to check if it is normal or denorm as there is a difference of implict 1
+    // Then need to adjust the exponent to align with the F8 exponent, in the meanwhile, shift
+    // The mantissa. Then for stochastic rounding, add rng to mantissa and truncate. And for
+    // RNE, no need to add rng. Then probably need to check whether there is carry and adjust
+    // exponent and mantissa again3
+
+    // For IEEE bias mode, the bias is 2^(k-1)-1 where k is the width of exponent bits
+    const int out_bias                  = (1 << (out_exp - 1)) - 1 + (negative_zero_nan ? 1 : 0);
+    const int out_denormal_act_exponent = 1 - out_bias; // actual exponent of f8 denormal
+    // act_exponent is the actual exponent of fp32/fp16 (after subtracting bias)
+    // out_exponent is the converted f8 exponent with bias encoding
+    // exponent_diff is the diff between fp32/fp16 exponent and f8 exponent,
+    // the difference needs to be adjusted and mantissa shifted
+    int act_exponent, out_exponent, exponent_diff;
+
+    if(exponent == 0)
+    { // fp32/fp16 is in denormal.
+        /* fp32 denormal is below 2^-127 so it is usually not a concern here, we mostly concern fp16
+here. In this case, f8 is usually in denormal. But there could be exceptions. fp16 denormal has
+exponent bias 15 while bf8 with NANOO has exponent bias 16. It means that there are some numbers in
+fp16 denormal but they are bf8 (NANOO) normals - smallest bf8 (NANOO) normal is 2^-15. fp16 numbers
+where exponent==0 (actual exponent -14) and highest bit of mantissa is 1 are bf8 (NANOO) normal.
+In this case, the fp16 mantissa should be shift left by 1 */
+        act_exponent  = exponent - bias + 1;
+        exponent_diff = out_denormal_act_exponent -
+                        act_exponent; // actual exponent is exponent-bias+1 as it is denormal
+    }
+    else
+    { // fp32/fp16 is normal with implicit 1
+        act_exponent = exponent - bias;
+        if(act_exponent <= out_denormal_act_exponent)
+        {
+            /* This is the case where fp32/fp16 is normal but it is in f8 denormal range.
+   For example fp8 nanoo mode, denormal exponent is -7, but if the fp32/fp16
+   actual exponent is -7, it is actually larger due to the implict 1,
+   Therefore it needs to be adjust to -6 and mantissa shift right by 1.
+   So for fp32/fp16, exponent -8 is the cut point to convert to fp8 nanoo */
+            exponent_diff = out_denormal_act_exponent - act_exponent;
+        }
+        else
+        { // both fp32/fp16 and f8 are in normal range
+            exponent_diff =
+                0; // exponent_diff=0 does not mean there is no difference for this case,
+            // act_exponent could be larger. Just that it does not need shift mantissa
+        }
+        mantissa += (1 << in_mant); // Add the implicit 1 into mantissa
+    }
+
+    bool midpoint = (mantissa & ((1 << (in_mant - out_mant + exponent_diff)) - 1)) ==
+                    (1 << (in_mant - out_mant + exponent_diff - 1));
+    /* This part is a bit tricky. The judgment of whether it is a tie needs to be done before we
+ shift right as shift right could rip off some residual part and make something not midpoint look
+ like midpoint. For example, the fp16 number 0x1002 (0 00100 0000000010), it is larger than
+ midpoint, but after shift right by 4 bits, it would look like midpoint. */
+
+    if(exponent_diff > 0)
+        mantissa >>= exponent_diff;
+    else if(exponent_diff == -1)
+        mantissa <<= -exponent_diff;
+    bool implicit_one = mantissa & (1 << in_mant);
+    // if there is no implict 1, it  means the f8 is denormal and need to adjust to denorm exponent
+    out_exponent =
+        (act_exponent + exponent_diff) /*actual f8 exponent*/ + out_bias - (implicit_one ? 0 : 1);
+
+    // Now we have the exponent and mantissa adjusted
+    bool odd =
+        mantissa &
+        (1 << (in_mant - out_mant)); // if the least significant bit that is not truncated is 1
+    mantissa += (stoch ? rng : (midpoint ? (odd ? mantissa : mantissa - 1) : mantissa)) & drop_mask;
+
+    // Now we deal with overflow
+    if(out_exponent == 0)
+    {
+        if((1 << in_mant) & mantissa)
+        {
+            out_exponent = 1; // denormal overflow to become normal, promote exponent
+            // No need to make 1 implicit now as it will be addressed later
+        }
+    }
+    else
+    {
+        if((1 << (in_mant + 1)) & mantissa)
+        {
+            mantissa >>= 1;
+            out_exponent++;
+            // No need to make 1 implicit now as it will be addressed later
+        }
+    }
+
+    mantissa >>= (in_mant - out_mant);
+
+    if(out_exponent > max_exp)
+    {
+        if(clip)
+        {
+            mantissa     = (1 << out_mant) - 1;
+            out_exponent = max_exp;
+        }
+        else
+        {
+            return __builtin_bit_cast(Y, static_cast<uint8_t>(signed_inf));
+        }
+    }
+
+    // check if x is 0.0 or -0.0
+    if(out_exponent == 0 && mantissa == 0)
+        return __builtin_bit_cast(
+            Y, static_cast<uint8_t>(negative_zero_nan ? 0 : (sign << (out_exp + out_mant))));
+    mantissa &= (1 << out_mant) - 1;
+    return __builtin_bit_cast(Y,
+                              static_cast<uint8_t>((sign << (out_exp + out_mant)) |
+                                                   (out_exponent << out_mant) | mantissa));
+}
+
+template <typename X, typename Y, bool negative_zero_nan>
+CK_TILE_HOST_DEVICE Y run_cast_from_f8(X x)
+{
+    // fp8/bf8 exponent/mantissa layout
+    constexpr int in_exp  = numeric_traits<X>::exp;
+    constexpr int in_mant = numeric_traits<X>::mant;
+
+    // resulting type exponent/mantissa layout
+    constexpr int out_exp  = numeric_traits<Y>::exp;
+    constexpr int out_mant = numeric_traits<Y>::mant;
+    uint8_t x_raw          = __builtin_bit_cast(uint8_t, x);
+
+    // prepare the codes
+    constexpr uint8_t nan_code = 0x80;
+    Y Inf, NegInf, NaN, Neg0;
+    using T_bitwise = typename numeric_traits<Y>::bitwise_type;
+
+    constexpr T_bitwise Inf_bitwise    = numeric_traits<Y>::Inf;
+    constexpr T_bitwise NegInf_bitwise = numeric_traits<Y>::NegInf;
+    constexpr T_bitwise NaN_bitwise    = numeric_traits<Y>::NaN;
+    constexpr T_bitwise Neg0_bitwise   = numeric_traits<Y>::Neg0;
+
+    Inf    = *(reinterpret_cast<const Y*>(&Inf_bitwise));
+    NegInf = *(reinterpret_cast<const Y*>(&NegInf_bitwise));
+    NaN    = *(reinterpret_cast<const Y*>(&NaN_bitwise));
+    Neg0   = *(reinterpret_cast<const Y*>(&Neg0_bitwise));
+
+    // check if x is 0.0
+    if(x_raw == 0)
+        return static_cast<Y>(0);
+
+    // unpack the input
+    uint32_t sign     = x_raw >> (in_exp + in_mant);
+    uint32_t mantissa = x_raw & ((1 << in_mant) - 1);
+    int exponent      = (x_raw & 0x7F) >> in_mant;
+
+    constexpr int exp_low_cutoff =
+        (1 << (out_exp - 1)) - (1 << (in_exp - 1)) + 1 - (negative_zero_nan ? 1 : 0);
+    T_bitwise retval;
+
+    if constexpr(negative_zero_nan)
+    {
+        if(x_raw == nan_code)
+            return NaN;
+    }
+    else
+    {
+        if(x_raw == nan_code)
+            return Neg0;
+        if(exponent == ((1 << in_exp) - 1))
+            return (mantissa == 0) ? (sign ? NegInf : Inf) : NaN;
+    }
+
+    if((numeric_traits<Y>::mant == 10) && (numeric_traits<X>::mant == 2) && !negative_zero_nan)
+    {
+        retval = x_raw;
+        retval <<= 8;
+        return *(reinterpret_cast<const Y*>(&retval));
+    }
+
+    // subnormal input
+    if(exponent == 0)
+    {
+        // guaranteed mantissa!=0 since cases 0x0 and 0x80 are handled above
+        int sh = 1 + clz(mantissa) - (32 - in_mant);
+        mantissa <<= sh;
+        exponent += 1 - sh;
+        mantissa &= ((1 << in_mant) - 1);
+    }
+    exponent += exp_low_cutoff - 1;
+    mantissa <<= out_mant - in_mant;
+
+    // subnormal output (occurs when T=half, we=5, negative_zero_nan=true)
+    if(exponent <= 0)
+    {
+        mantissa |= 1 << out_mant;
+        mantissa >>= 1 - exponent;
+        exponent = 0;
+    }
+
+    retval = (sign << (out_exp + out_mant)) | (exponent << out_mant) | mantissa;
+    return *(reinterpret_cast<const Y*>(&retval));
+}
+
+template <typename X, typename Y, bool negative_zero_nan, bool clip, bool stoch>
+CK_TILE_HOST_DEVICE Y cast_to_f8(X x, uint32_t rng)
+{
+    // check datatypes
+    constexpr bool is_half  = std::is_same<X, half_t>::value;
+    constexpr bool is_float = std::is_same<X, float>::value;
+    static_assert(is_half || is_float, "Only half and float can be casted.");
+
+    return run_cast_to_f8<X, Y, negative_zero_nan, clip, stoch>(x, rng);
+}
+
+template <typename X, typename Y, bool negative_zero_nan>
+CK_TILE_HOST_DEVICE Y cast_from_f8(X x)
+{
+    // check datatype
+    constexpr bool is_half  = std::is_same<Y, half_t>::value;
+    constexpr bool is_float = std::is_same<Y, float>::value;
+    static_assert(is_half || is_float, "only half and float are supported.");
+
+    return run_cast_from_f8<X, Y, negative_zero_nan>(x);
+}
+} // namespace impl
+
+CK_TILE_HOST_DEVICE fp8_raw_t float_to_fp8_sr_raw(float x)
+{
+    constexpr int seed = 42;
+    uint32_t rng       = prand_generator_t<float, seed>{}(reinterpret_cast<uintptr_t>(&x), x);
+#if defined(__gfx940__) || defined(__gfx941__) || defined(__gfx942__)
+    float max_fp8 = 240.0f;
+    x             = x > max_fp8 ? max_fp8 : (x < -max_fp8 ? -max_fp8 : x);
+    union
+    {
+        float fval;
+        uint32_t i32val;
+        uint8_t i8val[4]; // not endian independent
+    } val;
+    val.fval      = x;
+    uint32_t ival = 0;
+    ival          = __builtin_amdgcn_cvt_sr_fp8_f32(val.fval, rng, ival, 0); // 0 pos
+    val.i32val    = ival;
+    return val.i8val[0]; // little endian
+#else
+    constexpr bool negative_zero_nan = true;
+    constexpr bool clip              = true;
+    constexpr fp8_rounding_mode rm   = fp8_rounding_mode::stochastic;
+    return bit_cast<fp8_raw_t>(impl::cast_to_f8<float,
+                                                fp8_t,
+                                                negative_zero_nan,
+                                                clip,
+                                                (rm == fp8_rounding_mode::stochastic)>(x, rng));
+#endif
+}
+
+CK_TILE_HOST_DEVICE bf8_raw_t float_to_bf8_sr_raw(float x)
+{
+    constexpr int seed = 42;
+    uint32_t rng       = prand_generator_t<float, seed>{}(reinterpret_cast<uintptr_t>(&x), x);
+#if defined(__gfx940__) || defined(__gfx941__) || defined(__gfx942__)
+    union
+    {
+        float fval;
+        uint32_t i32val;
+        uint8_t i8val[4]; // not endian independent
+    } val;
+    val.fval      = x;
+    uint32_t ival = 0;
+    ival          = __builtin_amdgcn_cvt_sr_bf8_f32(val.fval, rng, ival, 0); // 0 pos
+    val.i32val    = ival;
+    return val.i8val[0]; // little endian
+#else
+    constexpr bool negative_zero_nan = true;
+    constexpr bool clip              = true;
+    constexpr fp8_rounding_mode rm   = fp8_rounding_mode::stochastic;
+    return bit_cast<bf8_raw_t>(impl::cast_to_f8<float,
+                                                bf8_t,
+                                                negative_zero_nan,
+                                                clip,
+                                                (rm == fp8_rounding_mode::stochastic)>(x, rng));
+#endif
+}
+
+CK_TILE_HOST_DEVICE fp8_raw_t float_to_fp8_rtn_raw(float x)
+{
+#if defined(__gfx940__) || defined(__gfx941__) || defined(__gfx942__)
+    float max_fp8 = 240.0f;
+    x             = x > max_fp8 ? max_fp8 : (x < -max_fp8 ? -max_fp8 : x);
+    union
+    {
+        float fval;
+        uint32_t i32val;
+        uint8_t i8val[4]; // not endian independent
+    } val;
+    val.fval      = x;
+    uint32_t ival = 0;
+    ival       = __builtin_amdgcn_cvt_pk_fp8_f32(val.fval, val.fval, ival, false); // false -> WORD0
+    val.i32val = ival;
+    return val.i8val[0];
+#else
+    constexpr bool negative_zero_nan = true;
+    constexpr bool clip              = true;
+    constexpr fp8_rounding_mode rm   = fp8_rounding_mode::standard;
+    constexpr uint32_t rng           = 0;
+    return bit_cast<fp8_raw_t>(impl::cast_to_f8<float,
+                                                fp8_t,
+                                                negative_zero_nan,
+                                                clip,
+                                                (rm == fp8_rounding_mode::stochastic)>(x, rng));
+#endif
+}
+CK_TILE_HOST_DEVICE bf8_raw_t float_to_bf8_rtn_raw(float x)
+{
+#if defined(__gfx940__) || defined(__gfx941__) || defined(__gfx942__)
+    union
+    {
+        float fval;
+        uint32_t i32val;
+        uint8_t i8val[4]; // not endian independent
+    } val;
+    val.fval      = x;
+    uint32_t ival = 0;
+    ival       = __builtin_amdgcn_cvt_pk_bf8_f32(val.fval, val.fval, ival, false); // false -> WORD0
+    val.i32val = ival;
+    return val.i8val[0];
+#else
+    constexpr bool negative_zero_nan = true;
+    constexpr bool clip              = true;
+    constexpr fp8_rounding_mode rm   = fp8_rounding_mode::standard;
+    constexpr uint32_t rng           = 0;
+    return bit_cast<bf8_raw_t>(impl::cast_to_f8<float,
+                                                bf8_t,
+                                                negative_zero_nan,
+                                                clip,
+                                                (rm == fp8_rounding_mode::stochastic)>(x, rng));
+#endif
+}
+
+// clang-format off
+template<fp8_rounding_mode rounding>
+CK_TILE_HOST_DEVICE fp8_raw_t float_to_fp8_raw(float x, constant<rounding>)
+{
+    if      constexpr (rounding == fp8_rounding_mode::standard)   return float_to_fp8_rtn_raw(x);
+    else if constexpr (rounding == fp8_rounding_mode::stochastic) return float_to_fp8_sr_raw(x);
+    else return fp8_raw_t{0};
+}
+
+template<fp8_rounding_mode rounding>
+CK_TILE_HOST_DEVICE bf8_raw_t float_to_bf8_raw(float x, constant<rounding>)
+{
+    if      constexpr (rounding == fp8_rounding_mode::standard)   return float_to_bf8_rtn_raw(x);
+    else if constexpr (rounding == fp8_rounding_mode::stochastic) return float_to_bf8_sr_raw(x);
+    else return bf8_raw_t{0};
+}
+
+CK_TILE_HOST_DEVICE float fp8_to_float_raw(fp8_raw_t x)
+{
+#if defined(__gfx940__) || defined(__gfx941__) || defined(__gfx942__)
+    float fval;
+    uint32_t i32val = static_cast<uint32_t>(x);
+    fval            = __builtin_amdgcn_cvt_f32_fp8(i32val, 0);
+    // asm volatile("v_cvt_f32_fp8 %0, %1 src0_sel:BYTE_0" : "=v"(fval) : "v"(i32val));
+    return fval;
+#else
+    constexpr bool negative_zero_nan = true;
+    return impl::cast_from_f8<fp8_t, float, negative_zero_nan>(bit_cast<fp8_t>(x));
+#endif
+}
+
+CK_TILE_HOST_DEVICE float bf8_to_float_raw(bf8_raw_t x)
+{
+#if defined(__gfx940__) || defined(__gfx941__) || defined(__gfx942__)
+    float fval;
+    uint32_t i32val = static_cast<uint32_t>(x);
+    fval            = __builtin_amdgcn_cvt_f32_bf8(i32val, 0);
+    // asm volatile("v_cvt_f32_bf8 %0, %1 src0_sel:BYTE_0" : "=v"(fval) : "v"(i32val));
+    return fval;
+#else
+    constexpr bool negative_zero_nan = true;
+    return impl::cast_from_f8<bf8_t, float, negative_zero_nan>(bit_cast<bf8_t>(x));
+#endif
+}
+
+template<fp8_rounding_mode rounding = static_cast<fp8_rounding_mode>(CK_TILE_FLOAT_TO_FP8_DEFAULT)>
+CK_TILE_HOST_DEVICE fp8_t float_to_fp8(float x, constant<rounding> = {})
+{
+    return bit_cast<fp8_t>(float_to_fp8_raw(x, constant<rounding>{}));
+}
+
+template<fp8_rounding_mode rounding = static_cast<fp8_rounding_mode>(CK_TILE_FLOAT_TO_FP8_DEFAULT)>
+CK_TILE_HOST_DEVICE bf8_t float_to_bf8(float x, constant<rounding> = {})
+{
+    return bit_cast<bf8_t>(float_to_bf8_raw(x, constant<rounding>{}));
+}
+
+CK_TILE_HOST_DEVICE float fp8_to_float(fp8_t x)
+{
+    return fp8_to_float_raw(bit_cast<fp8_raw_t>(x));
+}
+
+CK_TILE_HOST_DEVICE float bf8_to_float(bf8_t x)
+{
+    return bf8_to_float_raw(bit_cast<bf8_raw_t>(x));
+}
+
+// clang-format on
+
+template <typename T>
+struct numeric_traits;
+
+template <>
+struct numeric_traits<fp8_t>
+{
+    static constexpr int exp  = 4;
+    static constexpr int mant = 3;
+#if defined(__gfx940__) || defined(__gfx941__) || defined(__gfx942__)
+    static constexpr int bias = 8;
+#else
+    static constexpr int bias = 7;
+#endif
+};
+
+template <>
+struct numeric_traits<bf8_t>
+{
+    static constexpr int exp  = 5;
+    static constexpr int mant = 2;
+#if defined(__gfx940__) || defined(__gfx941__) || defined(__gfx942__)
+    static constexpr int bias = 16;
+#else
+    static constexpr int bias = 15; // IEEE
+#endif
+};
+
+template <class T>
+struct numeric;
+
+template <>
+struct numeric<fp8_t>
+{
+    // minimum finite value, or minimum positive normalized value for float
+    CK_TILE_HOST_DEVICE static constexpr fp8_t min()
+    {
+        return bit_cast<fp8_t>(static_cast<fp8_raw_t>(0x08));
+    }
+
+    // minumum finite value
+    CK_TILE_HOST_DEVICE static constexpr fp8_t lowest()
+    {
+        return bit_cast<fp8_t>(static_cast<fp8_raw_t>(0xff));
+    }
+
+    // maximum finite value
+    CK_TILE_HOST_DEVICE static constexpr fp8_t max()
+    {
+        return bit_cast<fp8_t>(static_cast<fp8_raw_t>(0x7f));
+    }
+
+    // difference between 1.0 and next value representable by float
+    CK_TILE_HOST_DEVICE static constexpr fp8_t epsilon()
+    {
+        return bit_cast<fp8_t>(static_cast<fp8_raw_t>(0x20));
+    }
+
+    // maximum rounding error
+    // bin :  7 6543 210
+    // bits:  s eeee mmm
+    //        0 0110 000 (0.5)
+    //
+    CK_TILE_HOST_DEVICE static constexpr fp8_t round_error()
+    {
+        return bit_cast<fp8_t>(static_cast<fp8_raw_t>(0x30));
+    }
+
+    // positive infinity value
+    CK_TILE_HOST_DEVICE static constexpr fp8_t infinity()
+    {
+        return bit_cast<fp8_t>(static_cast<fp8_raw_t>(0x80));
+    }
+
+    // quiet NaN
+    CK_TILE_HOST_DEVICE static constexpr fp8_t quiet_NaN()
+    {
+        return bit_cast<fp8_t>(static_cast<fp8_raw_t>(0x80));
+    }
+
+    // signaling NaN
+    CK_TILE_HOST_DEVICE static constexpr fp8_t signaling_NaN()
+    {
+        return bit_cast<fp8_t>(static_cast<fp8_raw_t>(0x80));
+    }
+
+    // smallest positive subnormal value
+    CK_TILE_HOST_DEVICE static constexpr fp8_t denorm_min()
+    {
+        return bit_cast<fp8_t>(static_cast<fp8_raw_t>(0x01));
+    }
+
+    CK_TILE_HOST_DEVICE static constexpr fp8_t zero()
+    {
+        return bit_cast<fp8_t>(static_cast<fp8_raw_t>(0));
+    }
+};
+
+template <>
+struct numeric<bf8_t>
+{
+    // minimum finite value, or minimum positive normalized value for float
+    CK_TILE_HOST_DEVICE static constexpr bf8_t min()
+    {
+        return bit_cast<bf8_t>(static_cast<bf8_raw_t>(0x04));
+    }
+
+    // minumum finite value
+    CK_TILE_HOST_DEVICE static constexpr bf8_t lowest()
+    {
+        return bit_cast<bf8_t>(static_cast<bf8_raw_t>(0xff));
+    }
+
+    // maximum finite value
+    CK_TILE_HOST_DEVICE static constexpr bf8_t max()
+    {
+        return bit_cast<bf8_t>(static_cast<bf8_raw_t>(0x7f));
+    }
+
+    // difference between 1.0 and next value representable by float
+    CK_TILE_HOST_DEVICE static constexpr bf8_t epsilon()
+    {
+        return bit_cast<bf8_t>(static_cast<bf8_raw_t>(0x34));
+    }
+
+    // maximum rounding error
+    // bin :  7 65432 10
+    // bits:  s eeeee mm
+    //        0 01110 00 (0.5)
+    //
+    CK_TILE_HOST_DEVICE static constexpr bf8_t round_error()
+    {
+        return bit_cast<bf8_t>(static_cast<bf8_raw_t>(0x38));
+    }
+
+    // positive infinity value
+    CK_TILE_HOST_DEVICE static constexpr bf8_t infinity()
+    {
+        return bit_cast<bf8_t>(static_cast<bf8_raw_t>(0x80));
+    }
+
+    // quiet NaN
+    CK_TILE_HOST_DEVICE static constexpr bf8_t quiet_NaN()
+    {
+        return bit_cast<bf8_t>(static_cast<bf8_raw_t>(0x80));
+    }
+
+    // signaling NaN
+    CK_TILE_HOST_DEVICE static constexpr bf8_t signaling_NaN()
+    {
+        return bit_cast<bf8_t>(static_cast<bf8_raw_t>(0x80));
+    }
+
+    // smallest positive subnormal value
+    CK_TILE_HOST_DEVICE static constexpr bf8_t denorm_min()
+    {
+        return bit_cast<bf8_t>(static_cast<bf8_raw_t>(0x01));
+    }
+
+    CK_TILE_HOST_DEVICE static constexpr bf8_t zero()
+    {
+        return bit_cast<bf8_t>(static_cast<bf8_raw_t>(0));
+    }
+};
+
+#if CK_TILE_USE_CUSTOM_DATA_TYPE
+CK_TILE_ARITHMETIC_USING_FLOAT(CK_TILE_HOST_DEVICE, fp8_t)
+CK_TILE_ARITHMETIC_USING_FLOAT(CK_TILE_HOST_DEVICE, bf8_t)
+#endif
+
+// math
+CK_TILE_HOST_DEVICE
+fp8_t abs(const fp8_t& x)
+{
+    return bit_cast<fp8_t>(static_cast<fp8_raw_t>(bit_cast<fp8_raw_t>(x) & 0x7f));
+}
+
+CK_TILE_HOST_DEVICE
+bool isnan(const fp8_t& x)
+{
+    uint8_t xx = bit_cast<fp8_raw_t>(x);
+    return xx == 0x80; // TODO: NANOO
+}
+
+CK_TILE_DEVICE
+fp8_t sqrt(fp8_t x) { return static_cast<fp8_t>(__builtin_amdgcn_sqrtf(static_cast<float>(x))); };
+
+CK_TILE_DEVICE
+fp8_t exp(fp8_t x) { return static_cast<fp8_t>(__expf(static_cast<float>(x))); };
+
+CK_TILE_DEVICE
+fp8_t exp2(fp8_t x) { return static_cast<fp8_t>(exp2f(static_cast<float>(x))); };
+
+CK_TILE_DEVICE
+fp8_t log(fp8_t x) { return static_cast<fp8_t>(__logf(static_cast<float>(x))); };
+
+CK_TILE_HOST_DEVICE
+bf8_t abs(const bf8_t& x)
+{
+    return bit_cast<bf8_t>(static_cast<fp8_raw_t>(bit_cast<bf8_raw_t>(x) & 0x7f));
+}
+
+CK_TILE_HOST_DEVICE
+bool isnan(const bf8_t& x)
+{
+    uint8_t xx = bit_cast<bf8_raw_t>(x);
+    return xx == 0x80; // TODO: NANOO
+}
+
+CK_TILE_DEVICE
+bf8_t sqrt(bf8_t x) { return static_cast<bf8_t>(__builtin_amdgcn_sqrtf(static_cast<float>(x))); };
+
+CK_TILE_DEVICE
+bf8_t exp(bf8_t x) { return static_cast<bf8_t>(__expf(static_cast<float>(x))); };
+
+CK_TILE_DEVICE
+bf8_t exp2(bf8_t x) { return static_cast<bf8_t>(exp2f(static_cast<float>(x))); };
+
+CK_TILE_DEVICE
+bf8_t log(bf8_t x) { return static_cast<bf8_t>(__logf(static_cast<float>(x))); };
+
+} // namespace ck_tile

include/ck_tile/core/numeric/half.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2024, Advanced Micro Devices, Inc. All rights reserved.
+
+#include "ck_tile/core/config.hpp"
+#include "ck_tile/core/utility/bit_cast.hpp"
+#include "ck_tile/core/numeric/numeric.hpp"
+#include <hip/hip_fp16.h>
+
+#pragma once
+
+namespace ck_tile {
+
+using fp16_hip_t = _Float16; // most of hip internal function use this type
+using fp16_raw_t = uint16_t;
+
+CK_TILE_HOST_DEVICE
+constexpr float fp16_to_float_hip(const fp16_hip_t& x);
+
+CK_TILE_HOST_DEVICE
+constexpr double fp16_to_double_hip(const fp16_hip_t& x);
+
+CK_TILE_HOST_DEVICE
+constexpr fp16_hip_t float_to_fp16_hip(const float& x);
+
+CK_TILE_HOST_DEVICE
+constexpr fp16_hip_t double_to_fp16_hip(const double& x);
+
+#if CK_TILE_USE_CUSTOM_DATA_TYPE
+// HIP use fp16_hip_t as interchangable data type for float16
+struct alignas(2) half_t
+{
+    using raw_type = fp16_raw_t;
+    raw_type data;
+
+    CK_TILE_HOST_DEVICE
+    static constexpr half_t bit_cast(raw_type x)
+    {
+        half_t y;
+        y.data = x;
+        return y;
+    }
+
+    CK_TILE_HOST_DEVICE
+    constexpr fp16_hip_t to_fp16() const { return ck_tile::bit_cast<fp16_hip_t>(data); }
+
+    // constructor
+    constexpr half_t() : data{} {}
+
+    // construct from HIP half
+    CK_TILE_HOST_DEVICE
+    explicit constexpr half_t(const fp16_hip_t& x) : data(ck_tile::bit_cast<raw_type>(x)) {}
+
+    // construct from float
+    CK_TILE_HOST_DEVICE
+    explicit constexpr half_t(const float& x) : half_t(float_to_fp16_hip(x)) {}
+
+    // construct from double
+    CK_TILE_HOST_DEVICE
+    explicit constexpr half_t(const double& x) : half_t(double_to_fp16_hip(x)) {}
+
+    // construct from int
+    CK_TILE_HOST_DEVICE
+    explicit constexpr half_t(const int& x) : half_t(static_cast<fp16_hip_t>(__int2half_rn(x))) {}
+
+    // construct from unsigned int
+    CK_TILE_HOST_DEVICE
+    explicit constexpr half_t(const unsigned int& x)
+        : half_t(static_cast<fp16_hip_t>(__uint2half_rn(x)))
+    {
+    }
+
+    // cast to float
+    CK_TILE_HOST_DEVICE
+    explicit constexpr operator float() const { return fp16_to_float_hip(to_fp16()); }
+
+    // cast to double
+    CK_TILE_HOST_DEVICE
+    explicit constexpr operator double() const { return fp16_to_double_hip(to_fp16()); }
+
+    // cast to int
+    CK_TILE_HOST_DEVICE
+    explicit constexpr operator int() const
+    {
+        return static_cast<int>(fp16_to_float_hip(to_fp16()));
+    }
+
+    CK_TILE_HOST_DEVICE
+    explicit constexpr operator fp16_hip_t() const { return ck_tile::bit_cast<fp16_hip_t>(data); }
+
+    // internal access
+    CK_TILE_HOST_DEVICE
+    constexpr raw_type& get() { return data; }
+
+    CK_TILE_HOST_DEVICE
+    constexpr raw_type get() const { return data; }
+};
+
+template <typename>
+struct native_t;
+
+template <>
+struct native_t<half_t>
+{
+    using type = _Float16;
+};
+
+using fp16_t     = half_t;
+using fp16_raw_t = typename half_t::raw_type;
+#else
+using fp16_t     = _Float16;
+using half_t     = _Float16;
+using fp16_raw_t = ushort;
+#endif
+
+// conversions
+CK_TILE_HOST_DEVICE
+constexpr float fp16_to_float_hip(const fp16_hip_t& x)
+{
+    // return __half2float(x);
+    return static_cast<float>(x);
+}
+
+CK_TILE_HOST_DEVICE
+constexpr double fp16_to_double_hip(const fp16_hip_t& x)
+{
+    return static_cast<double>(fp16_to_float_hip(x));
+}
+
+CK_TILE_HOST_DEVICE
+constexpr fp16_hip_t float_to_fp16_hip(const float& x)
+{
+    return __float2half(x);
+    // return static_cast<fp16_hip_t>(x);
+}
+
+CK_TILE_HOST_DEVICE
+constexpr fp16_hip_t double_to_fp16_hip(const double& x)
+{
+    // return __float2half(x);
+    return static_cast<fp16_hip_t>(x);
+}
+
+CK_TILE_HOST_DEVICE
+constexpr float fp16_to_float(const half_t& x) { return static_cast<float>(x); }
+
+CK_TILE_HOST_DEVICE
+constexpr float fp16_to_double(const half_t& x) { return static_cast<float>(x); }
+
+CK_TILE_HOST_DEVICE
+constexpr half_t float_to_fp16(const float& x) { return static_cast<half_t>(x); }
+
+CK_TILE_HOST_DEVICE
+constexpr half_t double_to_fp16(const double& x) { return static_cast<half_t>(x); }
+
+// limits
+template <class T>
+struct numeric;
+
+template <>
+struct numeric<half_t>
+{
+    // minimum finite value, or minimum positive normalized value for float
+    CK_TILE_HOST_DEVICE static constexpr half_t min()
+    {
+        return bit_cast<half_t>(static_cast<fp16_raw_t>(0x0400));
+    }
+
+    // minumum finite value
+    CK_TILE_HOST_DEVICE static constexpr half_t lowest()
+    {
+        return bit_cast<half_t>(static_cast<fp16_raw_t>(0xFBFF));
+    }
+
+    // maximum finite value
+    CK_TILE_HOST_DEVICE static constexpr half_t max()
+    {
+        return bit_cast<half_t>(static_cast<fp16_raw_t>(0x7BFF));
+    }
+
+    // difference between 1.0 and next value representable by float
+    CK_TILE_HOST_DEVICE static constexpr half_t epsilon()
+    {
+        return bit_cast<half_t>(static_cast<fp16_raw_t>(0x1800));
+    }
+
+    // maximum rounding error
+    // bin :  f edcba 9876543210
+    // bits:  s eeeee mmmmmmmmmm
+    //        0 01110 0000000000 (0.5)
+    //
+    CK_TILE_HOST_DEVICE static constexpr half_t round_error()
+    {
+        return bit_cast<half_t>(static_cast<fp16_raw_t>(0x3800));
+    }
+
+    // positive infinity value
+    CK_TILE_HOST_DEVICE static constexpr half_t infinity()
+    {
+        return bit_cast<half_t>(static_cast<fp16_raw_t>(0x7C00));
+    }
+
+    // quiet NaN
+    CK_TILE_HOST_DEVICE static constexpr half_t quiet_NaN()
+    {
+        return bit_cast<half_t>(static_cast<fp16_raw_t>(0x7FFF));
+    }
+
+    // signaling NaN
+    CK_TILE_HOST_DEVICE static constexpr half_t signaling_NaN()
+    {
+        return bit_cast<half_t>(static_cast<fp16_raw_t>(0x7FFF));
+    }
+
+    // smallest positive subnormal value
+    CK_TILE_HOST_DEVICE static constexpr half_t denorm_min()
+    {
+        return bit_cast<half_t>(static_cast<fp16_raw_t>(0x0001));
+    }
+
+    CK_TILE_HOST_DEVICE static constexpr half_t zero()
+    {
+        return bit_cast<half_t>(static_cast<fp16_raw_t>(0));
+    }
+};
+
+template <typename T>
+struct numeric_traits;
+
+template <>
+struct numeric_traits<half_t>
+{
+    static constexpr int exp            = 5;
+    static constexpr int mant           = 10;
+    static constexpr int bias           = 15;
+    static constexpr uint16_t nan_mask  = 0x7C00;
+    static constexpr uint16_t head_mask = 0xFC00;
+    static constexpr uint16_t mant_mask = 0x3FF;
+    static constexpr uint16_t exp_mask  = 0x1F;
+    static constexpr uint32_t Inf       = 0x7C00;
+    static constexpr uint32_t NegInf    = 0xFC00;
+    static constexpr uint32_t NaN       = 0x7C01;
+    static constexpr uint32_t Neg0      = 0x8000;
+    using bitwise_type                  = uint16_t;
+};
+
+#if CK_TILE_USE_CUSTOM_DATA_TYPE
+// arithmetic
+CK_TILE_DEVICE bool operator==(const half_t& x, const half_t& y)
+{
+    return __heq(x.to_fp16(), y.to_fp16());
+}
+
+CK_TILE_DEVICE
+bool operator!=(const half_t& x, const half_t& y) { return __hne(x.to_fp16(), y.to_fp16()); }
+
+CK_TILE_DEVICE
+bool operator<(const half_t& x, const half_t& y) { return __hlt(x.to_fp16(), y.to_fp16()); }
+
+CK_TILE_DEVICE
+bool operator<=(const half_t& x, const half_t& y) { return __hle(x.to_fp16(), y.to_fp16()); }
+
+CK_TILE_DEVICE
+bool operator>(const half_t& x, const half_t& y) { return __hgt(x.to_fp16(), y.to_fp16()); }
+
+CK_TILE_DEVICE
+bool operator>=(const half_t& x, const half_t& y) { return __hge(x.to_fp16(), y.to_fp16()); }
+
+#if 0
+CK_TILE_DEVICE
+half_t operator+(const half_t& x, const half_t& y)
+{
+    return half_t(__hadd(x.to_fp16(), y.to_fp16()));
+}
+
+CK_TILE_DEVICE
+half_t operator-(const half_t& x) { return half_t(__hneg(x.to_fp16())); }
+
+CK_TILE_DEVICE
+half_t operator-(const half_t& x, const half_t& y)
+{
+    return half_t(__hsub(x.to_fp16(), y.to_fp16()));
+}
+
+CK_TILE_DEVICE
+half_t operator*(const half_t& x, const half_t& y)
+{
+    return half_t(__hmul(x.to_fp16(), y.to_fp16()));
+}
+
+CK_TILE_DEVICE
+half_t operator/(const half_t& x, const half_t& y)
+{
+    return half_t(__hdiv(x.to_fp16(), y.to_fp16()));
+}
+
+CK_TILE_DEVICE
+half_t& operator+=(half_t& x, const half_t& y)
+{
+    x = half_t(__hadd(x.to_fp16(), y.to_fp16()));
+    return x;
+}
+
+CK_TILE_DEVICE
+half_t& operator-=(half_t& x, const half_t& y)
+{
+    x = half_t(__hsub(x.to_fp16(), y.to_fp16()));
+    return x;
+}
+
+CK_TILE_DEVICE
+half_t& operator*=(half_t& x, const half_t& y)
+{
+    x = half_t(__hmul(x.to_fp16(), y.to_fp16()));
+    return x;
+}
+
+CK_TILE_DEVICE
+half_t& operator/=(half_t& x, const half_t& y)
+{
+    x = half_t(__hdiv(x.to_fp16(), y.to_fp16()));
+    return x;
+}
+
+CK_TILE_DEVICE
+half_t& operator++(half_t& x)
+{
+    x = half_t(__hadd(x.to_fp16(), half_t(1.0f).to_fp16()));
+    return x;
+}
+
+CK_TILE_DEVICE
+half_t& operator--(half_t& x)
+{
+    x = half_t(__hsub(x.to_fp16(), half_t(1.0f).to_fp16()));
+    return x;
+}
+
+CK_TILE_DEVICE
+half_t operator++(half_t& x, int)
+{
+    half_t y(x);
+    x = half_t(__hadd(x.to_fp16(), half_t(1.0f).to_fp16()));
+    return y;
+}
+
+CK_TILE_DEVICE
+half_t operator--(half_t& x, int)
+{
+    half_t y(x);
+    x = half_t(__hsub(x.to_fp16(), half_t(1.0f).to_fp16()));
+    return y;
+}
+#endif
+
+#if CK_TILE_USE_CUSTOM_DATA_TYPE
+CK_TILE_ARITHMETIC_USING_FLOAT(CK_TILE_HOST, half_t)
+#endif
+
+// math
+CK_TILE_HOST_DEVICE
+half_t abs(const half_t& x) { return bit_cast<half_t>(x.get() & 0x7fff); }
+
+CK_TILE_HOST_DEVICE
+bool isnan(const half_t& x)
+{
+    uint16_t xx = x.get();
+    return (xx & 0x7FFF) > 0x7C00;
+}
+
+CK_TILE_DEVICE
+half_t sqrt(half_t x)
+{
+    return static_cast<half_t>(__builtin_amdgcn_sqrtf(static_cast<float>(x)));
+};
+
+CK_TILE_DEVICE
+half_t exp(half_t x) { return static_cast<half_t>(__expf(static_cast<float>(x))); };
+
+CK_TILE_DEVICE
+half_t exp2(half_t x) { return static_cast<half_t>(exp2f(static_cast<float>(x))); };
+
+CK_TILE_DEVICE
+half_t log(half_t x) { return static_cast<half_t>(__logf(static_cast<float>(x))); };
+#endif
+} // namespace ck_tile

include/ck_tile/core/numeric/integer.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2024, Advanced Micro Devices, Inc. All rights reserved.
+
+#pragma once
+#include <stdint.h>
+
+namespace ck_tile {
+
+using index_t      = int32_t;
+using long_index_t = int64_t;
+using int8_t       = int8_t;
+
+} // namespace ck_tile

include/ck_tile/core/numeric/integral_constant.hpp

+// 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/numeric/integer.hpp"
+
+namespace ck_tile {
+
+template <auto v>
+struct constant
+{
+    using value_type                  = decltype(v);
+    using type                        = constant; // using injected-class-name
+    static constexpr value_type value = v;
+    CK_TILE_HOST_DEVICE constexpr operator value_type() const noexcept { return value; }
+    CK_TILE_HOST_DEVICE constexpr value_type operator()() const noexcept { return value; }
+    CK_TILE_HOST_DEVICE static constexpr bool is_static() { return true; }
+};
+
+template <typename T, T v>
+struct integral_constant : constant<v>
+{
+    using value_type         = T;
+    using type               = integral_constant; // using injected-class-name
+    static constexpr T value = v;
+    // constexpr CK_TILE_HOST_DEVICE operator   value_type() const noexcept { return value; }
+    // constexpr CK_TILE_HOST_DEVICE value_type operator()() const noexcept { return value; } //
+};
+
+template <index_t v>
+using number = constant<v>;
+
+template <long_index_t v>
+using long_number = constant<v>;
+
+template <bool b>
+using bool_constant = constant<b>;
+
+#define CK_TILE_LEFT_UNARY_OP(OP)                               \
+    template <auto x>                                           \
+    CK_TILE_HOST_DEVICE constexpr auto operator OP(constant<x>) \
+    {                                                           \
+        return constant<(OP x)>{};                              \
+    }
+
+#define CK_TILE_BINARY_OP(OP)                                                \
+    template <auto x, auto y>                                                \
+    CK_TILE_HOST_DEVICE constexpr auto operator OP(constant<x>, constant<y>) \
+    {                                                                        \
+        return constant<(x OP y)>{};                                         \
+    }
+
+CK_TILE_LEFT_UNARY_OP(+)
+CK_TILE_LEFT_UNARY_OP(-)
+CK_TILE_LEFT_UNARY_OP(~)
+CK_TILE_LEFT_UNARY_OP(!)
+CK_TILE_LEFT_UNARY_OP(*)
+
+CK_TILE_BINARY_OP(+)
+CK_TILE_BINARY_OP(-)
+CK_TILE_BINARY_OP(*)
+CK_TILE_BINARY_OP(/)
+CK_TILE_BINARY_OP(%)
+CK_TILE_BINARY_OP(&)
+CK_TILE_BINARY_OP(|)
+CK_TILE_BINARY_OP(^)
+CK_TILE_BINARY_OP(<<)
+CK_TILE_BINARY_OP(>>)
+CK_TILE_BINARY_OP(&&)
+CK_TILE_BINARY_OP(||)
+CK_TILE_BINARY_OP(==)
+CK_TILE_BINARY_OP(!=)
+CK_TILE_BINARY_OP(>)
+CK_TILE_BINARY_OP(<)
+CK_TILE_BINARY_OP(>=)
+CK_TILE_BINARY_OP(<=)
+
+#undef CK_TILE_LEFT_UNARY_OP
+#undef CK_TILE_BINARY_OP
+
+} // namespace ck_tile

include/ck_tile/core/numeric/math.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2023, Advanced Micro Devices, Inc. All rights reserved.
+
+#pragma once
+
+#include "ck_tile/core/config.hpp"
+#include "ck_tile/core/numeric/integer.hpp"
+#include "ck_tile/core/numeric/integral_constant.hpp"
+#include "ck_tile/core/utility/bit_cast.hpp"
+#include <type_traits>
+#include <stdint.h>
+#include <cmath>
+
+namespace ck_tile {
+
+template <typename Scale, Scale lhs>
+struct scales_c
+{
+    template <typename Right>
+    CK_TILE_HOST_DEVICE constexpr auto operator()(const Right& rhs) const -> decltype(lhs * rhs)
+    {
+        return lhs * rhs;
+    }
+};
+
+template <typename Scale>
+struct scales
+{
+    static_assert(std::is_copy_constructible_v<Scale>);
+
+    CK_TILE_HOST_DEVICE constexpr explicit scales(Scale lhs) : lhs_(lhs) {}
+
+    template <typename Right>
+    CK_TILE_HOST_DEVICE constexpr auto operator()(const Right& rhs) const
+        -> decltype(std::declval<const Scale&>() * rhs)
+    {
+        return lhs_ * rhs;
+    }
+
+    private:
+    Scale lhs_;
+};
+
+/// FIXME: create macro to replace '__host__ __device__' and nothing more
+template <typename Scale>
+__host__ __device__ scales(Scale)->scales<Scale>;
+
+template <typename Left = void, typename Right = Left>
+struct plus
+{
+    CK_TILE_HOST_DEVICE constexpr auto operator()(const Left& lhs, const Right& rhs) const
+        -> decltype(lhs + rhs)
+    {
+        return lhs + rhs;
+    }
+};
+
+template <>
+struct plus<void, void>
+{
+    template <typename Left, typename Right>
+    CK_TILE_HOST_DEVICE constexpr auto operator()(const Left& lhs, const Right& rhs) const
+        -> decltype(lhs + rhs)
+    {
+        return lhs + rhs;
+    }
+};
+
+/// FIXME: create macro to replace '__host__ __device__' and nothing more
+__host__ __device__ plus()->plus<void, void>;
+
+template <typename Left = void, typename Right = Left>
+struct minus
+{
+    CK_TILE_HOST_DEVICE constexpr auto operator()(const Left& lhs, const Right& rhs) const
+        -> decltype(lhs - rhs)
+    {
+        return lhs - rhs;
+    }
+};
+
+template <>
+struct minus<void, void>
+{
+    template <typename Left, typename Right>
+    CK_TILE_HOST_DEVICE constexpr auto operator()(const Left& lhs, const Right& rhs) const
+        -> decltype(lhs - rhs)
+    {
+        return lhs - rhs;
+    }
+};
+
+/// FIXME: create macro to replace '__host__ __device__' and nothing more
+__host__ __device__ minus()->minus<void, void>;
+
+template <typename Left = void, typename Right = Left>
+struct multiplies
+{
+    CK_TILE_HOST_DEVICE constexpr auto operator()(const Left& lhs, const Right& rhs) const
+        -> decltype(lhs * rhs)
+    {
+        return lhs * rhs;
+    }
+};
+
+template <>
+struct multiplies<void, void>
+{
+    template <typename Left, typename Right>
+    CK_TILE_HOST_DEVICE constexpr auto operator()(const Left& lhs, const Right& rhs) const
+        -> decltype(lhs * rhs)
+    {
+        return lhs * rhs;
+    }
+};
+
+/// FIXME: create macro to replace '__host__ __device__' and nothing more
+__host__ __device__ multiplies()->multiplies<void, void>;
+
+template <typename T>
+struct maximize
+{
+    CK_TILE_HOST_DEVICE constexpr T operator()(T a, T b) const { return a >= b ? a : b; }
+};
+
+template <typename T>
+struct minimize
+{
+    CK_TILE_HOST_DEVICE constexpr T operator()(T a, T b) const { return a <= b ? a : b; }
+};
+
+template <typename T>
+struct integer_divide_ceiler
+{
+    CK_TILE_HOST_DEVICE constexpr T operator()(T a, T b) const
+    {
+        static_assert(std::is_same<T, index_t>{} || std::is_same<T, int>{}, "wrong type");
+        return (a + b - number<1>{}) / b;
+    }
+};
+
+template <typename X, typename Y>
+CK_TILE_HOST_DEVICE constexpr auto integer_divide_floor(X x, Y y)
+{
+    return x / y;
+}
+
+template <typename X, typename Y>
+CK_TILE_HOST_DEVICE constexpr auto integer_divide_ceil(X x, Y y)
+{
+    return (x + y - number<1>{}) / y;
+}
+
+template <typename X, typename Y>
+CK_TILE_HOST_DEVICE constexpr auto integer_least_multiple(X x, Y y)
+{
+    return y * integer_divide_ceil(x, y);
+}
+
+template <typename T>
+CK_TILE_HOST_DEVICE constexpr T max(T x)
+{
+    return x;
+}
+
+template <typename T>
+CK_TILE_HOST constexpr T max(T x, T y)
+{
+    return x > y ? x : y;
+}
+
+template <typename T>
+CK_TILE_DEVICE constexpr T max(T x, T y)
+{
+    return x > y ? x : y;
+}
+
+template <>
+CK_TILE_DEVICE constexpr float max(float x, float y)
+{
+    return __builtin_fmaxf(x, y); // can resultin v_max3_f32
+}
+
+template <>
+CK_TILE_DEVICE constexpr double max(double x, double y)
+{
+    return __builtin_fmax(x, y); // maybe still v_max3_f32
+}
+
+template <index_t X>
+CK_TILE_HOST_DEVICE constexpr index_t max(number<X>, index_t y)
+{
+    return X > y ? X : y;
+}
+
+template <index_t Y>
+CK_TILE_HOST_DEVICE constexpr index_t max(index_t x, number<Y>)
+{
+    return x > Y ? x : Y;
+}
+
+template <typename X, typename... Ys>
+CK_TILE_HOST_DEVICE constexpr auto max(X x, Ys... ys)
+{
+    static_assert(sizeof...(Ys) > 0, "not enough argument");
+    return max(x, max(ys...));
+}
+
+template <typename T>
+CK_TILE_HOST_DEVICE constexpr T min(T x)
+{
+    return x;
+}
+
+template <typename T>
+CK_TILE_HOST constexpr T min(T x, T y)
+{
+    return x < y ? x : y;
+}
+
+template <typename T>
+CK_TILE_DEVICE constexpr T min(T x, T y)
+{
+    return x < y ? x : y;
+}
+
+template <>
+CK_TILE_DEVICE constexpr float min(float x, float y)
+{
+    return __builtin_fminf(x, y);
+}
+
+template <>
+CK_TILE_DEVICE constexpr double min(double x, double y)
+{
+    return __builtin_fmin(x, y);
+}
+
+template <index_t X>
+CK_TILE_HOST_DEVICE constexpr index_t min(number<X>, index_t y)
+{
+    return X < y ? X : y;
+}
+
+template <index_t Y>
+CK_TILE_HOST_DEVICE constexpr index_t min(index_t x, number<Y>)
+{
+    return x < Y ? x : Y;
+}
+
+template <typename X, typename... Ys>
+CK_TILE_HOST_DEVICE constexpr auto min(X x, Ys... ys)
+{
+    static_assert(sizeof...(Ys) > 0, "not enough argument");
+    return min(x, min(ys...));
+}
+
+template <typename T>
+CK_TILE_HOST_DEVICE constexpr T clamp(const T& x, const T& lowerbound, const T& upperbound)
+{
+    return min(max(x, lowerbound), upperbound);
+}
+
+CK_TILE_HOST int clz(uint32_t x) { return __builtin_clz(x); }
+CK_TILE_DEVICE int clz(uint32_t x) { return __clz(x); }
+
+// greatest common divisor, aka highest common factor
+CK_TILE_HOST_DEVICE constexpr index_t gcd(index_t x, index_t y)
+{
+    if(x < 0)
+    {
+        return gcd(-x, y);
+    }
+    else if(y < 0)
+    {
+        return gcd(x, -y);
+    }
+    else if(x == y || x == 0)
+    {
+        return y;
+    }
+    else if(y == 0)
+    {
+        return x;
+    }
+    else if(x > y)
+    {
+        return gcd(x % y, y);
+    }
+    else
+    {
+        return gcd(x, y % x);
+    }
+}
+
+template <index_t X, index_t Y>
+CK_TILE_HOST_DEVICE constexpr auto gcd(number<X>, number<Y>)
+{
+    constexpr auto r = gcd(X, Y);
+
+    return number<r>{};
+}
+
+template <typename X,
+          typename... Ys,
+          typename std::enable_if<sizeof...(Ys) >= 2, bool>::type = false>
+CK_TILE_HOST_DEVICE constexpr auto gcd(X x, Ys... ys)
+{
+    return gcd(x, gcd(ys...));
+}
+
+// least common multiple
+template <typename X, typename Y>
+CK_TILE_HOST_DEVICE constexpr auto lcm(X x, Y y)
+{
+    return (x * y) / gcd(x, y);
+}
+
+template <typename X,
+          typename... Ys,
+          typename std::enable_if<sizeof...(Ys) >= 2, bool>::type = false>
+CK_TILE_HOST_DEVICE constexpr auto lcm(X x, Ys... ys)
+{
+    return lcm(x, lcm(ys...));
+}
+
+template <typename Left = void, typename Right = Left>
+struct equal
+{
+    CK_TILE_HOST_DEVICE constexpr auto operator()(const Left& lhs, const Right& rhs) const
+        -> decltype(lhs == rhs)
+    {
+        return lhs == rhs;
+    }
+};
+
+template <>
+struct equal<void, void>
+{
+    template <typename Left, typename Right>
+    CK_TILE_HOST_DEVICE constexpr auto operator()(const Left& lhs, const Right& rhs) const
+        -> decltype(lhs == rhs)
+    {
+        return lhs == rhs;
+    }
+};
+
+/// FIXME: create macro to replace '__host__ __device__' and nothing more
+__host__ __device__ equal()->equal<void, void>;
+
+template <>
+struct equal<float, float>
+{
+    CK_TILE_HOST_DEVICE constexpr bool operator()(float lhs, float rhs) const
+    {
+        return bit_cast<uint32_t>(lhs) == bit_cast<uint32_t>(rhs);
+    }
+};
+
+template <>
+struct equal<double, double>
+{
+    CK_TILE_HOST_DEVICE constexpr bool operator()(double lhs, double rhs) const
+    {
+        return bit_cast<uint64_t>(lhs) == bit_cast<uint64_t>(rhs);
+    }
+};
+
+template <typename Left = void, typename Right = Left>
+struct less
+{
+    CK_TILE_HOST_DEVICE constexpr auto operator()(const Left& lhs, const Right& rhs) const
+        -> decltype(lhs < rhs)
+    {
+        return lhs < rhs;
+    }
+};
+
+template <>
+struct less<void, void>
+{
+    template <typename Left, typename Right>
+    CK_TILE_HOST_DEVICE constexpr auto operator()(const Left& lhs, const Right& rhs) const
+        -> decltype(lhs < rhs)
+    {
+        return lhs < rhs;
+    }
+};
+
+/// FIXME: create macro to replace '__host__ __device__' and nothing more
+__host__ __device__ less()->less<void, void>;
+
+template <typename Left = void, typename Right = Left>
+struct less_equal
+{
+    CK_TILE_HOST_DEVICE constexpr auto operator()(const Left& lhs, const Right& rhs) const
+        -> decltype(lhs <= rhs)
+    {
+        return lhs <= rhs;
+    }
+};
+
+template <>
+struct less_equal<void, void>
+{
+    template <typename Left, typename Right>
+    CK_TILE_HOST_DEVICE constexpr auto operator()(const Left& lhs, const Right& rhs) const
+        -> decltype(lhs <= rhs)
+    {
+        return lhs <= rhs;
+    }
+};
+
+/// FIXME: create macro to replace '__host__ __device__' and nothing more
+__host__ __device__ less_equal()->less_equal<void, void>;
+
+template <>
+struct less_equal<float, float>
+{
+    CK_TILE_HOST_DEVICE constexpr bool operator()(float lhs, float rhs) const
+    {
+        return lhs < rhs || bit_cast<uint32_t>(lhs) == bit_cast<uint32_t>(rhs);
+    }
+};
+
+template <>
+struct less_equal<double, double>
+{
+    CK_TILE_HOST_DEVICE constexpr bool operator()(double lhs, double rhs) const
+    {
+        return lhs < rhs || bit_cast<uint64_t>(lhs) == bit_cast<uint64_t>(rhs);
+    }
+};
+
+CK_TILE_HOST_DEVICE constexpr int32_t next_power_of_two(int32_t x)
+{
+    // TODO: x need to be 2 ~ 0x7fffffff. 0, 1, or larger than 0x7fffffff will compile fail
+    return 1 << (32 - clz(x - 1));
+}
+
+template <index_t X>
+CK_TILE_HOST_DEVICE constexpr auto next_power_of_two()
+{
+    constexpr index_t y = next_power_of_two(X);
+    return number<y>{};
+}
+
+template <index_t X>
+CK_TILE_HOST_DEVICE constexpr auto next_power_of_two(number<X>)
+{
+    constexpr index_t y = next_power_of_two(X);
+    return number<y>{};
+}
+
+CK_TILE_HOST_DEVICE constexpr int32_t integer_log2_floor(int32_t x)
+{
+    // TODO: x need to be 1 ~ 0x7fffffff
+    // __builtin_clz will produce unexpected result if x is 0;
+    return 31 - __builtin_clz(x);
+}
+
+CK_TILE_HOST_DEVICE constexpr bool is_power_of_two_integer(int32_t x)
+{
+    // TODO: x need to be 1 ~ 0x7fffffff
+    return x == (1 << integer_log2_floor(x));
+}
+
+#ifndef C_LOG2E
+#define C_LOG2E 1.44269504088896340736 // log2(e)
+#endif
+
+template <typename T>
+struct log2e;
+
+template <>
+struct log2e<double>
+{
+    static constexpr double value = C_LOG2E;
+};
+
+template <>
+struct log2e<float>
+{
+    static constexpr float value = C_LOG2E;
+};
+
+template <typename T = double>
+constexpr T log2e_v = log2e<T>::value;
+
+// math
+CK_TILE_HOST_DEVICE
+float abs(const float& x)
+{
+    union
+    {
+        float f32;
+        uint32_t u32;
+    } y;
+    y.f32 = x;
+    y.u32 = y.u32 & 0x7fffffff;
+    return y.f32;
+}
+
+CK_TILE_HOST_DEVICE
+bool isnan(const float& x)
+{
+    uint32_t xx = bit_cast<uint32_t>(x);
+    return (xx & 0x7fffffff) > 0x7F800000;
+}
+
+CK_TILE_HOST float sqrt(float x) { return std::sqrt(x); };
+
+CK_TILE_HOST double sqrt(double x) { return std::sqrt(x); };
+
+CK_TILE_DEVICE
+float sqrt(float x) { return __builtin_amdgcn_sqrtf(x); };
+
+CK_TILE_DEVICE
+double sqrt(double x) { return __builtin_amdgcn_sqrt(x); };
+
+CK_TILE_DEVICE
+float exp(float x) { return __expf(x); };
+
+CK_TILE_HOST
+float exp(float x) { return std::expf(x); }
+
+CK_TILE_DEVICE
+float exp2(float x) { return exp2f(x); };
+
+CK_TILE_HOST
+float exp2(float x) { return std::exp2f(x); };
+
+CK_TILE_DEVICE
+float log(float x) { return __logf(x); };
+
+CK_TILE_HOST
+float log(float x) { return std::logf(x); };
+
+} // namespace ck_tile

include/ck_tile/core/numeric/numeric.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2024, Advanced Micro Devices, Inc. All rights reserved.
+
+#pragma once
+
+#include "ck_tile/core/config.hpp"
+#include <limits>
+#include <stdint.h>
+
+namespace ck_tile {
+
+// this struct has the information of
+// 1. limit of a certain type, simliar to std::numeric_limits
+// 2. some pre-defined value, zero, one...
+//
+template <typename T>
+struct numeric
+{
+    // minimum finite value, or minimum positive normalized value for float
+    CK_TILE_HOST_DEVICE static constexpr T min() { return std::numeric_limits<T>::min(); }
+
+    // minumum finite value
+    CK_TILE_HOST_DEVICE static constexpr T lowest() { return std::numeric_limits<T>::lowest(); }
+
+    // maximum finite value
+    CK_TILE_HOST_DEVICE static constexpr T max() { return std::numeric_limits<T>::max(); }
+
+    // difference between 1.0 and next value representable by float
+    CK_TILE_HOST_DEVICE static constexpr T epsilon() { return std::numeric_limits<T>::epsilon(); }
+
+    // maximum rounding error
+    CK_TILE_HOST_DEVICE static constexpr T round_error()
+    {
+        return std::numeric_limits<T>::round_error();
+    }
+
+    // positive infinity value
+    CK_TILE_HOST_DEVICE static constexpr T infinity() { return std::numeric_limits<T>::infinity(); }
+
+    // quiet NaN
+    CK_TILE_HOST_DEVICE static constexpr T quiet_NaN()
+    {
+        return std::numeric_limits<T>::quiet_NaN();
+    }
+
+    // signaling NaN
+    CK_TILE_HOST_DEVICE static constexpr T signaling_NaN()
+    {
+        return std::numeric_limits<T>::signaling_NaN();
+    }
+
+    // smallest positive subnormal value
+    CK_TILE_HOST_DEVICE static constexpr T denorm_min()
+    {
+        return std::numeric_limits<T>::denorm_min();
+    }
+
+    CK_TILE_HOST_DEVICE static constexpr T zero() { return static_cast<T>(0); }
+
+    CK_TILE_HOST_DEVICE static constexpr T one() { return static_cast<T>(1); }
+
+#ifndef C_LOG2E
+#define C_LOG2E 1.44269504088896340736 // log2(e)
+#endif
+
+    CK_TILE_HOST_DEVICE static constexpr T log2e()
+    {
+        if constexpr(std::is_same_v<T, float> || std::is_same_v<T, double>)
+        {
+            return static_cast<T>(C_LOG2E);
+        }
+        else
+        {
+            return 0; // TODO: integer?
+        }
+    }
+};
+
+template <typename T>
+struct numeric_traits;
+
+template <>
+struct numeric_traits<float>
+{
+    static constexpr int exp            = 8;
+    static constexpr int mant           = 23;
+    static constexpr int bias           = 127;
+    static constexpr uint32_t nan_mask  = 0x7F800000;
+    static constexpr uint32_t head_mask = 0xFF800000;
+    static constexpr uint32_t mant_mask = 0x7FFFFF;
+    static constexpr uint32_t exp_mask  = 0xFF;
+    static constexpr uint32_t Inf       = 0x7F800000;
+    static constexpr uint32_t NegInf    = 0xFF800000;
+    static constexpr uint32_t NaN       = 0x7F800001;
+    static constexpr uint32_t Neg0      = 0x80000000;
+    using bitwise_type                  = uint32_t;
+};
+
+} // namespace ck_tile
+
+#define CK_TILE_ARITHMETIC_USING_FLOAT(attr_, type_)                 \
+    attr_ bool operator==(const type_& x, const type_& y)            \
+    {                                                                \
+        return static_cast<float>(x) == static_cast<float>(y);       \
+    }                                                                \
+    attr_ bool operator!=(const type_& x, const type_& y)            \
+    {                                                                \
+        return static_cast<float>(x) != static_cast<float>(y);       \
+    }                                                                \
+    attr_ bool operator<(const type_& x, const type_& y)             \
+    {                                                                \
+        return static_cast<float>(x) < static_cast<float>(y);        \
+    }                                                                \
+    attr_ bool operator<=(const type_& x, const type_& y)            \
+    {                                                                \
+        return static_cast<float>(x) <= static_cast<float>(y);       \
+    }                                                                \
+    attr_ bool operator>(const type_& x, const type_& y)             \
+    {                                                                \
+        return static_cast<float>(x) > static_cast<float>(y);        \
+    }                                                                \
+    attr_ bool operator>=(const type_& x, const type_& y)            \
+    {                                                                \
+        return static_cast<float>(x) >= static_cast<float>(y);       \
+    }                                                                \
+    attr_ type_ operator+(const type_& x, const type_& y)            \
+    {                                                                \
+        return type_(static_cast<float>(x) + static_cast<float>(y)); \
+    }                                                                \
+    attr_ type_ operator-(const type_& x)                            \
+    {                                                                \
+        constexpr uint32_t bits = sizeof(type_) * 8;                 \
+        constexpr uint32_t mask = 1 << (bits - 1);                   \
+        type_ y                 = x;                                 \
+        y.data ^= static_cast<typename type_::raw_type>(mask);       \
+        return y;                                                    \
+    }                                                                \
+    attr_ type_ operator-(const type_& x, const type_& y)            \
+    {                                                                \
+        return type_(static_cast<float>(x) - static_cast<float>(y)); \
+    }                                                                \
+    attr_ type_ operator*(const type_& x, const type_& y)            \
+    {                                                                \
+        return type_(static_cast<float>(x) * static_cast<float>(y)); \
+    }                                                                \
+    attr_ type_ operator/(const type_& x, const type_& y)            \
+    {                                                                \
+        return type_(static_cast<float>(x) / static_cast<float>(y)); \
+    }                                                                \
+    attr_ type_& operator+=(type_& x, const type_& y)                \
+    {                                                                \
+        x = type_(static_cast<float>(x) + static_cast<float>(y));    \
+        return x;                                                    \
+    }                                                                \
+    attr_ type_& operator-=(type_& x, const type_& y)                \
+    {                                                                \
+        x = type_(static_cast<float>(x) - static_cast<float>(y));    \
+        return x;                                                    \
+    }                                                                \
+    attr_ type_& operator*=(type_& x, const type_& y)                \
+    {                                                                \
+        x = type_(static_cast<float>(x) * static_cast<float>(y));    \
+        return x;                                                    \
+    }                                                                \
+    attr_ type_& operator/=(type_& x, const type_& y)                \
+    {                                                                \
+        x = type_(static_cast<float>(x) / static_cast<float>(y));    \
+        return x;                                                    \
+    }                                                                \
+    attr_ type_& operator++(type_& x)                                \
+    {                                                                \
+        x = type_(static_cast<float>(x) + 1.f);                      \
+        return x;                                                    \
+    }                                                                \
+    attr_ type_& operator--(type_& x)                                \
+    {                                                                \
+        x = type_(static_cast<float>(x) - 1.f);                      \
+        return x;                                                    \
+    }                                                                \
+    attr_ type_ operator++(type_& x, int)                            \
+    {                                                                \
+        type_ y(x);                                                  \
+        x = type_(static_cast<float>(x) + 1.f);                      \
+        return y;                                                    \
+    }                                                                \
+    attr_ type_ operator--(type_& x, int)                            \
+    {                                                                \
+        type_ y(x);                                                  \
+        x = type_(static_cast<float>(x) - 1.f);                      \
+        return y;                                                    \
+    }

include/ck_tile/core/numeric/type_convert.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2023, Advanced Micro Devices, Inc. All rights reserved.
+
+#pragma once
+
+#include <stdint.h>
+#include <tuple>
+#include <type_traits>
+#include "ck_tile/core/config.hpp"
+#include "ck_tile/core/numeric/half.hpp"
+#include "ck_tile/core/numeric/bfloat16.hpp"
+#include "ck_tile/core/numeric/float8.hpp"
+
+namespace ck_tile {
+
+#if CK_TILE_USE_CUSTOM_DATA_TYPE
+template <typename Y, typename X>
+CK_TILE_HOST_DEVICE constexpr remove_cvref_t<Y> type_convert(const X& x)
+{
+    return static_cast<Y>(x);
+}
+#else
+// Convert X to Y, both X and Y are non-const data types.
+template <typename Y,
+          typename X,
+          std::enable_if_t<!(std::is_const_v<Y> || std::is_const_v<X>), bool> = false>
+CK_TILE_HOST_DEVICE constexpr Y type_convert(X x)
+{
+    static_assert(!std::is_reference_v<Y> && !std::is_reference_v<X>);
+    return static_cast<Y>(x);
+}
+
+// Convert X to Y, either X or Y is a const data type.
+template <typename Y,
+          typename X,
+          std::enable_if_t<std::is_const_v<Y> || std::is_const_v<X>, bool> = false>
+CK_TILE_HOST_DEVICE constexpr Y type_convert(X x)
+{
+    static_assert(!std::is_reference_v<Y> && !std::is_reference_v<X>);
+
+    using non_const_y = std::remove_const_t<Y>;
+    using non_const_x = std::remove_const_t<X>;
+    return static_cast<Y>(type_convert<non_const_y, non_const_x>(x));
+}
+
+#define CK_TILE_TYPE_CONVERT(dtype_, dname_, stype_, sname_)                    \
+    template <>                                                                 \
+    CK_TILE_HOST_DEVICE constexpr dtype_ type_convert<dtype_, stype_>(stype_ x) \
+    {                                                                           \
+        return sname_##_to_##dname_(x);                                         \
+    }
+
+CK_TILE_TYPE_CONVERT(float, float, fp16_t, fp16)
+CK_TILE_TYPE_CONVERT(float, float, bf16_t, bf16)
+CK_TILE_TYPE_CONVERT(float, float, fp8_t, fp8)
+CK_TILE_TYPE_CONVERT(float, float, bf8_t, bf8)
+
+CK_TILE_TYPE_CONVERT(fp16_t, fp16, float, float)
+CK_TILE_TYPE_CONVERT(bf16_t, bf16, float, float)
+CK_TILE_TYPE_CONVERT(fp8_t, fp8, float, float)
+CK_TILE_TYPE_CONVERT(bf8_t, bf8, float, float)
+
+#undef CK_TILE_TYPE_CONVERT
+#endif
+
+} // namespace ck_tile

include/ck_tile/core/numeric/vector_type.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2023, Advanced Micro Devices, Inc. All rights reserved.
+
+#pragma once
+
+#include "ck_tile/core/config.hpp"
+#include "ck_tile/core/container/array.hpp"
+#include "ck_tile/core/numeric/integer.hpp"
+#include "ck_tile/core/numeric/integral_constant.hpp"
+#include "ck_tile/core/numeric/float8.hpp"
+#include "ck_tile/core/numeric/half.hpp"
+#include "ck_tile/core/numeric/bfloat16.hpp"
+#include "ck_tile/core/utility/type_traits.hpp"
+
+namespace ck_tile {
+
+// this structure is used to pick up the <base> type inside
+// using xxx = <base> __attribute__((ext_vector_type(N)));
+// because clang only allow native type + bool in this term (custom type will fail)
+// overload this structure to let proper <base> type
+
+template <typename T>
+struct native_t
+{
+    using type = remove_cvref_t<T>;
+};
+
+// we name this as ext_vector purposely, because clang ext_vector_type extention only accept literay
+// basic type to construct a ext_vector_type you must be very careful using this, or will have lot
+// of compiler errors e.g. struct A; using Ax2_t = A __attribute__((ext_vector_type(2)));  -> will
+// have compiler error
+namespace impl {
+template <typename T_, index_t N_>
+struct ext_vector
+{
+    static constexpr index_t N = N_;
+    using value_type           = typename native_t<remove_cvref_t<T_>>::type;
+    static_assert(!std::is_class_v<value_type>);
+    using type = value_type __attribute__((ext_vector_type(N))); // this is danguous
+};
+
+template <typename V_, index_t Vs_, index_t N_>
+struct ext_vector<V_ __attribute__((ext_vector_type(Vs_))), N_>
+{
+    static constexpr index_t N = Vs_ * N_;
+    using value_type           = typename native_t<remove_cvref_t<V_>>::type;
+    static_assert(!std::is_class_v<value_type>);
+    using type = value_type __attribute__((ext_vector_type(N))); // this is danguous
+};
+
+} // namespace impl
+
+template <typename T, index_t N>
+using ext_vector_t = typename impl::ext_vector<T, N>::type;
+
+// by default, any type will result in a vector_size=1 with scalar_type=T traits.
+// ... unless we have other vector_traits specialization
+template <typename T>
+struct vector_traits
+{
+    using scalar_type                    = remove_cvref_t<T>;
+    static constexpr index_t vector_size = 1;
+};
+
+// specialization for ext_vector_type()
+template <typename T, index_t N>
+struct vector_traits<T __attribute__((ext_vector_type(N)))>
+{
+    using scalar_type                    = T;
+    static constexpr index_t vector_size = N;
+};
+
+template <typename X, typename Y>
+using has_same_scalar_type = std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                                          typename vector_traits<remove_cvref_t<Y>>::scalar_type>;
+
+// below are some pre-defines of ext_vector_type
+// attention! 2 vector type could be just the same type
+// fp64
+using fp64_t   = double;
+using fp64x2_t = double __attribute__((ext_vector_type(2)));
+using fp64x4_t = double __attribute__((ext_vector_type(4)));
+
+// fp32
+using fp32_t    = float;
+using fp32x2_t  = float __attribute__((ext_vector_type(2)));
+using fp32x4_t  = float __attribute__((ext_vector_type(4)));
+using fp32x8_t  = float __attribute__((ext_vector_type(8)));
+using fp32x16_t = float __attribute__((ext_vector_type(16)));
+using fp32x32_t = float __attribute__((ext_vector_type(32)));
+using fp32x64_t = float __attribute__((ext_vector_type(64)));
+
+// fp16
+// using fp16_t = ...
+using fp16x2_t  = _Float16 __attribute__((ext_vector_type(2)));
+using fp16x4_t  = _Float16 __attribute__((ext_vector_type(4)));
+using fp16x8_t  = _Float16 __attribute__((ext_vector_type(8)));
+using fp16x16_t = _Float16 __attribute__((ext_vector_type(16)));
+using fp16x32_t = _Float16 __attribute__((ext_vector_type(32)));
+using fp16x64_t = _Float16 __attribute__((ext_vector_type(64)));
+
+// bf16
+// using bf16_t = ...
+using bf16x2_t  = bf16_raw_t __attribute__((ext_vector_type(2)));
+using bf16x4_t  = bf16_raw_t __attribute__((ext_vector_type(4)));
+using bf16x8_t  = bf16_raw_t __attribute__((ext_vector_type(8)));
+using bf16x16_t = bf16_raw_t __attribute__((ext_vector_type(16)));
+using bf16x32_t = bf16_raw_t __attribute__((ext_vector_type(32)));
+using bf16x64_t = bf16_raw_t __attribute__((ext_vector_type(64)));
+
+// i32
+// using int32_t = ...
+using int32x2_t  = int32_t __attribute__((ext_vector_type(2)));
+using int32x4_t  = int32_t __attribute__((ext_vector_type(4)));
+using int32x8_t  = int32_t __attribute__((ext_vector_type(8)));
+using int32x16_t = int32_t __attribute__((ext_vector_type(16)));
+using int32x32_t = int32_t __attribute__((ext_vector_type(32)));
+using int32x64_t = int32_t __attribute__((ext_vector_type(64)));
+
+// i16
+// using int16_t = ...
+using int16x2_t  = int16_t __attribute__((ext_vector_type(2)));
+using int16x4_t  = int16_t __attribute__((ext_vector_type(4)));
+using int16x8_t  = int16_t __attribute__((ext_vector_type(8)));
+using int16x16_t = int16_t __attribute__((ext_vector_type(16)));
+using int16x32_t = int16_t __attribute__((ext_vector_type(32)));
+using int16x64_t = int16_t __attribute__((ext_vector_type(64)));
+
+// u16
+// using uint16_t
+using uint16x2_t  = uint16_t __attribute__((ext_vector_type(2)));
+using uint16x4_t  = uint16_t __attribute__((ext_vector_type(4)));
+using uint16x8_t  = uint16_t __attribute__((ext_vector_type(8)));
+using uint16x16_t = uint16_t __attribute__((ext_vector_type(16)));
+using uint16x32_t = uint16_t __attribute__((ext_vector_type(32)));
+using uint16x64_t = uint16_t __attribute__((ext_vector_type(64)));
+
+// i8
+// using int8_t
+using int8x2_t  = int8_t __attribute((ext_vector_type(2)));
+using int8x4_t  = int8_t __attribute((ext_vector_type(4)));
+using int8x8_t  = int8_t __attribute((ext_vector_type(8)));
+using int8x16_t = int8_t __attribute((ext_vector_type(16)));
+using int8x32_t = int8_t __attribute((ext_vector_type(32)));
+using int8x64_t = int8_t __attribute((ext_vector_type(64)));
+
+#if CK_TILE_USE_CUSTOM_DATA_TYPE
+// f8
+// using fp8_t
+using fp8x2_t  = fp8_raw_t __attribute((ext_vector_type(2)));
+using fp8x4_t  = fp8_raw_t __attribute((ext_vector_type(4)));
+using fp8x8_t  = fp8_raw_t __attribute((ext_vector_type(8)));
+using fp8x16_t = fp8_raw_t __attribute((ext_vector_type(16)));
+using fp8x32_t = fp8_raw_t __attribute((ext_vector_type(32)));
+using fp8x64_t = fp8_raw_t __attribute((ext_vector_type(64)));
+
+// bf8
+// using bf8_t
+using bf8x2_t  = bf8_raw_t __attribute((ext_vector_type(2)));
+using bf8x4_t  = bf8_raw_t __attribute((ext_vector_type(4)));
+using bf8x8_t  = bf8_raw_t __attribute((ext_vector_type(8)));
+using bf8x16_t = bf8_raw_t __attribute((ext_vector_type(16)));
+using bf8x32_t = bf8_raw_t __attribute((ext_vector_type(32)));
+using bf8x64_t = bf8_raw_t __attribute((ext_vector_type(64)));
+#else
+// f8
+// using fp8_t
+using fp8x2_t  = fp8_t __attribute((ext_vector_type(2)));
+using fp8x4_t  = fp8_t __attribute((ext_vector_type(4)));
+using fp8x8_t  = fp8_t __attribute((ext_vector_type(8)));
+using fp8x16_t = fp8_t __attribute((ext_vector_type(16)));
+using fp8x32_t = fp8_t __attribute((ext_vector_type(32)));
+using fp8x64_t = fp8_t __attribute((ext_vector_type(64)));
+
+// bf8
+// using bf8_t
+using bf8x2_t  = bf8_t __attribute((ext_vector_type(2)));
+using bf8x4_t  = bf8_t __attribute((ext_vector_type(4)));
+using bf8x8_t  = bf8_t __attribute((ext_vector_type(8)));
+using bf8x16_t = bf8_t __attribute((ext_vector_type(16)));
+using bf8x32_t = bf8_t __attribute((ext_vector_type(32)));
+using bf8x64_t = bf8_t __attribute((ext_vector_type(64)));
+#endif
+
+} // namespace ck_tile

include/ck_tile/core/tensor/buffer_view.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2023, Advanced Micro Devices, Inc. All rights reserved.
+
+#pragma once
+
+#include "ck_tile/core/config.hpp"
+#include "ck_tile/core/arch/arch.hpp"
+#include "ck_tile/core/arch/amd_buffer_addressing.hpp"
+#include "ck_tile/core/container/array.hpp"
+#include "ck_tile/core/numeric/integer.hpp"
+#include "ck_tile/core/numeric/integral_constant.hpp"
+#include "ck_tile/core/numeric/float8.hpp"
+#include "ck_tile/core/numeric/half.hpp"
+#include "ck_tile/core/numeric/bfloat16.hpp"
+#include "ck_tile/core/utility/type_traits.hpp"
+
+namespace ck_tile {
+
+// T may be scalar or vector
+// X may be scalar or vector
+// T and X have same scalar type
+// X contains multiple T
+// FIXME: InvalidElementUseNumericalZeroValue and invalid_element_value_ should be a property of
+//        transforms of tensor_view/Tensor
+// FIXME: amd_buffer_coherence_enum is only meaningful for buffer addressing. Need to split
+// buffer_view definition for different memory address space (Global/GenericLds/Vgpr)
+template <address_space_enum BufferAddressSpace,
+          typename T,
+          typename BufferSizeType,
+          bool InvalidElementUseNumericalZeroValue,
+          amd_buffer_coherence_enum Coherence = amd_buffer_coherence_enum::coherence_default>
+struct buffer_view;
+
+// Address Space: generic
+// T may be scalar or vector
+// X may be scalar or vector
+// T and X have same scalar type
+// X contains multiple T
+// FIXME: InvalidElementUseNumericalZeroValue and invalid_element_value_ should be a property of
+//        transforms of tensor_view/Tensor
+template <typename T, typename BufferSizeType, bool InvalidElementUseNumericalZeroValue>
+struct buffer_view<address_space_enum::generic,
+                   T,
+                   BufferSizeType,
+                   InvalidElementUseNumericalZeroValue,
+                   amd_buffer_coherence_enum::coherence_default>
+{
+    using type = T;
+
+    T* p_data_ = nullptr;
+    BufferSizeType buffer_size_;
+    remove_cvref_t<T> invalid_element_value_ = T{0};
+
+    CK_TILE_HOST_DEVICE constexpr buffer_view()
+        : p_data_{}, buffer_size_{}, invalid_element_value_{}
+    {
+    }
+
+    CK_TILE_HOST_DEVICE constexpr buffer_view(T* p_data, BufferSizeType buffer_size)
+        : p_data_{p_data}, buffer_size_{buffer_size}, invalid_element_value_{0}
+    {
+    }
+
+    CK_TILE_HOST_DEVICE constexpr buffer_view(T* p_data,
+                                              BufferSizeType buffer_size,
+                                              T invalid_element_value)
+        : p_data_{p_data}, buffer_size_{buffer_size}, invalid_element_value_{invalid_element_value}
+    {
+    }
+
+    CK_TILE_DEVICE static constexpr address_space_enum get_address_space()
+    {
+        return address_space_enum::generic;
+    }
+
+    // i is offset of T
+    // FIXME: doesn't do is_valid check
+    CK_TILE_DEVICE constexpr const T& operator[](index_t i) const { return p_data_[i]; }
+
+    // i is offset of T
+    // FIXME: doesn't do is_valid check
+    CK_TILE_DEVICE constexpr T& operator()(index_t i) { return p_data_[i]; }
+
+    // i is offset of T, not X. i should be aligned to X
+    template <typename X,
+              bool oob_conditional_check = true,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE constexpr auto
+    get(index_t i, bool is_valid_element, bool_constant<oob_conditional_check> = {}) const
+    {
+        // X contains multiple T
+        constexpr index_t scalar_per_t_vector = vector_traits<remove_cvref_t<T>>::vector_size;
+
+        constexpr index_t scalar_per_x_vector = vector_traits<remove_cvref_t<X>>::vector_size;
+
+        static_assert(scalar_per_x_vector % scalar_per_t_vector == 0,
+                      "wrong! X should contain multiple T");
+
+        if(is_valid_element)
+        {
+#if CK_TILE_EXPERIMENTAL_USE_MEMCPY_FOR_VECTOR_ACCESS
+            X tmp;
+
+            __builtin_memcpy(&tmp, &(p_data_[i]), sizeof(X));
+
+            return tmp;
+#else
+            return *c_style_pointer_cast<const X*>(&p_data_[i]);
+#endif
+        }
+        else
+        {
+            if constexpr(InvalidElementUseNumericalZeroValue)
+            {
+                return X{numeric<remove_cvref_t<T>>::zero()};
+            }
+            else
+            {
+                return X{invalid_element_value_};
+            }
+        }
+    }
+
+    // i is offset of T, not X. i should be aligned to X
+    template <memory_operation_enum Op,
+              typename X,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE void update(index_t i, bool is_valid_element, const X& x)
+    {
+        if constexpr(Op == memory_operation_enum::set)
+        {
+            this->template set<X>(i, is_valid_element, x);
+        }
+        // FIXME: remove memory_operation_enum::add
+        else if constexpr(Op == memory_operation_enum::add)
+        {
+            auto tmp = this->template get<X>(i, is_valid_element);
+            this->template set<X>(i, is_valid_element, x + tmp);
+        }
+    }
+
+    // i is offset of T, not X. i should be aligned to X
+    template <typename X,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE void set(index_t i, bool is_valid_element, const X& x)
+    {
+        // X contains multiple T
+        constexpr index_t scalar_per_t_vector = vector_traits<remove_cvref_t<T>>::vector_size;
+
+        constexpr index_t scalar_per_x_vector = vector_traits<remove_cvref_t<X>>::vector_size;
+
+        static_assert(scalar_per_x_vector % scalar_per_t_vector == 0,
+                      "wrong! X should contain multiple T");
+
+        if(is_valid_element)
+        {
+#if CK_TILE_EXPERIMENTAL_USE_MEMCPY_FOR_VECTOR_ACCESS
+            X tmp = x;
+
+            __builtin_memcpy(&(p_data_[i]), &tmp, sizeof(X));
+#else
+            *c_style_pointer_cast<X*>(&p_data_[i]) = x;
+#endif
+        }
+    }
+
+    // FIXME: remove
+    CK_TILE_DEVICE static constexpr bool is_static_buffer() { return false; }
+
+    // FIXME: remove
+    CK_TILE_DEVICE static constexpr bool is_dynamic_buffer() { return true; }
+
+    CK_TILE_HOST_DEVICE void print() const
+    {
+        printf("buffer_view{");
+
+        // AddressSpace
+        printf("AddressSpace: generic, ");
+
+        // p_data_
+        printf("p_data_: %p, ", static_cast<void*>(const_cast<remove_cvref_t<T>*>(p_data_)));
+
+        // buffer_size_
+        printf("buffer_size_: ");
+        print(buffer_size_);
+        printf(", ");
+
+        // invalid_element_value_
+        printf("invalid_element_value_: ");
+        print(invalid_element_value_);
+
+        printf("}");
+    }
+};
+
+// Address Space: Global
+// T may be scalar or vector
+// X may be scalar or vector
+// T and X have same scalar type
+// X contains multiple T
+// FIXME: InvalidElementUseNumericalZeroValue and invalid_element_value_ should be a property of
+//        transforms of tensor_view/Tensor
+template <typename T,
+          typename BufferSizeType,
+          bool InvalidElementUseNumericalZeroValue,
+          amd_buffer_coherence_enum Coherence>
+struct buffer_view<address_space_enum::global,
+                   T,
+                   BufferSizeType,
+                   InvalidElementUseNumericalZeroValue,
+                   Coherence>
+{
+    using type = T;
+
+    T* p_data_ = nullptr;
+    BufferSizeType buffer_size_;
+    remove_cvref_t<T> invalid_element_value_ = T{0};
+
+    CK_TILE_HOST_DEVICE constexpr buffer_view()
+        : p_data_{}, buffer_size_{}, invalid_element_value_{}
+    {
+    }
+
+    CK_TILE_HOST_DEVICE constexpr buffer_view(T* p_data, BufferSizeType buffer_size)
+        : p_data_{p_data}, buffer_size_{buffer_size}, invalid_element_value_{0}
+    {
+    }
+
+    CK_TILE_HOST_DEVICE constexpr buffer_view(T* p_data,
+                                              BufferSizeType buffer_size,
+                                              T invalid_element_value)
+        : p_data_{p_data}, buffer_size_{buffer_size}, invalid_element_value_{invalid_element_value}
+    {
+    }
+
+    CK_TILE_DEVICE static constexpr address_space_enum get_address_space()
+    {
+        return address_space_enum::global;
+    }
+
+    // i is offset of T
+    // FIXME: doesn't do is_valid check
+    CK_TILE_DEVICE constexpr const T& operator[](index_t i) const { return p_data_[i]; }
+
+    // i is offset of T
+    // FIXME: doesn't do is_valid check
+    CK_TILE_DEVICE constexpr T& operator()(index_t i) { return p_data_[i]; }
+
+    // i is offset of T, not X. i should be aligned to X
+    template <typename X,
+              bool oob_conditional_check = true,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE constexpr auto
+    get(index_t i, bool is_valid_element, bool_constant<oob_conditional_check> = {}) const
+    {
+        // X contains multiple T
+        constexpr index_t scalar_per_t_vector = vector_traits<remove_cvref_t<T>>::vector_size;
+
+        constexpr index_t scalar_per_x_vector = vector_traits<remove_cvref_t<X>>::vector_size;
+
+        static_assert(scalar_per_x_vector % scalar_per_t_vector == 0,
+                      "wrong! X should contain multiple T");
+
+#if CK_TILE_USE_AMD_BUFFER_LOAD
+        bool constexpr use_amd_buffer_addressing = true;
+#else
+        bool constexpr use_amd_buffer_addressing = false;
+#endif
+
+        if constexpr(use_amd_buffer_addressing)
+        {
+            constexpr index_t t_per_x = scalar_per_x_vector / scalar_per_t_vector;
+
+            if constexpr(InvalidElementUseNumericalZeroValue)
+            {
+                return amd_buffer_load_invalid_element_return_zero<remove_cvref_t<T>,
+                                                                   t_per_x,
+                                                                   Coherence,
+                                                                   oob_conditional_check>(
+                    p_data_, i, is_valid_element, buffer_size_);
+            }
+            else
+            {
+                return amd_buffer_load_invalid_element_return_customized_value<
+                    remove_cvref_t<T>,
+                    t_per_x,
+                    Coherence,
+                    oob_conditional_check>(
+                    p_data_, i, is_valid_element, buffer_size_, invalid_element_value_);
+            }
+        }
+        else
+        {
+            if(is_valid_element)
+            {
+#if CK_TILE_EXPERIMENTAL_USE_MEMCPY_FOR_VECTOR_ACCESS
+                X tmp;
+
+                __builtin_memcpy(&tmp, &(p_data_[i]), sizeof(X));
+
+                return tmp;
+#else
+                return *c_style_pointer_cast<const X*>(&p_data_[i]);
+#endif
+            }
+            else
+            {
+                if constexpr(InvalidElementUseNumericalZeroValue)
+                {
+                    return X{numeric<remove_cvref_t<T>>::zero()};
+                }
+                else
+                {
+                    return X{invalid_element_value_};
+                }
+            }
+        }
+    }
+
+    // i is offset of T, not X. i should be aligned to X
+    template <typename X,
+              bool oob_conditional_check = true,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE constexpr auto
+    get_raw(remove_cvref_t<X>& dst, index_t i, bool is_valid_element) const
+    {
+        constexpr index_t scalar_per_t_vector = vector_traits<remove_cvref_t<T>>::vector_size;
+
+        constexpr index_t scalar_per_x_vector = vector_traits<remove_cvref_t<X>>::vector_size;
+
+        static_assert(scalar_per_x_vector % scalar_per_t_vector == 0,
+                      "wrong! X should contain multiple T");
+
+        constexpr index_t t_per_x = scalar_per_x_vector / scalar_per_t_vector;
+
+        amd_buffer_load_raw<remove_cvref_t<T>, t_per_x, Coherence, oob_conditional_check>(
+            dst, p_data_, i, buffer_size_, is_valid_element);
+    }
+
+    // i is offset of T, not X. i should be aligned to X
+    template <typename X,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE constexpr auto
+    async_get(remove_cvref_t<T>* smem, index_t i, bool /*is_valid_element*/) const
+    {
+        // X is vector of T
+        constexpr index_t scalar_per_t_vector = vector_traits<remove_cvref_t<T>>::vector_size;
+        constexpr index_t scalar_per_x_vector = vector_traits<remove_cvref_t<X>>::vector_size;
+
+        static_assert(scalar_per_x_vector % scalar_per_t_vector == 0,
+                      "wrong! X should contain multiple T");
+
+        constexpr index_t t_per_x = scalar_per_x_vector / scalar_per_t_vector;
+
+        amd_async_buffer_load_with_oob<remove_cvref_t<T>, t_per_x, Coherence>(
+            smem, p_data_, i, buffer_size_);
+    }
+
+    // i is offset of T, not X. i should be aligned to X
+    template <memory_operation_enum Op,
+              typename X,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE void update(index_t i, bool is_valid_element, const X& x)
+    {
+        if constexpr(Op == memory_operation_enum::set)
+        {
+            this->template set<X>(i, is_valid_element, x);
+        }
+        else if constexpr(Op == memory_operation_enum::atomic_add)
+        {
+            this->template atomic_add<X>(i, is_valid_element, x);
+        }
+        else if constexpr(Op == memory_operation_enum::atomic_max)
+        {
+            this->template atomic_max<X>(i, is_valid_element, x);
+        }
+        // FIXME: remove memory_operation_enum::add
+        else if constexpr(Op == memory_operation_enum::add)
+        {
+            auto tmp = this->template get<X>(i, is_valid_element);
+            this->template set<X>(i, is_valid_element, x + tmp);
+            // tmp += x;
+            // this->template set<X>(i, is_valid_element, tmp);
+        }
+    }
+
+    // i is offset of T, not X. i should be aligned to X
+    template <typename X,
+              bool oob_conditional_check = true,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE void set(index_t i, bool is_valid_element, const X& x)
+    {
+        // X contains multiple T
+        constexpr index_t scalar_per_t_vector = vector_traits<remove_cvref_t<T>>::vector_size;
+
+        constexpr index_t scalar_per_x_vector = vector_traits<remove_cvref_t<X>>::vector_size;
+
+        static_assert(scalar_per_x_vector % scalar_per_t_vector == 0,
+                      "wrong! X should contain multiple T");
+
+#if CK_TILE_USE_AMD_BUFFER_STORE
+        bool constexpr use_amd_buffer_addressing = true;
+#else
+        bool constexpr use_amd_buffer_addressing = false;
+#endif
+
+        if constexpr(use_amd_buffer_addressing)
+        {
+            constexpr index_t t_per_x = scalar_per_x_vector / scalar_per_t_vector;
+
+            amd_buffer_store<remove_cvref_t<T>, t_per_x, Coherence>(
+                x, p_data_, i, is_valid_element, buffer_size_);
+        }
+        else
+        {
+            if(is_valid_element)
+            {
+#if CK_TILE_EXPERIMENTAL_USE_MEMCPY_FOR_VECTOR_ACCESS
+                X tmp = x;
+
+                __builtin_memcpy(&(p_data_[i]), &tmp, sizeof(X));
+#else
+                *c_style_pointer_cast<X*>(&p_data_[i]) = x;
+#endif
+            }
+        }
+    }
+
+    // i is offset of T, not X. i should be aligned to X
+    template <typename X,
+              bool oob_conditional_check = true,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE void set_raw(index_t i, bool is_valid_element, const X& x)
+    {
+        // X contains multiple T
+        constexpr index_t scalar_per_t_vector = vector_traits<remove_cvref_t<T>>::vector_size;
+
+        constexpr index_t scalar_per_x_vector = vector_traits<remove_cvref_t<X>>::vector_size;
+
+        static_assert(scalar_per_x_vector % scalar_per_t_vector == 0,
+                      "wrong! X should contain multiple T");
+
+        constexpr index_t t_per_x = scalar_per_x_vector / scalar_per_t_vector;
+        amd_buffer_store_raw<remove_cvref_t<T>, t_per_x, Coherence, oob_conditional_check>(
+            x, p_data_, i, is_valid_element, buffer_size_);
+    }
+
+    template <typename X,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE void atomic_add(index_t i, bool is_valid_element, const X& x)
+    {
+        using scalar_t = typename vector_traits<remove_cvref_t<T>>::scalar_type;
+
+        // X contains multiple T
+        constexpr index_t scalar_per_t_vector = vector_traits<remove_cvref_t<T>>::vector_size;
+
+        constexpr index_t scalar_per_x_vector = vector_traits<remove_cvref_t<X>>::vector_size;
+
+        static_assert(scalar_per_x_vector % scalar_per_t_vector == 0,
+                      "wrong! X should contain multiple T");
+
+        static_assert(get_address_space() == address_space_enum::global, "only support global mem");
+
+#if CK_TILE_USE_AMD_BUFFER_ATOMIC_ADD_INTEGER && CK_TILE_USE_AMD_BUFFER_ATOMIC_ADD_FLOAT
+        bool constexpr use_amd_buffer_addressing =
+            std::is_same_v<remove_cvref_t<scalar_t>, int32_t> ||
+            std::is_same_v<remove_cvref_t<scalar_t>, float> ||
+            (std::is_same_v<remove_cvref_t<scalar_t>, half_t> && scalar_per_x_vector % 2 == 0);
+#elif CK_TILE_USE_AMD_BUFFER_ATOMIC_ADD_INTEGER && (!CK_TILE_USE_AMD_BUFFER_ATOMIC_ADD_FLOAT)
+        bool constexpr use_amd_buffer_addressing =
+            std::is_same_v<remove_cvref_t<scalar_t>, int32_t>;
+#elif(!CK_TILE_USE_AMD_BUFFER_ATOMIC_ADD_INTEGER) && CK_TILE_USE_AMD_BUFFER_ATOMIC_ADD_FLOAT
+        bool constexpr use_amd_buffer_addressing =
+            std::is_same_v<remove_cvref_t<scalar_t>, float> ||
+            (std::is_same_v<remove_cvref_t<scalar_t>, half_t> && scalar_per_x_vector % 2 == 0);
+#else
+        bool constexpr use_amd_buffer_addressing = false;
+#endif
+
+        if constexpr(use_amd_buffer_addressing)
+        {
+            constexpr index_t t_per_x = scalar_per_x_vector / scalar_per_t_vector;
+
+            amd_buffer_atomic_add<remove_cvref_t<T>, t_per_x>(
+                x, p_data_, i, is_valid_element, buffer_size_);
+        }
+        else
+        {
+            if(is_valid_element)
+            {
+                atomic_add<X>(c_style_pointer_cast<X*>(&p_data_[i]), x);
+            }
+        }
+    }
+
+    template <typename X,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE void atomic_max(index_t i, bool is_valid_element, const X& x)
+    {
+        // X contains multiple T
+        constexpr index_t scalar_per_t_vector = vector_traits<remove_cvref_t<T>>::vector_size;
+
+        constexpr index_t scalar_per_x_vector = vector_traits<remove_cvref_t<X>>::vector_size;
+
+        static_assert(scalar_per_x_vector % scalar_per_t_vector == 0,
+                      "wrong! X should contain multiple T");
+
+        static_assert(get_address_space() == address_space_enum::global, "only support global mem");
+
+#if CK_TILE_USE_AMD_BUFFER_ATOMIC_MAX_FLOAT64
+        using scalar_t = typename vector_traits<remove_cvref_t<T>>::scalar_type;
+        bool constexpr use_amd_buffer_addressing = std::is_same_v<remove_cvref_t<scalar_t>, double>;
+#else
+        bool constexpr use_amd_buffer_addressing = false;
+#endif
+
+        if constexpr(use_amd_buffer_addressing)
+        {
+            constexpr index_t t_per_x = scalar_per_x_vector / scalar_per_t_vector;
+
+            amd_buffer_atomic_max<remove_cvref_t<T>, t_per_x>(
+                x, p_data_, i, is_valid_element, buffer_size_);
+        }
+        else if(is_valid_element)
+        {
+            atomic_max<X>(c_style_pointer_cast<X*>(&p_data_[i]), x);
+        }
+    }
+
+    // FIXME: remove
+    CK_TILE_DEVICE static constexpr bool is_static_buffer() { return false; }
+
+    // FIXME: remove
+    CK_TILE_DEVICE static constexpr bool is_dynamic_buffer() { return true; }
+
+    CK_TILE_HOST_DEVICE void print() const
+    {
+        printf("buffer_view{");
+
+        // AddressSpace
+        printf("AddressSpace: Global, ");
+
+        // p_data_
+        printf("p_data_: %p, ", static_cast<void*>(const_cast<remove_cvref_t<T>*>(p_data_)));
+
+        // buffer_size_
+        printf("buffer_size_: ");
+        print(buffer_size_);
+        printf(", ");
+
+        // invalid_element_value_
+        printf("invalid_element_value_: ");
+        print(invalid_element_value_);
+
+        printf("}");
+    }
+};
+
+// Address Space: LDS
+// T may be scalar or vector
+// X may be scalar or vector
+// T and X have same scalar type
+// X contains multiple T
+// FIXME: InvalidElementUseNumericalZeroValue and invalid_element_value_ should be a property of
+//        transforms of tensor_view/Tensor
+template <typename T, typename BufferSizeType, bool InvalidElementUseNumericalZeroValue>
+struct buffer_view<address_space_enum::lds,
+                   T,
+                   BufferSizeType,
+                   InvalidElementUseNumericalZeroValue,
+                   amd_buffer_coherence_enum::coherence_default>
+{
+    using type = T;
+
+    T* p_data_ = nullptr;
+    BufferSizeType buffer_size_;
+    remove_cvref_t<T> invalid_element_value_ = T{0};
+
+    CK_TILE_HOST_DEVICE constexpr buffer_view()
+        : p_data_{}, buffer_size_{}, invalid_element_value_{}
+    {
+    }
+
+    CK_TILE_HOST_DEVICE constexpr buffer_view(T* p_data, BufferSizeType buffer_size)
+        : p_data_{p_data}, buffer_size_{buffer_size}, invalid_element_value_{0}
+    {
+    }
+
+    CK_TILE_HOST_DEVICE constexpr buffer_view(T* p_data,
+                                              BufferSizeType buffer_size,
+                                              T invalid_element_value)
+        : p_data_{p_data}, buffer_size_{buffer_size}, invalid_element_value_{invalid_element_value}
+    {
+    }
+
+    CK_TILE_DEVICE static constexpr address_space_enum get_address_space()
+    {
+        return address_space_enum::lds;
+    }
+
+    // i is offset of T
+    // FIXME: doesn't do is_valid check
+    CK_TILE_DEVICE constexpr const T& operator[](index_t i) const { return p_data_[i]; }
+
+    // i is offset of T
+    // FIXME: doesn't do is_valid check
+    CK_TILE_DEVICE constexpr T& operator()(index_t i) { return p_data_[i]; }
+
+    // i is offset of T, not X. i should be aligned to X
+    template <typename X,
+              bool oob_conditional_check = true,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE constexpr auto
+    get(index_t i, bool is_valid_element, bool_constant<oob_conditional_check> = {}) const
+    {
+        // X contains multiple T
+        constexpr index_t scalar_per_t_vector = vector_traits<remove_cvref_t<T>>::vector_size;
+
+        constexpr index_t scalar_per_x_vector = vector_traits<remove_cvref_t<X>>::vector_size;
+
+        static_assert(scalar_per_x_vector % scalar_per_t_vector == 0,
+                      "wrong! X should contain multiple T");
+
+        if(is_valid_element)
+        {
+#if CK_TILE_EXPERIMENTAL_USE_MEMCPY_FOR_VECTOR_ACCESS
+            X tmp;
+
+            __builtin_memcpy(&tmp, &(p_data_[i]), sizeof(X));
+
+            return tmp;
+#else
+            using buf_t = ext_vector_t<typename vector_traits<remove_cvref_t<T>>::scalar_type,
+                                       scalar_per_t_vector * scalar_per_x_vector>;
+            // using buf_t = ushort __attribute__((ext_vector_type(8)));
+            auto rtn = *c_style_pointer_cast<const buf_t*>(&p_data_[i]);
+            return bit_cast<X>(rtn);
+#endif
+        }
+        else
+        {
+            if constexpr(InvalidElementUseNumericalZeroValue)
+            {
+                return X{numeric<remove_cvref_t<T>>::zero()};
+            }
+            else
+            {
+                return X{invalid_element_value_};
+            }
+        }
+    }
+
+    // i is offset of T, not X. i should be aligned to X
+    template <memory_operation_enum Op,
+              typename X,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE void update(index_t i, bool is_valid_element, const X& x)
+    {
+        if constexpr(Op == memory_operation_enum::set)
+        {
+            this->template set<X>(i, is_valid_element, x);
+        }
+        // FIXME: remove memory_operation_enum::add
+        else if constexpr(Op == memory_operation_enum::add)
+        {
+            auto tmp = this->template get<X>(i, is_valid_element);
+            this->template set<X>(i, is_valid_element, x + tmp);
+        }
+    }
+
+    // i is offset of T, not X. i should be aligned to X
+    template <typename X,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE void set(index_t i, bool is_valid_element, const X& x)
+    {
+        // X contains multiple T
+        constexpr index_t scalar_per_t_vector = vector_traits<remove_cvref_t<T>>::vector_size;
+
+        constexpr index_t scalar_per_x_vector = vector_traits<remove_cvref_t<X>>::vector_size;
+
+        static_assert(scalar_per_x_vector % scalar_per_t_vector == 0,
+                      "wrong! X should contain multiple T");
+
+#if CK_TILE_WORKAROUND_SWDEV_XXXXXX_INT8_DS_WRITE_ISSUE
+        bool constexpr workaround_int8_ds_write_issue = true;
+#else
+        bool constexpr workaround_int8_ds_write_issue = false;
+#endif
+
+        if constexpr(std::is_same<typename vector_traits<remove_cvref_t<T>>::scalar_type,
+                                  int8_t>::value &&
+                     workaround_int8_ds_write_issue)
+        {
+            if(is_valid_element)
+            {
+                // HACK: compiler would lower IR "store<i8, 16> address_space(3)" into inefficient
+                // ISA, so I try to let compiler emit IR "store<i32, 4>" which would be lower to
+                // ds_write_b128
+                // TODO: remove this after compiler fix
+                static_assert((std::is_same<remove_cvref_t<T>, int8_t>::value &&
+                               std::is_same<remove_cvref_t<X>, int8_t>::value) ||
+                                  (std::is_same<remove_cvref_t<T>, int8_t>::value &&
+                                   std::is_same<remove_cvref_t<X>, int8x2_t>::value) ||
+                                  (std::is_same<remove_cvref_t<T>, int8_t>::value &&
+                                   std::is_same<remove_cvref_t<X>, int8x4_t>::value) ||
+                                  (std::is_same<remove_cvref_t<T>, int8_t>::value &&
+                                   std::is_same<remove_cvref_t<X>, int8x8_t>::value) ||
+                                  (std::is_same<remove_cvref_t<T>, int8_t>::value &&
+                                   std::is_same<remove_cvref_t<X>, int8x16_t>::value) ||
+                                  (std::is_same<remove_cvref_t<T>, int8x4_t>::value &&
+                                   std::is_same<remove_cvref_t<X>, int8x4_t>::value) ||
+                                  (std::is_same<remove_cvref_t<T>, int8x8_t>::value &&
+                                   std::is_same<remove_cvref_t<X>, int8x8_t>::value) ||
+                                  (std::is_same<remove_cvref_t<T>, int8x16_t>::value &&
+                                   std::is_same<remove_cvref_t<X>, int8x16_t>::value),
+                              "wrong! not implemented for this combination, please add "
+                              "implementation");
+
+                if constexpr(std::is_same<remove_cvref_t<T>, int8_t>::value &&
+                             std::is_same<remove_cvref_t<X>, int8_t>::value)
+                {
+                    // HACK: cast pointer of x is bad
+                    // TODO: remove this after compiler fix
+                    *c_style_pointer_cast<int8_t*>(&p_data_[i]) =
+                        *c_style_pointer_cast<const int8_t*>(&x);
+                }
+                else if constexpr(std::is_same<remove_cvref_t<T>, int8_t>::value &&
+                                  std::is_same<remove_cvref_t<X>, int8x2_t>::value)
+                {
+                    // HACK: cast pointer of x is bad
+                    // TODO: remove this after compiler fix
+                    *c_style_pointer_cast<int16_t*>(&p_data_[i]) =
+                        *c_style_pointer_cast<const int16_t*>(&x);
+                }
+                else if constexpr(std::is_same<remove_cvref_t<T>, int8_t>::value &&
+                                  std::is_same<remove_cvref_t<X>, int8x4_t>::value)
+                {
+                    // HACK: cast pointer of x is bad
+                    // TODO: remove this after compiler fix
+                    *c_style_pointer_cast<int32_t*>(&p_data_[i]) =
+                        *c_style_pointer_cast<const int32_t*>(&x);
+                }
+                else if constexpr(std::is_same<remove_cvref_t<T>, int8_t>::value &&
+                                  std::is_same<remove_cvref_t<X>, int8x8_t>::value)
+                {
+                    // HACK: cast pointer of x is bad
+                    // TODO: remove this after compiler fix
+                    *c_style_pointer_cast<int32x2_t*>(&p_data_[i]) =
+                        *c_style_pointer_cast<const int32x2_t*>(&x);
+                }
+                else if constexpr(std::is_same<remove_cvref_t<T>, int8_t>::value &&
+                                  std::is_same<remove_cvref_t<X>, int8x16_t>::value)
+                {
+                    // HACK: cast pointer of x is bad
+                    // TODO: remove this after compiler fix
+                    *c_style_pointer_cast<int32x4_t*>(&p_data_[i]) =
+                        *c_style_pointer_cast<const int32x4_t*>(&x);
+                }
+                else if constexpr(std::is_same<remove_cvref_t<T>, int8x4_t>::value &&
+                                  std::is_same<remove_cvref_t<X>, int8x4_t>::value)
+                {
+                    // HACK: cast pointer of x is bad
+                    // TODO: remove this after compiler fix
+                    *c_style_pointer_cast<int32_t*>(&p_data_[i]) =
+                        *c_style_pointer_cast<const int32_t*>(&x);
+                }
+                else if constexpr(std::is_same<remove_cvref_t<T>, int8x8_t>::value &&
+                                  std::is_same<remove_cvref_t<X>, int8x8_t>::value)
+                {
+                    // HACK: cast pointer of x is bad
+                    // TODO: remove this after compiler fix
+                    *c_style_pointer_cast<int32x2_t*>(&p_data_[i]) =
+                        *c_style_pointer_cast<const int32x2_t*>(&x);
+                }
+                else if constexpr(std::is_same<remove_cvref_t<T>, int8x16_t>::value &&
+                                  std::is_same<remove_cvref_t<X>, int8x16_t>::value)
+                {
+                    // HACK: cast pointer of x is bad
+                    // TODO: remove this after compiler fix
+                    *c_style_pointer_cast<int32x4_t*>(&p_data_[i]) =
+                        *c_style_pointer_cast<const int32x4_t*>(&x);
+                }
+            }
+        }
+        else
+        {
+            if(is_valid_element)
+            {
+#if CK_TILE_EXPERIMENTAL_USE_MEMCPY_FOR_VECTOR_ACCESS
+                X tmp = x;
+
+                __builtin_memcpy(&(p_data_[i]), &tmp, sizeof(X));
+#else
+                using buf_t = ext_vector_t<typename vector_traits<remove_cvref_t<T>>::scalar_type,
+                                           scalar_per_t_vector * scalar_per_x_vector>;
+
+                *c_style_pointer_cast<buf_t*>(&p_data_[i]) = reinterpret_cast<const buf_t&>(x);
+#endif
+            }
+        }
+    }
+
+    // FIXME: remove
+    CK_TILE_DEVICE static constexpr bool is_static_buffer() { return false; }
+
+    // FIXME: remove
+    CK_TILE_DEVICE static constexpr bool is_dynamic_buffer() { return true; }
+
+    CK_TILE_HOST_DEVICE void print() const
+    {
+        printf("buffer_view{");
+
+        // AddressSpace
+        printf("AddressSpace: Lds, ");
+
+        // p_data_
+        printf("p_data_: %p, ", static_cast<void*>(const_cast<remove_cvref_t<T>*>(p_data_)));
+
+        // buffer_size_
+        printf("buffer_size_: ");
+        print(buffer_size_);
+        printf(", ");
+
+        // invalid_element_value_
+        printf("invalid_element_value_: ");
+        print(invalid_element_value_);
+
+        printf("}");
+    }
+};
+
+// Address Space: Vgpr
+// T may be scalar or vector
+// X may be scalar or vector
+// T and X have same scalar type
+// X contains multiple T
+// FIXME: InvalidElementUseNumericalZeroValue and invalid_element_value_ should be a property of
+//        transforms of tensor_view/Tensor
+template <typename T, typename BufferSizeType, bool InvalidElementUseNumericalZeroValue>
+struct buffer_view<address_space_enum::vgpr,
+                   T,
+                   BufferSizeType,
+                   InvalidElementUseNumericalZeroValue,
+                   amd_buffer_coherence_enum::coherence_default>
+{
+    using type = T;
+
+    T* p_data_ = nullptr;
+    BufferSizeType buffer_size_;
+    remove_cvref_t<T> invalid_element_value_ = T{0};
+
+    CK_TILE_HOST_DEVICE constexpr buffer_view()
+        : p_data_{}, buffer_size_{}, invalid_element_value_{}
+    {
+    }
+
+    CK_TILE_HOST_DEVICE constexpr buffer_view(T* p_data, BufferSizeType buffer_size)
+        : p_data_{p_data}, buffer_size_{buffer_size}, invalid_element_value_{0}
+    {
+    }
+
+    CK_TILE_HOST_DEVICE constexpr buffer_view(T* p_data,
+                                              BufferSizeType buffer_size,
+                                              T invalid_element_value)
+        : p_data_{p_data}, buffer_size_{buffer_size}, invalid_element_value_{invalid_element_value}
+    {
+    }
+
+    CK_TILE_DEVICE static constexpr address_space_enum get_address_space()
+    {
+        return address_space_enum::vgpr;
+    }
+
+    // i is offset of T
+    // FIXME: doesn't do is_valid check
+    CK_TILE_DEVICE constexpr const T& operator[](index_t i) const { return p_data_[i]; }
+
+    // i is offset of T
+    // FIXME: doesn't do is_valid check
+    CK_TILE_DEVICE constexpr T& operator()(index_t i) { return p_data_[i]; }
+
+    // i is offset of T, not X. i should be aligned to X
+    template <typename X,
+              bool oob_conditional_check = true,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE constexpr auto
+    get(index_t i, bool is_valid_element, bool_constant<oob_conditional_check> = {}) const
+    {
+        // X contains multiple T
+        constexpr index_t scalar_per_t_vector = vector_traits<remove_cvref_t<T>>::vector_size;
+
+        constexpr index_t scalar_per_x_vector = vector_traits<remove_cvref_t<X>>::vector_size;
+
+        static_assert(scalar_per_x_vector % scalar_per_t_vector == 0,
+                      "wrong! X should contain multiple T");
+
+        if(is_valid_element)
+        {
+#if CK_TILE_EXPERIMENTAL_USE_MEMCPY_FOR_VECTOR_ACCESS
+            X tmp;
+
+            __builtin_memcpy(&tmp, &(p_data_[i]), sizeof(X));
+
+            return tmp;
+#else
+            return *c_style_pointer_cast<const X*>(&p_data_[i]);
+#endif
+        }
+        else
+        {
+            if constexpr(InvalidElementUseNumericalZeroValue)
+            {
+                return X{numeric<remove_cvref_t<T>>::zero()};
+            }
+            else
+            {
+                return X{invalid_element_value_};
+            }
+        }
+    }
+
+    // i is offset of T, not X. i should be aligned to X
+    template <memory_operation_enum Op,
+              typename X,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE void update(index_t i, bool is_valid_element, const X& x)
+    {
+        if constexpr(Op == memory_operation_enum::set)
+        {
+            this->template set<X>(i, is_valid_element, x);
+        }
+        // FIXME: remove memory_operation_enum::add
+        else if constexpr(Op == memory_operation_enum::add)
+        {
+            auto tmp = this->template get<X>(i, is_valid_element);
+            this->template set<X>(i, is_valid_element, x + tmp);
+        }
+    }
+
+    // i is offset of T, not X. i should be aligned to X
+    template <typename X,
+              typename std::enable_if<
+                  std::is_same<typename vector_traits<remove_cvref_t<X>>::scalar_type,
+                               typename vector_traits<remove_cvref_t<T>>::scalar_type>::value,
+                  bool>::type = false>
+    CK_TILE_DEVICE void set(index_t i, bool is_valid_element, const X& x)
+    {
+        // X contains multiple T
+        constexpr index_t scalar_per_t_vector = vector_traits<remove_cvref_t<T>>::vector_size;
+
+        constexpr index_t scalar_per_x_vector = vector_traits<remove_cvref_t<X>>::vector_size;
+
+        static_assert(scalar_per_x_vector % scalar_per_t_vector == 0,
+                      "wrong! X should contain multiple T");
+
+        if(is_valid_element)
+        {
+#if CK_TILE_EXPERIMENTAL_USE_MEMCPY_FOR_VECTOR_ACCESS
+            X tmp = x;
+
+            __builtin_memcpy(&(p_data_[i]), &tmp, sizeof(X));
+#else
+            *c_style_pointer_cast<X*>(&p_data_[i]) = x;
+#endif
+        }
+    }
+
+    // FIXME: remove
+    CK_TILE_DEVICE static constexpr bool is_static_buffer() { return false; }
+
+    // FIXME: remove
+    CK_TILE_DEVICE static constexpr bool is_dynamic_buffer() { return true; }
+
+    CK_TILE_HOST_DEVICE void print() const
+    {
+        printf("buffer_view{");
+
+        // AddressSpace
+        printf("AddressSpace: Vgpr, ");
+
+        // p_data_
+        printf("p_data_: %p, ", static_cast<void*>(const_cast<remove_cvref_t<T>*>(p_data_)));
+
+        // buffer_size_
+        printf("buffer_size_: ");
+        print(buffer_size_);
+        printf(", ");
+
+        // invalid_element_value_
+        printf("invalid_element_value_: ");
+        print(invalid_element_value_);
+
+        printf("}");
+    }
+};
+
+template <address_space_enum BufferAddressSpace,
+          amd_buffer_coherence_enum Coherence = amd_buffer_coherence_enum::coherence_default,
+          typename T,
+          typename BufferSizeType>
+CK_TILE_HOST_DEVICE constexpr auto make_buffer_view(T* p, BufferSizeType buffer_size)
+{
+    return buffer_view<BufferAddressSpace, T, BufferSizeType, true, Coherence>{p, buffer_size};
+}
+
+template <address_space_enum BufferAddressSpace,
+          amd_buffer_coherence_enum Coherence = amd_buffer_coherence_enum::coherence_default,
+          typename T,
+          typename BufferSizeType,
+          typename X,
+          typename std::enable_if<std::is_same<remove_cvref_t<T>, remove_cvref_t<X>>::value,
+                                  bool>::type = false>
+CK_TILE_HOST_DEVICE constexpr auto
+make_buffer_view(T* p, BufferSizeType buffer_size, X invalid_element_value)
+{
+    return buffer_view<BufferAddressSpace, T, BufferSizeType, false, Coherence>{
+        p, buffer_size, invalid_element_value};
+}
+
+} // namespace ck_tile

include/ck_tile/core/tensor/load_tile.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2023, Advanced Micro Devices, Inc. All rights reserved.
+
+#pragma once
+
+#include "ck_tile/core/config.hpp"
+#include "ck_tile/core/numeric/integer.hpp"
+#include "ck_tile/core/numeric/integral_constant.hpp"
+#include "ck_tile/core/algorithm/coordinate_transform.hpp"
+#include "ck_tile/core/container/container_helper.hpp"
+#include "ck_tile/core/numeric/math.hpp"
+#include "ck_tile/core/tensor/tile_window.hpp"
+#include "ck_tile/core/utility/type_traits.hpp"
+#include "ck_tile/core/tensor/tile_window.hpp"
+#include "ck_tile/core/tensor/null_tile_window.hpp"
+#include "ck_tile/core/tensor/null_tensor.hpp"
+
+namespace ck_tile {
+
+template <typename BottomTensorView_,
+          typename WindowLengths_,
+          typename TileDistribution_,
+          index_t NumCoord,
+          bool oob_conditional_check = true>
+CK_TILE_DEVICE auto load_tile(const tile_window_with_static_distribution<BottomTensorView_,
+                                                                         WindowLengths_,
+                                                                         TileDistribution_,
+                                                                         NumCoord>& tile_window,
+                              bool_constant<oob_conditional_check> = {})
+{
+    return tile_window.load(bool_constant<oob_conditional_check>{});
+}
+
+template <typename T,
+          typename BottomTensorView_,
+          typename WindowLengths_,
+          typename TileDistribution_,
+          index_t NumCoord,
+          bool oob_conditional_check = true>
+CK_TILE_DEVICE auto load_tile_raw(T& tile,
+                                  const tile_window_with_static_distribution<BottomTensorView_,
+                                                                             WindowLengths_,
+                                                                             TileDistribution_,
+                                                                             NumCoord>& tile_window,
+                                  bool_constant<oob_conditional_check> = {})
+{
+    tile_window.load_raw(tile, bool_constant<oob_conditional_check>{});
+}
+
+template <typename LdsTileWindow_,
+          typename BottomTensorView_,
+          typename WindowLengths_,
+          typename TileDistribution_,
+          index_t NumCoord>
+CK_TILE_DEVICE auto
+async_load_tile_raw(LdsTileWindow_&& lds_tile,
+                    const tile_window_with_static_distribution<BottomTensorView_,
+                                                               WindowLengths_,
+                                                               TileDistribution_,
+                                                               NumCoord>& tile_window)
+{
+    return tile_window.async_load(lds_tile);
+}
+
+CK_TILE_DEVICE auto async_load_fence(index_t cnt = 0)
+{
+    asm volatile("s_waitcnt vmcnt(%0)" : : "n"(cnt) : "memory");
+}
+
+template <typename WindowLengths>
+CK_TILE_DEVICE auto load_tile(const null_tile_window<WindowLengths>&)
+{
+    return null_tensor{};
+}
+
+template <typename T, typename WindowLengths>
+CK_TILE_DEVICE auto load_tile_raw(T& /*null_tile*/, const null_tile_window<WindowLengths>&)
+{
+}
+
+} // namespace ck_tile

include/ck_tile/core/tensor/null_tensor.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2023, Advanced Micro Devices, Inc. All rights reserved.
+
+#pragma once
+
+namespace ck_tile {
+
+struct null_tensor
+{
+};
+
+} // namespace ck_tile