decode.cpp

#include "common.h"
#include "vec.h"

namespace {

// [NOTE] TODO list for this kernel:
//   1. tune the value for BLOCK_N
//   2. planning for {batches, num_heads, num_kv_splits}
//      and use actual num_kv_splits for small seq length
//   3. try fast impl of `.tanh()`
//   4. provide amx kernel for index_gemm_kernel_nn when M = 16
//

inline void fill_stub(float* __restrict__ out, float val, int64_t size) {
  using Vec = at::vec::Vectorized<float>;
  const Vec data_vec(val);
  at::vec::map<float>([data_vec](Vec out) { return out = data_vec; }, out, out, size);
}

template <typename scalar_t>
inline void copy_stub(scalar_t* __restrict__ out, const float* __restrict__ acc, float s, int64_t size) {
  using bVec = at::vec::Vectorized<scalar_t>;
  using fVec = at::vec::Vectorized<float>;
  const fVec s_fvec = fVec(s);
  int64_t d = 0;
  for (; d <= size - bVec::size(); d += bVec::size()) {
    fVec a_fvec0 = fVec::loadu(acc + d) * s_fvec;
    fVec a_fvec1 = fVec::loadu(acc + d + fVec::size()) * s_fvec;
    bVec out_bvec = convert_from_float_ext<scalar_t>(a_fvec0, a_fvec1);
    out_bvec.store(out + d);
  }
  for (; d < size; ++d) {
    out[d] = static_cast<scalar_t>(acc[d] * s);
  }
}

template <typename scalar_t>
inline void copy_stub(scalar_t* __restrict__ out, const scalar_t* __restrict__ src, int64_t size) {
  using bVec = at::vec::Vectorized<scalar_t>;
  int64_t d = 0;
  for (; d <= size - bVec::size(); d += bVec::size()) {
    bVec out_bvec = bVec::loadu(src + d);
    out_bvec.store(out + d);
  }
  for (; d < size; ++d) {
    out[d] = src[d];
  }
}

// GEMM handles query @ key (indexed) x scale
//   A : [M, K]
//   B : [N, K] indexed
//   C : [M, N]
//
template <typename scalar_t, typename index_t, int BLOCK_M, int BLOCK_N>
struct tinygemm_kernel_nt {
  static inline void apply(
      const scalar_t* __restrict__ A,
      const scalar_t* __restrict__ B,
      float* __restrict__ C,
      const index_t* __restrict__ indices,
      float scale,
      int64_t lda,
      int64_t ldb,
      int64_t ldc,
      int64_t K,
      int64_t max_tokens) {
    for (int64_t m = 0; m < BLOCK_M; ++m) {
      for (int64_t n = 0; n < BLOCK_N; ++n) {
        float sum = 0.f;
        int64_t b_idx = indices[n];
        TORCH_CHECK(b_idx < max_tokens, "token index out of scope!");
        for (int64_t k = 0; k < K; ++k) {
          sum += scale * static_cast<float>(A[m * lda + k]) * static_cast<float>(B[b_idx * ldb + k]);
        }
        C[m * ldc + n] = sum;
      }
    }
  }
};

#if defined(CPU_CAPABILITY_AVX512)
template <typename index_t, int BLOCK_M, int BLOCK_N>
struct tinygemm_kernel_nt<at::BFloat16, index_t, BLOCK_M, BLOCK_N> {
  static inline void apply(
      const at::BFloat16* __restrict__ A,
      const at::BFloat16* __restrict__ B,
      float* __restrict__ C,
      const index_t* __restrict__ indices,
      float scale,
      int64_t lda,
      int64_t ldb,
      int64_t ldc,
      int64_t K,
      int64_t max_tokens) {
    constexpr int ROWS = BLOCK_M;
    constexpr int COLS = BLOCK_N;

    __m512bh va;
    __m512bh vb[COLS];
    __m512 vc[ROWS * COLS];
    __m512 vscale = _mm512_set1_ps(scale);

    auto loadc = [&](auto i) { vc[i] = _mm512_setzero_ps(); };
    Unroll<ROWS * COLS>{}(loadc);

    // for main loop
    auto compute = [&](auto i, int64_t k) {
      constexpr int row = i / COLS;
      constexpr int col = i % COLS;

      if constexpr (col == 0) {
        va = (__m512bh)(_mm512_loadu_si512(A + row * lda + k));
      }
      if constexpr (row == 0) {
        if constexpr (col + 1 < COLS) {
          int64_t b_idx_prefetch = indices[col + 1];
          _mm_prefetch(B + b_idx_prefetch * ldb + k, _MM_HINT_T0);
        }
        int64_t b_idx = indices[col];
        TORCH_CHECK(b_idx < max_tokens, "token index out of scope!");
        vb[col] = (__m512bh)(_mm512_loadu_si512(B + b_idx * ldb + k));
      }
      vc[i] = _mm512_dpbf16_ps(vc[i], va, vb[col]);
    };

    // for remainder
    auto compute2 = [&](auto i, int64_t k, __mmask32 mask) {
      constexpr int row = i / COLS;
      constexpr int col = i % COLS;

      if constexpr (col == 0) {
        va = (__m512bh)(_mm512_maskz_loadu_epi16(mask, A + row * lda + k));
      }
      if constexpr (row == 0) {
        int64_t b_idx = indices[col];
        TORCH_CHECK(b_idx < max_tokens, "token index out of scope!");
        vb[col] = (__m512bh)(_mm512_maskz_loadu_epi16(mask, B + b_idx * ldb + k));
      }
      vc[i] = _mm512_dpbf16_ps(vc[i], va, vb[col]);
    };

    int64_t k = 0;
    for (; k <= K - 32; k += 32) {
      Unroll<ROWS * COLS>{}(compute, k);
    }
    int64_t count = K - k;
    if (count > 0) {
      __mmask32 mask = (1ULL << count) - 1;
      Unroll<ROWS * COLS>{}(compute2, k, mask);
    }

    auto storec = [&](auto i) {
      constexpr int row = i / COLS;
      constexpr int col = i % COLS;
      C[row * ldc + col] = _mm512_reduce_add_ps(_mm512_mul_ps(vc[i], vscale));
    };
    Unroll<ROWS * COLS>{}(storec);
  }
};
#endif

#define LAUNCH_TINYGEMM_KERNEL_NT(MB_SIZE, NB_SIZE)               \
  tinygemm_kernel_nt<scalar_t, index_t, MB_SIZE, NB_SIZE>::apply( \
      A + mb_start * lda, B, C + mb_start * ldc + nb_start, indices + nb_start, scale, lda, ldb, ldc, K, max_tokens);

// this is used when N isn't multiple of 16,
// N corresponds to `head_size_v` which should be 16x
template <typename scalar_t, typename index_t>
inline void tinygemm_kernel_nn_scalar(
    const float* __restrict__ A,
    const scalar_t* __restrict__ B,
    float* __restrict__ C,
    const index_t* __restrict__ indices,
    const float* __restrict__ scale,
    int64_t M,
    int64_t N,
    int64_t K,
    int64_t lda,
    int64_t ldb,
    int64_t ldc,
    int64_t max_tokens) {
  for (int64_t m = 0; m < M; ++m) {
    for (int64_t n = 0; n < N; ++n) {
      C[m * ldc + n] *= scale[m];
      for (int64_t k = 0; k < K; ++k) {
        int64_t b_idx = indices[k];
        TORCH_CHECK(b_idx < max_tokens, "token index out of scope!");
        C[m * ldc + n] += A[m * lda + k] * static_cast<float>(B[b_idx * ldb + n]);
      }
    }
  }
}

// GEMM handles v' * scale + attn @ value (indexed)
//   A : [M, K]
//   B : [K, N] indexed
//   C ：[M, N]
//
template <typename scalar_t, typename index_t, int BLOCK_M, int BLOCK_N>
struct tinygemm_kernel_nn {
  static inline void apply(
      const float* __restrict__ A,
      const scalar_t* __restrict__ B,
      float* __restrict__ C,
      const index_t* __restrict__ indices,
      const float* __restrict__ scale,
      int64_t lda,
      int64_t ldb,
      int64_t ldc,
      int64_t K,
      int64_t max_tokens) {
    tinygemm_kernel_nn_scalar(A, B, C, indices, scale, BLOCK_M, BLOCK_N, K, lda, ldb, ldc, max_tokens);
  }
};

#if defined(CPU_CAPABILITY_AVX512)
template <typename index_t, int BLOCK_M, int BLOCK_N>
struct tinygemm_kernel_nn<at::BFloat16, index_t, BLOCK_M, BLOCK_N> {
  static inline void apply(
      const float* __restrict__ A,
      const at::BFloat16* __restrict__ B,
      float* __restrict__ C,
      const index_t* __restrict__ indices,
      const float* __restrict__ scale,
      int64_t lda,
      int64_t ldb,
      int64_t ldc,
      int64_t K,
      int64_t max_tokens) {
    constexpr int ROWS = BLOCK_M;
    constexpr int COLS = BLOCK_N / 16;

    __m512 va;
    __m512 vb[COLS];
    __m512 vc[ROWS * COLS];
    __m512 vscale;

    auto loadc = [&](auto i) {
      constexpr int row = i / COLS;
      constexpr int col = i % COLS;
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Warray-bounds"
      if constexpr (col == 0) {
        vscale = _mm512_set1_ps(scale[row]);
      }
#pragma GCC diagnostic pop
      vc[i] = _mm512_loadu_ps(C + row * ldc + col * 16);
      vc[i] = _mm512_mul_ps(vc[i], vscale);
    };
    Unroll<ROWS * COLS>{}(loadc);

    auto compute = [&](auto i, int64_t k) {
      constexpr int row = i / COLS;
      constexpr int col = i % COLS;

      if constexpr (col == 0) {
        va = _mm512_set1_ps(A[row * lda + k]);
      }
      if constexpr (row == 0) {
        if (k + 1 < K) {
          int64_t b_idx_prefetch = indices[k + 1];
          _mm_prefetch(B + b_idx_prefetch * ldb + col * 16, _MM_HINT_T0);
        }
        int64_t b_idx = indices[k];
        TORCH_CHECK(b_idx < max_tokens, "token index out of scope!");

        // for COLS = 2, 4, 6, 8 use 512 bit load
        // for COLS = 1, 3, 5, 7 use 256 bit load
        if constexpr (COLS % 2 == 0) {
          if constexpr (col % 2 == 0) {
            __m512i b16 = _mm512_loadu_si512(reinterpret_cast<const __m512i*>(B + b_idx * ldb + col * 16));
            vb[col + 0] = CVT_BF16_TO_FP32(_mm512_extracti32x8_epi32(b16, 0));
            vb[col + 1] = CVT_BF16_TO_FP32(_mm512_extracti32x8_epi32(b16, 1));
          }
        } else {
          __m256i b16 = _mm256_loadu_si256(reinterpret_cast<const __m256i*>(B + b_idx * ldb + col * 16));
          vb[col] = CVT_BF16_TO_FP32(b16);
        }
      }
      vc[i] = _mm512_fmadd_ps(va, vb[col], vc[i]);
    };

    for (int64_t k = 0; k < K; ++k) {
      Unroll<ROWS * COLS>{}(compute, k);
    }

    auto storec = [&](auto i) {
      constexpr int row = i / COLS;
      constexpr int col = i % COLS;
      _mm512_storeu_ps(C + row * ldc + col * 16, vc[i]);
    };
    Unroll<ROWS * COLS>{}(storec);
  }
};
#endif

#define LAUNCH_TINYGEMM_KERNEL_NN(MB_SIZE, NB_SIZE)               \
  tinygemm_kernel_nn<scalar_t, index_t, MB_SIZE, NB_SIZE>::apply( \
      A + mb_start * lda,                                         \
      B + nb_start,                                               \
      C + mb_start * ldc + nb_start,                              \
      indices,                                                    \
      scale + mb_start,                                           \
      lda,                                                        \
      ldb,                                                        \
      ldc,                                                        \
      K,                                                          \
      max_tokens);

template <typename scalar_t, typename index_t>
void index_gemm_kernel_nt(
    const scalar_t* __restrict__ A,
    const scalar_t* __restrict__ B,
    float* __restrict__ C,
    const index_t* __restrict__ indices,
    float scale,
    int64_t M,
    int64_t N,
    int64_t K,
    int64_t lda,
    int64_t ldb,
    int64_t ldc,
    int64_t max_tokens) {
  // pattern: 1-8-8
  if (M == 1) {
    constexpr int64_t BLOCK_N = 8;
    const int64_t NB = div_up(N, BLOCK_N);
    int64_t mb_start = 0, lda = 1, ldc = 1;

    for (int64_t nb = 0; nb < NB; ++nb) {
      int64_t nb_start = nb * BLOCK_N;
      int64_t nb_size = std::min(BLOCK_N, N - nb_start);

      switch (nb_size) {
        case 1:
          LAUNCH_TINYGEMM_KERNEL_NT(1, 1);
          break;
        case 2:
          LAUNCH_TINYGEMM_KERNEL_NT(1, 2);
          break;
        case 3:
          LAUNCH_TINYGEMM_KERNEL_NT(1, 3);
          break;
        case 4:
          LAUNCH_TINYGEMM_KERNEL_NT(1, 4);
          break;
        case 5:
          LAUNCH_TINYGEMM_KERNEL_NT(1, 5);
          break;
        case 6:
          LAUNCH_TINYGEMM_KERNEL_NT(1, 6);
          break;
        case 7:
          LAUNCH_TINYGEMM_KERNEL_NT(1, 7);
          break;
        case 8:
          LAUNCH_TINYGEMM_KERNEL_NT(1, 8);
          break;
        default:
          TORCH_CHECK(false, "Unexpected block size, 1x", "nb_size");
      }
    }
    return;
  }

  // pattern: 1-6-24
  constexpr int64_t BLOCK_M = 4;
  constexpr int64_t BLOCK_N = 6;
  const int64_t MB = div_up(M, BLOCK_M);
  const int64_t NB = div_up(N, BLOCK_N);

  for (int64_t mb = 0; mb < MB; ++mb) {
    int64_t mb_start = mb * BLOCK_M;
    int64_t mb_size = std::min(BLOCK_M, M - mb_start);
    for (int64_t nb = 0; nb < NB; ++nb) {
      int64_t nb_start = nb * BLOCK_N;
      int64_t nb_size = std::min(BLOCK_N, N - nb_start);

      switch (mb_size << 4 | nb_size) {
        // mb_size = 1
        case 0x11:
          LAUNCH_TINYGEMM_KERNEL_NT(1, 1);
          break;
        case 0x12:
          LAUNCH_TINYGEMM_KERNEL_NT(1, 2);
          break;
        case 0x13:
          LAUNCH_TINYGEMM_KERNEL_NT(1, 3);
          break;
        case 0x14:
          LAUNCH_TINYGEMM_KERNEL_NT(1, 4);
          break;
        case 0x15:
          LAUNCH_TINYGEMM_KERNEL_NT(1, 5);
          break;
        case 0x16:
          LAUNCH_TINYGEMM_KERNEL_NT(1, 6);
          break;
        // mb_size = 2
        case 0x21:
          LAUNCH_TINYGEMM_KERNEL_NT(2, 1);
          break;
        case 0x22:
          LAUNCH_TINYGEMM_KERNEL_NT(2, 2);
          break;
        case 0x23:
          LAUNCH_TINYGEMM_KERNEL_NT(2, 3);
          break;
        case 0x24:
          LAUNCH_TINYGEMM_KERNEL_NT(2, 4);
          break;
        case 0x25:
          LAUNCH_TINYGEMM_KERNEL_NT(2, 5);
          break;
        case 0x26:
          LAUNCH_TINYGEMM_KERNEL_NT(2, 6);
          break;
        // mb_size = 3
        case 0x31:
          LAUNCH_TINYGEMM_KERNEL_NT(3, 1);
          break;
        case 0x32:
          LAUNCH_TINYGEMM_KERNEL_NT(3, 2);
          break;
        case 0x33:
          LAUNCH_TINYGEMM_KERNEL_NT(3, 3);
          break;
        case 0x34:
          LAUNCH_TINYGEMM_KERNEL_NT(3, 4);
          break;
        case 0x35:
          LAUNCH_TINYGEMM_KERNEL_NT(3, 5);
          break;
        case 0x36:
          LAUNCH_TINYGEMM_KERNEL_NT(3, 6);
          break;
        // mb_size = 4
        case 0x41:
          LAUNCH_TINYGEMM_KERNEL_NT(4, 1);
          break;
        case 0x42:
          LAUNCH_TINYGEMM_KERNEL_NT(4, 2);
          break;
        case 0x43:
          LAUNCH_TINYGEMM_KERNEL_NT(4, 3);
          break;
        case 0x44:
          LAUNCH_TINYGEMM_KERNEL_NT(4, 4);
          break;
        case 0x45:
          LAUNCH_TINYGEMM_KERNEL_NT(4, 5);
          break;
        case 0x46:
          LAUNCH_TINYGEMM_KERNEL_NT(4, 6);
          break;
        default:
          TORCH_CHECK(false, "Unexpected block size, ", mb_size, "x", "nb_size");
      }
    }
  }
}

template <typename scalar_t, typename index_t>
void index_gemm_kernel_nn(
    const float* __restrict__ A,
    const scalar_t* __restrict__ B,
    float* __restrict__ C,
    const index_t* __restrict__ indices,
    float* __restrict__ scale,
    int64_t M,
    int64_t N,
    int64_t K,
    int64_t lda,
    int64_t ldb,
    int64_t ldc,
    int64_t max_tokens) {
  constexpr int kVecSize = 16;
  if ((N & (kVecSize - 1)) != 0) {
    tinygemm_kernel_nn_scalar(A, B, C, indices, scale, M, N, K, lda, ldb, ldc, max_tokens);
    return;
  }

  // pattern: 1-8-8
  if (M == 1) {
    constexpr int64_t BLOCK_N = 8 * kVecSize;
    const int64_t NB = div_up(N, BLOCK_N);
    int64_t mb_start = 0, lda = 1, ldc = 1;

    for (int64_t nb = 0; nb < NB; ++nb) {
      int64_t nb_start = nb * BLOCK_N;
      int64_t nb_size = std::min(BLOCK_N, N - nb_start);

      switch (nb_size >> 4) {
        case 1:
          LAUNCH_TINYGEMM_KERNEL_NN(1, 16);
          break;
        case 2:
          LAUNCH_TINYGEMM_KERNEL_NN(1, 32);
          break;
        case 3:
          LAUNCH_TINYGEMM_KERNEL_NN(1, 48);
          break;
        case 4:
          LAUNCH_TINYGEMM_KERNEL_NN(1, 64);
          break;
        case 5:
          LAUNCH_TINYGEMM_KERNEL_NN(1, 80);
          break;
        case 6:
          LAUNCH_TINYGEMM_KERNEL_NN(1, 96);
          break;
        case 7:
          LAUNCH_TINYGEMM_KERNEL_NN(1, 112);
          break;
        case 8:
          LAUNCH_TINYGEMM_KERNEL_NN(1, 128);
          break;
        default:
          TORCH_CHECK(false, "Unexpected block size, 1x", "nb_size");
      }
    }
    return;
  }

  constexpr int64_t BLOCK_M = 4;
  constexpr int64_t BLOCK_N = 6 * kVecSize;
  const int64_t MB = div_up(M, BLOCK_M);
  const int64_t NB = div_up(N, BLOCK_N);

  for (int64_t mb = 0; mb < MB; ++mb) {
    int64_t mb_start = mb * BLOCK_M;
    int64_t mb_size = std::min(BLOCK_M, M - mb_start);
    for (int64_t nb = 0; nb < NB; ++nb) {
      int64_t nb_start = nb * BLOCK_N;
      int64_t nb_size = std::min(BLOCK_N, N - nb_start);

      switch (mb_size << 4 | nb_size >> 4) {
        // mb_size = 1
        case 0x11:
          LAUNCH_TINYGEMM_KERNEL_NN(1, 16);
          break;
        case 0x12:
          LAUNCH_TINYGEMM_KERNEL_NN(1, 32);
          break;
        case 0x13:
          LAUNCH_TINYGEMM_KERNEL_NN(1, 48);
          break;
        case 0x14:
          LAUNCH_TINYGEMM_KERNEL_NN(1, 64);
          break;
        case 0x15:
          LAUNCH_TINYGEMM_KERNEL_NN(1, 80);
          break;
        case 0x16:
          LAUNCH_TINYGEMM_KERNEL_NN(1, 96);
          break;
        // mb_size = 2
        case 0x21:
          LAUNCH_TINYGEMM_KERNEL_NN(2, 16);
          break;
        case 0x22:
          LAUNCH_TINYGEMM_KERNEL_NN(2, 32);
          break;
        case 0x23:
          LAUNCH_TINYGEMM_KERNEL_NN(2, 48);
          break;
        case 0x24:
          LAUNCH_TINYGEMM_KERNEL_NN(2, 64);
          break;
        case 0x25:
          LAUNCH_TINYGEMM_KERNEL_NN(2, 80);
          break;
        case 0x26:
          LAUNCH_TINYGEMM_KERNEL_NN(2, 96);
          break;
        // mb_size = 3
        case 0x31:
          LAUNCH_TINYGEMM_KERNEL_NN(3, 16);
          break;
        case 0x32:
          LAUNCH_TINYGEMM_KERNEL_NN(3, 32);
          break;
        case 0x33:
          LAUNCH_TINYGEMM_KERNEL_NN(3, 48);
          break;
        case 0x34:
          LAUNCH_TINYGEMM_KERNEL_NN(3, 64);
          break;
        case 0x35:
          LAUNCH_TINYGEMM_KERNEL_NN(3, 80);
          break;
        case 0x36:
          LAUNCH_TINYGEMM_KERNEL_NN(3, 96);
          break;
        // mb_size = 4
        case 0x41:
          LAUNCH_TINYGEMM_KERNEL_NN(4, 16);
          break;
        case 0x42:
          LAUNCH_TINYGEMM_KERNEL_NN(4, 32);
          break;
        case 0x43:
          LAUNCH_TINYGEMM_KERNEL_NN(4, 48);
          break;
        case 0x44:
          LAUNCH_TINYGEMM_KERNEL_NN(4, 64);
          break;
        case 0x45:
          LAUNCH_TINYGEMM_KERNEL_NN(4, 80);
          break;
        case 0x46:
          LAUNCH_TINYGEMM_KERNEL_NN(4, 96);
          break;
        default:
          TORCH_CHECK(false, "Unexpected block size, ", mb_size, "x", "nb_size");
      }
    }
  }
}

template <typename scalar_t, typename index_t>
void decode_attention_kernel_impl(
    scalar_t* __restrict__ output,
    float* __restrict__ attn_logits,
    const scalar_t* __restrict__ query,
    scalar_t* __restrict__ k_buffer,
    scalar_t* __restrict__ v_buffer,
    const scalar_t* __restrict__ key,
    const scalar_t* __restrict__ value,
    const int64_t* __restrict__ loc,
    const index_t* __restrict__ req_to_token,
    const int64_t* __restrict__ req_pool_indices,
    const int64_t* __restrict__ seq_lens,
    int64_t batches,
    int64_t num_heads,
    int64_t head_size,
    int64_t head_size_v,
    int64_t num_kv_splits,
    int64_t k_strideN,
    int64_t k_strideH,
    int64_t v_strideN,
    int64_t v_strideH,
    int64_t nk_strideN,
    int64_t nk_strideH,
    int64_t nv_strideN,
    int64_t nv_strideH,
    float scaling,
    float logit_cap,
    int64_t max_num_reqs,
    int64_t max_context_len,
    int64_t max_total_num_tokens) {
  at::parallel_for(0, batches * num_heads, 0, [&](int64_t begin, int64_t end) {
    int64_t bs{0}, head_id{0};
    data_index_init(begin, bs, batches, head_id, num_heads);

    for (int64_t i = begin; i < end; i++) {
      int64_t loc_val = loc[bs];
      scalar_t* k_buffer_ptr = k_buffer + loc_val * k_strideN + head_id * k_strideH;
      scalar_t* v_buffer_ptr = v_buffer + loc_val * v_strideN + head_id * v_strideH;
      const scalar_t* new_key_ptr = key + bs * nk_strideN + head_id * nk_strideH;
      const scalar_t* new_value_ptr = value + bs * nv_strideN + head_id * nv_strideH;
      copy_stub<scalar_t>(k_buffer_ptr, new_key_ptr, head_size);
      copy_stub<scalar_t>(v_buffer_ptr, new_value_ptr, head_size_v);

      // move to the next index
      data_index_step(bs, batches, head_id, num_heads);
    }
  });

  using Vec = at::vec::Vectorized<float>;

  // block length for k_buffer and v_buffer
  constexpr int64_t BLOCK_N = 256;

  // strides
  const int64_t q_strideM = num_heads * head_size;
  const int64_t q_strideH = head_size;
  const int64_t l_stride1 = num_kv_splits * (head_size_v + 1);
  const int64_t l_stride2 = head_size_v + 1;

  const bool has_logit_cap = logit_cap > 0;
  float rlogit_cap = has_logit_cap ? 1 / logit_cap : 0.f;

  // parallel on [batches, num_heads, num_kv_splits]
  at::parallel_for(0, batches * num_heads * num_kv_splits, 0, [&](int64_t begin, int64_t end) {
    int64_t bs{0}, head_id{0}, kv_id{0};
    data_index_init(begin, bs, batches, head_id, num_heads, kv_id, num_kv_splits);

    // s_prime and s_delta
    alignas(64) float s_i[BLOCK_N];
    float* __restrict__ s_delta = s_i;

    for (int64_t i = begin; i < end; ++i) {
      // get query
      const scalar_t* __restrict__ q_ptr = query + bs * q_strideM + head_id * q_strideH;

      // get key/value
      int64_t seq_len_kv = seq_lens[bs];
      int64_t req_pool_id = req_pool_indices[bs];
      TORCH_CHECK(seq_len_kv <= max_context_len, "seq_len_kv out of scope!");
      TORCH_CHECK(req_pool_id < max_num_reqs, "req_pool_id out of scope!");

      const int64_t SPLIT_SIZE = div_up(seq_len_kv, num_kv_splits);
      const int64_t kv_start = kv_id * SPLIT_SIZE;
      const int64_t kv_end = std::min(kv_start + SPLIT_SIZE, seq_len_kv);

      float m_prime = -std::numeric_limits<float>::infinity();
      float s_prime = 0.f;

      // get v_prime, and init to zero
      float* __restrict__ v_prime = attn_logits + i * (head_size_v + 1);
      fill_stub(v_prime, 0.f, head_size_v);

      // loop over K and V sequence with BLOCK_N
      for (int64_t n = kv_start; n < kv_end; n += BLOCK_N) {
        int64_t n_size = std::min(BLOCK_N, kv_end - n);

        // calculate s_i <- scale * Q @ K
        index_gemm_kernel_nt<scalar_t, index_t>(
            /* A   */ q_ptr,
            /* B   */ k_buffer + head_id * k_strideH,
            /* C   */ s_i,
            /* ind */ req_to_token + req_pool_id * max_context_len + n,
            /* scl */ scaling,
            /* M   */ 1,
            /* N   */ n_size,
            /* K   */ head_size,
            /* lda */ 1,
            /* ldb */ k_strideN,
            /* ldc */ 1,
            /* mtt */ max_total_num_tokens);

        // TODO: `tanh` from torch uses sleef u10, going to be slow
        if (has_logit_cap) {
          at::vec::map<float>(
              [logit_cap, rlogit_cap](Vec x) { return Vec(logit_cap) * (x * Vec(rlogit_cap)).tanh(); },
              s_i,
              s_i,
              n_size);
        }

        // m_i: max value per row
        float m_i = at::vec::reduce_all<float>([](Vec& x, Vec& y) { return at::vec::maximum(x, y); }, s_i, n_size);
        m_i = std::max(m_i, m_prime);

        // m_delta <- exp(m' - m_i)
        float m_delta = std::exp(m_prime - m_i);

        // s_delta <- exp(s_i - m_i)
        at::vec::map<float>([m_i](Vec x) { return (x - Vec(m_i)).exp_u20(); }, s_delta, s_i, n_size);

        // s' <- s' * m_delta + sum(s_delta)
        s_prime *= m_delta;
        s_prime += at::vec::reduce_all<float>([](Vec& x, Vec& y) { return x + y; }, s_delta, n_size);

        m_prime = m_i;

        // calculate V' <- s_delta @ V + V' * m_delta
        index_gemm_kernel_nn<scalar_t, index_t>(
            /* A   */ s_delta,
            /* B   */ v_buffer + head_id * v_strideH,
            /* C   */ v_prime,
            /* ind */ req_to_token + req_pool_id * max_context_len + n,
            /* scl */ &m_delta,
            /* M   */ 1,
            /* N   */ head_size_v,
            /* K   */ n_size,
            /* lda */ 1,
            /* ldb */ v_strideN,
            /* ldc */ 1,
            /* mtt */ max_total_num_tokens);
      }  // loop with KV blocks

      // only update v' when kv_split_size > 0
      if (kv_end > kv_start) {
        float s = 1 / s_prime;
        at::vec::map<float>([s](Vec out) { return out * Vec(s); }, v_prime, v_prime, head_size_v);

        v_prime[head_size_v] = m_prime + std::log(s_prime);
      }

      // move to the next index
      data_index_step(bs, batches, head_id, num_heads, kv_id, num_kv_splits);
    }
  });

  // parallel on [batches, num_heads]
  at::parallel_for(0, batches * num_heads, 0, [&](int64_t begin, int64_t end) {
    // NB: here we use logits[b][h][0] as acc, since
    // for the first kv split (kv_id == 0):
    //   m_delta = std::exp(-inf) = 0
    //   e_logic = std::exp(0) = 1
    //   acc = acc * m_delta + tv * e_logic = tv
    for (int64_t i = begin; i < end; ++i) {
      float* __restrict__ acc = attn_logits + i * l_stride1;

      float s_prime = 0.f;
      float m_prime = -std::numeric_limits<scalar_t>::infinity();

      // update acc with from each kv_split
      for (int64_t kv_id = 0; kv_id < num_kv_splits; ++kv_id) {
        float* __restrict__ tv = acc + kv_id * l_stride2;
        const float tlogic = (acc + kv_id * l_stride2)[head_size_v];

        float m_i = std::max(tlogic, m_prime);
        float m_delta = std::exp(m_prime - m_i);
        float e_logic = std::exp(tlogic - m_i);
        if (kv_id != 0) {
          at::vec::map2<float>(
              [m_delta, e_logic](Vec x, Vec y) { return x * Vec(m_delta) + y * Vec(e_logic); },
              acc,
              acc,
              tv,
              head_size_v);
        }

        s_prime = s_prime * m_delta + e_logic;
        m_prime = m_i;
      }

      copy_stub<scalar_t>(output + i * head_size_v, acc, 1 / s_prime, head_size_v);
    }
  });
}

template <typename scalar_t, typename index_t>
void decode_attention_grouped_kernel_impl(
    scalar_t* __restrict__ output,
    float* __restrict__ attn_logits,
    const scalar_t* __restrict__ query,
    scalar_t* __restrict__ k_buffer,
    scalar_t* __restrict__ v_buffer,
    const scalar_t* __restrict__ key,
    const scalar_t* __restrict__ value,
    const int64_t* __restrict__ loc,
    const index_t* __restrict__ req_to_token,
    const int64_t* __restrict__ req_pool_indices,
    const int64_t* __restrict__ seq_lens,
    int64_t batches,
    int64_t num_heads,
    int64_t num_heads_kv,
    int64_t head_size,
    int64_t head_size_v,
    int64_t num_kv_splits,
    int64_t k_strideN,
    int64_t k_strideH,
    int64_t v_strideN,
    int64_t v_strideH,
    int64_t nk_strideN,
    int64_t nk_strideH,
    int64_t nv_strideN,
    int64_t nv_strideH,
    float scaling,
    float logit_cap,
    int64_t max_num_reqs,
    int64_t max_context_len,
    int64_t max_total_num_tokens) {
  at::parallel_for(0, batches * num_heads_kv, 0, [&](int64_t begin, int64_t end) {
    int64_t bs{0}, head_kv_id{0};
    data_index_init(begin, bs, batches, head_kv_id, num_heads_kv);

    for (int64_t i = begin; i < end; i++) {
      int64_t loc_val = loc[bs];
      scalar_t* k_buffer_ptr = k_buffer + loc_val * k_strideN + head_kv_id * k_strideH;
      scalar_t* v_buffer_ptr = v_buffer + loc_val * v_strideN + head_kv_id * v_strideH;
      const scalar_t* new_key_ptr = key + bs * nk_strideN + head_kv_id * nk_strideH;
      const scalar_t* new_value_ptr = value + bs * nv_strideN + head_kv_id * nv_strideH;
      copy_stub<scalar_t>(k_buffer_ptr, new_key_ptr, head_size);
      copy_stub<scalar_t>(v_buffer_ptr, new_value_ptr, head_size_v);

      // move to the next index
      data_index_step(bs, batches, head_kv_id, num_heads_kv);
    }
  });

  using Vec = at::vec::Vectorized<float>;

  // block length for k_buffer and v_buffer
  constexpr int64_t BLOCK_N = 256;
  // block length for heads
  // we parallel on [batches, divup(num_heads, BLOCK_H), num_kv_splits]
  // use smaller BLOCK_H when batches is small to utilize all cores
  constexpr int64_t kBLOCK_H = 16;
  const int64_t BLOCK_H = std::min(4 * batches, kBLOCK_H);

  // strides
  const int64_t q_strideM = num_heads * head_size;
  const int64_t q_strideH = head_size;
  const int64_t l_stride0 = num_heads * num_kv_splits * (head_size_v + 1);
  const int64_t l_stride1 = num_kv_splits * (head_size_v + 1);
  const int64_t l_stride2 = head_size_v + 1;

  const bool has_logit_cap = logit_cap > 0;
  float rlogit_cap = has_logit_cap ? 1 / logit_cap : 0.f;

  // partition the heads into blocks for parallel
  const int64_t num_groups = num_heads / num_heads_kv;
  const int64_t num_blocks = div_up(num_groups, BLOCK_H);

  // parallel on [batches, num_heads_kv, num_blocks, num_kv_splits]
  at::parallel_for(0, batches * num_heads_kv * num_blocks * num_kv_splits, 0, [&](int64_t begin, int64_t end) {
    int64_t bs{0}, head_kv_id{0}, block_id{0}, kv_id{0};
    data_index_init(begin, bs, batches, head_kv_id, num_heads_kv, block_id, num_blocks, kv_id, num_kv_splits);

    alignas(64) float s_i[BLOCK_H * BLOCK_N];
    float* __restrict__ s_delta = s_i;

    alignas(64) float s_prime[BLOCK_H];
    alignas(64) float m_prime[BLOCK_H];
    alignas(64) float m_delta[BLOCK_H];

    for (int64_t i = begin; i < end; ++i) {
      const int64_t h_start = head_kv_id * num_groups + block_id * BLOCK_H;
      const int64_t h_end = head_kv_id * num_groups + std::min(block_id * BLOCK_H + BLOCK_H, num_groups);
      const int64_t h_size = h_end - h_start;

      // get query
      const scalar_t* __restrict__ q_ptr = query + bs * q_strideM + h_start * q_strideH;

      int64_t seq_len_kv = seq_lens[bs];
      int64_t req_pool_id = req_pool_indices[bs];
      TORCH_CHECK(seq_len_kv <= max_context_len, "seq_len_kv out of scope!");
      TORCH_CHECK(req_pool_id < max_num_reqs, "req_pool_id out of scope!");

      const int64_t SPLIT_SIZE = div_up(seq_len_kv, num_kv_splits);
      const int64_t kv_start = kv_id * SPLIT_SIZE;
      const int64_t kv_end = std::min(kv_start + SPLIT_SIZE, seq_len_kv);

      fill_stub(s_prime, 0.f, BLOCK_H);
      fill_stub(m_prime, -std::numeric_limits<float>::infinity(), BLOCK_H);

      // get v_prime, and init to zero
      float* __restrict__ v_prime = attn_logits + bs * l_stride0 + h_start * l_stride1 + kv_id * l_stride2;
      for (int64_t h = 0; h < h_size; ++h) {
        fill_stub(v_prime + h * l_stride1, 0.f, head_size_v);
      }

      // loop over K and V sequence with BLOCK_N
      for (int64_t n = kv_start; n < kv_end; n += BLOCK_N) {
        int64_t n_size = std::min(BLOCK_N, kv_end - n);

        // calculate Q @ K
        index_gemm_kernel_nt<scalar_t, index_t>(
            /* A   */ q_ptr,
            /* B   */ k_buffer + head_kv_id * k_strideH,
            /* C   */ s_i,
            /* ind */ req_to_token + req_pool_id * max_context_len + n,
            /* scl */ scaling,
            /* M   */ h_size,
            /* N   */ n_size,
            /* K   */ head_size,
            /* lda */ q_strideH,
            /* ldb */ k_strideN,
            /* ldc */ BLOCK_N,
            /* mtt */ max_total_num_tokens);

        if (has_logit_cap) {
          at::vec::map<float>(
              [logit_cap, rlogit_cap](Vec x) { return Vec(logit_cap) * (x * Vec(rlogit_cap)).tanh(); },
              s_i,
              s_i,
              n_size);
        }

        // update the scaling coefficients
        for (int64_t h = 0; h < h_size; ++h) {
          // m_i: max value per row
          float m_i = at::vec::reduce_all<float>(
              [](Vec& x, Vec& y) { return at::vec::maximum(x, y); }, s_i + h * BLOCK_N, n_size);
          m_i = std::max(m_i, m_prime[h]);

          // m_delta <- exp(m' - m_i)
          m_delta[h] = std::exp(m_prime[h] - m_i);

          // s_delta <- exp(s_i - m_i)
          at::vec::map<float>(
              [m_i](Vec x) { return (x - Vec(m_i)).exp_u20(); }, s_delta + h * BLOCK_N, s_i + h * BLOCK_N, n_size);

          // s' <- s' * m_delta + sum(s_delta)
          s_prime[h] *= m_delta[h];
          s_prime[h] += at::vec::reduce_all<float>([](Vec& x, Vec& y) { return x + y; }, s_delta + h * BLOCK_N, n_size);

          m_prime[h] = m_i;
        }

        // calculate V' <- s_delta @ V + V' * m_delta
        index_gemm_kernel_nn<scalar_t, index_t>(
            /* A   */ s_delta,
            /* B   */ v_buffer + head_kv_id * v_strideH,
            /* C   */ v_prime,
            /* ind */ req_to_token + req_pool_id * max_context_len + n,
            /* scl */ m_delta,
            /* M   */ h_size,
            /* N   */ head_size_v,
            /* K   */ n_size,
            /* lda */ BLOCK_N,
            /* ldb */ v_strideN,
            /* ldc */ l_stride1,
            /* mtt */ max_total_num_tokens);
      }  // loop with KV blocks

      // only update v' when kv_split_size > 0
      if (kv_end > kv_start) {
        for (int64_t h = 0; h < h_size; ++h) {
          float s = 1 / s_prime[h];
          at::vec::map<float>(
              [s](Vec out) { return out * Vec(s); }, v_prime + h * l_stride1, v_prime + h * l_stride1, head_size_v);
          (v_prime + h * l_stride1)[head_size_v] = m_prime[h] + std::log(s_prime[h]);
        }
      }

      // move to the next index
      data_index_step(bs, batches, head_kv_id, num_heads_kv, block_id, num_blocks, kv_id, num_kv_splits);
    }
  });

  // parallel on [batches, num_heads]
  at::parallel_for(0, batches * num_heads, 0, [&](int64_t begin, int64_t end) {
    // NB: same as above
    for (int64_t i = begin; i < end; ++i) {
      float* __restrict__ acc = attn_logits + i * l_stride1;

      float s_prime = 0.f;
      float m_prime = -std::numeric_limits<scalar_t>::infinity();

      // update acc with from each kv_split
      for (int64_t kv_id = 0; kv_id < num_kv_splits; ++kv_id) {
        float* __restrict__ tv = acc + kv_id * l_stride2;
        const float tlogic = (acc + kv_id * l_stride2)[head_size_v];

        float m_i = std::max(tlogic, m_prime);
        float m_delta = std::exp(m_prime - m_i);
        float e_logic = std::exp(tlogic - m_i);
        if (kv_id != 0) {
          at::vec::map2<float>(
              [m_delta, e_logic](Vec x, Vec y) { return x * Vec(m_delta) + y * Vec(e_logic); },
              acc,
              acc,
              tv,
              head_size_v);
        }

        s_prime = s_prime * m_delta + e_logic;
        m_prime = m_i;
      }

      copy_stub<scalar_t>(output + i * head_size_v, acc, 1 / s_prime, head_size_v);
    }
  });
}

}  // anonymous namespace

// query:            [num_tokens, num_heads, head_size]
// output:           [num_tokens, num_heads, head_size]
// k_buffer:         [max_total_num_tokens, num_heads, head_size]
// v_buffer:         [max_total_num_tokens, num_heads, head_size_v]
// attn_logits:      [num_seqs, num_heads, num_kv_splits, head_size_v + 1]
// req_to_token:     [max_num_reqs, max_context_len] int32 or int64
// req_pool_indices: [num_seqs] int64
// seq_lens:         [num_seqs] int64
//
void decode_attention_cpu(
    at::Tensor& query,
    at::Tensor& k_buffer,
    at::Tensor& v_buffer,
    at::Tensor& output,
    at::Tensor& key,
    at::Tensor& value,
    at::Tensor& loc,
    at::Tensor& attn_logits,
    at::Tensor& req_to_token,
    at::Tensor& req_pool_indices,
    at::Tensor& seq_lens,
    double sm_scale,
    double logit_cap) {
  RECORD_FUNCTION(
      "sgl-kernel::decode_attention_cpu",
      std::vector<c10::IValue>(
          {query, output, k_buffer, v_buffer, attn_logits, req_to_token, req_pool_indices, seq_lens}));

  CHECK_INPUT(query);
  CHECK_LAST_DIM_CONTIGUOUS_INPUT(k_buffer);
  CHECK_LAST_DIM_CONTIGUOUS_INPUT(v_buffer);
  // for MLA, key and value shares the same storage and value could be non-contiguous
  CHECK_LAST_DIM_CONTIGUOUS_INPUT(key);
  CHECK_LAST_DIM_CONTIGUOUS_INPUT(value);
  CHECK_DIM(3, query);
  CHECK_DIM(3, k_buffer);
  CHECK_DIM(3, v_buffer);
  CHECK_DIM(3, key);
  CHECK_DIM(3, value);
  CHECK_DIM(1, loc);

  int64_t num_seqs = seq_lens.size(0);
  int64_t max_num_reqs = req_to_token.size(0);
  int64_t max_context_len = req_to_token.size(1);
  int64_t max_total_num_tokens = k_buffer.size(0);

  int64_t num_heads = query.size(1);
  int64_t num_heads_kv = k_buffer.size(1);
  int64_t head_size = query.size(2);
  int64_t head_size_v = v_buffer.size(2);

  int64_t num_kv_splits = attn_logits.size(2);

  CHECK_EQ(loc.numel(), num_seqs);
  CHECK_EQ(attn_logits.size(0), num_seqs);
  CHECK_EQ(attn_logits.size(1), num_heads);
  CHECK_EQ(attn_logits.size(3), head_size_v + 1);
  CHECK_EQ(attn_logits.scalar_type(), at::kFloat);

  // strides for k_buffer and v_buffer
  int64_t k_strideN = k_buffer.stride(0);
  int64_t k_strideH = k_buffer.stride(1);
  int64_t v_strideN = v_buffer.stride(0);
  int64_t v_strideH = v_buffer.stride(1);
  // strides for new key and value
  int64_t nk_strideN = key.stride(0);
  int64_t nk_strideH = key.stride(1);
  int64_t nv_strideN = value.stride(0);
  int64_t nv_strideH = value.stride(1);

  // check index data types
  const auto index_dtype = req_to_token.scalar_type();
  TORCH_CHECK(
      index_dtype == at::kInt || index_dtype == at::kLong,
      "decode: expect req_to_token to be int32 or int64, got ",
      index_dtype);
  TORCH_CHECK(seq_lens.scalar_type() == at::kLong, "decode: expect req_lens to be int64, got ", seq_lens.scalar_type());
  TORCH_CHECK(
      req_pool_indices.scalar_type() == at::kLong,
      "decode: expect req_pool_indices to be int64, got ",
      req_pool_indices.scalar_type());

  AT_DISPATCH_REDUCED_FLOATING_TYPES(query.scalar_type(), "decode_attention_kernel", [&] {
    AT_DISPATCH_INDEX_TYPES(index_dtype, "decode_attention_indices", [&] {
      if (num_heads == num_heads_kv) {
        // MHA
        decode_attention_kernel_impl<scalar_t, index_t>(
            output.data_ptr<scalar_t>(),
            attn_logits.data_ptr<float>(),
            query.data_ptr<scalar_t>(),
            k_buffer.data_ptr<scalar_t>(),
            v_buffer.data_ptr<scalar_t>(),
            key.data_ptr<scalar_t>(),
            value.data_ptr<scalar_t>(),
            loc.data_ptr<int64_t>(),
            req_to_token.data_ptr<index_t>(),
            req_pool_indices.data_ptr<int64_t>(),
            seq_lens.data_ptr<int64_t>(),
            num_seqs,
            num_heads,
            head_size,
            head_size_v,
            num_kv_splits,
            k_strideN,
            k_strideH,
            v_strideN,
            v_strideH,
            nk_strideN,
            nv_strideH,
            nv_strideN,
            nv_strideH,
            sm_scale,
            logit_cap,
            max_num_reqs,
            max_context_len,
            max_total_num_tokens);
      } else {
        // GQA/MQA/MLA
        decode_attention_grouped_kernel_impl<scalar_t, index_t>(
            output.data_ptr<scalar_t>(),
            attn_logits.data_ptr<float>(),
            query.data_ptr<scalar_t>(),
            k_buffer.data_ptr<scalar_t>(),
            v_buffer.data_ptr<scalar_t>(),
            key.data_ptr<scalar_t>(),
            value.data_ptr<scalar_t>(),
            loc.data_ptr<int64_t>(),
            req_to_token.data_ptr<index_t>(),
            req_pool_indices.data_ptr<int64_t>(),
            seq_lens.data_ptr<int64_t>(),
            num_seqs,
            num_heads,
            num_heads_kv,
            head_size,
            head_size_v,
            num_kv_splits,
            k_strideN,
            k_strideH,
            v_strideN,
            v_strideH,
            nk_strideN,
            nk_strideH,
            nv_strideN,
            nv_strideH,
            sm_scale,
            logit_cap,
            max_num_reqs,
            max_context_len,
            max_total_num_tokens);
      }
    });
  });
}