flash_attn_mma_split_kv.cu

#include "utils.h"

// Write FlashAttention-2 from scratch using Tensor Cores with MMA PTX instruction.
// The input is Q,K,V, 4D tensor with shape [batch_size, num_heads, seq_len, head_dim].
// The output is O, a 4D tensor with shape [batch_size, num_heads, seq_len, head_dim].

// The FlashAttention-2 algorithm is described in the following paper:
// https://arxiv.org/pdf/2307.08691

// Q,K,V,O: [batch_size, num_heads, seq_len, head_dim], [B,H,N,d]
// each block processes Q_tile with shape [Br,d] and full K,V with shape [N,d]
// Currently, we only support Br = Bc = 64.
template<
         const int kHeadDim,          // Headdim, 32,64,128     
         const int kMmaAtomM,         // MMA Atom M, 16
         const int kMmaAtomN,         // MMA Atom N, 8
         const int kMmaAtomK,         // MMA Atom K, 16
         const int kMmaTileSeqLenQ,   // 2, more MMA(warp), M=16*2=32, Q@K^T=[Br(M), d(K)]@[d(K),  Bc(N)]  
         const int kMmaTileSeqLenK,   // 4, more MMA(warp), N=8*4= 32, Q@K^T=[Br(M), d(K)]@[d(K),  Bc(N)]    
         const int kMmaTileSeqLenP,   // 2, more MMA(warp), M=16*2=32, P@V  =[Br(M),Bc(K)]@[Bc(K), d(N) ]
         const int kMmaTileHeadDimV,  // 4, more MMA(warp), N=8*4= 32, P@V  =[Br(M),Bc(K)]@[Bc(K), d(N) ]       
         const int kWarpTileSeqLenQ,  // 2, more values, M, Br=32*2=64, matmul M 
         const int kWarpTileSeqLenK,  // 2, more values, N, Bc=32*2=64, matmul N
         const int kWarpTileSeqLenP,  // 2, more values, M, Br=32*2=64, matmul M
         const int kWarpTileHeadDimV, // 2, more values, N, d=32*(1|2|3|4|...)=32|64|96|128|...
         const int kStage, 
         const int kPad
         >
__global__ void __launch_bounds__(
  WARP_SIZE * kMmaTileSeqLenQ * kMmaTileSeqLenK) 
flash_attn_mma_stages_split_kv_kernel(half* Q, 
                                      half* K, 
                                      half* V, 
                                      half* O, 
                                      int QKV_seqlen,
                                      int QKV_head) {
  // Matmul Layout: Q[Br,d]@K^T[d,Bc] NT, P[Br,Bc]@V[Bc,d] NN.
  // NOTE: K[Bc,d] with row major means K^T[d,Bc] in col major.
  static_assert(kMmaAtomM == 16 && kMmaAtomN == 8 && kMmaAtomK == 16); // m16n8k16
  static_assert(kMmaTileSeqLenQ  == 2 && kMmaTileSeqLenK  == 4); // Q@K^T
  static_assert(kMmaTileSeqLenP  == 2 && kMmaTileHeadDimV == 4); // P@V
  static_assert(kWarpTileSeqLenQ == 2 && kWarpTileSeqLenK == 2); // Q@K^T
  // e.g, kWarpTileHeadDimV: 1->d 32, 2->d 64, 3->d 96, 4-> d 128, ..., etc.
  static_assert(kWarpTileSeqLenP == 2 && kWarpTileHeadDimV == (
    kHeadDim / (kMmaAtomN * kMmaTileHeadDimV))); // P@V
  static_assert(kStage > 0 && kStage < 3); // 1,2
  static_assert(kPad >= 0 && kPad % 8 == 0); // 0,8,16
  constexpr int Br = kMmaAtomM * kMmaTileSeqLenQ * kWarpTileSeqLenQ; // 16*2*2=64
  constexpr int Bc = kMmaAtomN * kMmaTileSeqLenK * kWarpTileSeqLenK; // 8*4*2=64
  constexpr int kNumThreads = WARP_SIZE * kMmaTileSeqLenQ * kMmaTileSeqLenK; // 32*2*4=256, num threads
  // Now, N must be mutliples of Bc(32/64) for KV tiling across seqlen.
  const int Tc = div_ceil(QKV_seqlen, Bc); // Tc K^T_tile[d,Bc]
  const float scale = 1.0f / sqrt((float) kHeadDim);
  
  // grid(div_ceil(QKV_seqlen, Br), QKV_batch * QKV_head), (x,y,z)
  const int QKV_batch_id = blockIdx.y / QKV_head; // Batch size
  const int QKV_head_id  = blockIdx.y % QKV_head; // Head num
  const int Q_tile_id    = blockIdx.x;            // Q tile_id, range [0, Tr]
  const int O_tile_id    = Q_tile_id;       // O tile_id, same as Q.
  const int tid          = threadIdx.x;     // within block
  const int warp_id      = tid / WARP_SIZE; // 0~7 warp_id within block
  const int lane_id      = tid % WARP_SIZE; // 0~31
  const int warp_QP      = warp_id % 2;     // 0,1
  const int warp_KV      = warp_id / 2;     // 0,1,2,3
  // The layout of 8 MMA(2x4) [before] kWarpTileSeqLenQxkWarpTileSeqLenK(2x2) -> 16x2,8x4=32x32:
  // |  [32,32]  | warp_KV 0 | warp_KV 1 | warp_KV 2 | warp_KV 3 |
  // | warp_QP 0 |-- MMA 0 --|-- MMA 2 --|-- MMA 4 --|-- MMA 6 --|
  // | warp_QP 1 |-- MMA 1 --|-- MMA 3 --|-- MMA 5 --|-- MMA 7 --|
  // The layout of 8 MMA(2x4)  [after] kWarpTileSeqLenQxkWarpTileSeqLenK(2x2) -> 32x2,32x2=64x64: 
  // |  [64,64]  |    warp_KV 0    |    warp_KV 1    |    warp_KV 2    |    warp_KV 3    |
  // | warp_QP 0 |-- MMA 0,MMA 0 --|-- MMA 2,MMA 2 --|-- MMA 4,MMA 4 --|-- MMA 6,MMA 6 --|
  // | warp_QP 0 |-- MMA 0,MMA 0 --|-- MMA 2,MMA 2 --|-- MMA 4,MMA 4 --|-- MMA 6,MMA 6 --|
  // | warp_QP 1 |-- MMA 1,MMA 1 --|-- MMA 3,MMA 2 --|-- MMA 5,MMA 5 --|-- MMA 7,MMA 7 --|
  // | warp_QP 1 |-- MMA 1,MMA 1 --|-- MMA 3,MMA 2 --|-- MMA 5,MMA 5 --|-- MMA 7,MMA 7 --|
  // gridDim.y = head_num, gridDim.z = N/Br = Tr.
  const int Q_gmem_offset = ((QKV_batch_id * QKV_head * QKV_seqlen * kHeadDim) + 
                             (QKV_head_id * QKV_seqlen * kHeadDim)); // Q [seqlen,d]
  const int K_gmem_offset = ((QKV_batch_id * QKV_head * QKV_seqlen * kHeadDim) + 
                             (QKV_head_id * QKV_seqlen * kHeadDim)); // K [seqlen,d] 
  const int V_gmem_offset = Q_gmem_offset; // V [seqlen,d]
  const int O_gmem_offset = Q_gmem_offset; // O [seqlen,d]

  // Mapping Q gmem -> tid -> smem, Q[Br,d]=[64,64 or 128], 256 threads.
  int load_smem_Q_Br = (tid / (kNumThreads / Br)); // Br 64, tid / 4, row 0~64
  int load_smem_Q_d  = (tid % (kNumThreads / Br)) * (kHeadDim / (kNumThreads / Br)); // (tid % 4) * 16, 0,16,32,48
  // Mapping K gmem -> tid -> smem, K[Bc,d]=[64 or 128,64], 128 threads.
  int load_smem_K_Bc = (tid / (kNumThreads / Bc)); // Bc 64, tid / 2, row 0~64
  int load_smem_K_d  = (tid % (kNumThreads / Bc)) * (kHeadDim / (kNumThreads / Bc)); // (tid % 4) * 16, 0,16,32,48
  // Mapping V gmem -> tid -> smem, V[Bc,d]=[64,64 or 128], 256 threads.
  int load_smem_V_Bc = (tid / (kNumThreads / Bc)); // Bc 64, tid / 4, row 0~64
  int load_smem_V_d  = (tid % (kNumThreads / Bc)) * (kHeadDim / (kNumThreads / Bc)); // (tid % 4) * 16, 0,16,32,48
  // global Q row of current head for tile [Br,d] per block.
  int load_gmem_Q_Br = Q_tile_id * Br + load_smem_Q_Br; 
  if (load_gmem_Q_Br >= QKV_seqlen) return;
  // KV tile gmem load index starts from 0 and increments with 
  // each iteration as we loop over seqlen.
  int load_gmem_K_Bc_offset = 0; 
  int load_gmem_V_Bc_offset = 0; 

  // Shared memory for Q,K,V,S, we don not need additional smem for O 
  // collective store which perform via registers reuse and warp shuffle.
  extern __shared__ half smem[];
  constexpr int Q_tile_size  = Br * (kHeadDim + kPad); // 64*64=4096, ~8192 bytes=8M
  constexpr int KV_tile_size = Bc * (kHeadDim + kPad); // 64*64=4096, ~8192 bytes=8M
  constexpr int S_tile_size  = Br * (Bc + kPad); // 64*64=4096, ~8192 bytes=8M
  // K multi-stages: currently, only apply multi stages for K across seq_len.
  half* Q_tile_smem = smem; // 8M/16M
  half* K_tile_smem = Q_tile_smem + Q_tile_size; // 8M/16M
  half* V_tile_smem = K_tile_smem + kStage * KV_tile_size; 
  half* S_tile_smem = V_tile_smem + KV_tile_size; // for temp S=Q@K^T
  // stage 2, no shared KV smem, Br=Bc=64,  d=64: 8M+(8M)*2+8M   =32M,  shared KV smem: 24M
  // stage 2, no shared KV smem, Br=Bc=64, d=128: 16M+(16M)*2+16M=64M,  shared KV smem: 48M
  // stage 2, no shared KV smem, Br=Bc=64, d=256: 32M+(32M)*2+32M=128M, shared KV smem: 96M
  // stage 1, no shared KV smem, Br=Bc=64, d=256: 32M+(32M)*1+32M=96M,  shared KV smem: 64M
 
  uint32_t smem_Q_base_ptr = __cvta_generic_to_shared(Q_tile_smem);
  uint32_t smem_K_base_ptr = __cvta_generic_to_shared(K_tile_smem);
  uint32_t smem_V_base_ptr = __cvta_generic_to_shared(V_tile_smem);
  uint32_t smem_S_base_ptr = __cvta_generic_to_shared(S_tile_smem);

  // --------------------- Registers/SMEM for thread block -------------------------
  // block m_old, l_old, store in lane, use float to keep precision.
  float lane_block_row_max_old[kWarpTileSeqLenQ][2];
  float lane_block_row_sum_old[kWarpTileSeqLenQ][2];
  fill_2D_regs<float, kWarpTileSeqLenQ, 2>(lane_block_row_max_old, -INFINITY);
  fill_2D_regs<float, kWarpTileSeqLenQ, 2>(lane_block_row_sum_old, 0.0f);
  // m[Br], l[Br], for the output of P[Br,Bc]=Q[Br,d]@K^T[d,Bc],
  // 64x(4)x4=1024 bytes, 1M+1M=2M. For now, I have choose to 
  // add 1 to reduce bank conflicts, may boost 2~3 TFLOPS.
  __shared__ float block_row_max_new_smem[Br][kMmaTileSeqLenK + 1]; 
  __shared__ float block_row_sum_new_smem[Br][kMmaTileSeqLenK + 1];

  // ---------------------- Registers for S=Q@K^T/O=P@V ----------------------------
  // registers for QKV, S=Q[Br,d]@K[Bc,d]=[Br,Bc] and O=P[Br,Bc]@V[Bc,d]=[Br,d].
  uint32_t R_Q[kWarpTileSeqLenQ][ 4];
  uint32_t R_K[kWarpTileSeqLenK][ 2];
  uint32_t R_V[kWarpTileHeadDimV][2];
  // registers for current tile_K_seqlen within, [64,64] = S_tile[Br,Bc]
  // = Q_tile[Br,d] * K[Bc,d], each thread hold 2x32 bits regs.
  uint32_t R_S[kWarpTileSeqLenQ][kWarpTileSeqLenK][ 2]; // [2][2][2]
  // registers for tile_K_seqlen O=PV[Br,d]=P@V, [2][2/4][2], 8 or 16 regs.
  // TODO: may reuse R_D as R_O? kWarpTileSeqLenP=kWarpTileSeqLenQ.
  uint32_t R_O[kWarpTileSeqLenP][kWarpTileHeadDimV][2]; // [2][2/4][2]
  // registers final Output [D]=final rescale(R_O), [2][2/4][2], 8 or 16 regs.
  uint32_t R_D[kWarpTileSeqLenP][kWarpTileHeadDimV][2]; // [2][2/4][2]
  fill_3D_regs<uint32_t, kWarpTileSeqLenQ, kWarpTileSeqLenK,  2>(R_S, 0);
  fill_3D_regs<uint32_t, kWarpTileSeqLenP, kWarpTileHeadDimV, 2>(R_D, 0);
  fill_3D_regs<uint32_t, kWarpTileSeqLenP, kWarpTileHeadDimV, 2>(R_O, 0);
  
  // load Q from gmem -> smem, only load once.
  {
    int load_gmem_Q_d = load_smem_Q_d;
    int load_gmem_Q_addr = (Q_gmem_offset + load_gmem_Q_Br * kHeadDim + load_gmem_Q_d);
    uint32_t load_smem_Q_ptr = (smem_Q_base_ptr + (
      load_smem_Q_Br * (kHeadDim + kPad) + load_smem_Q_d) * sizeof(half));
    #pragma unroll
    for (int i = 0; i < (kHeadDim / (kNumThreads / Br)); i += 8) {
      CP_ASYNC_CG(load_smem_Q_ptr + i * 2, &Q[load_gmem_Q_addr + i], 16);
    }
    CP_ASYNC_COMMIT_GROUP();
  }

  // load K from gmem -> smem, (kStage - 1) K^T tiles, [d,Bc]
  if constexpr (kStage > 1) {
    #pragma unroll
    for (int stage = 0; stage < (kStage - 1); ++stage) {
      // update the offset of n according to stages
      load_gmem_K_Bc_offset = stage * Bc; // e.g (0~3)*64=(0,64,128,192,...)
      int load_gmem_K_Bc = load_gmem_K_Bc_offset + load_smem_K_Bc; // < seqlen
      int load_gmem_K_d  = load_smem_K_d; // K [Bc,d] from [seqlen,d]
      int load_gmem_K_addr = (
        K_gmem_offset + load_gmem_K_Bc * kHeadDim + load_gmem_K_d);
      uint32_t load_smem_K_ptr = (
        smem_K_base_ptr + (stage * KV_tile_size + 
                           load_smem_K_Bc * (kHeadDim + kPad) + 
                           load_smem_K_d) * sizeof(half)
      );
      #pragma unroll
      for (int i = 0; i < (kHeadDim / (kNumThreads / Bc)); i += 8) {
        CP_ASYNC_CG(load_smem_K_ptr + i * 2, &K[load_gmem_K_addr + i], 16);
      }
      CP_ASYNC_COMMIT_GROUP();
    }
  }

  // wait Q and at least (kStage - 1) for K ready.
  if constexpr (kStage > 1) {
    CP_ASYNC_WAIT_GROUP(kStage - 2); // s2->0, s3->1, s4->2
    __syncthreads(); 
  }

  // <loop over K seqlen>: for K^T[d,seqlen] with K^T_tile[d,Bc]
  // tile_K_seqlen: compute S_tile[Br,Bc] = Q@K^T = Q_tile[Br,d] * K^T[d,Bc]
  #pragma unroll 1
  for (int tile_K_seqlen = 0; tile_K_seqlen < Tc; ++tile_K_seqlen) { 
    // TODO: process last tile_K_seqlen ? pad to multiple of 8.
    // s2 tn 0->0, 1->1, 2->0; s3 tn 0->0, 1->1, 2->2, 3->0;
    int smem_sel      = (tile_K_seqlen) % kStage;   
    // s2 tn 0->1, 1->0, 2->1; s3 tn 0->2, 1->0, 2->1, 3->2;  
    int smem_sel_next = (tile_K_seqlen + (kStage - 1)) % kStage;

    // multi stages pipeling gmem -> smem
    // NOTE: kStage must be > 1 for pipeling. For s1, smem_sel 
    // and smem_sel_next will always equal 0, thus, we can not 
    // prefetch KV from gmem to smem before tile_K_seqlen MMA done.

    if constexpr (kStage > 1) {
      // First, prefetch curr V tile_K_seqlen [Bc,d] (no stages)
      {
        load_gmem_V_Bc_offset = tile_K_seqlen * Bc; // e.g (0~3)*64=(0,64,128,192,...)
        int load_gmem_V_Bc = load_gmem_V_Bc_offset + load_smem_V_Bc;
        int load_gmem_V_d  = load_smem_V_d;
        int load_gmem_V_addr = (
          V_gmem_offset + load_gmem_V_Bc * kHeadDim + load_gmem_V_d);
        uint32_t load_smem_V_ptr = (
          smem_V_base_ptr + (load_smem_V_Bc * (kHeadDim + kPad) + 
                             load_smem_V_d) * sizeof(half)
        );
        #pragma unroll
        for (int i = 0; i < (kHeadDim / (kNumThreads / Bc)); i += 8) {
          CP_ASYNC_CG(load_smem_V_ptr + i * 2, &V[load_gmem_V_addr + i], 16);
        }
        CP_ASYNC_COMMIT_GROUP();
      }

      // Then, prefetch next stage K (tile_K_seqlen + 1) [d,Bc]
      if ((tile_K_seqlen + 1) < Tc) {
        load_gmem_K_Bc_offset = (tile_K_seqlen + 1) * Bc; // e.g (0~3)*64=(0,64,128,192,...)
        int load_gmem_K_Bc = load_gmem_K_Bc_offset + load_smem_K_Bc; // < seqlen
        int load_gmem_K_d  = load_smem_K_d; // K [Bc,d] from [seqlen,d]
        int load_gmem_K_addr = (
          K_gmem_offset + load_gmem_K_Bc * kHeadDim + load_gmem_K_d);
        uint32_t load_smem_K_ptr = (
          smem_K_base_ptr + (smem_sel_next * KV_tile_size + 
                             load_smem_K_Bc * (kHeadDim + kPad) + 
                             load_smem_K_d) * sizeof(half)
        );
        #pragma unroll
        for (int i = 0; i < (kHeadDim / (kNumThreads / Bc)); i += 8) {
          CP_ASYNC_CG(load_smem_K_ptr + i * 2, &K[load_gmem_K_addr + i], 16);
        }
        CP_ASYNC_COMMIT_GROUP();
      }
    } else {
      // If no stages, kStage = 1, we have to load current K tile
      // from gmem to smem and have to wait it ready for Q@K^T MMA.

      // First, prefetch curr K tile_K_seqlen [d,Bc] (no stages)
      {
        load_gmem_K_Bc_offset = tile_K_seqlen * Bc; // e.g (0~3)*64=(0,64,128,192,...)
        int load_gmem_K_Bc = load_gmem_K_Bc_offset + load_smem_K_Bc; // < seqlen
        int load_gmem_K_d  = load_smem_K_d; // K [Bc,d] from [seqlen,d]
        int load_gmem_K_addr = (
          K_gmem_offset + load_gmem_K_Bc * kHeadDim + load_gmem_K_d);
        uint32_t load_smem_K_ptr = (
          smem_K_base_ptr + (smem_sel * KV_tile_size + 
                             load_smem_K_Bc * (kHeadDim + kPad) + 
                             load_smem_K_d) * sizeof(half)
        );
        #pragma unroll
        for (int i = 0; i < (kHeadDim / (kNumThreads / Bc)); i += 8) {
          CP_ASYNC_CG(load_smem_K_ptr + i * 2, &K[load_gmem_K_addr + i], 16);
        }

        CP_ASYNC_COMMIT_GROUP();
      }

      // Then, prefetch curr K tile_K_seqlen [d,Bc] (no stages)
      {
        load_gmem_V_Bc_offset = tile_K_seqlen * Bc; // e.g (0~3)*64=(0,64,128,192,...)
        int load_gmem_V_Bc = load_gmem_V_Bc_offset + load_smem_V_Bc;
        int load_gmem_V_d  = load_smem_V_d;
        int load_gmem_V_addr = (
          V_gmem_offset + load_gmem_V_Bc * kHeadDim + load_gmem_V_d);
        uint32_t load_smem_V_ptr = (
          smem_V_base_ptr + (load_smem_V_Bc * (kHeadDim + kPad) + 
                             load_smem_V_d) * sizeof(half)
        );
        #pragma unroll
        for (int i = 0; i < (kHeadDim / (kNumThreads / Bc)); i += 8) {
          CP_ASYNC_CG(load_smem_V_ptr + i * 2, &V[load_gmem_V_addr + i], 16);
        }
        CP_ASYNC_COMMIT_GROUP();
      }

      // Wait curr Q and K tile ready and let curr V tile copy async.
      CP_ASYNC_WAIT_GROUP(1); 
      __syncthreads(); 
    }
    
    // <loop over K d>: tile_K_d, kMmaAtomK = 16, K_tile_d[kMmaAtomK,Bc]
    // Matmul with NT layout, Q row major, K^T col major. 
    // NOTE: K[Bc,d] with row major means K^T[d,Bc] in col major.
    // S_tile[Br,Bc]=Q_tile[Br,d]@K[Bc,d]
    fill_3D_regs<uint32_t, kWarpTileSeqLenQ, kWarpTileSeqLenK, 2>(R_S, 0);
    #pragma unroll
    for (int tile_K_d = 0; tile_K_d < (kHeadDim / kMmaAtomK); ++tile_K_d) {
      // smem -> reg, load m16k16 smem Q, offset d according tile_K_d.
      // ldmatrix.x4 for Q_tile_smem.
      #pragma unroll
      for (int i = 0; i < kWarpTileSeqLenQ; ++i) { // Q[Br,d]=[M,K]
        int warp_smem_Q_Br = warp_QP * (kMmaAtomM * kWarpTileSeqLenQ) + i * kMmaAtomM;
        int lane_smem_Q_Br = warp_smem_Q_Br + lane_id % 16; // 0~15
        int lane_smem_Q_d  = tile_K_d * kMmaAtomK + (lane_id / 16) * 8; // 0,8
        uint32_t lane_smem_Q_ptr = (
            smem_Q_base_ptr + (lane_smem_Q_Br * (kHeadDim + kPad) + 
                               lane_smem_Q_d) * sizeof(half)
        );
        LDMATRIX_X4(R_Q[i][0], R_Q[i][1], R_Q[i][2], R_Q[i][3], 
                    lane_smem_Q_ptr); // now, R_Q
      }

      // smem -> reg, load k16n8 from smem K, offset d according tile_K_d.
      // ldmatrix.x2 for K_tile_smem, [Bc,kMmaAtomK] from [Bc,d]=[K,N]
      #pragma unroll
      for (int j = 0; j < kWarpTileSeqLenK; ++j) {
        // load k16n8 via ldmatrix.x2 from K_tile_smem[Bc,d]. 
        // K[Bc,d] with row major means K^T[d,Bc] in col major.
        int warp_smem_K_Bc = warp_KV * (kMmaAtomN * kWarpTileSeqLenK) + j * kMmaAtomN;
        int lane_smem_K_Bc = warp_smem_K_Bc + lane_id % 8; // 0~7
        int lane_smem_K_d = tile_K_d * kMmaAtomK + ((lane_id / 8) % 2) * 8; // 0,8
        uint32_t lane_smem_K_ptr = (
            smem_K_base_ptr + (smem_sel * KV_tile_size + 
                               lane_smem_K_Bc * (kHeadDim + kPad) + 
                               lane_smem_K_d) * sizeof(half)
        );
        LDMATRIX_X2(R_K[j][0], R_K[j][1], lane_smem_K_ptr); // R_K
      } // end for kWarpTileSeqLenK

      // MMA compute
      #pragma unroll
      for (int i = 0; i < kWarpTileSeqLenQ; ++i) {
        #pragma unroll
        for (int j = 0; j < kWarpTileSeqLenK; ++j) {
          HMMA16816(R_S[i][j][0], R_S[i][j][1], 
                    R_Q[i][0],    R_Q[i][1],    R_Q[i][2], R_Q[i][3], 
                    R_K[j][0],    R_K[j][1], 
                    R_S[i][j][0], R_S[i][j][1]);
        }
      }
    } // end loop over d, S=Q@K^T
    __syncthreads();

    // The layout of 8 MMA m16n8k16 (2x4)  [after] kWarpTileQPxkWarpTileKV(2x2) -> 32x2,32x2=64x64: 
    // |  [64,64]  |    warp_KV 0    |    warp_KV 1    |    warp_KV 2    |    warp_KV 3    |
    // | warp_QP 0 |-- MMA 0,MMA 0 --|-- MMA 2,MMA 2 --|-- MMA 4,MMA 4 --|-- MMA 6,MMA 6 --| row max
    // | warp_QP 0 |-- MMA 0,MMA 0 --|-- MMA 2,MMA 2 --|-- MMA 4,MMA 4 --|-- MMA 6,MMA 6 --| row max
    // | warp_QP 1 |-- MMA 1,MMA 1 --|-- MMA 3,MMA 3 --|-- MMA 5,MMA 5 --|-- MMA 7,MMA 7 --| row max
    // | warp_QP 1 |-- MMA 1,MMA 1 --|-- MMA 3,MMA 3 --|-- MMA 5,MMA 5 --|-- MMA 7,MMA 7 --| row max

    // Online safe softmax, warp/block reduce max/sum, row wise
    // warp 0/2/4/6, [0][2] row 0~15,  col 0/8/16/32, max, [1][2] row 16~31, col 0/8/16/32, max
    // warp 1/3/5/7, [0][2] row 32~47, col 0/8/16/32, max, [1][2] row 48~61, col 0/8/16/32, max
    float lane_row_max_new[kWarpTileSeqLenQ][2]; 
    float lane_row_sum_new[kWarpTileSeqLenQ][2]; 
    fill_2D_regs<float, kWarpTileSeqLenQ, 2>(lane_row_max_new, -INFINITY);
    fill_2D_regs<float, kWarpTileSeqLenQ, 2>(lane_row_sum_new, 0.0f);

    // Row max for [Br,Bc] tile, Thread -> Warp -> Block.
    #pragma unroll
    for (int i = 0; i < kWarpTileSeqLenQ; ++i) {
      // Thread level reduce max across kWarpTileSeqLenK dim, namely Bc.
      #pragma unroll
      for (int j = 0; j < kWarpTileSeqLenK; ++j) {
        // reference: https://docs.nvidia.com/cuda/parallel-thread-execution/index.html
        // #matrix-fragments-for-mma-m16n8k16-with-floating-point-type
        // The layout of the fragments held by different threads for C. (m16n8k16)
        // Row\Col  0    1    2    3    4    5    6    7
        // 0        T0: {c0, c1}  T1: {c0, c1}  T2: {c0, c1}  T3: {c0, c1}
        // 1        T4: {c0, c1}  T5: {c0, c1}  T6: {c0, c1}  T7: {c0, c1}
        // 2        ...
        // ...
        // 7        T28: {c0, c1}  T29: {c0, c1}  T30: {c0, c1}  T31: {c0, c1}
        // 8        T0: {c2, c3}   T1: {c2, c3}   T2: {c2, c3}   T3: {c2, c3}
        // 9        T4: {c2, c3}   T5: {c2, c3}   T6: {c2, c3}   T7: {c2, c3}
        // 10       ...
        // ...
        // 15       T28: {c2, c3}  T29: {c2, c3}  T30: {c2, c3}  T31: {c2, c3}
        float2 t_reg_S_0 = __half22float2(HALF2(R_S[i][j][0])); // 0~7  {c0, c1}
        float2 t_reg_S_1 = __half22float2(HALF2(R_S[i][j][1])); // 8~15 {c2, c3}
        // This should be the row max after S = (Q @ K^T) / sqrt(d)
        float tmp_max_0 = max(t_reg_S_0.x, t_reg_S_0.y) * scale;
        float tmp_max_1 = max(t_reg_S_1.x, t_reg_S_1.y) * scale;
        lane_row_max_new[i][0] = max(lane_row_max_new[i][0], tmp_max_0);
        lane_row_max_new[i][1] = max(lane_row_max_new[i][1], tmp_max_1);
      } // end for kWarpTileSeqLenK

      // Warp level reduce max, warp_size = 4
      // Each thread contains the maximum of 2 rows of Br, 
      // and only the values of T0, T4, ..., T28 are used.
      // Br, row_id = warp_QP<0|1> * 32 + i<0|1> * 16 + 0 * 8 + (lane / 4) <0~7>
      lane_row_max_new[i][0] = warp_reduce_max<float, 4>(lane_row_max_new[i][0]);
      // Br, row_id = warp_QP<0|1> * 32 + i<0|1> * 16 + 1 * 8 + (lane / 4) <8~15>
      lane_row_max_new[i][1] = warp_reduce_max<float, 4>(lane_row_max_new[i][1]);

      if (lane_id % 4 == 0) { // only need T0,T4,...,T28
        block_row_max_new_smem[ // Br, row_id, 0~7,  16~23, 32~39, 48~55
          warp_QP * 32 + i * 16 + 0 * 8 + (lane_id / 4)][warp_KV] = lane_row_max_new[i][0];
        block_row_max_new_smem[ // Br, row_id, 8~15, 24~31, 40~47, 56~63
          warp_QP * 32 + i * 16 + 1 * 8 + (lane_id / 4)][warp_KV] = lane_row_max_new[i][1];
      }
    } // end for kWarpTileSeqLenQ
    __syncthreads();

    // Block level reduce max, row wise, 64x4=256. Warp reduce operation
    // is faster than atomaicMaxFloat in my tests.
    float wrp_row_max_new = (
      block_row_max_new_smem[tid / kMmaTileSeqLenK][tid % kMmaTileSeqLenK]); // [0~63][0~4]
    float blk_row_max_new = warp_reduce_max<float, 4>(wrp_row_max_new);
    block_row_max_new_smem[tid / kMmaTileSeqLenK][tid % kMmaTileSeqLenK] = (
      blk_row_max_new);
    __syncthreads();

    // Exp sum and mul scale_factor for [Br,Bc] tile, Thread -> Warp -> Block.
    #pragma unroll
    for (int i = 0; i < kWarpTileSeqLenQ; ++i) {
      // Use latest global row max without update.
      // Br 0, row_id, 0~7,  16~23, 32~39, 48~55; 
      float block_row_max_new_0 = block_row_max_new_smem[
        warp_QP * 32 + i * 16 + 0 * 8 + (lane_id / 4)][0]; 
      // Br 1, row_id, 8~15, 24~31, 40~47, 56~63;
      float block_row_max_new_1 = block_row_max_new_smem[
        warp_QP * 32 + i * 16 + 1 * 8 + (lane_id / 4)][0];
  
      float block_row_max_old_0 = lane_block_row_max_old[i][0];
      float block_row_max_old_1 = lane_block_row_max_old[i][1];
      // Apply m_new = max(m_old, m_new) here.
      block_row_max_new_0 = max(block_row_max_old_0, block_row_max_new_0);
      block_row_max_new_1 = max(block_row_max_old_1, block_row_max_new_1);

      #pragma unroll
      for (int j = 0; j < kWarpTileSeqLenK; ++j) {
        float2 t_reg_S_0 = __half22float2(HALF2(R_S[i][j][0])); // 0~7  {c0, c1}
        float2 t_reg_S_1 = __half22float2(HALF2(R_S[i][j][1])); // 8~15 {c2, c3}
        // P = Exp(S - m_new), fmaf(x, y, z) = x * y + z;
        t_reg_S_0.x = __expf(__fmaf_rn(t_reg_S_0.x, scale, - block_row_max_new_0));
        t_reg_S_0.y = __expf(__fmaf_rn(t_reg_S_0.y, scale, - block_row_max_new_0));
        t_reg_S_1.x = __expf(__fmaf_rn(t_reg_S_1.x, scale, - block_row_max_new_1));
        t_reg_S_1.y = __expf(__fmaf_rn(t_reg_S_1.y, scale, - block_row_max_new_1));
        lane_row_sum_new[i][0] += (t_reg_S_0.x + t_reg_S_0.y);
        lane_row_sum_new[i][1] += (t_reg_S_1.x + t_reg_S_1.y);
        // Update R_S for P[Br,Bc] = Exp(S-m), point wise.
        HALF2(R_S[i][j][0]) = __float22half2_rn(t_reg_S_0);
        HALF2(R_S[i][j][1]) = __float22half2_rn(t_reg_S_1);
      } // end for kWarpTileSeqLenK

      // Warp level reduce sum, warp_size = 4
      lane_row_sum_new[i][0] = warp_reduce_sum<float, 4>(lane_row_sum_new[i][0]);
      lane_row_sum_new[i][1] = warp_reduce_sum<float, 4>(lane_row_sum_new[i][1]);

      if (lane_id % 4 == 0) { // only need T0,T4,...,T28
        block_row_sum_new_smem[ // Br, row_id, 0~7,  16~23, 32~39, 48~55
          warp_QP * 32 + i * 16 + 0 * 8 + (lane_id / 4)][warp_KV] = lane_row_sum_new[i][0];
        block_row_sum_new_smem[ // Br, row_id, 8~15, 24~31, 40~47, 56~63
          warp_QP * 32 + i * 16 + 1 * 8 + (lane_id / 4)][warp_KV] = lane_row_sum_new[i][1];
      }
    } // end for kWarpTileSeqLenQ
    __syncthreads();

    // Block level reduce sum, row wise, 64x4=256. Warp reduce operation
    // is faster than atomaicAdd float in my tests.
    float wrp_row_sum_new = (
      block_row_sum_new_smem[tid / kMmaTileSeqLenK][tid % kMmaTileSeqLenK]); // [0~63][0~4]
    float blk_row_sum_new = warp_reduce_sum<float, 4>(wrp_row_sum_new);
    block_row_sum_new_smem[tid / kMmaTileSeqLenK][tid % kMmaTileSeqLenK] = (
      blk_row_sum_new);
    __syncthreads();
    
    // Retile warp for [Br,d], kWarpTileHeadDimV: 1=32/(4*8); 2=64/(4*8); 4=128/(4*8).
    // Compute P[Br,Bc] @ V[Bc,d] = [Br,d] = [64, 64/128], partion Attention.

    // If headdim=<32>, then, kWarpTileHeadDimV = 1, the layout of 8 MMA m16n8k16 (2x4) after 
    // kWarpTileSeqLenPxkWarpTileHeadDimV(2x1) tiling to (32x2,32x1)=(64x32), will look like: 
    // |  [64,32]  | warp_KV 0 | warp_KV 1 | warp_KV 2 | warp_KV 3 |
    // | warp_QP 0 |-- MMA 0 --|-- MMA 2 --|-- MMA 4 --|-- MMA 6 --|
    // | warp_QP 0 |-- MMA 0 --|-- MMA 2 --|-- MMA 4 --|-- MMA 6 --|
    // | warp_QP 1 |-- MMA 1 --|-- MMA 3 --|-- MMA 5 --|-- MMA 7 --|
    // | warp_QP 1 |-- MMA 1 --|-- MMA 3 --|-- MMA 5 --|-- MMA 7 --|

    // If headdim=<64>, then, kWarpTileHeadDimV = 2, the layout of 8 MMA m16n8k16 (2x4) after 
    // kWarpTileSeqLenPxkWarpTileHeadDimV(2x2) tiling to (32x2,32x2)=(64x64), will look like: 
    // |  [64,64]  |    warp_KV 0    |    warp_KV 1    |    warp_KV 2    |    warp_KV 3    |
    // | warp_QP 0 |-- MMA 0,MMA 0 --|-- MMA 2,MMA 2 --|-- MMA 4,MMA 4 --|-- MMA 6,MMA 6 --|
    // | warp_QP 0 |-- MMA 0,MMA 0 --|-- MMA 2,MMA 2 --|-- MMA 4,MMA 4 --|-- MMA 6,MMA 6 --|
    // | warp_QP 1 |-- MMA 1,MMA 1 --|-- MMA 3,MMA 3 --|-- MMA 5,MMA 5 --|-- MMA 7,MMA 7 --|
    // | warp_QP 1 |-- MMA 1,MMA 1 --|-- MMA 3,MMA 3 --|-- MMA 5,MMA 5 --|-- MMA 7,MMA 7 --|

    // If headdim=<128>, then, kWarpTileHeadDimV = 4, the layout of 8 MMA m16n8k16 (2x4) after 
    // kWarpTileSeqLenPxkWarpTileHeadDimV(2x2x2) tiling to (32x2,32x2x2)=(64x64x2), will look like: 
    // | [64,64x2] |         warp_KV 0           |           warp_KV 1         |           warp_KV 2         |          warp_KV 3          |
    // | warp_QP 0 |-- MMA 0,MMA 0,MMA 0,MMA 0 --|-- MMA 2,MMA 2,MMA 2,MMA 2 --|-- MMA 4,MMA 4,MMA 4,MMA 4 --|-- MMA 6,MMA 6,MMA 6,MMA 6 --|
    // | warp_QP 0 |-- MMA 0,MMA 0,MMA 0,MMA 0 --|-- MMA 2,MMA 2,MMA 2,MMA 2 --|-- MMA 4,MMA 4,MMA 4,MMA 4 --|-- MMA 6,MMA 6,MMA 6,MMA 6 --|
    // | warp_QP 1 |-- MMA 1,MMA 1,MMA 1,MMA 1 --|-- MMA 3,MMA 3,MMA 3,MMA 3 --|-- MMA 5,MMA 5,MMA 5,MMA 5 --|-- MMA 7,MMA 7,MMA 7,MMA 7 --|
    // | warp_QP 1 |-- MMA 1,MMA 1,MMA 1,MMA 1 --|-- MMA 3,MMA 3,MMA 3,MMA 3 --|-- MMA 5,MMA 5,MMA 5,MMA 5 --|-- MMA 7,MMA 7,MMA 7,MMA 7 --|
    
    // Write R_P(R_S) to P_smem [Br,Bc]
    // store S[Br,Bc] of [seqlen,seqlen] [64,64]
    #pragma unroll 
    for (int i = 0; i < kWarpTileSeqLenQ; ++i) {
      #pragma unroll 
      for (int j = 0; j < kWarpTileSeqLenK; ++j) {
        R_Q[0][0] = R_S[i][j][0]; R_Q[1][0] = R_S[i][j][1]; // warp_size 4
        R_Q[0][1] = __shfl_sync((0xffffffff), R_S[i][j][0], lane_id + 1, 4);
        R_Q[0][2] = __shfl_sync((0xffffffff), R_S[i][j][0], lane_id + 2, 4);
        R_Q[0][3] = __shfl_sync((0xffffffff), R_S[i][j][0], lane_id + 3, 4);
        R_Q[1][1] = __shfl_sync((0xffffffff), R_S[i][j][1], lane_id + 1, 4);
        R_Q[1][2] = __shfl_sync((0xffffffff), R_S[i][j][1], lane_id + 2, 4);
        R_Q[1][3] = __shfl_sync((0xffffffff), R_S[i][j][1], lane_id + 3, 4);
  
        // st.global.v4 128 bits.
        if (lane_id % 4 == 0) {
          // (0/1)*32 + (0/1)*16=(0,16,32,48), + 0~7 -> 0~56
          int store_warp_regs_S_Br = warp_QP * (kMmaAtomM * kWarpTileSeqLenQ) + i * kMmaAtomM;
          int store_lane_smem_S_Br = store_warp_regs_S_Br + lane_id / 4; // 0~7
          // (0~3)*16 + (0/1)*8=(0,8,16,24,...,48,56)
          int store_warp_regs_S_Bc = warp_KV * (kMmaAtomN * kWarpTileSeqLenK) + j * kMmaAtomN;
          int store_lane_smem_S_Bc = store_warp_regs_S_Bc; // (0~3)*16+(0/8)
          int store_smem_S_addr_0 = ((store_lane_smem_S_Br + 0) * (Bc + kPad) + store_lane_smem_S_Bc);
          int store_smem_S_addr_1 = ((store_lane_smem_S_Br + 8) * (Bc + kPad) + store_lane_smem_S_Bc);
          LDST128BITS(S_tile_smem[store_smem_S_addr_0]) = LDST128BITS(R_Q[0][0]);
          LDST128BITS(S_tile_smem[store_smem_S_addr_1]) = LDST128BITS(R_Q[1][0]);
        }
      } // end for kWarpTileHeadDimV
    } // end for kWarpTileSeqLenQ
    __syncthreads(); 

    // Compute P[Br,Bc] @ V[Bc,d] = [Br,d] = [64, 64/128], partion Attention.
    // Here, we have to wait V ready before compute O = P @ V
    if constexpr (kStage > 1) {
      // NOTE: For kStage > 1, we have send V mem issues before K
      if ((tile_K_seqlen + 1) < Tc) {
        CP_ASYNC_WAIT_GROUP(1); 
      } else {
        CP_ASYNC_WAIT_GROUP(0);
      }
    } else {
      CP_ASYNC_WAIT_GROUP(0);
    }
    __syncthreads(); 
    
    // <loop over V Bc>: P[Br,Bc]@V[Bc,d]=[Br,d]=[64,64/128], partion Attention.
    // Matmul with NN layout: P[Br,Bc] row major, V[Bc,d] row major.
    // Make sure to clear the states in R_O before MMA for P@V for each step.
    fill_3D_regs<uint32_t, kWarpTileSeqLenP, kWarpTileHeadDimV, 2>(R_O, 0);
    #pragma unroll
    for (int tile_V_Bc = 0; tile_V_Bc < (Bc / kMmaAtomK); ++tile_V_Bc) {
      // smem -> reg, load m16k16 smem Q, offset d according tile_K_d.
      // ldmatrix.x4 from S_tile_smem.
      #pragma unroll
      for (int i = 0; i < kWarpTileSeqLenP; ++i) { // S[Br,Bc]=[M,K]
        int warp_smem_S_Br = warp_QP * (kMmaAtomM * kWarpTileSeqLenP) + i * kMmaAtomM;
        int lane_smem_S_Br = warp_smem_S_Br + lane_id % 16; // 0~15
        int lane_smem_S_Bc  = tile_V_Bc * kMmaAtomK + (lane_id / 16) * 8; // 0,8
        uint32_t lane_smem_S_ptr = (
            smem_S_base_ptr + (lane_smem_S_Br * (Bc + kPad) + 
                               lane_smem_S_Bc) * sizeof(half)
        );
        LDMATRIX_X4(R_Q[i][0], R_Q[i][1], R_Q[i][2], R_Q[i][3], 
                    lane_smem_S_ptr); // now, R_P
      }

      // Load k16n8 V from smem -> regs, R_KV, ldmatrix.x2.trans.
      #pragma unroll
      for (int j = 0; j < kWarpTileHeadDimV; ++j) { 
        int warp_smem_V_d  = warp_KV * (kMmaAtomN * kWarpTileHeadDimV) + j * kMmaAtomN; // d, matmaul N
        int lane_smem_V_Bc = tile_V_Bc * kMmaAtomK + lane_id % 16; // 0~15; Bc, matmul K
        int lane_smem_V_d  = warp_smem_V_d; // 0
        uint32_t lane_smem_V_ptr = (
          smem_V_base_ptr + (lane_smem_V_Bc * (kHeadDim + kPad) + 
                             lane_smem_V_d) * sizeof(half)
        );
        LDMATRIX_X2_T(R_V[j][0], R_V[j][1], lane_smem_V_ptr); // R_V
      }
        
      // NOTE: Values for P[Br,Bc] already in R_S registers, can we use these 
      // registers for P(A) matrix directly ? How to do that ?
      // according to the A matrix layout for MMA m16n8k16 instruction. 
      // reference: https://docs.nvidia.com/cuda/parallel-thread-execution/index.html
      // #matrix-fragments-for-mma-m16n8k16-with-floating-point-type
      // The layout of the fragments held by different threads for A matrix with .f16.
      // R\C  0    1    2    3    4    5    6    7    8    9   10   11   12   13   14   15
      // 0    T0: {a0, a1}  T1: {a0, a1}  T2: {a0, a1}  T3: {a0, a1}  T0: {a4, a5}  T1: {a4, a5}  T2: {a4, a5}  T3: {a4, a5}
      // 1    T4: {a0, a1}  T5: {a0, a1}  T6: {a0, a1}  T7: {a0, a1}  T4: {a4, a5}  T5: {a4, a5}  T6: {a4, a5}  T7: {a4, a5}
      // 2    (dashed arrow pointing right)
      // ...
      // 7    T28: {a0, a1}  T29: {a0, a1}  T30: {a0, a1}  T31: {a0, a1}  T28: {a4, a5}  T29: {a4, a5}  T30: {a4, a5}  T31: {a4, a5}
      // 8    T0: {a2, a3}   T1: {a2, a3}   T2: {a2, a3}   T3: {a2, a3}   T0: {a6, a7}   T1: {a6, a7}   T2: {a6, a7}   T3: {a6, a7}
      // 9    T4: {a2, a3}   T5: {a2, a3}   T6: {a2, a3}   T7: {a2, a3}   T4: {a6, a7}   T5: {a6, a7}   T6: {a6, a7}   T7: {a6, a7}
      // 10   (dashed arrow pointing right)
      // ...
      // 15   T28: {a2, a3}  T29: {a2, a3}  T30: {a2, a3}  T31: {a2, a3}  T28: {a6, a7}  T29: {a6, a7}  T30: {a6, a7}  T31: {a6, a7}
      #pragma unroll
      for (int i = 0; i < kWarpTileSeqLenP; ++i) { // kWarpTileSeqLenQ=2
        #pragma unroll
        for (int j = 0; j < kWarpTileHeadDimV; ++j) { // kWarpTileHeadDimV=1,2,3,4,...
          // tile_V_Bc = 0, all curr MMAs(0~7) need P[:,  0:16]; but only stored in MMA 0, MMA 1, S_P
          // tile_V_Bc = 1, all curr MMAs(0~7) need P[:, 16:32]; but only stored in MMA 2, MMA 3, S_P 
          // tile_V_Bc = 2, all curr MMAs(0~7) need P[:, 32:48]; but only stored in MMA 4, MMA 5, S_P 
          // tile_V_Bc = 3, all curr MMAs(0~7) need P[:, 48:64]; but only stored in MMA 6, MMA 7, S_P 
          // We have to comm across warps to get right values for MMA inner loop, namely,
          // we have to use shared memory to collect values from other warps. Thus, S_P
          // can not use as A matrix in MMA in split_kv mode.
          HMMA16816(R_O[i][j][0], R_O[i][j][1], 
                    R_Q[i][0],    R_Q[i][1],    R_Q[i][2], R_Q[i][3], 
                    R_V[j][0],    R_V[j][1],
                    R_O[i][j][0], R_O[i][j][1]);
        }
      }
    } // end for V Bc.
    __syncthreads(); 

    // Rescale O -> Update row sum Exp -> then, Update row max.
    #pragma unroll
    for (int i = 0; i < kWarpTileSeqLenP; ++i) { // kWarpTileSeqLenQ=kWarpTileSeqLenP
      // m = max(m_old, m_new), l = exp(m_old - m) * l_old + l_new (FA2 paper)
      // Br 0, row_id, 0~7,  16~23, 32~39, 48~55; Br 1, row_id, 8~15, 24~31, 40~47, 56~63
      float block_row_max_new_0 = block_row_max_new_smem[
        warp_QP * 32 + i * 16 + 0 * 8 + (lane_id / 4)][0];
      float block_row_max_new_1 = block_row_max_new_smem[
        warp_QP * 32 + i * 16 + 1 * 8 + (lane_id / 4)][0];
      float block_row_sum_new_0 = block_row_sum_new_smem[
        warp_QP * 32 + i * 16 + 0 * 8 + (lane_id / 4)][0];
      float block_row_sum_new_1 = block_row_sum_new_smem[
        warp_QP * 32 + i * 16 + 1 * 8 + (lane_id / 4)][0];
      
      float block_row_max_old_0 = lane_block_row_max_old[i][0];
      float block_row_max_old_1 = lane_block_row_max_old[i][1];
      // NOTE: max(-inf, val) = val.
      block_row_max_new_0 = max(block_row_max_old_0, block_row_max_new_0);
      block_row_max_new_1 = max(block_row_max_old_1, block_row_max_new_1);   
      // Avoid inf value while using m_old for rescaling O.
      block_row_max_old_0 = (tile_K_seqlen > 0 ? block_row_max_old_0 : 
                                                 block_row_max_new_0);                                       
      block_row_max_old_1 = (tile_K_seqlen > 0 ? block_row_max_old_1 : 
                                                 block_row_max_new_1);  

      // rescale factor for O and l, exp(m_old - m)
      float rescale_o_factor_0 = __expf(block_row_max_old_0 - block_row_max_new_0);
      float rescale_o_factor_1 = __expf(block_row_max_old_1 - block_row_max_new_1);
      // 0. Rescale O: Online rescaling O each tile_K_seqlen step, need m_new, m_old.
      // m = max(m_old, m_new), O_new[Br,d] = exp(m_old - m) * O_old + P@V
      #pragma unroll
      for (int j = 0; j < kWarpTileHeadDimV; ++j) {
        float2 t_reg_O_0 = __half22float2(HALF2(R_O[i][j][0])); // 0~7  {c0, c1}
        float2 t_reg_O_1 = __half22float2(HALF2(R_O[i][j][1])); // 8~15 {c2, c3}
        float2 t_reg_D_0 = __half22float2(HALF2(R_D[i][j][0])); // 0~7  {c0, c1}
        float2 t_reg_D_1 = __half22float2(HALF2(R_D[i][j][1])); // 8~15 {c2, c3}
        // Note that the formula in the FA2 paper is incorrect; here, 
        // the inverse of the exp function should not be taken, as it 
        // would result in an error during rescaling, namely, you have
        // use exp(m_old - m_new), not 1/(m_old - m_new).
        // O_new[Br,d] = exp(m_old - m_new) * O_old + P@V
        t_reg_D_0.x = __fmaf_rn(rescale_o_factor_0, t_reg_D_0.x, t_reg_O_0.x);
        t_reg_D_0.y = __fmaf_rn(rescale_o_factor_0, t_reg_D_0.y, t_reg_O_0.y);
        t_reg_D_1.x = __fmaf_rn(rescale_o_factor_1, t_reg_D_1.x, t_reg_O_1.x);
        t_reg_D_1.y = __fmaf_rn(rescale_o_factor_1, t_reg_D_1.y, t_reg_O_1.y);
        HALF2(R_D[i][j][0]) = __float22half2_rn(t_reg_D_0);
        HALF2(R_D[i][j][1]) = __float22half2_rn(t_reg_D_1);
      } // end for kWarpTileHeadDimV.

      // Now, we can update m, l after O has been scaled.
      // 1. First, update block row sum Exp for each lane which
      // need both m_new and m_old.
      float block_row_sum_old_0 = lane_block_row_sum_old[i][0];
      float block_row_sum_old_1 = lane_block_row_sum_old[i][1];
      // Update l = exp(m_old - m_new) * l_old + row_sum(P).
      lane_block_row_sum_old[i][0] = (__fmaf_rn(
        rescale_o_factor_0, block_row_sum_old_0, block_row_sum_new_0));
      lane_block_row_sum_old[i][1] = (__fmaf_rn(
        rescale_o_factor_1, block_row_sum_old_1, block_row_sum_new_1));
      // 2. Then, update block row max for each lane.
      lane_block_row_max_old[i][0] = block_row_max_new_0;
      lane_block_row_max_old[i][1] = block_row_max_new_1;
    }

    // NOTE: After compute P @ V, we have to wait next K tile ready in smem.
    // do not need to wait any things if kStage == 1.
    if constexpr (kStage > 1) {
      if ((tile_K_seqlen + 1) < Tc) {
        CP_ASYNC_WAIT_GROUP(0); 
      }
      __syncthreads(); 
    }

  } // end loop over N
  __syncthreads();

  // Finaly, we still have to rescale O once more.
  // O_output(D) = ( 1/l_final ) * O_final (FA2 paper)
  #pragma unroll
  for (int i = 0; i < kWarpTileSeqLenP; ++i) {
    float rescale_factor_0 = __frcp_rn(lane_block_row_sum_old[i][0]);
    float rescale_factor_1 = __frcp_rn(lane_block_row_sum_old[i][1]);
    #pragma unroll
    for (int j = 0; j < kWarpTileHeadDimV; ++j) {
      float2 t_reg_D_0 = __half22float2(HALF2(R_D[i][j][0])); // 0~7  {c0, c1}
      float2 t_reg_D_1 = __half22float2(HALF2(R_D[i][j][1])); // 8~15 {c2, c3}
      t_reg_D_0.x = rescale_factor_0 * t_reg_D_0.x;
      t_reg_D_0.y = rescale_factor_0 * t_reg_D_0.y;
      t_reg_D_1.x = rescale_factor_1 * t_reg_D_1.x;
      t_reg_D_1.y = rescale_factor_1 * t_reg_D_1.y;
      HALF2(R_D[i][j][0]) = __float22half2_rn(t_reg_D_0);
      HALF2(R_D[i][j][1]) = __float22half2_rn(t_reg_D_1);
    }
  }

  // Store O(D): Write O[Br,d] from regs -> gmem, collective store 
  // with reg reuse & warp shuffle. need R[2][4], may reuse 
  // R_Q[kWarpTileSeqLenQ][4]=[2][4].
  #pragma unroll
  for (int i = 0; i < kWarpTileSeqLenP; ++i) {
    #pragma unroll
    for (int j = 0; j < kWarpTileHeadDimV; ++j) {
      static_assert(kWarpTileSeqLenQ >= 2);
      R_Q[0][0] = R_D[i][j][0]; R_Q[1][0] = R_D[i][j][1]; // warp_size 4
      R_Q[0][1] = __shfl_sync((0xffffffff), R_D[i][j][0], lane_id + 1, 4);
      R_Q[0][2] = __shfl_sync((0xffffffff), R_D[i][j][0], lane_id + 2, 4);
      R_Q[0][3] = __shfl_sync((0xffffffff), R_D[i][j][0], lane_id + 3, 4);
      R_Q[1][1] = __shfl_sync((0xffffffff), R_D[i][j][1], lane_id + 1, 4);
      R_Q[1][2] = __shfl_sync((0xffffffff), R_D[i][j][1], lane_id + 2, 4);
      R_Q[1][3] = __shfl_sync((0xffffffff), R_D[i][j][1], lane_id + 3, 4);

      // st.global.v4 128 bits. [Br,d]
      if (lane_id % 4 == 0) {
        // (0/1)*32 + (0/1)*16=(0,16,32,48), + 0~7 -> 0~56
        int store_warp_regs_O_Br = warp_QP * (kMmaAtomM * kWarpTileSeqLenP ) + i * kMmaAtomM;
        int store_lane_gmem_O_Br = O_tile_id * Br + store_warp_regs_O_Br + lane_id / 4; // 0~7
        // (0~3)*16 + (0/1)*8=(0,8,16,24,...,48,56)
        int store_warp_regs_O_d = warp_KV * (kMmaAtomN * kWarpTileHeadDimV) + j * kMmaAtomN;
        int store_lane_gmem_O_d = store_warp_regs_O_d; // (0~3)*16+(0/8)
        int store_gmem_O_addr_0 = (O_gmem_offset + (store_lane_gmem_O_Br + 0) * kHeadDim + 
                                   store_lane_gmem_O_d);
        int store_gmem_O_addr_1 = (O_gmem_offset + (store_lane_gmem_O_Br + 8) * kHeadDim + 
                                   store_lane_gmem_O_d);
        LDST128BITS(O[store_gmem_O_addr_0]) = LDST128BITS(R_Q[0][0]);
        LDST128BITS(O[store_gmem_O_addr_1]) = LDST128BITS(R_Q[1][0]);
      }
    } // end for kWarpTileHeadDimV
  } // end for kWarpTileSeqLenQ
}


template<const int kHeadDim, const int kStage>
void launch_flash_attn_mma_stages_split_kv(
  torch::Tensor Q, torch::Tensor K, torch::Tensor V, torch::Tensor O) {
  // Now: fixed tile BrxBc=64x64
  // TODO: dynamic tile size for Br, Bc according to kHeadDim and shared memory size.
  constexpr int kMmaAtomM = 16;
  constexpr int kMmaAtomN = 8;
  constexpr int kMmaAtomK = 16;
  constexpr int kMmaTileSeqLenQ = 2;
  constexpr int kMmaTileSeqLenP = 2;
  constexpr int kMmaTileSeqLenK = 4;
  constexpr int kMmaTileHeadDimV = 4;
  constexpr int kWarpTileSeqLenQ = 2;
  constexpr int kWarpTileSeqLenP = 2;
  constexpr int kWarpTileSeqLenK = 2;
  constexpr int kWarpTileHeadDimV = (kHeadDim / (kMmaAtomN*kMmaTileHeadDimV));
  constexpr int Br = kMmaAtomM * kMmaTileSeqLenQ * kWarpTileSeqLenQ; // 16*2*2=64
  constexpr int Bc = kMmaAtomN * kMmaTileSeqLenK * kWarpTileSeqLenK; // 8*4*2=64
  constexpr int kNumThreads = WARP_SIZE * kMmaTileSeqLenQ * kMmaTileSeqLenK;
  constexpr int kPad = 8;

  // static int kMaxSramPerBlock;
  // cudaDeviceGetAttribute(&kMaxSramPerBlock, cudaDevAttrMaxSharedMemoryPerBlock, 0);
  // Calculate SRAM size needed per block, Q,K,V,S smem size
  const int smem_max_size = ((Br * (kHeadDim + kPad)) + 
                             (kStage * Bc * (kHeadDim + kPad)) + 
                             (Bc * (kHeadDim + kPad)) + 
                             (Br * (Bc + kPad))) * sizeof(half);

  const int QKV_batch  = Q.size(0); 
  const int QKV_head   = Q.size(1);
  const int QKV_seqlen = Q.size(2); // QKV_seqlen
  assert(QKV_seqlen % Bc == 0); // multiple of Bc=64

  // TODO: How to apply block swizzle to improve L2 Cache hit rate?
  // NOTE: reorder (B,H,Tr) -> (Tr,B*H) seems can improve L2 Cache hit rate. 
  // This might be because SM schedules blocks starting from the x-dimension. 
  // Placing Tr at the forefront ensures that identical KV pairs are placed 
  // in consecutive scheduling queues, thereby improving L2 Cache hit rates.
  // Tr(=N/Br), batch_size x num_heads
  dim3 grid(div_ceil(QKV_seqlen, Br), QKV_batch * QKV_head); 
  dim3 block(kNumThreads); // 4/8 warps per block

  cudaFuncSetAttribute(
    flash_attn_mma_stages_split_kv_kernel<
      kHeadDim, 
      kMmaAtomM, 
      kMmaAtomN, 
      kMmaAtomK, 
      kMmaTileSeqLenQ, 
      kMmaTileSeqLenK, 
      kMmaTileSeqLenP, 
      kMmaTileHeadDimV, 
      kWarpTileSeqLenQ, 
      kWarpTileSeqLenK, 
      kWarpTileSeqLenP, 
      kWarpTileHeadDimV, 
      kStage, 
      kPad
    >,
    cudaFuncAttributeMaxDynamicSharedMemorySize,
    // kMaxSramPerBlock
    98304
  );

  flash_attn_mma_stages_split_kv_kernel<
    kHeadDim, 
    kMmaAtomM, 
    kMmaAtomN, 
    kMmaAtomK, 
    kMmaTileSeqLenQ,  
    kMmaTileSeqLenK,
    kMmaTileSeqLenP, 
    kMmaTileHeadDimV, 
    kWarpTileSeqLenQ, 
    kWarpTileSeqLenK, 
    kWarpTileSeqLenP, 
    kWarpTileHeadDimV, 
    kStage, 
    kPad
  ><<<grid, block, smem_max_size>>>(
    reinterpret_cast<half*>(Q.data_ptr()),
    reinterpret_cast<half*>(K.data_ptr()),
    reinterpret_cast<half*>(V.data_ptr()),
    reinterpret_cast<half*>(O.data_ptr()),
    QKV_seqlen,
    QKV_head
  );
}

void flash_attn_mma_stages_split_kv(torch::Tensor Q, 
                                    torch::Tensor K, 
                                    torch::Tensor V, 
                                    torch::Tensor O, 
                                    int stages) {
  CHECK_TORCH_TENSOR_DTYPE(Q, torch::kHalf) // Q [B,H,N,D]
  CHECK_TORCH_TENSOR_DTYPE(K, torch::kHalf) // K [B,H,N,D]
  CHECK_TORCH_TENSOR_DTYPE(V, torch::kHalf) // V [B,H,N,D]
  CHECK_TORCH_TENSOR_DTYPE(O, torch::kHalf) // O [B,H,N,D]
  const int d = Q.size(3); // B, H, N, d

  if (stages > 1) {
    switch (d)
    {
    case 32:
      launch_flash_attn_mma_stages_split_kv<32,  2>(Q, K, V, O);
      break;
    case 64:
      launch_flash_attn_mma_stages_split_kv<64,  2>(Q, K, V, O);
      break;
    case 96:
      launch_flash_attn_mma_stages_split_kv<96,  2>(Q, K, V, O);
      break;
    case 128:
      launch_flash_attn_mma_stages_split_kv<128, 2>(Q, K, V, O);
      break;
    default:
      throw std::runtime_error("headdim not support!");
      break;
    }
  } else {
    switch (d)
    {
    case 32:
      launch_flash_attn_mma_stages_split_kv<32,  1>(Q, K, V, O);
      break;
    case 64:
      launch_flash_attn_mma_stages_split_kv<64,  1>(Q, K, V, O);
      break;
    case 96:
      launch_flash_attn_mma_stages_split_kv<96,  1>(Q, K, V, O);
      break;
    case 128:
      launch_flash_attn_mma_stages_split_kv<128, 1>(Q, K, V, O);
      break;
    default:
      throw std::runtime_error("headdim not support!");
      break;
    }
  }
}