/*
Kernels for layernorm backward pass.

Compile example:
nvcc -O3 --use_fast_math -lcublas -lcublasLt layernorm_backward.cu -o layernorm_backward

version 1 is naive port from CPU code to kernel: parallelizes over B,T, loops over C
./layernorm_backward 1

version 2 moves a lot of reduction to shared memory over global memory
./layernorm_backward 2
*/

#include <stdio.h>
#include <stdlib.h>
#include <cuda_runtime.h>
#include <assert.h>
#include <cooperative_groups.h>
#include <cooperative_groups/reduce.h>

#define ENABLE_BF16
#include "common.h"

// ----------------------------------------------------------------------------
// CPU code reference

void layernorm_forward_cpu(float* out, float* mean, float* rstd,
                       const float* inp, const float* weight, const float* bias,
                       int B, int T, int C) {
    // reference: https://pytorch.org/docs/stable/generated/torch.nn.LayerNorm.html
    // both inp and out are (B,T,C) of the activations
    // mean and rstd are (B,T) buffers, to be used later in backward pass
    // at each position (b,t) of the input, the C-dimensional vector
    // of activations gets normalized, then scaled and shifted
    float eps = 1e-5f;
    for (int b = 0; b < B; b++) {
        for (int t = 0; t < T; t++) {
            // seek to the input position inp[b,t,:]
            const float* x = inp + b * T * C + t * C;
            // calculate the mean
            float m = 0.0f;
            for (int i = 0; i < C; i++) {
                m += x[i];
            }
            m = m/C;
            // calculate the variance (without any bias correction)
            float v = 0.0f;
            for (int i = 0; i < C; i++) {
                float xshift = x[i] - m;
                v += xshift * xshift;
            }
            v = v/C;
            // calculate the rstd (reciprocal standard deviation)
            float s = 1.0f / sqrtf(v + eps);
            // seek to the output position in out[b,t,:]
            float* out_bt = out + b * T * C + t * C;
            for (int i = 0; i < C; i++) {
                float n = (s * (x[i] - m)); // normalize
                float o = n * weight[i] + bias[i]; // scale and shift
                out_bt[i] = o; // write
            }
            // cache the mean and rstd for the backward pass later
            mean[b * T + t] = m;
            rstd[b * T + t] = s;
        }
    }
}

void layernorm_backward_cpu(float* dinp, float* dweight, float* dbias,
                        const float* dout, const float* inp, const float* weight, const float* mean, const float* rstd,
                        int B, int T, int C) {
    for (int b = 0; b < B; b++) {
        for (int t = 0; t < T; t++) {
            const float* dout_bt = dout + b * T * C + t * C;
            const float* inp_bt = inp + b * T * C + t * C;
            float* dinp_bt = dinp + b * T * C + t * C;
            const float mean_bt = mean[b * T + t];
            const float rstd_bt = rstd[b * T + t];

            // first: two reduce operations
            float dnorm_mean = 0.0f;
            float dnorm_norm_mean = 0.0f;
            for (int i = 0; i < C; i++) {
                float norm_bti = (inp_bt[i] - mean_bt) * rstd_bt;
                float dnorm_i = weight[i] * dout_bt[i];
                dnorm_mean += dnorm_i;
                dnorm_norm_mean += dnorm_i * norm_bti;
            }
            dnorm_mean = dnorm_mean / C;
            dnorm_norm_mean = dnorm_norm_mean / C;

            // now iterate again and accumulate all the gradients
            for (int i = 0; i < C; i++) {
                float norm_bti = (inp_bt[i] - mean_bt) * rstd_bt;
                float dnorm_i = weight[i] * dout_bt[i];
                // gradient contribution to bias
                dbias[i] += dout_bt[i];
                // gradient contribution to weight
                dweight[i] += norm_bti * dout_bt[i];
                // gradient contribution to input
                float dval = 0.0f;
                dval += dnorm_i; // term 1
                dval -= dnorm_mean; // term 2
                dval -= norm_bti * dnorm_norm_mean; // term 3
                dval *= rstd_bt; // final scale
                dinp_bt[i] += dval;
            }
        }
    }
}

// ----------------------------------------------------------------------------
// GPU kernels

// GPU helper functions for atomicAdd on smaller than 32-bit types
#ifdef ENABLE_BF16
__device__ void atomicAddX(__nv_bfloat16* addr, __nv_bfloat16 val) {
    uintptr_t ptr_val = reinterpret_cast<uintptr_t>(addr);
    __nv_bfloat162* ptr_bf16 = reinterpret_cast<__nv_bfloat162*>(ptr_val & ~uintptr_t(0x3));

    // Prepare the value to add, setting the other half to zero
    __nv_bfloat162 add_val = (ptr_val & 0x3) ? __halves2bfloat162(__ushort_as_bfloat16(0), val)
                                             : __halves2bfloat162(val, __ushort_as_bfloat16(0));
    atomicAdd(ptr_bf16, add_val);
}
#endif
#ifdef ENABLE_FP16
__device__ void atomicAddX(half* addr, half val) {
    uintptr_t ptr_val = reinterpret_cast<uintptr_t>(addr);
    half2* ptr_fp16 = reinterpret_cast<half2*>(ptr_val & ~uintptr_t(0x3));

    // Prepare the value to add, setting the other half to zero
    half2 add_val = (ptr_val & 0x3) ? __halves2half2(__ushort_as_half(0), val)
                                    : __halves2half2(val, __ushort_as_half(0));
    atomicAdd(ptr_fp16, add_val);
}
#endif
__device__ void atomicAddX(float* addr, float val) {
    atomicAdd(addr, val);
}

// super naive kernel that just parallelizes over B,T and loops over C
__global__ void layernorm_backward_kernel1(float* dinp, float* dweight, float* dbias,
                        const float* dout, const float* inp, const float* weight, const float* mean, const float* rstd,
                        int B, int T, int C) {
    int idx = blockIdx.x * blockDim.x + threadIdx.x;
    if (idx >= B*T) return;
    int b = idx / T;
    int t = idx % T;

    const float* dout_bt = dout + b * T * C + t * C;
    const float* inp_bt = inp + b * T * C + t * C;
    float* dinp_bt = dinp + b * T * C + t * C;
    const float mean_bt = mean[b * T + t];
    const float rstd_bt = rstd[b * T + t];

    // first: two reduce operations
    float dnorm_mean = 0.0f;
    float dnorm_norm_mean = 0.0f;
    for (int i = 0; i < C; i++) {
        float norm_bti = (inp_bt[i] - mean_bt) * rstd_bt;
        float dnorm_i = weight[i] * dout_bt[i];
        dnorm_mean += dnorm_i;
        dnorm_norm_mean += dnorm_i * norm_bti;
    }
    dnorm_mean = dnorm_mean / C;
    dnorm_norm_mean = dnorm_norm_mean / C;

    // now iterate again and accumulate all the gradients
    for (int i = 0; i < C; i++) {
        float norm_bti = (inp_bt[i] - mean_bt) * rstd_bt;
        float dnorm_i = weight[i] * dout_bt[i];
        // gradient contribution to bias
        atomicAdd(&dbias[i], dout_bt[i]);
        // gradient contribution to weight
        atomicAdd(&dweight[i], norm_bti * dout_bt[i]);
        // gradient contribution to input
        float dval = 0.0f;
        dval += dnorm_i; // term 1
        dval -= dnorm_mean; // term 2
        dval -= norm_bti * dnorm_norm_mean; // term 3
        dval *= rstd_bt; // final scale
        dinp_bt[i] += dval;
    }
}

// uses shared memory instead for the reduces
template <typename Tdinp, typename Tparams, typename Tdout, typename Trest>
__global__ void layernorm_backward_kernel2(Tdinp* dinp, Tparams* dweight, Tparams* dbias,
                        const Tdout* dout, const Trest* inp, const Tparams* weight, const Trest* mean, const Trest* rstd,
                        int B, int T, int C) {
    extern __shared__ float shared[]; // size = 2 * C

    namespace cg = cooperative_groups;
    cg::thread_block block = cg::this_thread_block();
    cg::thread_block_tile<32> warp = cg::tiled_partition<32>(block);
    int idx = blockIdx.x * warp.meta_group_size() + warp.meta_group_rank();
    int N = B * T;
    if(idx >= N) { return; } // thread guards

    int b = idx / T;
    int t = idx % T;

    const Tdout* dout_bt = dout + b * T * C + t * C;
    const Trest* inp_bt = inp + b * T * C + t * C;
    Tdinp* dinp_bt = dinp + b * T * C + t * C;
    const float mean_bt = (float)mean[b * T + t];
    const float rstd_bt = (float)rstd[b * T + t];

    // the first half of shared memory is bias, second is weight
    float* dbias_shared = shared;
    float* dweight_shared = shared + C;

    // init shared memory to zero
    #pragma unroll
    for(int i = threadIdx.x; i < C; i+= blockDim.x){
       dbias_shared[i] = 0.0f;
       dweight_shared[i] = 0.0f;
    }
    __syncthreads();

    // first: two reduce operations
    float dnorm_mean = 0.0f;
    float dnorm_norm_mean = 0.0f;
    for (int i = warp.thread_rank(); i < C; i  += warp.size()) {
        float norm_bti = ((float)inp_bt[i] - mean_bt) * rstd_bt;
        float dnorm_i = (float)weight[i] * (float)dout_bt[i];
        dnorm_mean += dnorm_i;
        dnorm_norm_mean += dnorm_i * norm_bti;
    }
    dnorm_mean = cg::reduce(warp, dnorm_mean, cg::plus<float>{});
    dnorm_norm_mean = cg::reduce(warp, dnorm_norm_mean, cg::plus<float>{});
    dnorm_mean = dnorm_mean / C;
    dnorm_norm_mean = dnorm_norm_mean / C;

    // now iterate again and accumulate all the gradients
    for (int i = warp.thread_rank(); i < C; i += warp.size()) {
        float norm_bti = ((float)inp_bt[i] - mean_bt) * rstd_bt;
        float dnorm_i = (float)weight[i] * (float)dout_bt[i];
        // gradient contribution to bias
        atomicAdd(&dbias_shared[i], (float)dout_bt[i]);
        // gradient contribution to weight
        atomicAdd(&dweight_shared[i], norm_bti * (float)dout_bt[i]);
        // gradient contribution to input
        float dval = 0.0f;
        dval += dnorm_i; // term 1
        dval -= dnorm_mean; // term 2
        dval -= norm_bti * dnorm_norm_mean; // term 3
        dval *= rstd_bt; // final scale
        dinp_bt[i] = (Tdinp)((float)dinp_bt[i] + dval);
    }
    __syncthreads();

    // write to global memory
    for(int i = threadIdx.x; i < C; i+= blockDim.x) {
        atomicAddX(&dbias[i], (Tparams)dbias_shared[i]);
        atomicAddX(&dweight[i], (Tparams)dweight_shared[i]);
    }
}

// kernel2 is 1 threadblock for all Cs on 32 BTs (assuming threadblock size of 1024 threads = 32 warps)
// To minimise the amount of atomicAdds, we will aim for 1 threadblock per SM, processing (total BTs / threadblocks) BTs
template <typename Tdinp, typename Tparams, typename Tdout, typename Trest>
__global__ void layernorm_backward_kernel3(Tdinp* dinp, Tparams* dweight, Tparams* dbias,
                        const Tdout* dout, const Trest* inp, const Tparams* weight, const Trest* mean, const Trest* rstd,
                        int B, int T, int C) {
    extern __shared__ float shared[]; // size = 2 * C

    namespace cg = cooperative_groups;
    cg::thread_block block = cg::this_thread_block();
    cg::thread_block_tile<32> warp = cg::tiled_partition<32>(block);
    int base_idx = blockIdx.x * warp.meta_group_size() + warp.meta_group_rank();

    // the first half of shared memory is bias, second is weight
    float* dbias_shared = shared;
    float* dweight_shared = shared + C;

    // init shared memory to zero
    #pragma unroll 4
    for(int i = threadIdx.x; i < C; i+= blockDim.x){
       dbias_shared[i] = 0.0f;
       dweight_shared[i] = 0.0f;
    }
    __syncthreads();

    int warps_in_grid = gridDim.x * warp.meta_group_size();
    for (int idx = base_idx; idx < B * T; idx += warps_in_grid) {
        int b = idx / T;
        int t = idx % T;

        const Tdout* dout_bt = dout + b * T * C + t * C;
        const Trest* inp_bt = inp + b * T * C + t * C;
        Tdinp* dinp_bt = dinp + b * T * C + t * C;
        const float mean_bt = (float)mean[b * T + t];
        const float rstd_bt = (float)rstd[b * T + t];

        // first: two reduce operations
        float dnorm_mean = 0.0f;
        float dnorm_norm_mean = 0.0f;
        for (int i = warp.thread_rank(); i < C; i  += warp.size()) {
            float norm_bti = ((float)inp_bt[i] - mean_bt) * rstd_bt;
            float dnorm_i = (float)weight[i] * (float)dout_bt[i];
            dnorm_mean += dnorm_i;
            dnorm_norm_mean += dnorm_i * norm_bti;
        }
        dnorm_mean = cg::reduce(warp, dnorm_mean, cg::plus<float>{});
        dnorm_norm_mean = cg::reduce(warp, dnorm_norm_mean, cg::plus<float>{});
        dnorm_mean = dnorm_mean / C;
        dnorm_norm_mean = dnorm_norm_mean / C;

        // now iterate again and accumulate all the gradients
        for (int i = warp.thread_rank(); i < C; i += warp.size()) {
            float dout_i = (float)__ldcs(&dout_bt[i]);
            float norm_bti = ((float)__ldcs(&inp_bt[i]) - mean_bt) * rstd_bt;
            float dnorm_i = (float)weight[i] * dout_i;
            // gradient contribution to bias
            atomicAdd(&dbias_shared[i], dout_i);
            // gradient contribution to weight
            atomicAdd(&dweight_shared[i], norm_bti * dout_i);
            // gradient contribution to input
            float dval = 0.0f;
            dval += dnorm_i; // term 1
            dval -= dnorm_mean; // term 2
            dval -= norm_bti * dnorm_norm_mean; // term 3
            dval *= rstd_bt; // final scale
            dinp_bt[i] = (Tdinp)((float)dinp_bt[i] + dval);
        }
    }
    __syncthreads();

    for(int i = threadIdx.x; i < C; i+= blockDim.x) {
        atomicAddX(&dbias[i], (Tparams)dbias_shared[i]);
        atomicAddX(&dweight[i], (Tparams)dweight_shared[i]);
    }
}

// atomicCAS version of kernel3
template <typename Tdinp, typename Tparams, typename Tdout, typename Trest>
__global__ void layernorm_backward_kernel4(Tdinp* dinp, Tparams* dweight, Tparams* dbias,
                        const Tdout* dout, const Trest* inp, const Tparams* weight, const Trest* mean, const Trest* rstd,
                        int B, int T, int C) {
    extern __shared__ float shared[]; // size = 2 * C

    namespace cg = cooperative_groups;
    cg::thread_block block = cg::this_thread_block();
    cg::thread_block_tile<32> warp = cg::tiled_partition<32>(block);
    int base_idx = blockIdx.x * warp.meta_group_size() + warp.meta_group_rank();

    // the first half of shared memory is bias, second is weight
    float* dbias_shared = shared;
    float* dweight_shared = shared + C;

    // init shared memory to zero
    #pragma unroll 4
    for(int i = threadIdx.x; i < C; i+= blockDim.x){
       dbias_shared[i] = 0.0f;
       dweight_shared[i] = 0.0f;
    }
    __syncthreads();

    int warps_in_grid = gridDim.x * warp.meta_group_size();
    for (int idx = base_idx; idx < B * T; idx += warps_in_grid) {
        int b = idx / T;
        int t = idx % T;

        const Tdout* dout_bt = dout + b * T * C + t * C;
        const Trest* inp_bt = inp + b * T * C + t * C;
        Tdinp* dinp_bt = dinp + b * T * C + t * C;
        const float mean_bt = (float)mean[b * T + t];
        const float rstd_bt = (float)rstd[b * T + t];

        // first: two reduce operations
        float dnorm_mean = 0.0f;
        float dnorm_norm_mean = 0.0f;
        for (int i = warp.thread_rank(); i < C; i  += warp.size()) {
            float norm_bti = ((float)inp_bt[i] - mean_bt) * rstd_bt;
            float dnorm_i = (float)weight[i] * (float)dout_bt[i];
            dnorm_mean += dnorm_i;
            dnorm_norm_mean += dnorm_i * norm_bti;
        }
        dnorm_mean = cg::reduce(warp, dnorm_mean, cg::plus<float>{});
        dnorm_norm_mean = cg::reduce(warp, dnorm_norm_mean, cg::plus<float>{});
        dnorm_mean = dnorm_mean / C;
        dnorm_norm_mean = dnorm_norm_mean / C;

        // now iterate again and accumulate all the gradients
        for (int i = warp.thread_rank(); i < C; i += warp.size()) {
            float dout_i = (float)__ldcs(&dout_bt[i]);
            float norm_bti = ((float)__ldcs(&inp_bt[i]) - mean_bt) * rstd_bt;
            float dnorm_i = (float)weight[i] * dout_i;
            // gradient contribution to bias
            atomicAdd(&dbias_shared[i], dout_i);
            // gradient contribution to weight
            atomicAdd(&dweight_shared[i], norm_bti * dout_i);
            // gradient contribution to input
            float dval = 0.0f;
            dval += dnorm_i; // term 1
            dval -= dnorm_mean; // term 2
            dval -= norm_bti * dnorm_norm_mean; // term 3
            dval *= rstd_bt; // final scale
            dinp_bt[i] = (Tdinp)((float)dinp_bt[i] + dval);
        }
    }
    __syncthreads();

    __nv_bfloat162* dbiasVec2 = reinterpret_cast<__nv_bfloat162*>(dbias);
    __nv_bfloat162* dweightVec2 = reinterpret_cast<__nv_bfloat162*>(dweight);

    // write to global memory
    for(int i = threadIdx.x; i < C/2; i+= blockDim.x) {
        __nv_bfloat162 add_dbias = __halves2bfloat162((__nv_bfloat16)dbias_shared[i*2], (__nv_bfloat16)dbias_shared[i*2+1]);
        __nv_bfloat162 add_dweight = __halves2bfloat162((__nv_bfloat16)dweight_shared[i*2], (__nv_bfloat16)dweight_shared[i*2+1]);

        // Get the current value from L2 cache
        __nv_bfloat162 current_dbias = __ldcg(&dbiasVec2[i]);
        __nv_bfloat162 current_dweight = __ldcg(&dweightVec2[i]);

        // Add the two values
        __nv_bfloat162 new_dbias = add_dbias + current_dbias;
        __nv_bfloat162 new_dweight = add_dweight + current_dweight;

        // Write the result back to L2 cache using 32-bit integer atomic compare and exchange
        uint current_dbias32b = *reinterpret_cast<uint*>(&current_dbias);
        uint current_dweight32b = *reinterpret_cast<uint*>(&current_dweight);

        uint new_dbias32b = *reinterpret_cast<uint*>(&new_dbias);
        uint new_dweight32b = *reinterpret_cast<uint*>(&new_dweight);

        uint old_dbias32b = atomicCAS((uint*)&dbiasVec2[i], current_dbias32b, new_dbias32b);
        uint old_dweight32b = atomicCAS((uint*)&dweightVec2[i], current_dweight32b, new_dweight32b);

        // If the value has changed between read and atomic, we need to try again
        while (old_dbias32b != current_dbias32b) {
            current_dbias32b = old_dbias32b;
            new_dbias = *reinterpret_cast<__nv_bfloat162*>(&current_dbias32b) + add_dbias;
            new_dbias32b = *reinterpret_cast<uint*>(&new_dbias);
            old_dbias32b = atomicCAS((uint*)&dbiasVec2[i], current_dbias32b, new_dbias32b);
        }

        while (old_dweight32b != current_dweight32b) {
            current_dweight32b = old_dweight32b;
            new_dweight = *reinterpret_cast<__nv_bfloat162*>(&current_dweight32b) + add_dweight;
            new_dweight32b = *reinterpret_cast<uint*>(&new_dweight);
            old_dweight32b = atomicCAS((uint*)&dweightVec2[i], current_dweight32b, new_dweight32b);
        }
    }
}

// FP32 scratchpad per threadgroup, zero atomics except atomicAdd on uint for the flag (based on kernel3)
template <typename Tdinp, typename Tparams, typename Tdout, typename Trest>
__global__ void layernorm_backward_kernel5(Tdinp* dinp, Tparams* dweight, Tparams* dbias, float* scratch,
                        const Tdout* dout, const Trest* inp, const Tparams* weight, const Trest* mean, const Trest* rstd,
                        int B, int T, int C) {
    extern __shared__ float shared[]; // size = 2 * C + 1

    namespace cg = cooperative_groups;
    cg::thread_block block = cg::this_thread_block();
    cg::thread_block_tile<32> warp = cg::tiled_partition<32>(block);
    int base_idx = blockIdx.x * warp.meta_group_size() + warp.meta_group_rank();

    // the first half of shared memory is bias, second is weight
    float* dbias_shared = shared;
    float* dweight_shared = shared + C;

    // init shared memory to zero
    #pragma unroll 4
    for(int i = threadIdx.x; i < C; i+= blockDim.x){
       dbias_shared[i] = 0.0f;
       dweight_shared[i] = 0.0f;
    }
    uint *tmp_flag = (uint*)(shared + C*2);
    __syncthreads();

    int warps_in_grid = gridDim.x * warp.meta_group_size();
    for (int idx = base_idx; idx < B * T; idx += warps_in_grid) {
        int b = idx / T;
        int t = idx % T;

        const Tdout* dout_bt = dout + b * T * C + t * C;
        const Trest* inp_bt = inp + b * T * C + t * C;
        Tdinp* dinp_bt = dinp + b * T * C + t * C;
        const float mean_bt = (float)mean[b * T + t];
        const float rstd_bt = (float)rstd[b * T + t];

        // first: two reduce operations
        float dnorm_mean = 0.0f;
        float dnorm_norm_mean = 0.0f;
        for (int i = warp.thread_rank(); i < C; i  += warp.size()) {
            float norm_bti = ((float)inp_bt[i] - mean_bt) * rstd_bt;
            float dnorm_i = (float)weight[i] * (float)dout_bt[i];
            dnorm_mean += dnorm_i;
            dnorm_norm_mean += dnorm_i * norm_bti;
        }
        dnorm_mean = cg::reduce(warp, dnorm_mean, cg::plus<float>{});
        dnorm_norm_mean = cg::reduce(warp, dnorm_norm_mean, cg::plus<float>{});
        dnorm_mean = dnorm_mean / C;
        dnorm_norm_mean = dnorm_norm_mean / C;

        // now iterate again and accumulate all the gradients
        for (int i = warp.thread_rank(); i < C; i += warp.size()) {
            float dout_i = (float)__ldcs(&dout_bt[i]);
            float norm_bti = ((float)__ldcs(&inp_bt[i]) - mean_bt) * rstd_bt;
            float dnorm_i = (float)weight[i] * dout_i;
            // gradient contribution to bias
            atomicAdd(&dbias_shared[i], dout_i);
            // gradient contribution to weight
            atomicAdd(&dweight_shared[i], norm_bti * dout_i);
            // gradient contribution to input
            float dval = 0.0f;
            dval += dnorm_i; // term 1
            dval -= dnorm_mean; // term 2
            dval -= norm_bti * dnorm_norm_mean; // term 3
            dval *= rstd_bt; // final scale
            dinp_bt[i] = (Tdinp)((float)dinp_bt[i] + dval);
        }
    }
    __syncthreads();

    float* scratch_dbias = scratch;
    float* scratch_dweight = scratch + C * gridDim.x;
    uint* scratchFlag = (uint*)(scratch + (2 * C * gridDim.x));

    for(int i = threadIdx.x; i < C; i+= blockDim.x) {
        scratch_dbias[i + C*blockIdx.x] = dbias_shared[i];
        scratch_dweight[i + C*blockIdx.x] = dweight_shared[i];
    }
    __threadfence();
    __syncthreads();
    if (threadIdx.x == 0) {
        *tmp_flag = atomicAdd(scratchFlag, 1);
    }
    __syncthreads();
    if (*tmp_flag == gridDim.x-1) {
        // last block to finish, accumulate the scratchpad
        for (int i = threadIdx.x; i < C; i += blockDim.x) {
            float dbias_sum = 0.0f;
            float dweight_sum = 0.0f;
            #pragma unroll 8
            for (int j = 0; j < gridDim.x; j++) {
                dbias_sum += scratch_dbias[i + j*C];
                dweight_sum += scratch_dweight[i + j*C];
            }
            dbias[i] = (Tparams)((float)dbias[i] + dbias_sum);
            dweight[i] = (Tparams)((float)dweight[i] + dweight_sum);
        }
    }
}

// single FP32 scratchpad shared by all the threadblocks (based on kernels 3 & 5)
template <typename Tdinp, typename Tparams, typename Tdout, typename Trest>
__global__ void layernorm_backward_kernel6(Tdinp* dinp, Tparams* dweight, Tparams* dbias, float* scratch,
                        const Tdout* dout, const Trest* inp, const Tparams* weight, const Trest* mean, const Trest* rstd,
                        int B, int T, int C) {
    extern __shared__ float shared[]; // size = 2 * C + 1

    namespace cg = cooperative_groups;
    cg::thread_block block = cg::this_thread_block();
    cg::thread_block_tile<32> warp = cg::tiled_partition<32>(block);
    int base_idx = blockIdx.x * warp.meta_group_size() + warp.meta_group_rank();

    // the first half of shared memory is bias, second is weight
    float* dbias_shared = shared;
    float* dweight_shared = shared + C;

    // init shared memory to zero
    #pragma unroll 4
    for(int i = threadIdx.x; i < C; i+= blockDim.x){
       dbias_shared[i] = 0.0f;
       dweight_shared[i] = 0.0f;
    }
    uint *tmp_flag = (uint*)(shared + C*2);
    __syncthreads();

    int warps_in_grid = gridDim.x * warp.meta_group_size();
    for (int idx = base_idx; idx < B * T; idx += warps_in_grid) {
        int b = idx / T;
        int t = idx % T;

        const Tdout* dout_bt = dout + b * T * C + t * C;
        const Trest* inp_bt = inp + b * T * C + t * C;
        Tdinp* dinp_bt = dinp + b * T * C + t * C;
        const float mean_bt = (float)mean[b * T + t];
        const float rstd_bt = (float)rstd[b * T + t];

        // first: two reduce operations
        float dnorm_mean = 0.0f;
        float dnorm_norm_mean = 0.0f;
        for (int i = warp.thread_rank(); i < C; i  += warp.size()) {
            float norm_bti = ((float)inp_bt[i] - mean_bt) * rstd_bt;
            float dnorm_i = (float)weight[i] * (float)dout_bt[i];
            dnorm_mean += dnorm_i;
            dnorm_norm_mean += dnorm_i * norm_bti;
        }
        dnorm_mean = cg::reduce(warp, dnorm_mean, cg::plus<float>{});
        dnorm_norm_mean = cg::reduce(warp, dnorm_norm_mean, cg::plus<float>{});
        dnorm_mean = dnorm_mean / C;
        dnorm_norm_mean = dnorm_norm_mean / C;

        // now iterate again and accumulate all the gradients
        for (int i = warp.thread_rank(); i < C; i += warp.size()) {
            float dout_i = (float)__ldcs(&dout_bt[i]);
            float norm_bti = ((float)__ldcs(&inp_bt[i]) - mean_bt) * rstd_bt;
            float dnorm_i = (float)weight[i] * dout_i;
            // gradient contribution to bias
            atomicAdd(&dbias_shared[i], dout_i);
            // gradient contribution to weight
            atomicAdd(&dweight_shared[i], norm_bti * dout_i);
            // gradient contribution to input
            float dval = 0.0f;
            dval += dnorm_i; // term 1
            dval -= dnorm_mean; // term 2
            dval -= norm_bti * dnorm_norm_mean; // term 3
            dval *= rstd_bt; // final scale
            dinp_bt[i] = (Tdinp)((float)dinp_bt[i] + dval);
        }
    }

    // Accumulate into a FP32 scratchpad
    // BF16 atomics are potentially much slower... and this is more precise!
    __syncthreads();
    float* scratch_dbias = scratch;
    float* scratch_dweight = scratch + C;
    uint* scratchFlag = (uint*)(scratch + (2 * C));
    for(int i = threadIdx.x; i < C; i+= blockDim.x) {
        atomicAdd(&scratch_dbias[i], dbias_shared[i]);
        atomicAdd(&scratch_dweight[i], dweight_shared[i]);
    }
    __syncthreads();
    if (threadIdx.x == 0) {
        *tmp_flag = atomicAdd(scratchFlag, 1);
    }
    __syncthreads();
    if (*tmp_flag == gridDim.x-1) {
        for(int i = threadIdx.x; i < C; i+= blockDim.x) {
            // todo - potentially do stochastic rounding here as well
            dbias[i] = (Tparams)scratch_dbias[i];
            dweight[i] = (Tparams)scratch_dweight[i];
        }
    }
}


// Same as kernel 6 but without cooperative groups or templates
__global__ void layernorm_backward_kernel7(floatX* dinp, floatX* dweight, floatX* dbias, float* scratch,
                        const floatX* dout, const floatX* inp, const floatX* weight, const floatX* mean, const floatX* rstd,
                        int B, int T, int C) {
    extern __shared__ float shared[]; // size = 2 * C + 1
    int warpId = threadIdx.x / warpSize; // warp index within a block
    int warpsInBlock = blockDim.x / warpSize;
    int base_idx = blockIdx.x * warpsInBlock + warpId;
    int warpThreadIdx = threadIdx.x % warpSize; // Thread index within the warp
    int warps_in_grid = gridDim.x * warpsInBlock;

    // the first half of shared memory is bias, second is weight
    float* dbias_shared = shared;
    float* dweight_shared = shared + C;

    // init shared memory to zero
    #pragma unroll 4
    for(int i = threadIdx.x; i < C; i+= blockDim.x){
       dbias_shared[i] = 0.0f;
       dweight_shared[i] = 0.0f;
    }
    uint *tmp_flag = (uint*)(shared + C*2);
    __syncthreads();

    for (int idx = base_idx; idx < B * T; idx += warps_in_grid) {
        int b = idx / T;
        int t = idx % T;

        const floatX* dout_bt = dout + b * T * C + t * C;
        const floatX* inp_bt = inp + b * T * C + t * C;
        floatX* dinp_bt = dinp + b * T * C + t * C;
        const float mean_bt = (float)mean[b * T + t];
        const float rstd_bt = (float)rstd[b * T + t];

        // first: two reduce operations
        float dnorm_mean = 0.0f;
        float dnorm_norm_mean = 0.0f;
        for (int i = warpThreadIdx; i < C; i  += warpSize) {
            float norm_bti = ((float)inp_bt[i] - mean_bt) * rstd_bt;
            float dnorm_i = (float)weight[i] * (float)dout_bt[i];
            dnorm_mean += dnorm_i;
            dnorm_norm_mean += dnorm_i * norm_bti;
        }
        dnorm_mean = warpReduceSum(dnorm_mean);
        dnorm_norm_mean = warpReduceSum(dnorm_norm_mean);

        dnorm_mean = dnorm_mean / C;
        dnorm_norm_mean = dnorm_norm_mean / C;

        // now iterate again and accumulate all the gradients
        for (int i = warpThreadIdx; i < C; i += warpSize) {
            float dout_i = (float)__ldcs(&dout_bt[i]);
            float norm_bti = ((float)__ldcs(&inp_bt[i]) - mean_bt) * rstd_bt;
            float dnorm_i = (float)weight[i] * dout_i;
            // gradient contribution to bias
            atomicAdd(&dbias_shared[i], dout_i);
            // gradient contribution to weight
            atomicAdd(&dweight_shared[i], norm_bti * dout_i);
            // gradient contribution to input
            float dval = 0.0f;
            dval += dnorm_i; // term 1
            dval -= dnorm_mean; // term 2
            dval -= norm_bti * dnorm_norm_mean; // term 3
            dval *= rstd_bt; // final scale
            dinp_bt[i] = (floatX)((float)dinp_bt[i] + dval);
        }
    }

    // Accumulate into a FP32 scratchpad
    // BF16 atomics are potentially much slower... and this is more precise!
    __syncthreads();
    float* scratch_dbias = scratch;
    float* scratch_dweight = scratch + C;
    uint* scratchFlag = (uint*)(scratch + (2 * C));
    for(int i = threadIdx.x; i < C; i+= blockDim.x) {
        atomicAdd(&scratch_dbias[i], dbias_shared[i]);
        atomicAdd(&scratch_dweight[i], dweight_shared[i]);
    }
    __syncthreads();
    if (threadIdx.x == 0) {
        *tmp_flag = atomicAdd(scratchFlag, 1);
    }
    __syncthreads();
    if (*tmp_flag == gridDim.x-1) {
        for(int i = threadIdx.x; i < C; i+= blockDim.x) {
            // todo - potentially do stochastic rounding here as well
            dbias[i] = (floatX)scratch_dbias[i];
            dweight[i] = (floatX)scratch_dweight[i];
        }
    }
}

__global__ void __launch_bounds__(1024, MAX_1024_THREADS_BLOCKS)
                layernorm_backward_kernel8(floatX* dinp, floatX* dweight, floatX* dbias, float* scratch,
                                            const floatX* dout, const floatX* inp, const floatX* weight,
                                            const floatX* mean, const floatX* rstd,
                                            int B, int T, int C) {
    extern __shared__ float shared[]; // size = 2 * C + 1
    int warpId = threadIdx.x / warpSize; // warp index within a block
    int warpsInBlock = blockDim.x / warpSize; //number of warps in block
    int baseIdx = blockIdx.x * warpsInBlock + warpId;
    int warpThreadIdx = threadIdx.x % warpSize; // Thread index within the warp
    int warpsInGrid = gridDim.x * warpsInBlock;
    int C_per_iteration = warpSize * x128::size;
    int iterations_C = C / C_per_iteration;

    // the first half of shared memory is bias, second is weight
    float* dbias_shared = shared;
    float* dweight_shared = shared + C;

    // init shared memory to zero
    for(int i = threadIdx.x; i < C; i+= blockDim.x){
       dbias_shared[i] = 0.0f;
       dweight_shared[i] = 0.0f;
    }
    unsigned int *tmp_flag = (unsigned int*)(shared + C*2);
    __syncthreads();

    for (int idx = baseIdx; idx < B * T; idx += warpsInGrid) {
        int b = idx / T;
        int t = idx % T;

        const floatX* dout_bt = dout + b * T * C + t * C;
        const floatX* inp_bt = inp + b * T * C + t * C;
        floatX* dinp_bt = dinp + b * T * C + t * C;
        const float mean_bt = (float)mean[b * T + t];
        const float rstd_bt = (float)rstd[b * T + t];

        // first: two reduce operations
        float dnorm_mean = 0.0f;
        float dnorm_norm_mean = 0.0f;
        for (int i = warpThreadIdx * x128::size; i < C; i += warpSize * x128::size) {
            x128 dout128_i   = load128(dout_bt + i);
            x128 inp128_i    = load128(inp_bt  + i);
            x128 weight128_i = load128(weight  + i);
            for (int k = 0; k < x128::size; k++) {
                float norm_bti = ((float)inp128_i[k] - mean_bt) * rstd_bt;
                float dnorm_i = (float)weight128_i[k] * (float)dout128_i[k];
                dnorm_mean += dnorm_i;
                dnorm_norm_mean += dnorm_i * norm_bti;
            }
        }
        dnorm_mean = warpReduceSum(dnorm_mean) / C;
        dnorm_norm_mean = warpReduceSum(dnorm_norm_mean) / C;

        // now iterate again and accumulate all the gradients
        // unfortunately we cannot use the same index for x128 arrays and shared memory
        // as atomics can only be 32-bit rather than 128-bit (at least pre-SM90/Hopper)
        // so this would result in an 8-way bank conflict, and kill performance
        // so instead, we use a shared memory friendly index, and reorder before the final write
        for (int i = 0; i < iterations_C; i++) {
            int global_index = (warpThreadIdx * x128::size) + (i * C_per_iteration);
            int shared_index = warpThreadIdx + (i * C_per_iteration);
            x128 dout128   = load128cs(dout_bt + global_index);
            x128 inp128    = load128cs(inp_bt  + global_index);
            x128 dinp128   = load128(dinp_bt   + global_index);
            x128 weight128 = load128(weight    + global_index);

            for (int x = 0; x < x128::size; x++) {
                float dout_i = (float)dout128[x];
                float norm_bti = ((float)inp128[x] - mean_bt) * rstd_bt;
                float dnorm_i = (float)weight128[x] * dout_i;
                // gradient contribution to bias (using shared memory friendly index)
                atomicAdd(&dbias_shared[shared_index + x*warpSize], dout_i);
                // gradient contribution to weight (using shared memory friendly index)
                atomicAdd(&dweight_shared[shared_index + x*warpSize], norm_bti * dout_i);
                // gradient contribution to input
                float dval = 0.0f;
                dval += dnorm_i; // term 1
                dval -= dnorm_mean; // term 2
                dval -= norm_bti * dnorm_norm_mean; // term 3
                dval *= rstd_bt; // final scale
                dinp128[x] = (floatX)((float)dinp128[x] + dval);
            }
            // cache in L2 as this is read by the next kernel, but bypass L1 to minimise thrashing
            store128cg(dinp_bt + global_index, dinp128);
        }
    }
    // Accumulate into a FP32 scratchpad
    // BF16 atomics are potentially much slower... and this is more precise!
    // todo - could potentially avoid the extra copy if floatX is FP32, fairly negligible though
    __syncthreads();
    float* scratch_dbias = scratch;
    float* scratch_dweight = scratch + C;
    unsigned int* scratchFlag = (unsigned int*)(scratch + (2 * C));
    for(int i = threadIdx.x; i < C; i+= blockDim.x) {
        // global atomics in the same "shared memory banking friendly" order
        atomicAdd(&scratch_dbias[i], dbias_shared[i]);
        atomicAdd(&scratch_dweight[i], dweight_shared[i]);
    }
    __syncthreads();
    if (threadIdx.x == 0) {
        *tmp_flag = atomicInc(scratchFlag, gridDim.x);
    }
    __syncthreads();
    if (*tmp_flag == gridDim.x-1) {
        for (int i = warpId; i < iterations_C; i += warpsInBlock) {
            // reorder from atomic/shared memory-friendly index to real global memory index
            // and convert from float/FP32 to floatX/BF16 for the final write
            int global_index = (warpThreadIdx * x128::size) + (i * C_per_iteration);
            int shared_index = warpThreadIdx + (i * C_per_iteration);

            x128 dbias128 = load128(dbias + global_index);
            x128 dweight128 = load128(dweight + global_index);
            for (int x = 0; x < x128::size; x++) {
                float s_db = scratch_dbias[shared_index + x*warpSize];
                float s_dw = scratch_dweight[shared_index + x*warpSize];
                dbias128[x] = (floatX)(s_db + (float)dbias128[x]);
                dweight128[x] = (floatX)(s_dw + (float)dweight128[x]);
            }
            store128(dbias + global_index, dbias128);
            store128(dweight + global_index, dweight128);
        }
    }
}

// ----------------------------------------------------------------------------
// kernel launchers

void layernorm_backward1(float* dinp, float* dweight, float* dbias,
                        const float* dout, const float* inp, const float* weight, const float* mean, const float* rstd,
                        int B, int T, int C, const int block_size) {
    const int N = B * T;
    const int grid_size = ceil_div(N, block_size);
    layernorm_backward_kernel1<<<grid_size, block_size>>>(dinp, dweight, dbias, dout, inp, weight, mean, rstd, B, T, C);
}

template <typename Tdinp, typename Tparams, typename Tdout, typename Trest>
void layernorm_backward2(Tdinp* dinp, Tparams* dweight, Tparams* dbias,
                        const Tdout* dout, const Trest* inp, const Tparams* weight, const Trest* mean, const Trest* rstd,
                        int B, int T, int C, int block_size) {
    const int N = B * T;
    const int grid_size = ceil_div(32*N, block_size);
    size_t shared_mem_size = 2 * C * sizeof(float);
    layernorm_backward_kernel2<<<grid_size, block_size, shared_mem_size>>>(dinp, dweight, dbias, dout, inp, weight, mean, rstd, B, T, C);
}

template <typename Tdinp, typename Tparams, typename Tdout, typename Trest>
void layernorm_backward3(Tdinp* dinp, Tparams* dweight, Tparams* dbias,
                        const Tdout* dout, const Trest* inp, const Tparams* weight, const Trest* mean, const Trest* rstd,
                        int B, int T, int C, int block_size) {
    const int grid_size = (1024/block_size) * cuda_num_SMs;
    size_t shared_mem_size = 2 * C * sizeof(float);
    layernorm_backward_kernel3<<<grid_size, block_size, shared_mem_size>>>(dinp, dweight, dbias, dout, inp, weight, mean, rstd, B, T, C);
}

template <typename Tdinp, typename Tparams, typename Tdout, typename Trest>
void layernorm_backward4(Tdinp* dinp, Tparams* dweight, Tparams* dbias,
                        const Tdout* dout, const Trest* inp, const Tparams* weight, const Trest* mean, const Trest* rstd,
                        int B, int T, int C, int block_size) {
        const int grid_size = (1024/block_size) * cuda_num_SMs;
        size_t shared_mem_size = 2 * C * sizeof(float);
        layernorm_backward_kernel4<<<grid_size, block_size, shared_mem_size>>>(dinp, dweight, dbias, dout, inp, weight, mean, rstd, B, T, C);
}

template <typename Tdinp, typename Tparams, typename Tdout, typename Trest>
void layernorm_backward5(Tdinp* dinp, Tparams* dweight, Tparams* dbias, float* scratch,
                        const Tdout* dout, const Trest* inp, const Tparams* weight, const Trest* mean, const Trest* rstd,
                        int B, int T, int C, int block_size) {
        const int grid_size = 1 * cuda_num_SMs; // only support 1 block per SM for simplicity, 1024 threads is best anyway
        size_t shared_mem_size = (2 * C + 1) * sizeof(float);
        cudaMemset(scratch, 0, (grid_size * 2 * C + 1) * sizeof(float));
        layernorm_backward_kernel5<<<grid_size, block_size, shared_mem_size>>>(dinp, dweight, dbias, scratch, dout, inp, weight, mean, rstd, B, T, C);
}

template <typename Tdinp, typename Tparams, typename Tdout, typename Trest>
void layernorm_backward6(Tdinp* dinp, Tparams* dweight, Tparams* dbias, float* scratch,
                        const Tdout* dout, const Trest* inp, const Tparams* weight, const Trest* mean, const Trest* rstd,
                        int B, int T, int C, int block_size) {
        const int grid_size = (1024/block_size) * cuda_num_SMs;
        size_t shared_mem_size = (2 * C + 1) * sizeof(float);

        // Including this as part of the timing until we can parallelise it
        // It should fully hide the cost and improve kernel perf by >5% if done in parallel using CUDA streams
        cudaMemset(scratch, 0, (1 + 2 * C) * sizeof(float));

        layernorm_backward_kernel6<<<grid_size, block_size, shared_mem_size>>>(dinp, dweight, dbias, scratch, dout, inp, weight, mean, rstd, B, T, C);
}

template <typename Tdinp, typename Tparams, typename Tdout, typename Trest>
void layernorm_backward7(Tdinp* dinp, Tparams* dweight, Tparams* dbias, float* scratch,
                        const Tdout* dout, const Trest* inp, const Tparams* weight, const Trest* mean, const Trest* rstd,
                        int B, int T, int C, int block_size) {
        const int grid_size = (1024/block_size) * cuda_num_SMs;
        size_t shared_mem_size = (2 * C + 1) * sizeof(float);

        // Including this as part of the timing until we can parallelise it
        // It should fully hide the cost and improve kernel perf by >5% if done in parallel using CUDA streams
        cudaMemset(scratch, 0, (1 + 2 * C) * sizeof(float));

        layernorm_backward_kernel7<<<grid_size, block_size, shared_mem_size>>>(dinp, dweight, dbias, scratch, dout, inp, weight, mean, rstd, B, T, C);
}

template <typename Tdinp, typename Tparams, typename Tdout, typename Trest>
void layernorm_backward8(Tdinp* dinp, Tparams* dweight, Tparams* dbias, float* scratch,
                        const Tdout* dout, const Trest* inp, const Tparams* weight, const Trest* mean, const Trest* rstd,
                        int B, int T, int C, int block_size) {
        const int grid_size = (1024/block_size) * cuda_num_SMs;
        size_t shared_mem_size = (2 * C + 1) * sizeof(float);

        // Including this as part of the timing until we can parallelise it
        // It should fully hide the cost and improve kernel perf by >5% if done in parallel using CUDA streams
        cudaMemset(scratch, 0, (1 + 2 * C) * sizeof(float));

        layernorm_backward_kernel8<<<grid_size, block_size, shared_mem_size>>>(dinp, dweight, dbias, scratch, dout, inp, weight, mean, rstd, B, T, C);
}

// kernel version dispatch
void layernorm_backward(int kernel_num,
                        floatX* dinp, floatX* dweight, floatX* dbias, float* scratch,
                        const floatX* dout, const floatX* inp, const floatX* weight, const floatX* mean, const floatX* rstd,
                        int B, int T, int C,
                        const int block_size) {
    switch (kernel_num) {
#if !defined(ENABLE_BF16) && !defined(ENABLE_FP16)
        case 1:
            layernorm_backward1(dinp, dweight, dbias, dout, inp, weight, mean, rstd, B, T, C, block_size);
            break;
#endif
        case 2:
            layernorm_backward2(dinp, dweight, dbias, dout, inp, weight, mean, rstd, B, T, C, block_size);
            break;
        case 3:
            layernorm_backward3(dinp, dweight, dbias, dout, inp, weight, mean, rstd, B, T, C, block_size);
            break;
#if defined(ENABLE_BF16)
        case 4:
            layernorm_backward4(dinp, dweight, dbias, dout, inp, weight, mean, rstd, B, T, C, block_size);
            break;
#endif
        case 5:
            layernorm_backward5(dinp, dweight, dbias, scratch, dout, inp, weight, mean, rstd, B, T, C, block_size);
            break;
        case 6:
            layernorm_backward6(dinp, dweight, dbias, scratch, dout, inp, weight, mean, rstd, B, T, C, block_size);
            break;
        case 7:
            layernorm_backward7(dinp, dweight, dbias, scratch, dout, inp, weight, mean, rstd, B, T, C, block_size);
            break;
        case 8:
            layernorm_backward8(dinp, dweight, dbias, scratch, dout, inp, weight, mean, rstd, B, T, C, block_size);
            break;
    default:
            printf("Invalid kernel number\n");
            exit(1);
    }
    cudaCheck(cudaGetLastError());
}

// ----------------------------------------------------------------------------

int main(int argc, char **argv) {
    setup_main();

    int B = 8;
    int T = 1024;
    int C = 768;

    // first do the forward pass in CPU
    float* out = (float*)malloc(B * T * C * sizeof(float));
    float* mean = (float*)malloc(B * T * sizeof(float));
    float* rstd = (float*)malloc(B * T * sizeof(float));
    float* inp = make_random_float(B * T * C);
    float* weight = make_random_float(C);
    float* bias = make_random_float(C);
    layernorm_forward_cpu(out, mean, rstd, inp, weight, bias, B, T, C);

    // now do the backward pass, again on CPU
    float *dout = make_random_float(B * T * C);
    float *dinp = make_zeros_float(B * T * C);
    float *dweight = make_zeros_float(C);
    float *dbias = make_zeros_float(C);
    layernorm_backward_cpu(dinp, dweight, dbias, dout, inp, weight, mean, rstd, B, T, C);

    // the above calculations act as the reference
    // now let's do the same on the GPU

    // read kernel_num from command line
    int kernel_num = 2;
    if (argc > 1) {
        kernel_num = atoi(argv[1]);
    }
    printf("Using kernel %d\n", kernel_num);

    // move all the variables we need for backward pass onto the GPU
    floatX* d_dinp;
    floatX* d_dweight;
    floatX* d_dbias;
    floatX* d_dout;
    floatX* d_inp;
    floatX* d_weight;
    floatX* d_mean;
    floatX* d_rstd;
    float* d_scratch;
    cudaCheck(cudaMalloc(&d_dinp, B * T * C * sizeof(floatX)));
    cudaCheck(cudaMalloc(&d_dweight, C * sizeof(floatX)));
    cudaCheck(cudaMalloc(&d_dbias, C * sizeof(floatX)));
    cudaCheck(cudaMalloc(&d_dout, B * T * C * sizeof(floatX)));
    cudaCheck(cudaMalloc(&d_inp, B * T * C * sizeof(floatX)));
    cudaCheck(cudaMalloc(&d_weight, C * sizeof(floatX)));
    cudaCheck(cudaMalloc(&d_mean, B * T * sizeof(floatX)));
    cudaCheck(cudaMalloc(&d_rstd, B * T * sizeof(floatX)));
    cudaCheck(cudaMalloc(&d_scratch, cuda_num_SMs * (2 * C + 1) * sizeof(float)));
    // copy over the "inputs" to the backward call
    cudaCheck(memcpy_convert(d_dout, dout, B * T * C));
    cudaCheck(memcpy_convert(d_inp, inp, B * T * C));
    cudaCheck(memcpy_convert(d_weight, weight, C));
    cudaCheck(memcpy_convert(d_mean, mean, B * T));
    cudaCheck(memcpy_convert(d_rstd, rstd, B * T));

    // launch the kernel
    int block_sizes[] = {32, 64, 128, 256, 512, 768, 1024};
    for (int j = 0; j < sizeof(block_sizes) / sizeof(int); j++) {
        int block_size = block_sizes[j];
        // init the "outputs" of the backward call to zeros
        cudaCheck(cudaMemset(d_dinp, 0, B * T * C * sizeof(floatX)));
        cudaCheck(cudaMemset(d_dweight, 0, C * sizeof(floatX)));
        cudaCheck(cudaMemset(d_dbias, 0, C * sizeof(floatX)));

        layernorm_backward(kernel_num, d_dinp, d_dweight, d_dbias, d_scratch, d_dout, d_inp, d_weight, d_mean, d_rstd,
                           B, T, C, block_size);

        // check the correctness of the kernel
        float error_threshold_dinp = sizeof(floatX) == 4 ? 1e-3f : 1e-1f; // allow larger errors for BF16/FP16
        float error_threshold_dparams = sizeof(floatX) == 4 ? 1e-3f : 5e-1f; // much, much larger...
        printf("Checking correctness...\n");
        printf("dinp:\n");
        validate_result(d_dinp, dinp, "dinp", B * T * C, error_threshold_dinp);
        printf("dweight:\n");
        validate_result(d_dweight, dweight, "dweight", C, error_threshold_dparams);
        printf("dbias:\n");
        validate_result(d_dbias, dbias, "dbias", C, error_threshold_dparams);

        printf("All results match for block_size=%d.\n\n", block_size);
    }

    // now time the kernel
    for (int j = 0; j < sizeof(block_sizes) / sizeof(int); j++) {
        int block_size = block_sizes[j];
        int repeat_times = 100;
        float elapsed_time = benchmark_kernel(repeat_times, layernorm_backward, kernel_num,
                                              d_dinp, d_dweight, d_dbias, d_scratch, d_dout, d_inp, d_weight, d_mean, d_rstd,
                                              B, T, C, block_size);
        printf("block_size %4d time %.4f ms\n", block_size, elapsed_time);
    }

    // cleanups
    free(out);
    free(mean);
    free(rstd);
    free(inp);
    free(weight);
    free(bias);
    free(dout);
    free(dinp);
    free(dweight);
    free(dbias);
    cudaCheck(cudaFree(d_dinp));
    cudaCheck(cudaFree(d_dweight));
    cudaCheck(cudaFree(d_dbias));
    cudaCheck(cudaFree(d_dout));
    cudaCheck(cudaFree(d_inp));
    cudaCheck(cudaFree(d_weight));
    cudaCheck(cudaFree(d_mean));
    cudaCheck(cudaFree(d_rstd));
    cudaCheck(cudaFree(d_scratch));
    return 0;
}