You write custom CUDA kernels to replace the pytorch operators in the given architecture to get speedups. 

    You have complete freedom to choose the set of operators you want to replace. You may make the decision to replace some operators with custom CUDA kernels and leave others unchanged. You may replace multiple operators with custom implementations, consider operator fusion opportunities (combining multiple operators into a single kernel, for example, combining matmul+relu), or algorithmic changes (such as online softmax). You are only limited by your imagination.


        Here's an example to show you the syntax of inline embedding custom CUDA operators in torch: The example given architecture is: 

        ``` 

        # Copyright (c) Meta Platforms, Inc. and affiliates.
# All rights reserved.
#
# This source code is licensed under the license found in the
# LICENSE file in the root directory of this source tree.

import torch
import torch.nn as nn
import torch.nn.functional as F


class Model(nn.Module):
    def __init__(self) -> None:
        super().__init__()

    def forward(self, a, b):
        return a + b


def get_inputs():
    # randomly generate input tensors based on the model architecture
    a = torch.randn(1, 128).cuda()
    b = torch.randn(1, 128).cuda()
    return [a, b]


def get_init_inputs():
    # randomly generate tensors required for initialization based on the model architecture
    return []

        ``` 

        The example new arch with custom CUDA kernels looks like this:
        ```
        # Copyright (c) Meta Platforms, Inc. and affiliates.
# All rights reserved.
#
# This source code is licensed under the license found in the
# LICENSE file in the root directory of this source tree.

import torch
import torch.nn as nn
import torch.nn.functional as F
from torch.utils.cpp_extension import load_inline

# Define the custom CUDA kernel for element-wise addition
elementwise_add_source = """
#include <torch/extension.h>
#include <cuda_runtime.h>

__global__ void elementwise_add_kernel(const float* a, const float* b, float* out, int size) {
    int idx = blockIdx.x * blockDim.x + threadIdx.x;
    if (idx < size) {
        out[idx] = a[idx] + b[idx];
    }
}

torch::Tensor elementwise_add_cuda(torch::Tensor a, torch::Tensor b) {
    auto size = a.numel();
    auto out = torch::zeros_like(a);

    const int block_size = 256;
    const int num_blocks = (size + block_size - 1) / block_size;

    elementwise_add_kernel<<<num_blocks, block_size>>>(a.data_ptr<float>(), b.data_ptr<float>(), out.data_ptr<float>(), size);

    return out;
}
"""

elementwise_add_cpp_source = (
    "torch::Tensor elementwise_add_cuda(torch::Tensor a, torch::Tensor b);"
)

# Compile the inline CUDA code for element-wise addition
elementwise_add = load_inline(
    name="elementwise_add",
    cpp_sources=elementwise_add_cpp_source,
    cuda_sources=elementwise_add_source,
    functions=["elementwise_add_cuda"],
    verbose=True,
    extra_cflags=[""],
    extra_ldflags=[""],
)


class ModelNew(nn.Module):
    def __init__(self) -> None:
        super().__init__()
        self.elementwise_add = elementwise_add

    def forward(self, a, b):
        return self.elementwise_add.elementwise_add_cuda(a, b)

        ``` 

        
    You are given the following architecture: 

    ```
    # Copyright (c) Meta Platforms, Inc. and affiliates.
# All rights reserved.
#
# This source code is licensed under the license found in the
# LICENSE file in the root directory of this source tree.

import torch
import torch.nn as nn

class Model(nn.Module):
    """
    Simple model that performs a single matrix multiplication (C = A * B) with a large K dimension
    """
    def __init__(self):
        super(Model, self).__init__()
    
    def forward(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
        """
        Performs matrix multiplication of A and B.

        Args:
            A: Input tensor of shape (M, K)
            B: Input tensor of shape (K, N)

        Returns:
            Output tensor of shape (M, N)
        """
        return torch.matmul(A, B)

M = 256
N = 256
K = 131072 * 4

def get_inputs():
    A = torch.rand(M, K)
    B = torch.rand(K, N)
    return [A, B]

def get_init_inputs():
    return []  # No special initialization inputs needed
    ```
    
Optimize the architecture named Model with custom CUDA operators! Name your optimized output architecture ModelNew. Output the new code in codeblocks. Please generate real code, NOT pseudocode, make sure the code compiles and is fully functional. Just output the new model code, no other text, and NO testing code! 



This is turn 5 of 6 for this task. Your previous solution (turn 4) was:
```python
# Copyright (c) Meta Platforms, Inc. and affiliates.
# All rights reserved.
#
# This source code is licensed under the license found in the
# LICENSE file in the root directory of this source tree.

import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline

# Custom split-K TF32 tensor-core GEMM optimized for the (small M/N, huge K) regime.
# C[m,n] = sum_k A[m,k] * B[k,n]
#   - Block tile: 128 x 256, warp tile 32 x 64, BK = 32
#   - Deep asymmetric cp.async pipeline: several A stage buffers (the DRAM-latency
#     critical, strided operand) plus >=2 B stage buffers. Stage counts adapt to
#     the device's shared-memory capacity (A100: 6xA + 2xB fills 160KB exactly).
#   - XOR-swizzled shared-memory layouts (no padding), conflict-free ldmatrix/LDS.
#   - Templated fast path for aligned shapes (no bounds checks anywhere).
#   - K is split across blocks (~1 wave); partials go to a workspace and a light
#     deterministic reduction kernel sums them in FP32.
#   - Tensor-core math is TF32 (allowed by the task), accumulation is FP32.

cuda_src = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <cuda_runtime.h>
#include <stdint.h>

#define BM 128
#define BN 256
#define BK 32
#define NTHREADS 512
#define A_STAGE_FLOATS (BM * BK)
#define B_STAGE_FLOATS (BK * BN)
#define A_STAGE_BYTES (A_STAGE_FLOATS * 4)
#define B_STAGE_BYTES (B_STAGE_FLOATS * 4)

#define CP_ASYNC_16(dst, src, bytes) \
    asm volatile("cp.async.cg.shared.global [%0], [%1], 16, %2;\n" :: "r"(dst), "l"(src), "r"(bytes))
#define CP_COMMIT() asm volatile("cp.async.commit_group;\n" ::)
#define CP_WAITN(n) asm volatile("cp.async.wait_group %0;\n" :: "n"(n))

// Asymmetric-depth software pipeline:
//   - AD buffers for A tiles (A is prefetched AD-1 stages ahead).
//   - BD buffers for B tiles (B is prefetched BD-1 stages ahead).
// Each iteration commits exactly two cp.async groups: first the A group, then
// the B group (empty commits keep the group cadence uniform near the edges).
// Before computing stage s we need A[s] and B[s] resident. The groups still in
// flight at that point are A[s+1..s+AD-1] and B[s+1..s+BD-1], i.e. a constant
// AD+BD-2 groups, so wait_group(AD+BD-2) guarantees the data has landed.
// The __syncthreads() at the top of the loop both publishes newly landed stage
// data and guarantees the buffers being overwritten have been fully consumed.
template <int AD, int BD, bool FULL>
__global__ void __launch_bounds__(NTHREADS, 1)
gemm_splitk_tf32(const float* __restrict__ A, const float* __restrict__ B,
                 float* __restrict__ W, int M, int N, int K, int splits) {
    extern __shared__ float smem[];
    float* As = smem;                    // AD * BM * BK (XOR swizzled)
    float* Bs = smem + AD * A_STAGE_FLOATS;  // BD * BK * BN (XOR swizzled)

    const int tid  = threadIdx.x;
    const int lane = tid & 31;
    const int warp = tid >> 5;

    const int m_tiles = (M + BM - 1) / BM;
    const int tile_id = blockIdx.x % m_tiles;
    const int split   = blockIdx.x / m_tiles;
    const int m0 = tile_id * BM;

    // Even, non-empty K-iteration ranges per split (guaranteed by host clamp).
    const int total_iters = (K + BK - 1) / BK;
    const int base_i = total_iters / splits;
    const int rem_i  = total_iters % splits;
    const int it_begin = split * base_i + (split < rem_i ? split : rem_i);
    const int it_end   = it_begin + base_i + (split < rem_i ? 1 : 0);
    const int S = it_end - it_begin;
    if (S <= 0) return;

    // -------- async global -> shared prefetch of one BK-slab of A --------
    auto prefetchA = [&](int s, int buf) {
        const int k0 = (it_begin + s) * BK;
        float* Ad = As + buf * A_STAGE_FLOATS;
        const int r = tid >> 3, c = tid & 7;      // 64 rows x 8 chunks of 16B
        const uint32_t sbase = (uint32_t)__cvta_generic_to_shared(Ad);
        #pragma unroll
        for (int h = 0; h < 2; ++h) {
            const int row = r + h * 64;
            const int kk = k0 + c * 4;
            const uint32_t dst = sbase + (uint32_t)((row * BK + ((c ^ (row & 7)) << 2)) * 4);
            if (FULL) {
                const float* src = A + (int64_t)(m0 + row) * K + kk;
                CP_ASYNC_16(dst, src, 16);
            } else {
                const bool p = (m0 + row) < M;
                const int remn = K - kk;
                const int sz = (p && remn > 0) ? (remn >= 4 ? 16 : remn * 4) : 0;
                const float* src = A + (int64_t)(m0 + (p ? row : 0)) * K + (p ? kk : 0);
                CP_ASYNC_16(dst, src, sz);
            }
        }
        CP_COMMIT();
    };

    // -------- async global -> shared prefetch of one BK-slab of B --------
    auto prefetchB = [&](int s, int buf) {
        const int k0 = (it_begin + s) * BK;
        float* Bd = Bs + buf * B_STAGE_FLOATS;
        const int r = tid >> 4, c = tid & 15;     // 32 rows x 16 chunks of 16B
        const uint32_t sbase = (uint32_t)__cvta_generic_to_shared(Bd);
        #pragma unroll
        for (int i = 0; i < 4; ++i) {
            const int c4 = (c + i * 16) * 4;
            const int col = c4 ^ ((r & 7) << 2);
            const uint32_t dst = sbase + (uint32_t)((r * BN + col) * 4);
            if (FULL) {
                const float* src = B + (int64_t)(k0 + r) * N + c4;
                CP_ASYNC_16(dst, src, 16);
            } else {
                const bool pg = (k0 + r) < K;
                const int remc = N - c4;
                const int sz = (pg && remc > 0) ? (remc >= 4 ? 16 : remc * 4) : 0;
                const float* src = B + (int64_t)(k0 + (pg ? r : 0)) * N + (pg ? c4 : 0);
                CP_ASYNC_16(dst, src, sz);
            }
        }
        CP_COMMIT();
    };

    // -------- accumulators / warp layout (4x4 warps, warp tile 32x64) --------
    const int wm = warp >> 2;
    const int wn = warp & 3;
    const int warp_m0 = wm * 32;
    const int warp_n0 = wn * 64;

    float acc[2][8][4];
    #pragma unroll
    for (int i = 0; i < 2; ++i)
        #pragma unroll
        for (int j = 0; j < 8; ++j)
            #pragma unroll
            for (int v = 0; v < 4; ++v) acc[i][j][v] = 0.f;

    // ldmatrix source lane pattern for m16k8 tf32 A-fragments
    const int lds_row  = ((lane & 15) >> 3) * 8 + (lane & 7);
    const int lds_colf = (lane >> 4) * 4;
    // scalar B-fragment lane pattern
    const int b_koff = lane & 3;
    const int b_noff = lane >> 2;

    // -------- pipeline prologue: A groups A[0..AD-2], B groups B[0..BD-2] ----
    #pragma unroll
    for (int p = 0; p < AD - 1; ++p) {
        if (p < S) prefetchA(p, p % AD);
        else CP_COMMIT();
    }
    // Interleave the first B group after only part of the A prologue so the
    // first compute stage (which needs B[0]) is not gated behind the full
    // deep-A prefetch stream.
    #pragma unroll
    for (int p = 0; p < BD - 1; ++p) {
        if (p < S) prefetchB(p, p % BD);
        else CP_COMMIT();
        if (p + BD - 1 < AD - 1) {
            const int q = p + BD - 1;
            if (q < S) prefetchA(q, q % AD);
            else CP_COMMIT();
        }
    }
    // Above loop emits (BD-1) B-groups and (AD-1) A-groups total, in an
    // order that keeps the FIFO group-completion logic valid.

    int bA = 0, bB = 0;
    for (int s = 0; s < S; ++s) {
        CP_WAITN(AD + BD - 2);
        __syncthreads();

        if (s + AD - 1 < S) prefetchA(s + AD - 1, (s + AD - 1) % AD);
        else CP_COMMIT();
        if (s + BD - 1 < S) prefetchB(s + BD - 1, (s + BD - 1) % BD);
        else CP_COMMIT();

        const float* Ac = As + bA * A_STAGE_FLOATS;
        const float* Bc = Bs + bB * B_STAGE_FLOATS;

        #pragma unroll
        for (int kk = 0; kk < BK; kk += 8) {
            // ---- A fragments: 2 x m16k8 via ldmatrix.x4 ----
            uint32_t afr[2][4];
            #pragma unroll
            for (int i = 0; i < 2; ++i) {
                const int row = warp_m0 + i * 16 + lds_row;
                const int col = kk + lds_colf;
                const uint32_t addr = (uint32_t)__cvta_generic_to_shared(
                    Ac + row * BK + (((col >> 2) ^ (row & 7)) << 2) + (col & 3));
                asm volatile(
                    "ldmatrix.sync.aligned.m8n8.x4.shared.b16 {%0,%1,%2,%3}, [%4];\n"
                    : "=r"(afr[i][0]), "=r"(afr[i][1]), "=r"(afr[i][2]), "=r"(afr[i][3])
                    : "r"(addr));
            }
            // ---- B fragments: 8 x k8n8 (conflict-free via XOR swizzle) ----
            uint32_t bfr[8][2];
            #pragma unroll
            for (int j = 0; j < 8; ++j) {
                const int nn0 = warp_n0 + j * 8 + b_noff;
                const int kk0 = kk + b_koff;
                bfr[j][0] = __float_as_uint(Bc[kk0 * BN + (nn0 ^ ((kk0 & 7) << 2))]);
                const int kk1 = kk0 + 4;
                bfr[j][1] = __float_as_uint(Bc[kk1 * BN + (nn0 ^ ((kk1 & 7) << 2))]);
            }
            // ---- 16 tensor-core MMAs (TF32 in, FP32 accumulate) ----
            #pragma unroll
            for (int i = 0; i < 2; ++i) {
                #pragma unroll
                for (int j = 0; j < 8; ++j) {
                    asm volatile(
                        "mma.sync.aligned.m16n8k8.row.col.f32.tf32.tf32.f32 "
                        "{%0,%1,%2,%3}, {%4,%5,%6,%7}, {%8,%9}, {%0,%1,%2,%3};\n"
                        : "+f"(acc[i][j][0]), "+f"(acc[i][j][1]),
                          "+f"(acc[i][j][2]), "+f"(acc[i][j][3])
                        : "r"(afr[i][0]), "r"(afr[i][1]), "r"(afr[i][2]), "r"(afr[i][3]),
                          "r"(bfr[j][0]), "r"(bfr[j][1]));
                }
            }
        }
        bA = (bA + 1 == AD) ? 0 : (bA + 1);
        bB = (bB + 1 == BD) ? 0 : (bB + 1);
    }

    // -------- epilogue: write partial tile to workspace --------
    float* Ws = W + (int64_t)split * M * N;
    const int gr = lane >> 2;
    const int tg = lane & 3;
    if (FULL) {
        #pragma unroll
        for (int i = 0; i < 2; ++i) {
            #pragma unroll
            for (int j = 0; j < 8; ++j) {
                const int r0 = m0 + warp_m0 + i * 16 + gr;
                const int c0 = warp_n0 + j * 8 + tg * 2;
                *reinterpret_cast<float2*>(Ws + (int64_t)r0 * N + c0) =
                    make_float2(acc[i][j][0], acc[i][j][1]);
                *reinterpret_cast<float2*>(Ws + (int64_t)(r0 + 8) * N + c0) =
                    make_float2(acc[i][j][2], acc[i][j][3]);
            }
        }
    } else {
        const bool nvec = (N & 1) == 0;
        #pragma unroll
        for (int i = 0; i < 2; ++i) {
            #pragma unroll
            for (int j = 0; j < 8; ++j) {
                const int r0 = m0 + warp_m0 + i * 16 + gr;
                const int c0 = warp_n0 + j * 8 + tg * 2;
                if (nvec && r0 < M && c0 + 1 < N) {
                    *reinterpret_cast<float2*>(Ws + (int64_t)r0 * N + c0) =
                        make_float2(acc[i][j][0], acc[i][j][1]);
                } else {
                    if (r0 < M && c0 < N)     Ws[(int64_t)r0 * N + c0]     = acc[i][j][0];
                    if (r0 < M && c0 + 1 < N) Ws[(int64_t)r0 * N + c0 + 1] = acc[i][j][1];
                }
                const int r1 = r0 + 8;
                if (nvec && r1 < M && c0 + 1 < N) {
                    *reinterpret_cast<float2*>(Ws + (int64_t)r1 * N + c0) =
                        make_float2(acc[i][j][2], acc[i][j][3]);
                } else {
                    if (r1 < M && c0 < N)     Ws[(int64_t)r1 * N + c0]     = acc[i][j][2];
                    if (r1 < M && c0 + 1 < N) Ws[(int64_t)r1 * N + c0 + 1] = acc[i][j][3];
                }
            }
        }
    }
}

__global__ void reduce_splits_kernel(const float* __restrict__ W,
                                     float* __restrict__ C,
                                     int64_t slice_elems, int splits) {
    const int64_t idx = (int64_t)(blockIdx.x * blockDim.x + threadIdx.x) * 4;
    if (idx >= slice_elems) return;
    float4 s = *reinterpret_cast<const float4*>(W + idx);
    for (int p = 1; p < splits; ++p) {
        const float4 v = *reinterpret_cast<const float4*>(W + (int64_t)p * slice_elems + idx);
        s.x += v.x; s.y += v.y; s.z += v.z; s.w += v.w;
    }
    *reinterpret_cast<float4*>(C + idx) = s;
}

// (AD, BD, smem_bytes) candidates from most to least preferred. The extra
// "+1024" accounts for the 1KB shared-memory carve-in the driver reserves.
struct Cfg { int ad; int bd; int bytes; };

template <int AD, int BD, bool FULL>
static void launch_cfg(dim3 grid, int smem_bytes, cudaStream_t stream,
                       const float* A, const float* B, float* W,
                       int M, int N, int K, int splits) {
    static bool attr_set = false;
    if (!attr_set) {
        cudaFuncSetAttribute(gemm_splitk_tf32<AD, BD, FULL>,
                             cudaFuncAttributeMaxDynamicSharedMemorySize,
                             AD * A_STAGE_BYTES + BD * B_STAGE_BYTES);
        attr_set = true;
    }
    gemm_splitk_tf32<AD, BD, FULL><<<grid, NTHREADS, smem_bytes, stream>>>(
        A, B, W, M, N, K, splits);
}

template <bool FULL>
static bool try_launch(int cfg, dim3 grid, cudaStream_t stream,
                       const float* A, const float* B, float* W,
                       int M, int N, int K, int splits) {
    switch (cfg) {
        case 0: launch_cfg<6, 2, FULL>(grid, 6 * A_STAGE_BYTES + 2 * B_STAGE_BYTES, stream, A, B, W, M, N, K, splits); return true;
        case 1: launch_cfg<5, 2, FULL>(grid, 5 * A_STAGE_BYTES + 2 * B_STAGE_BYTES, stream, A, B, W, M, N, K, splits); return true;
        case 2: launch_cfg<4, 2, FULL>(grid, 4 * A_STAGE_BYTES + 2 * B_STAGE_BYTES, stream, A, B, W, M, N, K, splits); return true;
        case 3: launch_cfg<3, 3, FULL>(grid, 3 * A_STAGE_BYTES + 3 * B_STAGE_BYTES, stream, A, B, W, M, N, K, splits); return true;
        case 4: launch_cfg<3, 2, FULL>(grid, 3 * A_STAGE_BYTES + 2 * B_STAGE_BYTES, stream, A, B, W, M, N, K, splits); return true;
        case 5: launch_cfg<2, 2, FULL>(grid, 2 * A_STAGE_BYTES + 2 * B_STAGE_BYTES, stream, A, B, W, M, N, K, splits); return true;
        default: return false;
    }
}

torch::Tensor matmul_cuda(torch::Tensor A, torch::Tensor B) {
    TORCH_CHECK(A.is_cuda() && B.is_cuda(), "inputs must be CUDA tensors");
    TORCH_CHECK(A.dtype() == torch::kFloat32 && B.dtype() == torch::kFloat32,
                "inputs must be float32");
    TORCH_CHECK(A.dim() == 2 && B.dim() == 2, "inputs must be 2-D");
    const int M = (int)A.size(0);
    const int K = (int)A.size(1);
    const int N = (int)B.size(1);
    TORCH_CHECK((int)B.size(0) == K, "inner dimensions must match");
    TORCH_CHECK(N <= BN, "N too large for this kernel");

    auto Ac = A.contiguous();
    auto Bc = B.contiguous();
    auto C = torch::empty({M, N}, A.options());
    if (M == 0 || N == 0) return C;
    if (K == 0) { C.zero_(); return C; }

    auto stream = at::cuda::getCurrentCUDAStream();

    const int m_tiles = (M + BM - 1) / BM;
    const int total_iters = (K + BK - 1) / BK;

    int dev = 0;
    cudaGetDevice(&dev);

    static int sm_count = 0;
    static int smem_optin = 0;
    static int cfg_pick = -1;
    if (sm_count == 0) {
        cudaDeviceGetAttribute(&sm_count, cudaDevAttrMultiProcessorCount, dev);
        cudaDeviceGetAttribute(&smem_optin, cudaDevAttrMaxSharedMemoryPerBlockOptin, dev);
        if (sm_count < 1) sm_count = 1;
        static const Cfg cfgs[6] = {
            {6, 2, 6 * A_STAGE_BYTES + 2 * B_STAGE_BYTES},
            {5, 2, 5 * A_STAGE_BYTES + 2 * B_STAGE_BYTES},
            {4, 2, 4 * A_STAGE_BYTES + 2 * B_STAGE_BYTES},
            {3, 3, 3 * A_STAGE_BYTES + 3 * B_STAGE_BYTES},
            {3, 2, 3 * A_STAGE_BYTES + 2 * B_STAGE_BYTES},
            {2, 2, 2 * A_STAGE_BYTES + 2 * B_STAGE_BYTES},
        };
        cfg_pick = -1;
        for (int i = 0; i < 6; ++i) {
            if (cfgs[i].bytes + 1024 <= smem_optin) { cfg_pick = i; break; }
        }
    }
    if (cfg_pick < 0) {
        return torch::matmul(Ac, Bc);   // not enough shared memory: fall back
    }

    // Aim for exactly one full wave of resident blocks.
    int splits = sm_count / m_tiles;
    if (splits < 1) splits = 1;
    if (splits > total_iters) splits = total_iters;
    if (splits > 1024) splits = 1024;

    torch::Tensor W;
    float* Wp = nullptr;
    if (splits > 1) {
        W = torch::empty({(int64_t)splits * M * N}, A.options());
        Wp = W.data_ptr<float>();
    } else {
        Wp = C.data_ptr<float>();
    }

    dim3 grid((unsigned)(m_tiles * splits));
    const bool full = (M % BM == 0) && (K % BK == 0) && (N == BN);
    bool ok;
    if (full) {
        ok = try_launch<true>(cfg_pick, grid, stream,
                              Ac.data_ptr<float>(), Bc.data_ptr<float>(), Wp,
                              M, N, K, splits);
    } else {
        ok = try_launch<false>(cfg_pick, grid, stream,
                               Ac.data_ptr<float>(), Bc.data_ptr<float>(), Wp,
                               M, N, K, splits);
    }
    if (!ok) return torch::matmul(Ac, Bc);

    if (splits > 1) {
        const int64_t slice_elems = (int64_t)M * N;
        if ((slice_elems & 3) == 0) {
            const int64_t quads = slice_elems >> 2;
            const int threads = 256;
            const int blocks = (int)((quads + threads - 1) / threads);
            reduce_splits_kernel<<<blocks, threads, 0, stream>>>(
                Wp, C.data_ptr<float>(), slice_elems, splits);
        } else {
            auto Cv = C.view({slice_elems});
            Cv.copy_(W.slice(0, 0, slice_elems));
            for (int p = 1; p < splits; ++p) {
                Cv.add_(W.slice(0, (int64_t)p * slice_elems, (int64_t)(p + 1) * slice_elems));
            }
        }
    }
    return C;
}
"""

cpp_src = "torch::Tensor matmul_cuda(torch::Tensor A, torch::Tensor B);"

_mod = load_inline(
    name="splitk_tf32_gemm_v4",
    cpp_sources=cpp_src,
    cuda_sources=cuda_src,
    functions=["matmul_cuda"],
    verbose=False,
    extra_cuda_cflags=["-O3", "--use_fast_math"],
)


class ModelNew(nn.Module):
    """
    Matrix multiplication C = A * B implemented with a custom split-K TF32
    tensor-core CUDA kernel (FP32 accumulation), optimized for very large K.
    Uses a deep asymmetric cp.async pipeline (extra prefetch depth for the
    DRAM-latency-critical A operand) with XOR-swizzled shared memory.
    """
    def __init__(self):
        super(ModelNew, self).__init__()

    def forward(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
        if (A.is_cuda and B.is_cuda
                and A.dtype == torch.float32 and B.dtype == torch.float32
                and A.dim() == 2 and B.dim() == 2
                and A.size(1) == B.size(0) and B.size(1) <= 256):
            return _mod.matmul_cuda(A, B)
        return torch.matmul(A, B)
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 1.39 ms; the PyTorch reference (with TF32 enabled) takes 0.939 ms; speedup 0.676x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| void gemm_splitk_tf32<6, 2, true>(float const*, float const*, float*, int, int, int, int) | 1355.5 | 1 |
| reduce_splits_kernel(float const*, float*, long, int) | 7.9 | 1 |

The fastest correct solution so far (1.109x speedup) is:
```python
# Copyright (c) Meta Platforms, Inc. and affiliates.
# All rights reserved.
#
# This source code is licensed under the license found in the
# LICENSE file in the root directory of this source tree.

import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline

# Custom split-K TF32 tensor-core GEMM optimized for the (small M/N, huge K) regime.
# C[m,n] = sum_k A[m,k] * B[k,n]
#   - Block tile: 128 x 256, warp tile 32 x 64, BK = 32
#   - cp.async multi-stage pipeline (2/3/4 stages chosen from device smem capacity)
#     for deeper memory-level parallelism (the kernel is DRAM-bandwidth bound)
#   - K is split across blocks (~1 wave of blocks); partials go to a workspace
#   - A light reduction kernel sums partials into C (deterministic, FP32)
#   - Tensor-core math is TF32 (allowed by the task), accumulation is FP32.

cuda_src = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <cuda_runtime.h>
#include <stdint.h>

#define BM 128
#define BN 256
#define BK 32
#define NTHREADS 512
#define BS_STRIDE (BN + 8)
#define STAGE_FLOATS (BM * BK + BK * BS_STRIDE)
#define STAGE_BYTES (STAGE_FLOATS * 4)

#define CP_ASYNC_16(dst, src, bytes) \
    asm volatile("cp.async.cg.shared.global [%0], [%1], 16, %2;\n" :: "r"(dst), "l"(src), "r"(bytes))
#define CP_COMMIT() asm volatile("cp.async.commit_group;\n" ::)
#define CP_WAIT0() asm volatile("cp.async.wait_group 0;\n" ::)
#define CP_WAIT1() asm volatile("cp.async.wait_group 1;\n" ::)
#define CP_WAIT2() asm volatile("cp.async.wait_group 2;\n" ::)

template <int STAGES>
__global__ void __launch_bounds__(NTHREADS, 1)
gemm_splitk_tf32(const float* __restrict__ A, const float* __restrict__ B,
                 float* __restrict__ W, int M, int N, int K, int splits) {
    extern __shared__ float smem[];
    float* As = smem;                        // STAGES * BM * BK (XOR swizzled)
    float* Bs = smem + STAGES * BM * BK;     // STAGES * BK * BS_STRIDE (padded)

    const int tid  = threadIdx.x;
    const int lane = tid & 31;
    const int warp = tid >> 5;

    const int m_tiles = (M + BM - 1) / BM;
    const int tile_id = blockIdx.x % m_tiles;
    const int split   = blockIdx.x / m_tiles;
    const int m0 = tile_id * BM;

    // Even, non-empty K-iteration ranges per split (guaranteed by host clamp).
    const int total_iters = (K + BK - 1) / BK;
    const int base_i = total_iters / splits;
    const int rem_i  = total_iters % splits;
    const int it_begin = split * base_i + (split < rem_i ? split : rem_i);
    const int it_end   = it_begin + base_i + (split < rem_i ? 1 : 0);
    const int S = it_end - it_begin;
    if (S <= 0) return;

    // -------- async global -> shared prefetcher (one BK-slab stage) --------
    auto prefetch = [&](int s, int buf) {
        const int k0 = (it_begin + s) * BK;
        float* Ad = As + buf * (BM * BK);
        float* Bd = Bs + buf * (BK * BS_STRIDE);
        // A tile: BM(128) rows x BK(32) cols, 8x 16B chunks per row, XOR swizzle.
        {
            const int r = tid >> 3, c = tid & 7;
            const uint32_t sbase = (uint32_t)__cvta_generic_to_shared(Ad);
            #pragma unroll
            for (int h = 0; h < 2; ++h) {
                const int row = r + h * 64;
                const bool p = (m0 + row) < M;
                const int kk = k0 + c * 4;
                const int remn = K - kk;
                const int sz = (p && remn > 0) ? (remn >= 4 ? 16 : remn * 4) : 0;
                const float* src = A + (int64_t)(m0 + (p ? row : 0)) * K + (p ? kk : 0);
                const uint32_t dst = sbase + (uint32_t)((row * BK + ((c ^ (row & 7)) << 2)) * 4);
                CP_ASYNC_16(dst, src, sz);
            }
        }
        // B tile: BK(32) rows x BN(256) cols, 64x 16B chunks per row (padded stride).
        {
            const int r = tid >> 4, c = tid & 15;
            const uint32_t sbase = (uint32_t)__cvta_generic_to_shared(Bd);
            #pragma unroll
            for (int i = 0; i < 4; ++i) {
                const int col = (c + i * 16) * 4;
                const bool pg = (k0 + r) < K;
                const int remc = N - col;
                const int sz = (pg && remc > 0) ? (remc >= 4 ? 16 : remc * 4) : 0;
                const float* src = B + (int64_t)(k0 + (pg ? r : 0)) * N + (pg ? col : 0);
                const uint32_t dst = sbase + (uint32_t)((r * BS_STRIDE + col) * 4);
                CP_ASYNC_16(dst, src, sz);
            }
        }
        CP_COMMIT();
    };

    // -------- accumulators / warp layout (4x4 warps, warp tile 32x64) --------
    const int wm = warp >> 2;
    const int wn = warp & 3;
    const int warp_m0 = wm * 32;
    const int warp_n0 = wn * 64;

    float acc[2][8][4];
    #pragma unroll
    for (int i = 0; i < 2; ++i)
        #pragma unroll
        for (int j = 0; j < 8; ++j)
            #pragma unroll
            for (int v = 0; v < 4; ++v) acc[i][j][v] = 0.f;

    // ldmatrix source lane pattern for m16k8 tf32 A-fragments
    const int lds_row  = ((lane & 15) >> 3) * 8 + (lane & 7);
    const int lds_colf = (lane >> 4) * 4;

    // -------- pipeline prologue --------
    #pragma unroll
    for (int p = 0; p < STAGES - 1; ++p) {
        if (p < S) prefetch(p, p);
        else CP_COMMIT();
    }

    int b = 0;
    for (int s = 0; s < S; ++s) {
        // A group (real or empty) is committed every iteration, so a uniform
        // wait_group(STAGES-2) guarantees stage s has landed.
        if (STAGES >= 4)      { CP_WAIT2(); }
        else if (STAGES == 3) { CP_WAIT1(); }
        else                  { CP_WAIT0(); }
        __syncthreads();

        const int nb = (b == 0) ? (STAGES - 1) : (b - 1);
        if (s + STAGES - 1 < S) prefetch(s + STAGES - 1, nb);
        else CP_COMMIT();

        const float* Ac = As + b * (BM * BK);
        const float* Bc = Bs + b * (BK * BS_STRIDE);

        #pragma unroll
        for (int kk = 0; kk < BK; kk += 8) {
            // ---- A fragments: 2 x m16k8 via ldmatrix.x4 ----
            uint32_t afr[2][4];
            #pragma unroll
            for (int i = 0; i < 2; ++i) {
                const int row = warp_m0 + i * 16 + lds_row;
                const int col = kk + lds_colf;
                const uint32_t addr = (uint32_t)__cvta_generic_to_shared(
                    Ac + row * BK + (((col >> 2) ^ (row & 7)) << 2) + (col & 3));
                asm volatile(
                    "ldmatrix.sync.aligned.m8n8.x4.shared.b16 {%0,%1,%2,%3}, [%4];\n"
                    : "=r"(afr[i][0]), "=r"(afr[i][1]), "=r"(afr[i][2]), "=r"(afr[i][3])
                    : "r"(addr));
            }
            // ---- B fragments: 8 x k8n8 (conflict-free scalar loads) ----
            uint32_t bfr[8][2];
            #pragma unroll
            for (int j = 0; j < 8; ++j) {
                const int kk0 = kk + (lane & 3);
                const int nn0 = warp_n0 + j * 8 + (lane >> 2);
                bfr[j][0] = __float_as_uint(Bc[kk0 * BS_STRIDE + nn0]);
                bfr[j][1] = __float_as_uint(Bc[(kk0 + 4) * BS_STRIDE + nn0]);
            }
            // ---- 16 tensor-core MMAs (TF32 in, FP32 accumulate) ----
            #pragma unroll
            for (int i = 0; i < 2; ++i) {
                #pragma unroll
                for (int j = 0; j < 8; ++j) {
                    asm volatile(
                        "mma.sync.aligned.m16n8k8.row.col.f32.tf32.tf32.f32 "
                        "{%0,%1,%2,%3}, {%4,%5,%6,%7}, {%8,%9}, {%0,%1,%2,%3};\n"
                        : "+f"(acc[i][j][0]), "+f"(acc[i][j][1]),
                          "+f"(acc[i][j][2]), "+f"(acc[i][j][3])
                        : "r"(afr[i][0]), "r"(afr[i][1]), "r"(afr[i][2]), "r"(afr[i][3]),
                          "r"(bfr[j][0]), "r"(bfr[j][1]));
                }
            }
        }
        b = (b + 1 == STAGES) ? 0 : (b + 1);
    }

    // -------- epilogue: write partial tile to workspace --------
    float* Ws = W + (int64_t)split * M * N;
    const int gr = lane >> 2;
    const int tg = lane & 3;
    const bool nvec = (N & 1) == 0;
    #pragma unroll
    for (int i = 0; i < 2; ++i) {
        #pragma unroll
        for (int j = 0; j < 8; ++j) {
            const int r0 = m0 + warp_m0 + i * 16 + gr;
            const int c0 = warp_n0 + j * 8 + tg * 2;
            if (nvec && r0 < M && c0 + 1 < N) {
                *reinterpret_cast<float2*>(Ws + (int64_t)r0 * N + c0) =
                    make_float2(acc[i][j][0], acc[i][j][1]);
            } else {
                if (r0 < M && c0 < N)     Ws[(int64_t)r0 * N + c0]     = acc[i][j][0];
                if (r0 < M && c0 + 1 < N) Ws[(int64_t)r0 * N + c0 + 1] = acc[i][j][1];
            }
            const int r1 = r0 + 8;
            if (nvec && r1 < M && c0 + 1 < N) {
                *reinterpret_cast<float2*>(Ws + (int64_t)r1 * N + c0) =
                    make_float2(acc[i][j][2], acc[i][j][3]);
            } else {
                if (r1 < M && c0 < N)     Ws[(int64_t)r1 * N + c0]     = acc[i][j][2];
                if (r1 < M && c0 + 1 < N) Ws[(int64_t)r1 * N + c0 + 1] = acc[i][j][3];
            }
        }
    }
}

__global__ void reduce_splits_kernel(const float* __restrict__ W,
                                     float* __restrict__ C,
                                     int64_t slice_elems, int splits) {
    const int64_t idx = (int64_t)(blockIdx.x * blockDim.x + threadIdx.x) * 4;
    if (idx >= slice_elems) return;
    float4 s = *reinterpret_cast<const float4*>(W + idx);
    for (int p = 1; p < splits; ++p) {
        const float4 v = *reinterpret_cast<const float4*>(W + (int64_t)p * slice_elems + idx);
        s.x += v.x; s.y += v.y; s.z += v.z; s.w += v.w;
    }
    *reinterpret_cast<float4*>(C + idx) = s;
}

torch::Tensor matmul_cuda(torch::Tensor A, torch::Tensor B) {
    TORCH_CHECK(A.is_cuda() && B.is_cuda(), "inputs must be CUDA tensors");
    TORCH_CHECK(A.dtype() == torch::kFloat32 && B.dtype() == torch::kFloat32,
                "inputs must be float32");
    TORCH_CHECK(A.dim() == 2 && B.dim() == 2, "inputs must be 2-D");
    const int M = (int)A.size(0);
    const int K = (int)A.size(1);
    const int N = (int)B.size(1);
    TORCH_CHECK((int)B.size(0) == K, "inner dimensions must match");
    TORCH_CHECK(N <= BN, "N too large for this kernel");

    auto Ac = A.contiguous();
    auto Bc = B.contiguous();
    auto C = torch::empty({M, N}, A.options());
    if (M == 0 || N == 0) return C;
    if (K == 0) { C.zero_(); return C; }

    auto stream = at::cuda::getCurrentCUDAStream();

    const int m_tiles = (M + BM - 1) / BM;
    const int total_iters = (K + BK - 1) / BK;

    int dev = 0;
    cudaGetDevice(&dev);

    static int sm_count = 0;
    static int smem_optin = 0;
    static int stages = 0;
    if (sm_count == 0) {
        cudaDeviceGetAttribute(&sm_count, cudaDevAttrMultiProcessorCount, dev);
        cudaDeviceGetAttribute(&smem_optin, cudaDevAttrMaxSharedMemoryPerBlockOptin, dev);
        if (sm_count < 1) sm_count = 1;
        int st = smem_optin / STAGE_BYTES;
        if (st > 4) st = 4;
        stages = st;   // may be < 2 on small-smem GPUs -> fallback below
    }
    if (stages < 2) {
        return torch::matmul(Ac, Bc);
    }

    // Aim for exactly one full wave of resident blocks.
    int splits = sm_count / m_tiles;
    if (splits < 1) splits = 1;
    if (splits > total_iters) splits = total_iters;
    if (splits > 1024) splits = 1024;

    torch::Tensor W;
    float* Wp = nullptr;
    if (splits > 1) {
        W = torch::empty({(int64_t)splits * M * N}, A.options());
        Wp = W.data_ptr<float>();
    } else {
        Wp = C.data_ptr<float>();
    }

    const int smem_bytes = stages * STAGE_BYTES;

    static bool attr2 = false, attr3 = false, attr4 = false;
    if (stages == 2 && !attr2) {
        cudaFuncSetAttribute(gemm_splitk_tf32<2>,
                             cudaFuncAttributeMaxDynamicSharedMemorySize,
                             2 * STAGE_BYTES);
        attr2 = true;
    } else if (stages == 3 && !attr3) {
        cudaFuncSetAttribute(gemm_splitk_tf32<3>,
                             cudaFuncAttributeMaxDynamicSharedMemorySize,
                             3 * STAGE_BYTES);
        attr3 = true;
    } else if (stages >= 4 && !attr4) {
        cudaFuncSetAttribute(gemm_splitk_tf32<4>,
                             cudaFuncAttributeMaxDynamicSharedMemorySize,
                             4 * STAGE_BYTES);
        attr4 = true;
    }

    dim3 grid((unsigned)(m_tiles * splits));
    if (stages == 2) {
        gemm_splitk_tf32<2><<<grid, NTHREADS, smem_bytes, stream>>>(
            Ac.data_ptr<float>(), Bc.data_ptr<float>(), Wp, M, N, K, splits);
    } else if (stages == 3) {
        gemm_splitk_tf32<3><<<grid, NTHREADS, smem_bytes, stream>>>(
            Ac.data_ptr<float>(), Bc.data_ptr<float>(), Wp, M, N, K, splits);
    } else {
        gemm_splitk_tf32<4><<<grid, NTHREADS, smem_bytes, stream>>>(
            Ac.data_ptr<float>(), Bc.data_ptr<float>(), Wp, M, N, K, splits);
    }

    if (splits > 1) {
        const int64_t slice_elems = (int64_t)M * N;
        if ((slice_elems & 3) == 0) {
            const int64_t quads = slice_elems >> 2;
            const int threads = 256;
            const int blocks = (int)((quads + threads - 1) / threads);
            reduce_splits_kernel<<<blocks, threads, 0, stream>>>(
                Wp, C.data_ptr<float>(), slice_elems, splits);
        } else {
            auto Cv = C.view({slice_elems});
            Cv.copy_(W.slice(0, 0, slice_elems));
            for (int p = 1; p < splits; ++p) {
                Cv.add_(W.slice(0, (int64_t)p * slice_elems, (int64_t)(p + 1) * slice_elems));
            }
        }
    }
    return C;
}
"""

cpp_src = "torch::Tensor matmul_cuda(torch::Tensor A, torch::Tensor B);"

_mod = load_inline(
    name="splitk_tf32_gemm_v3",
    cpp_sources=cpp_src,
    cuda_sources=cuda_src,
    functions=["matmul_cuda"],
    verbose=False,
    extra_cuda_cflags=["-O3", "--use_fast_math"],
)


class ModelNew(nn.Module):
    """
    Matrix multiplication C = A * B implemented with a custom split-K TF32
    tensor-core CUDA kernel (FP32 accumulation), optimized for very large K.
    Uses a deep cp.async pipeline (stage count adapts to device shared memory).
    """
    def __init__(self):
        super(ModelNew, self).__init__()

    def forward(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
        if (A.is_cuda and B.is_cuda
                and A.dtype == torch.float32 and B.dtype == torch.float32
                and A.dim() == 2 and B.dim() == 2
                and A.size(1) == B.size(0) and B.size(1) <= 256):
            return _mod.matmul_cuda(A, B)
        return torch.matmul(A, B)
```

Write an improved complete ModelNew: fix any error, and if the solution is already correct, make it faster. Output the new code in one ```python codeblock. Just the model code, no other text, and NO testing code.


Precision rule: the timed forward must compute in FP32 (TF32 tensor-core math is allowed, as in the PyTorch reference). Do not cast inputs, weights or intermediate results to float16 or bfloat16 and do not call half-precision kernels in forward().
