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)
    """
    def __init__(self):
        super(Model, self).__init__()
    
    def forward(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
        """
        Performs matrix multiplication.

        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.T)

M = 1024 * 2
K = 4096 * 2
N = 2048 * 2

def get_inputs():
    A = torch.rand(M, K)
    B = torch.rand(N, K)
    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 4 of 6 for this task. Your previous solution (turn 3) 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

# High-performance TF32 tensor-core GEMM: C = A @ B^T
#   A: (M, K) row-major, B: (N, K) row-major, C: (M, N) row-major
#
# Optimizations over a naive tensor-core GEMM:
#   * 128x128x32 CTA tile, 4 warps, each computing a 64x64 warp tile
#   * mma.sync.aligned.m16n8k8.row.col.f32.tf32.tf32.f32 instructions
#   * ldmatrix.sync.aligned.m8n8.x4.shared.b16 to fetch the tf32 fragments
#     from shared memory (1 instruction per 16x8 fragment instead of 4 LDS.32)
#   * cp.async 16B global->shared copies with a multi-stage (up to 5) pipeline
#   * 128B-row XOR-swizzled shared-memory layout: conflict-free cp.async
#     stores and conflict-free ldmatrix loads
#   * register double buffering of A/B fragments across k-steps
#   * CTA rasterization swizzle for L2 locality
cuda_source = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <cuda_runtime.h>
#include <cstdint>

#define BM 128
#define BN 128
#define BK 32
#define A_STAGE_FLOATS (BM * BK)
#define STAGE_FLOATS (A_STAGE_FLOATS + BN * BK)
#define THREADS 128

__device__ __forceinline__ uint32_t smem_u32(const void* p) {
    return static_cast<uint32_t>(__cvta_generic_to_shared(p));
}

__device__ __forceinline__ void cp_async16(uint32_t dst, const void* src) {
#if __CUDA_ARCH__ >= 800
    asm volatile("cp.async.cg.shared.global [%0], [%1], 16;\n" :: "r"(dst), "l"(src));
#endif
}

__device__ __forceinline__ void cp_commit() {
#if __CUDA_ARCH__ >= 800
    asm volatile("cp.async.commit_group;\n");
#endif
}

template <int N>
__device__ __forceinline__ void cp_wait() {
#if __CUDA_ARCH__ >= 800
    asm volatile("cp.async.wait_group %0;\n" :: "n"(N));
#endif
}

__device__ __forceinline__ void ldsm_x4(uint32_t& r0, uint32_t& r1,
                                        uint32_t& r2, uint32_t& r3,
                                        uint32_t addr) {
#if __CUDA_ARCH__ >= 800
    asm volatile("ldmatrix.sync.aligned.m8n8.x4.shared.b16 {%0,%1,%2,%3}, [%4];\n"
                 : "=r"(r0), "=r"(r1), "=r"(r2), "=r"(r3) : "r"(addr));
#endif
}

__device__ __forceinline__ void mma_tf32(
        float& d0, float& d1, float& d2, float& d3,
        uint32_t a0, uint32_t a1, uint32_t a2, uint32_t a3,
        uint32_t b0, uint32_t b1) {
#if __CUDA_ARCH__ >= 800
    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"(d0), "+f"(d1), "+f"(d2), "+f"(d3)
        : "r"(a0), "r"(a1), "r"(a2), "r"(a3), "r"(b0), "r"(b1));
#endif
}

template <int STAGES>
__global__ void __launch_bounds__(THREADS)
gemm_nt_tf32_ldsm_kernel(const float* __restrict__ A,
                         const float* __restrict__ B,
                         float* __restrict__ C,
                         int M, int N, int K) {
#if __CUDA_ARCH__ >= 800
    extern __shared__ float smem[];
    float* smA = smem;                        // STAGES * 4096 floats
    float* smB = smem + STAGES * A_STAGE_FLOATS;

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

    // ---- CTA rasterization swizzle (improves L2 reuse) ----
    const int num_m = M / BM;
    const int num_n = N / BN;
    const int GRP = 8;
    int pid = blockIdx.x;
    int per_group = GRP * num_m;
    int group = pid / per_group;
    int rem = pid - group * per_group;
    int first_n = group * GRP;
    int gsz = min(num_n - first_n, GRP);
    int pm = rem / gsz;
    int pn = first_n + (rem % gsz);
    const int block_m = pm * BM;
    const int block_n = pn * BN;

    const int warp_m = (warp >> 1) * 64;
    const int warp_n = (warp & 1) * 64;

    const int k_tiles = K / BK;

    // Global->shared copy thread mapping: 128 threads x 16B = 16 rows of 128B
    const int ldg_row  = tid >> 3;   // 0..15
    const int ldg_unit = tid & 7;    // 16B unit within a 128B row (0..7)

    auto load_stage = [&](int stage, int kt) {
        const int k0 = kt * BK;
        float* dA = smA + stage * A_STAGE_FLOATS;
        float* dB = smB + stage * A_STAGE_FLOATS;
        #pragma unroll
        for (int i = 0; i < 8; ++i) {
            int row = ldg_row + i * 16;
            int sw  = ((ldg_unit ^ (row & 7)) << 2);
            cp_async16(smem_u32(dA + row * BK + sw),
                       A + (size_t)(block_m + row) * K + k0 + ldg_unit * 4);
        }
        #pragma unroll
        for (int i = 0; i < 8; ++i) {
            int row = ldg_row + i * 16;
            int sw  = ((ldg_unit ^ (row & 7)) << 2);
            cp_async16(smem_u32(dB + row * BK + sw),
                       B + (size_t)(block_n + row) * K + k0 + ldg_unit * 4);
        }
    };

    // Accumulators: warp tile 64x64 -> 4 x 8 fragments of m16n8
    float acc[4][8][4];
    #pragma unroll
    for (int i = 0; i < 4; ++i)
        #pragma unroll
        for (int j = 0; j < 8; ++j)
            #pragma unroll
            for (int r = 0; r < 4; ++r) acc[i][j][r] = 0.0f;

    uint32_t fa[2][4][4];
    uint32_t fb[2][8][2];

    const int lr = lane >> 2;   // groupID           (0..7)
    const int lc = lane & 3;    // threadID in group (0..3)

    // Per-lane row offsets for ldmatrix (invariant across the mainloop).
    // A: lanes 0-15 address rows (lane&15) of the 16-row fragment using 16B
    //    unit u0; lanes 16-31 address the same rows with unit u0+1.
    const int a_lane_row = lane & 15;
    const int a_lane_cu  = lane >> 4;          // 0 or 1
    // B: two 8-row fragments per ldmatrix.x4; unit u0/u0+1 alternate per
    //    8-lane group.
    const int b_lane_row = (lane & 7) + ((lane & 16) >> 1);   // 0..15
    const int b_lane_cu  = (lane >> 3) & 1;                   // 0 or 1

    auto load_frags = [&](int stage, int kk, int buf) {
        const float* pA = smA + stage * A_STAGE_FLOATS;
        const float* pB = smB + stage * A_STAGE_FLOATS;
        const int u0 = kk * 2;
        #pragma unroll
        for (int i = 0; i < 4; ++i) {
            int row = warp_m + i * 16 + a_lane_row;
            int sw  = ((u0 + a_lane_cu) ^ (row & 7)) << 2;
            ldsm_x4(fa[buf][i][0], fa[buf][i][1], fa[buf][i][2], fa[buf][i][3],
                    smem_u32(pA + row * BK + sw));
        }
        #pragma unroll
        for (int j2 = 0; j2 < 4; ++j2) {
            int row = warp_n + j2 * 16 + b_lane_row;
            int sw  = ((u0 + b_lane_cu) ^ (row & 7)) << 2;
            ldsm_x4(fb[buf][2 * j2][0], fb[buf][2 * j2][1],
                    fb[buf][2 * j2 + 1][0], fb[buf][2 * j2 + 1][1],
                    smem_u32(pB + row * BK + sw));
        }
    };

    // ---- Prologue: fill STAGES-1 pipeline stages ----
    #pragma unroll
    for (int s = 0; s < STAGES - 1; ++s) {
        if (s < k_tiles) load_stage(s, s);
        cp_commit();
    }
    cp_wait<STAGES - 2>();
    __syncthreads();

    load_frags(0, 0, 0);

    int read_stage  = 0;
    int write_stage = STAGES - 1;

    for (int t = 0; t < k_tiles; ++t) {
        #pragma unroll
        for (int kk = 0; kk < BK / 8; ++kk) {
            if (kk == BK / 8 - 1) {
                int tn = t + STAGES - 1;
                if (tn < k_tiles) load_stage(write_stage, tn);
                cp_commit();
                cp_wait<STAGES - 2>();
                __syncthreads();
                load_frags(read_stage == STAGES - 1 ? 0 : read_stage + 1, 0, 0);
            } else {
                load_frags(read_stage, kk + 1, (kk + 1) & 1);
            }
            const int cb = kk & 1;
            #pragma unroll
            for (int i = 0; i < 4; ++i)
                #pragma unroll
                for (int j = 0; j < 8; ++j)
                    mma_tf32(acc[i][j][0], acc[i][j][1], acc[i][j][2], acc[i][j][3],
                             fa[cb][i][0], fa[cb][i][1], fa[cb][i][2], fa[cb][i][3],
                             fb[cb][j][0], fb[cb][j][1]);
        }
        read_stage  = (read_stage  == STAGES - 1) ? 0 : read_stage + 1;
        write_stage = (write_stage == STAGES - 1) ? 0 : write_stage + 1;
    }

    // ---- Epilogue: direct vectorized stores to global memory ----
    #pragma unroll
    for (int i = 0; i < 4; ++i) {
        int row = block_m + warp_m + i * 16 + lr;
        #pragma unroll
        for (int j = 0; j < 8; ++j) {
            int col = block_n + warp_n + j * 8 + lc * 2;
            *reinterpret_cast<float2*>(C + (size_t)row * N + col) =
                make_float2(acc[i][j][0], acc[i][j][1]);
            *reinterpret_cast<float2*>(C + (size_t)(row + 8) * N + col) =
                make_float2(acc[i][j][2], acc[i][j][3]);
        }
    }
#endif
}

template <int STAGES>
static void launch_gemm(const float* A, const float* B, float* C,
                        int M, int N, int K, int blocks, cudaStream_t stream) {
    static bool configured = false;
    const size_t smem = (size_t)STAGES * STAGE_FLOATS * sizeof(float);
    if (!configured) {
        cudaFuncSetAttribute(gemm_nt_tf32_ldsm_kernel<STAGES>,
                             cudaFuncAttributeMaxDynamicSharedMemorySize,
                             (int)smem);
        configured = true;
    }
    gemm_nt_tf32_ldsm_kernel<STAGES><<<blocks, THREADS, smem, stream>>>(A, B, C, M, N, K);
}

torch::Tensor gemm_nt_tf32_ldsm(torch::Tensor A, torch::Tensor B) {
    TORCH_CHECK(A.is_cuda() && B.is_cuda(), "A and B must be CUDA tensors");
    TORCH_CHECK(A.scalar_type() == at::kFloat && B.scalar_type() == at::kFloat,
                "A and B must be float32");
    TORCH_CHECK(A.dim() == 2 && B.dim() == 2, "A and B must be 2D");
    TORCH_CHECK(A.size(1) == B.size(1), "inner (K) dimensions must match");

    const int64_t M = A.size(0);
    const int64_t K = A.size(1);
    const int64_t N = B.size(0);

    auto prop = at::cuda::getCurrentDeviceProperties();
    const bool fast = (prop->major >= 8) && A.is_contiguous() && B.is_contiguous()
                      && (M % BM == 0) && (N % BN == 0) && (K % BK == 0)
                      && M > 0 && N > 0 && K > 0;

    if (!fast) {
        // Generic fallback (edge shapes, non-contiguous, or pre-Ampere GPU).
        return at::matmul(A, B.t());
    }

    auto C = at::empty({M, N}, A.options());
    const int blocks = (int)((M / BM) * (N / BN));
    auto stream = at::cuda::getCurrentCUDAStream();

    const int stage_bytes = STAGE_FLOATS * (int)sizeof(float);
    const int optin = (int)prop->sharedMemPerBlockOptin;

    if (optin >= 5 * stage_bytes) {
        launch_gemm<5>(A.data_ptr<float>(), B.data_ptr<float>(), C.data_ptr<float>(),
                       (int)M, (int)N, (int)K, blocks, stream);
    } else if (optin >= 4 * stage_bytes) {
        launch_gemm<4>(A.data_ptr<float>(), B.data_ptr<float>(), C.data_ptr<float>(),
                       (int)M, (int)N, (int)K, blocks, stream);
    } else if (optin >= 3 * stage_bytes) {
        launch_gemm<3>(A.data_ptr<float>(), B.data_ptr<float>(), C.data_ptr<float>(),
                       (int)M, (int)N, (int)K, blocks, stream);
    } else {
        launch_gemm<2>(A.data_ptr<float>(), B.data_ptr<float>(), C.data_ptr<float>(),
                       (int)M, (int)N, (int)K, blocks, stream);
    }
    return C;
}
"""

cpp_source = "torch::Tensor gemm_nt_tf32_ldsm(torch::Tensor A, torch::Tensor B);"

# Compile the inline CUDA extension
_gemm_ext = load_inline(
    name="gemm_nt_tf32_ldsm_ext",
    cpp_sources=cpp_source,
    cuda_sources=cuda_source,
    functions=["gemm_nt_tf32_ldsm"],
    extra_cuda_cflags=["-O3"],
    verbose=False,
)


class ModelNew(nn.Module):
    """
    Optimized model that performs C = A @ B^T with a custom TF32
    tensor-core CUDA kernel (ldmatrix fragment loads, cp.async
    multi-stage pipeline, FP32 storage and FP32 accumulation).
    """
    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):
            return _gemm_ext.gemm_nt_tf32_ldsm(A, B)
        return torch.matmul(A, B.T)
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 1.24 ms; the PyTorch reference (with TF32 enabled) takes 1.15 ms; speedup 0.927x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| void gemm_nt_tf32_ldsm_kernel<5>(float const*, float const*, float*, int, int, int) | 1213.1 | 1 |

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().
