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 2 of 6 for this task. Your previous solution (turn 1) was:
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline

_cpp_source = r"""
#include <torch/extension.h>
torch::Tensor splitk_matmul(torch::Tensor a, torch::Tensor b);
"""

_cuda_source = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <c10/cuda/CUDAGuard.h>
#include <c10/cuda/CUDAException.h>
#include <cuda.h>
#include <cuda_runtime.h>
#include <mma.h>
#include <algorithm>
#include <climits>

namespace {

constexpr int TILE = 64;
constexpr int KTILE = 32;

__global__ void splitk_tf32_kernel(
    const float* __restrict__ a,
    const float* __restrict__ b,
    float* __restrict__ partial,
    int m, int n, int k, int chunk,
    int tiles_m, int tiles_n)
{
#if defined(__CUDA_ARCH__) && __CUDA_ARCH__ >= 800
    namespace wmma = nvcuda::wmma;

    __shared__ __align__(32) float ah[TILE * KTILE];
    __shared__ __align__(32) float al[TILE * KTILE];
    __shared__ __align__(32) float bh[KTILE * TILE];
    __shared__ __align__(32) float bl[KTILE * TILE];

    const int tid = threadIdx.x;
    const int warp = tid >> 5;
    const int warp_row = (warp >> 1) * 32;
    const int warp_col = (warp & 1) * 32;
    const int row_base = blockIdx.y * TILE;
    const int col_base = blockIdx.x * TILE;
    const int begin = blockIdx.z * chunk;
    const int end = min(begin + chunk, k);

    wmma::fragment<wmma::accumulator, 16, 16, 8, float> acc[2][2];

    #pragma unroll
    for (int i = 0; i < 2; ++i) {
        #pragma unroll
        for (int j = 0; j < 2; ++j) {
            wmma::fill_fragment(acc[i][j], 0.0f);
        }
    }

    for (int kb = begin; kb < end; kb += KTILE) {
        #pragma unroll
        for (int p = tid; p < TILE * KTILE; p += 128) {
            const int r = p / KTILE;
            const int c = p % KTILE;
            float x = 0.0f;
            if (row_base + r < m && kb + c < end) {
                x = a[static_cast<int64_t>(row_base + r) * k + kb + c];
            }
            const float h = wmma::__float_to_tf32(x);
            ah[p] = h;
            al[p] = wmma::__float_to_tf32(x - h);
        }

        #pragma unroll
        for (int p = tid; p < KTILE * TILE; p += 128) {
            const int r = p / TILE;
            const int c = p % TILE;
            float x = 0.0f;
            if (kb + r < end && col_base + c < n) {
                x = b[static_cast<int64_t>(kb + r) * n + col_base + c];
            }
            const float h = wmma::__float_to_tf32(x);
            bh[p] = h;
            bl[p] = wmma::__float_to_tf32(x - h);
        }
        __syncthreads();

        #pragma unroll
        for (int kk = 0; kk < KTILE; kk += 8) {
            wmma::fragment<
                wmma::matrix_a, 16, 16, 8,
                wmma::precision::tf32, wmma::row_major> a_hi[2], a_lo[2];
            wmma::fragment<
                wmma::matrix_b, 16, 16, 8,
                wmma::precision::tf32, wmma::row_major> b_hi[2], b_lo[2];

            #pragma unroll
            for (int i = 0; i < 2; ++i) {
                const int offset = (warp_row + i * 16) * KTILE + kk;
                wmma::load_matrix_sync(a_hi[i], ah + offset, KTILE);
                wmma::load_matrix_sync(a_lo[i], al + offset, KTILE);
            }

            #pragma unroll
            for (int j = 0; j < 2; ++j) {
                const int offset = kk * TILE + warp_col + j * 16;
                wmma::load_matrix_sync(b_hi[j], bh + offset, TILE);
                wmma::load_matrix_sync(b_lo[j], bl + offset, TILE);
            }

            // Retain the two first-order TF32 rounding corrections.
            #pragma unroll
            for (int i = 0; i < 2; ++i) {
                #pragma unroll
                for (int j = 0; j < 2; ++j) {
                    wmma::mma_sync(acc[i][j], a_lo[i], b_hi[j], acc[i][j]);
                    wmma::mma_sync(acc[i][j], a_hi[i], b_lo[j], acc[i][j]);
                    wmma::mma_sync(acc[i][j], a_hi[i], b_hi[j], acc[i][j]);
                }
            }
        }
        __syncthreads();
    }

    const int64_t tile_index =
        (static_cast<int64_t>(blockIdx.z) * tiles_m + blockIdx.y)
        * tiles_n + blockIdx.x;
    float* dst = partial + tile_index * TILE * TILE;

    #pragma unroll
    for (int i = 0; i < 2; ++i) {
        #pragma unroll
        for (int j = 0; j < 2; ++j) {
            wmma::store_matrix_sync(
                dst + (warp_row + i * 16) * TILE + warp_col + j * 16,
                acc[i][j], TILE, wmma::mem_row_major);
        }
    }
#endif
}

__global__ void reduce_splitk_kernel(
    const float* __restrict__ partial,
    float* __restrict__ out,
    int m, int n, int splits, int tiles_m, int tiles_n)
{
    const int64_t index =
        static_cast<int64_t>(blockIdx.x) * blockDim.x + threadIdx.x;
    if (index >= static_cast<int64_t>(m) * n) {
        return;
    }

    const int row = index / n;
    const int col = index % n;
    const int64_t tile = static_cast<int64_t>(row / TILE) * tiles_n + col / TILE;
    const int64_t offset =
        tile * TILE * TILE + (row % TILE) * TILE + col % TILE;
    const int64_t stride =
        static_cast<int64_t>(tiles_m) * tiles_n * TILE * TILE;

    float s0 = 0.0f;
    float s1 = 0.0f;
    float s2 = 0.0f;
    float s3 = 0.0f;
    int s = 0;

    for (; s + 3 < splits; s += 4) {
        s0 += partial[offset + static_cast<int64_t>(s) * stride];
        s1 += partial[offset + static_cast<int64_t>(s + 1) * stride];
        s2 += partial[offset + static_cast<int64_t>(s + 2) * stride];
        s3 += partial[offset + static_cast<int64_t>(s + 3) * stride];
    }
    float sum = (s0 + s1) + (s2 + s3);
    for (; s < splits; ++s) {
        sum += partial[offset + static_cast<int64_t>(s) * stride];
    }
    out[index] = sum;
}

}  // namespace

torch::Tensor splitk_matmul(torch::Tensor a, torch::Tensor b) {
    TORCH_CHECK(a.is_cuda() && b.is_cuda(), "Expected CUDA tensors");
    TORCH_CHECK(a.device() == b.device(), "Inputs must share a device");
    TORCH_CHECK(a.dim() == 2 && b.dim() == 2, "Expected matrices");
    TORCH_CHECK(a.size(1) == b.size(0), "Incompatible matrix dimensions");
    TORCH_CHECK(
        a.scalar_type() == torch::kFloat32 &&
        b.scalar_type() == torch::kFloat32,
        "Expected float32 tensors");
    TORCH_CHECK(a.is_contiguous() && b.is_contiguous(), "Expected contiguous inputs");

    c10::cuda::CUDAGuard device_guard(a.device());

    if (at::cuda::getCurrentDeviceProperties()->major < 8) {
        return at::matmul(a, b);
    }

    TORCH_CHECK(
        a.size(0) <= INT_MAX && a.size(1) <= INT_MAX && b.size(1) <= INT_MAX,
        "Matrix dimensions exceed kernel limits");

    const int m = static_cast<int>(a.size(0));
    const int k = static_cast<int>(a.size(1));
    const int n = static_cast<int>(b.size(1));

    if (m == 0 || n == 0 || k == 0) {
        return torch::zeros({m, n}, a.options());
    }

    const int tiles_m = (m + TILE - 1) / TILE;
    const int tiles_n = (n + TILE - 1) / TILE;
    const int requested_splits = static_cast<int>(
        std::min<int64_t>(128, (static_cast<int64_t>(k) + 4095) / 4096));
    const int chunk = static_cast<int>(
        ((static_cast<int64_t>(k) + requested_splits - 1) /
         requested_splits + KTILE - 1) / KTILE * KTILE);
    const int splits = static_cast<int>(
        (static_cast<int64_t>(k) + chunk - 1) / chunk);

    auto partial = torch::empty(
        {splits, tiles_m, tiles_n, TILE, TILE}, a.options());
    auto out = torch::empty({m, n}, a.options());

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

    splitk_tf32_kernel<<<dim3(tiles_n, tiles_m, splits), 128, 0, stream>>>(
        a.data_ptr<float>(), b.data_ptr<float>(), partial.data_ptr<float>(),
        m, n, k, chunk, tiles_m, tiles_n);
    C10_CUDA_KERNEL_LAUNCH_CHECK();

    const int64_t elements = static_cast<int64_t>(m) * n;
    reduce_splitk_kernel<<<(elements + 255) / 256, 256, 0, stream>>>(
        partial.data_ptr<float>(), out.data_ptr<float>(),
        m, n, splits, tiles_m, tiles_n);
    C10_CUDA_KERNEL_LAUNCH_CHECK();

    return out;
}
"""

_extension = load_inline(
    name="large_k_splitk_tf32_corrected",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["splitk_matmul"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
    verbose=False,
)


class ModelNew(nn.Module):
    def __init__(self):
        super().__init__()

    def forward(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
        if (
            not A.is_cuda
            or not B.is_cuda
            or A.dtype != torch.float32
            or B.dtype != torch.float32
            or A.ndim != 2
            or B.ndim != 2
            or (torch.is_grad_enabled() and (A.requires_grad or B.requires_grad))
        ):
            return torch.matmul(A, B)
        return _extension.splitk_matmul(A.contiguous(), B.contiguous())
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 8.94 ms; the PyTorch reference (with TF32 enabled) takes 0.939 ms; speedup 0.105x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| (anonymous namespace)::splitk_tf32_kernel(float const*, float const*, float*, int, int, in | 8948.9 | 1 |
| (anonymous namespace)::reduce_splitk_kernel(float const*, float*, int, int, int, int, int) | 33.0 | 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.
