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 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 matmul_nt_cuda(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>

namespace wmma = nvcuda::wmma;

__global__ void matmul_nt_kernel(
    const float* __restrict__ a,
    const float* __restrict__ b,
    float* __restrict__ c,
    int M, int N, int K)
{
#if defined(__CUDA_ARCH__) && __CUDA_ARCH__ >= 800
    constexpr int TILE = 64;
    constexpr int BK = 32;

    __shared__ __align__(32) float storage[4 * TILE * BK];
    float* ah = storage;
    float* al = ah + TILE * BK;
    float* bh = al + TILE * BK;
    float* bl = bh + TILE * BK;

    const int tid = threadIdx.x;
    const int warp = tid / 32;
    const int tile_m = blockIdx.y * TILE;
    const int tile_n = blockIdx.x * TILE;
    const int warp_m = (warp / 2) * 32;
    const int warp_n = (warp % 2) * 32;

    using AFragment = wmma::fragment<
        wmma::matrix_a, 16, 16, 8,
        wmma::precision::tf32, wmma::row_major>;
    using BFragment = wmma::fragment<
        wmma::matrix_b, 16, 16, 8,
        wmma::precision::tf32, wmma::col_major>;
    using Accumulator = wmma::fragment<
        wmma::accumulator, 16, 16, 8, float>;

    Accumulator main_acc[2][2];
    Accumulator correction_acc[2][2];

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

    for (int base_k = 0; base_k < K; base_k += BK) {
        for (int index = tid; index < TILE * BK; index += 128) {
            const int row = index / BK;
            const int k = base_k + index % BK;

            float av = 0.0f;
            float bv = 0.0f;
            if (tile_m + row < M && k < K)
                av = a[static_cast<int64_t>(tile_m + row) * K + k];
            if (tile_n + row < N && k < K)
                bv = b[static_cast<int64_t>(tile_n + row) * K + k];

            const float av_hi = __float_to_tf32(av);
            const float bv_hi = __float_to_tf32(bv);
            ah[index] = av_hi;
            al[index] = __float_to_tf32(av - av_hi);
            bh[index] = bv_hi;
            bl[index] = __float_to_tf32(bv - bv_hi);
        }
        __syncthreads();

        #pragma unroll
        for (int kk = 0; kk < BK; kk += 8) {
            AFragment a_hi[2], a_lo[2];
            BFragment b_hi[2], b_lo[2];

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

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

            #pragma unroll
            for (int i = 0; i < 2; ++i) {
                #pragma unroll
                for (int j = 0; j < 2; ++j) {
                    wmma::mma_sync(
                        main_acc[i][j], a_hi[i], b_hi[j],
                        main_acc[i][j]);
                    wmma::mma_sync(
                        correction_acc[i][j], a_hi[i], b_lo[j],
                        correction_acc[i][j]);
                    wmma::mma_sync(
                        correction_acc[i][j], a_lo[i], b_hi[j],
                        correction_acc[i][j]);
                }
            }
        }
        __syncthreads();
    }

    // Reuse the input staging memory for the completed output tile.
    #pragma unroll
    for (int i = 0; i < 2; ++i) {
        #pragma unroll
        for (int j = 0; j < 2; ++j) {
            #pragma unroll
            for (int e = 0; e < Accumulator::num_elements; ++e)
                main_acc[i][j].x[e] += correction_acc[i][j].x[e];

            wmma::store_matrix_sync(
                storage + (warp_m + i * 16) * TILE + warp_n + j * 16,
                main_acc[i][j], TILE, wmma::mem_row_major);
        }
    }
    __syncthreads();

    for (int index = tid; index < TILE * TILE; index += 128) {
        const int row = tile_m + index / TILE;
        const int col = tile_n + index % TILE;
        if (row < M && col < N)
            c[static_cast<int64_t>(row) * N + col] = storage[index];
    }
#endif
}

torch::Tensor matmul_nt_cuda(torch::Tensor a, torch::Tensor b) {
    TORCH_CHECK(a.is_cuda() && b.is_cuda(), "Expected CUDA tensors");
    TORCH_CHECK(a.device() == b.device(), "Tensor devices must match");
    TORCH_CHECK(a.scalar_type() == at::kFloat &&
                b.scalar_type() == at::kFloat, "Expected float32 tensors");
    TORCH_CHECK(a.dim() == 2 && b.dim() == 2, "Expected matrices");
    TORCH_CHECK(a.size(1) == b.size(1), "Inner dimensions must match");
    TORCH_CHECK(a.is_contiguous() && b.is_contiguous(),
                "Expected contiguous tensors");
    TORCH_CHECK(a.size(0) <= INT_MAX && b.size(0) <= INT_MAX &&
                a.size(1) <= INT_MAX, "Matrix dimensions are too large");

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

    const int M = static_cast<int>(a.size(0));
    const int N = static_cast<int>(b.size(0));
    const int K = static_cast<int>(a.size(1));

    if (at::cuda::getDeviceProperties(a.get_device())->major < 8 ||
        K == 0 || M > 65535 * 64) {
        return at::matmul(a, b.transpose(0, 1));
    }

    auto out = torch::empty({M, N}, a.options());
    if (M == 0 || N == 0)
        return out;

    const dim3 grid(
        (static_cast<unsigned int>(N) + 63) / 64,
        (static_cast<unsigned int>(M) + 63) / 64);

    matmul_nt_kernel<<<
        grid, 128, 0, at::cuda::getCurrentCUDAStream(a.get_device())
    >>>(
        a.data_ptr<float>(), b.data_ptr<float>(), out.data_ptr<float>(),
        M, N, K);

    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return out;
}
"""

_matmul_extension = load_inline(
    name="custom_matmul_nt_tf32_split_v1",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["matmul_nt_cuda"],
    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 (torch.is_grad_enabled() and (A.requires_grad or B.requires_grad))
        ):
            return torch.matmul(A, B.T)

        return _matmul_extension.matmul_nt_cuda(
            A.contiguous(), B.contiguous()
        )
```
Result of evaluating it:
The solution failed to compile. Error output:
```
RuntimeError: Error building extension 'custom_matmul_nt_tf32_split_v1': [1/3] c++ -MMD -MF main.o.d -DTORCH_EXTENSION_NAME=custom_matmul_nt_tf32_split_v1 -DTORCH_API_INCLUDE_EXTENSION_H -isystem /opt/kbv/v/lib/python3.12/site-packages/torch/include -isystem /opt/kbv/v/lib/python3.12/site-packages/torch/include/torch/csrc/api/include -isystem /opt/kbv/v/lib/python3.12/site-packages/nvidia/cu13/include -isystem /usr/include/python3.12 -fPIC -std=c++20 -O3 -c /srv/jobs/1791725147764592662-3456539-0/build/custom_matmul_nt_tf32_split_v1/main.cpp -o main.o 
[2/3] /opt/kbv/v/lib/python3.12/site-packages/nvidia/cu13/bin/nvcc -MD -MF cuda.cuda.o.d -DTORCH_EXTENSION_NAME=custom_matmul_nt_tf32_split_v1 -DTORCH_API_INCLUDE_EXTENSION_H -isystem /opt/kbv/v/lib/python3.12/site-packages/torch/include -isystem /opt/kbv/v/lib/python3.12/site-packages/torch/include/torch/csrc/api/include -isystem /opt/kbv/v/lib/python3.12/site-packages/nvidia/cu13/include -isystem /usr/include/python3.12 -D__CUDA_NO_HALF_OPERATORS__ -D__CUDA_NO_HALF_CONVERSIONS__ -D__CUDA_NO_BFLOAT16_CONVERSIONS__ -D__CUDA_NO_HALF2_OPERATORS__ --expt-relaxed-constexpr -gencode=arch=compute_80,code=sm_80 --compiler-options '-fPIC' -O3 -std=c++20 -c /srv/jobs/1791725147764592662-3456539-0/build/custom_matmul_nt_tf32_split_v1/cuda.cu -o cuda.cuda.o 
FAILED: [code=1] cuda.cuda.o 
/opt/kbv/v/lib/python3.12/site-packages/nvidia/cu13/bin/nvcc -MD -MF cuda.cuda.o.d -DTORCH_EXTENSION_NAME=custom_matmul_nt_tf32_split_v1 -DTORCH_API_INCLUDE_EXTENSION_H -isystem /opt/kbv/v/lib/python3.12/site-packages/torch/include -isystem /opt/kbv/v/lib/python3.12/site-packages/torch/include/torch/csrc/api/include -isystem /opt/kbv/v/lib/python3.12/site-packages/nvidia/cu13/include -isystem /usr/include/python3.12 -D__CUDA_NO_HALF_OPERATORS__ -D__CUDA_NO_HALF_CONVERSIONS__ -D__CUDA_NO_BFLOAT16_CONVERSIONS__ -D__CUDA_NO_HALF2_OPERATORS__ --expt-relaxed-constexpr -gencode=arch=compute_80,code=sm_80 --compiler-options '-fPIC' -O3 -std=c++20 -c /srv/jobs/1791725147764592662-3456539-0/build/custom_matmul_nt_tf32_split_v1/cuda.cu -o cuda.cuda.o 
/srv/jobs/1791725147764592662-3456539-0/build/custom_matmul_nt_tf32_split_v1/cuda.cu(71): error: identifier "__float_to_tf32" is undefined
              const float av_hi = __float_to_tf32(av);
                                  ^

1 error detected in the compilation of "/srv/jobs/1791725147764592662-3456539-0/build/custom_matmul_nt_tf32_split_v1/cuda.cu".
ninja: build stopped: subcommand failed.

```

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.
