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):
    """
    Performs a transposed 2D convolution with square input and square kernel.

    Args:
        in_channels (int): Number of channels in the input tensor.
        out_channels (int): Number of channels produced by the convolution.
        kernel_size (int): Size of the square convolution kernel.
        stride (int, optional): Stride of the convolution. Defaults to 1.
        padding (int, optional): Padding applied to the input. Defaults to 0.
        output_padding (int, optional): Additional size added to one side of the output shape. Defaults to 0.
        groups (int, optional): Number of blocked connections from input channels to output channels. Defaults to 1.
        bias (bool, optional): If `True`, adds a learnable bias to the output. Defaults to `False`.
    """
    def __init__(self, in_channels: int, out_channels: int, kernel_size: int, stride: int = 1, padding: int = 0, output_padding: int = 0, groups: int = 1, bias: bool = False):
        super(Model, self).__init__()
        self.conv_transpose2d = nn.ConvTranspose2d(in_channels, out_channels, kernel_size, stride=stride, padding=padding, output_padding=output_padding, groups=groups, bias=bias)

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        """
        Performs the transposed 2D convolution.

        Args:
            x (torch.Tensor): Input tensor of shape (batch_size, in_channels, height, width).

        Returns:
            torch.Tensor: Output tensor of shape (batch_size, out_channels, height_out, width_out).
        """
        return self.conv_transpose2d(x)

# Test code
batch_size = 8
in_channels = 64  # double channels for heavier compute
out_channels = 64
kernel_size = 3
# larger square input
height = 1024
width = 1024

def get_inputs():
    x = torch.rand(batch_size, in_channels, height, width)
    return [x]

def get_init_inputs():
    return [in_channels, out_channels, kernel_size]  # Provide in_channels, out_channels, kernel_size for initialization
    ```
    
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
import torch.nn.functional as F
from torch.utils.cpp_extension import load_inline

cuda_src = r"""
#include <torch/extension.h>
#include <cuda_runtime.h>
#include <mma.h>

using namespace nvcuda;

#define TH 8
#define TW 8
#define TILE_M 64
#define TILE_N 64
#define KSTEP 8

__global__ void __launch_bounds__(256) tconv2d_implicit_gemm_tf32(
    const float* __restrict__ in,
    const float* __restrict__ wre,
    float* __restrict__ out,
    int Bn, int Cin, int Cout, int H, int W, int Ho, int Wo)
{
#if defined(__CUDA_ARCH__) && (__CUDA_ARCH__ >= 800)
    extern __shared__ float smem[];
    const int rows = TH + 2;
    const int cols = TW + 2;
    const int IN_SIZE = Cin * rows * cols;
    const int CS_SIZE = TILE_M * TILE_N;
    float* base = smem;
    float* As = base + (IN_SIZE > CS_SIZE ? IN_SIZE : CS_SIZE);
    float* Bs = As + TILE_M * 16;

    const int ntiles = Cout / TILE_N;
    const int b = blockIdx.z / ntiles;
    const int n0 = (blockIdx.z % ntiles) * TILE_N;
    const int y0 = blockIdx.y * TH;
    const int x0 = blockIdx.x * TW;

    const int tid = threadIdx.x;

    for (int idx = tid; idx < Cin * rows * cols; idx += blockDim.x) {
        int cin = idx / (rows * cols);
        int r = idx % (rows * cols);
        int iy = r / cols;
        int ix = r % cols;
        int gy = y0 + iy - 2;
        int gx = x0 + ix - 2;
        float v = 0.f;
        if (gy >= 0 && gy < H && gx >= 0 && gx < W) {
            v = in[((long long)(b * Cin + cin) * H + gy) * W + gx];
        }
        base[(cin * rows + iy) * cols + ix] = v;
    }
    __syncthreads();

    wmma::fragment<wmma::accumulator, 16, 8, 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);

    const int warp = tid >> 5;
    const int warp_m = warp >> 2;
    const int warp_n = warp & 3;

    const int KSTEPS = (9 * Cin) / KSTEP;

    for (int ks = 0; ks < KSTEPS; ks++) {
        int k0 = ks * KSTEP;
        int tap = k0 / Cin;
        int cin0 = k0 - tap * Cin;
        int ky = tap / 3;
        int kx = tap % 3;
        int dy = 2 - ky;
        int dx = 2 - kx;

        #pragma unroll
        for (int idx = tid; idx < TILE_M * KSTEP; idx += blockDim.x) {
            int m = idx >> 3;
            int kk = idx & 7;
            int ty = m >> 3;
            int tx = m & 7;
            As[m * 16 + kk] = base[((cin0 + kk) * rows + (ty + dy)) * cols + (tx + dx)];
        }
        const float* wptr = wre + (long long)k0 * Cout + n0;
        #pragma unroll
        for (int idx = tid; idx < KSTEP * TILE_N; idx += blockDim.x) {
            int kk = idx >> 6;
            int n = idx & 63;
            Bs[kk * 72 + n] = wptr[kk * Cout + n];
        }
        __syncthreads();

        #pragma unroll
        for (int mi = 0; mi < 2; mi++) {
            wmma::fragment<wmma::matrix_a, 16, 8, 8, wmma::precision::tf32, wmma::col_major> afrag;
            wmma::load_matrix_sync(afrag, &Bs[warp_m * 32 + mi * 16], 72);
            #pragma unroll
            for (int ni = 0; ni < 2; ni++) {
                wmma::fragment<wmma::matrix_b, 16, 8, 8, wmma::precision::tf32, wmma::col_major> bfrag;
                wmma::load_matrix_sync(bfrag, &As[(warp_n * 16 + ni * 8) * 16], 16);
                wmma::mma_sync(acc[mi][ni], afrag, bfrag, acc[mi][ni]);
            }
        }
        __syncthreads();
    }

    bool interior = (y0 + TH <= Ho) && (x0 + TW <= Wo);
    if (interior) {
        #pragma unroll
        for (int mi = 0; mi < 2; mi++) {
            int cn = n0 + warp_m * 32 + mi * 16;
            #pragma unroll
            for (int ni = 0; ni < 2; ni++) {
                int pm = warp_n * 16 + ni * 8;
                int ty = pm >> 3;
                float* optr = out + ((long long)(b * Cout + cn) * Ho + (y0 + ty)) * Wo + x0;
                wmma::store_matrix_sync(optr, acc[mi][ni], (long long)Ho * Wo, wmma::mem_row_major);
            }
        }
    } else {
        __syncthreads();
        float* Cs = base;
        #pragma unroll
        for (int mi = 0; mi < 2; mi++) {
            #pragma unroll
            for (int ni = 0; ni < 2; ni++) {
                wmma::store_matrix_sync(&Cs[(warp_m * 32 + mi * 16) * 64 + warp_n * 16 + ni * 8],
                                        acc[mi][ni], 64, wmma::mem_row_major);
            }
        }
        __syncthreads();
        for (int idx = tid; idx < TILE_M * TILE_N; idx += blockDim.x) {
            int n = idx >> 6;
            int m = idx & 63;
            int ty = m >> 3;
            int tx = m & 7;
            int y = y0 + ty;
            int x = x0 + tx;
            if (y < Ho && x < Wo) {
                out[((long long)(b * Cout + n0 + n) * Ho + y) * Wo + x] = Cs[n * 64 + m];
            }
        }
    }
#endif
}

void tconv2d_forward(torch::Tensor in, torch::Tensor wre, torch::Tensor out,
                     int64_t Bn, int64_t Cin, int64_t Cout,
                     int64_t H, int64_t W, int64_t Ho, int64_t Wo)
{
    dim3 grid((Wo + TW - 1) / TW, (Ho + TH - 1) / TH, Bn * (Cout / TILE_N));
    dim3 block(256);
    int in_size = (int)(Cin * (TH + 2) * (TW + 2));
    int cs_size = TILE_M * TILE_N;
    int smem = ((in_size > cs_size ? in_size : cs_size) + TILE_M * 16 + KSTEP * 72) * sizeof(float);
    if (smem > 48 * 1024) {
        cudaFuncSetAttribute(tconv2d_implicit_gemm_tf32, cudaFuncAttributeMaxDynamicSharedMemorySize, smem);
    }
    tconv2d_implicit_gemm_tf32<<<grid, block, smem>>>(
        in.data_ptr<float>(), wre.data_ptr<float>(), out.data_ptr<float>(),
        (int)Bn, (int)Cin, (int)Cout, (int)H, (int)W, (int)Ho, (int)Wo);
}
"""

cpp_src = "void tconv2d_forward(torch::Tensor in, torch::Tensor wre, torch::Tensor out, int64_t Bn, int64_t Cin, int64_t Cout, int64_t H, int64_t W, int64_t Ho, int64_t Wo);"

_mod = load_inline(
    name="tconv2d_implicit_gemm",
    cpp_sources=cpp_src,
    cuda_sources=cuda_src,
    functions=["tconv2d_forward"],
    verbose=False,
    extra_cuda_cflags=["-O3"],
)


class ModelNew(nn.Module):
    def __init__(self, in_channels: int, out_channels: int, kernel_size: int, stride: int = 1, padding: int = 0, output_padding: int = 0, groups: int = 1, bias: bool = False):
        super(ModelNew, self).__init__()
        self.conv_transpose2d = nn.ConvTranspose2d(in_channels, out_channels, kernel_size, stride=stride, padding=padding, output_padding=output_padding, groups=groups, bias=bias)
        self.kernel_size = kernel_size
        self.stride = stride
        self.padding = padding
        self.output_padding = output_padding
        self.groups = groups
        self._use_custom = (
            stride == 1 and padding == 0 and output_padding == 0 and groups == 1
            and kernel_size == 3 and (in_channels % 8 == 0) and (out_channels % 64 == 0)
            and not bias
        )
        self._wre = None
        self._wflip = None
        self._path = None  # 0: custom kernel, 1: conv2d-flipped, 2: native
        if not self._use_custom:
            self._path = 2

    def _custom_run(self, x):
        w = self.conv_transpose2d.weight
        if self._wre is None or self._wre.device != x.device:
            k = self.kernel_size
            ci = w.shape[0]
            co = w.shape[1]
            self._wre = w.permute(2, 3, 0, 1).reshape(k * k * ci, co).contiguous()
        x = x.contiguous()
        Bn, Cin, H, W = x.shape
        Cout = w.shape[1]
        Ho = (H - 1) * self.stride - 2 * self.padding + self.kernel_size + self.output_padding
        Wo = (W - 1) * self.stride - 2 * self.padding + self.kernel_size + self.output_padding
        out = torch.empty((Bn, Cout, Ho, Wo), device=x.device, dtype=x.dtype)
        _mod.tconv2d_forward(x, self._wre, out, Bn, Cin, Cout, H, W, Ho, Wo)
        return out

    def _conv2d_run(self, x):
        w = self.conv_transpose2d.weight
        if self._wflip is None or self._wflip.device != x.device:
            self._wflip = w.flip([2, 3]).permute(1, 0, 2, 3).contiguous()
        return F.conv2d(x, self._wflip, None, 1, self.kernel_size - 1)

    def _autotune(self, x):
        ref = self.conv_transpose2d(x)
        candidates = [(2, lambda: self.conv_transpose2d(x))]
        try:
            out_c = self._custom_run(x)
            if torch.allclose(out_c, ref, atol=1e-2, rtol=1e-2):
                candidates.append((0, lambda: self._custom_run(x)))
        except Exception:
            pass
        try:
            out_f = self._conv2d_run(x)
            if torch.allclose(out_f, ref, atol=1e-2, rtol=1e-2):
                candidates.append((1, lambda: self._conv2d_run(x)))
        except Exception:
            pass
        best_path, best_t = 2, float("inf")
        for path, fn in candidates:
            try:
                fn()
                torch.cuda.synchronize()
                start = torch.cuda.Event(enable_timing=True)
                end = torch.cuda.Event(enable_timing=True)
                start.record()
                for _ in range(3):
                    fn()
                end.record()
                torch.cuda.synchronize()
                t = start.elapsed_time(end)
                if t < best_t:
                    best_t, best_path = t, path
            except Exception:
                continue
        self._path = best_path

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        if self._path is None:
            self._autotune(x)
        if self._path == 0:
            return self._custom_run(x)
        elif self._path == 1:
            return self._conv2d_run(x)
        return self.conv_transpose2d(x)
```
Result of evaluating it:
The solution failed to compile. Error output:
```
...[truncated]
tconv2d_implicit_gemm': [1/3] c++ -MMD -MF main.o.d -DTORCH_EXTENSION_NAME=tconv2d_implicit_gemm -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 -c /srv/jobs/1791681942239349082-526805-0/build/tconv2d_implicit_gemm/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=tconv2d_implicit_gemm -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/1791681942239349082-526805-0/build/tconv2d_implicit_gemm/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=tconv2d_implicit_gemm -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/1791681942239349082-526805-0/build/tconv2d_implicit_gemm/cuda.cu -o cuda.cuda.o 
/srv/jobs/1791681942239349082-526805-0/build/tconv2d_implicit_gemm/cuda.cu(56): error: incomplete type "nvcuda::wmma::fragment<nvcuda::wmma::accumulator, 16, 8, 8, float, void> [2][2]" is not allowed
      wmma::fragment<wmma::accumulator, 16, 8, 8, float> acc[2][2];
                                                         ^

/srv/jobs/1791681942239349082-526805-0/build/tconv2d_implicit_gemm/cuda.cu(61): error: no instance of function template "nvcuda::wmma::fill_fragment" matches the argument list
            argument types are: (<error-type>, float)
              wmma::fill_fragment(acc[i][j], 0.0f);
              ^
/opt/kbv/v/lib/python3.12/site-packages/nvidia/cu13/include/crt/mma.h(260): note #3327-D: candidate function template "nvcuda::wmma::fill_fragment" failed deduction
      static __attribute__((device)) __inline__ void fill_fragment(__frag_base<FragEleType, size, packed_size>& f,
                                                     ^

/srv/jobs/1791681942239349082-526805-0/build/tconv2d_implicit_gemm/cuda.cu(97): error: incomplete type "nvcuda::wmma::fragment<nvcuda::wmma::matrix_a, 16, 8, 8, nvcuda::wmma::precision::tf32, nvcuda::wmma::col_major>" is not allowed
              wmma::fragment<wmma::matrix_a, 16, 8, 8, wmma::precision::tf32, wmma::col_major> afrag;
                                                                                               ^

/srv/jobs/1791681942239349082-526805-0/build/tconv2d_implicit_gemm/cuda.cu(101): error: incomplete type "nvcuda::wmma::fragment<nvcuda::wmma::matrix_b, 16, 8, 8, nvcuda::wmma::precision::tf32, nvcuda::wmma::col_major>" is not allowed
                  wmma::fragment<wmma::matrix_b, 16, 8, 8, wmma::precision::tf32, wmma::col_major> bfrag;
                                                                                                   ^

4 errors detected in the compilation of "/srv/jobs/1791681942239349082-526805-0/build/tconv2d_implicit_gemm/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.


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().
