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 3D transposed convolution operation with square input and square kernel,
    and supports padding, dilation, and stride.

    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 convolution kernel (square kernel, so only one value needed).
        stride (int, optional): Stride of the convolution. Defaults to 1.
        padding (int, optional): Padding applied to the input. Defaults to 0.
        dilation (int, optional): Spacing between kernel elements. 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, dilation: int = 1, bias: bool = False):
        super(Model, self).__init__()
        self.conv_transpose3d = nn.ConvTranspose3d(in_channels, out_channels, kernel_size=(kernel_size, kernel_size, kernel_size), stride=stride, padding=padding, dilation=dilation, bias=bias)

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        """
        Performs the 3D transposed convolution.

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

        Returns:
            torch.Tensor: Output tensor of shape (batch_size, out_channels, depth_out, height_out, width_out).
        """
        return self.conv_transpose3d(x)

# Test code
batch_size = 16
in_channels = 32
out_channels = 64
kernel_size = 3
depth = 16
height = 32
width = 32
stride = 2
padding = 1
dilation = 2

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

def get_init_inputs():
    return [in_channels, out_channels, kernel_size, stride, padding, dilation]
    ```
    
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 6 of 6 for this task. Your previous solution (turn 5) was:
```python
import torch
import torch.nn as nn
import torch.nn.functional as F
from torch.utils.cpp_extension import load_inline


_cpp_source = r"""
#include <torch/extension.h>
#include <vector>

std::vector<torch::Tensor> prepare_transpose3d(
    torch::Tensor input, torch::Tensor weight);
torch::Tensor expand_transpose3d(torch::Tensor input);
"""

_cuda_source = r"""
#include <torch/extension.h>
#include <c10/cuda/CUDAGuard.h>
#include <c10/cuda/CUDAStream.h>
#include <c10/cuda/CUDAException.h>
#include <cuda_runtime.h>
#include <algorithm>
#include <cstdint>
#include <vector>

__global__ void prepare_kernel(
    const float* __restrict__ input,
    const float* __restrict__ weight,
    float* __restrict__ converted_input,
    float* __restrict__ converted_weight,
    int in_channels,
    int out_channels,
    int channel_tiles,
    int input_blocks,
    int64_t weight_count)
{
    if (blockIdx.x < input_blocks) {
        __shared__ float tile[32][33];

        const int lane = threadIdx.x & 31;
        const int row = threadIdx.x >> 5;
        const int spatial_tile = blockIdx.x % 512;
        const int channel_tile = (blockIdx.x / 512) % channel_tiles;
        const int batch = blockIdx.x / (512 * channel_tiles);
        const int spatial = spatial_tile * 32 + lane;

        #pragma unroll
        for (int j = 0; j < 32; j += 8) {
            const int channel = channel_tile * 32 + row + j;
            if (channel < in_channels) {
                tile[row + j][lane] =
                    input[(int64_t(batch) * in_channels + channel) *
                          16384 + spatial];
            }
        }

        __syncthreads();

        const int channel = channel_tile * 32 + lane;
        if (channel < in_channels) {
            #pragma unroll
            for (int j = 0; j < 32; j += 8) {
                const int destination_spatial =
                    spatial_tile * 32 + row + j;
                converted_input[
                    (int64_t(batch) * 16384 + destination_spatial) *
                    in_channels + channel] = tile[lane][row + j];
            }
        }
    } else {
        const int64_t i =
            int64_t(blockIdx.x - input_blocks) * 256 + threadIdx.x;

        if (i < weight_count) {
            const int ic = i % in_channels;
            const int k = (i / in_channels) % 27;
            const int oc = i / (int64_t(in_channels) * 27);

            converted_weight[i] =
                weight[(int64_t(ic) * out_channels + oc) * 27 + 26 - k];
        }
    }
}

__device__ __forceinline__ void streaming_store(
    float4* address, float4 value)
{
    asm volatile(
        "st.global.cs.v4.f32 [%0], {%1, %2, %3, %4};"
        :
        : "l"(address), "f"(value.x), "f"(value.y),
          "f"(value.z), "f"(value.w)
        : "memory");
}

template<int FixedChannelVectors>
__global__ void expand_kernel(
    const float4* __restrict__ input,
    float4* __restrict__ output,
    uint32_t main_vectors,
    uint32_t total_vectors,
    uint32_t runtime_channel_vectors)
{
    const uint32_t channel_vectors =
        FixedChannelVectors ? FixedChannelVectors : runtime_channel_vectors;
    const uint32_t step = gridDim.x * blockDim.x * 4;

    for (uint32_t base = blockIdx.x * blockDim.x * 4 + threadIdx.x;
         base < total_vectors;
         base += step) {
        #pragma unroll
        for (int j = 0; j < 4; ++j) {
            const uint32_t v = base + j * blockDim.x;
            if (v < total_vectors) {
                float4 value = make_float4(0.f, 0.f, 0.f, 0.f);

                if (v < main_vectors) {
                    const uint32_t spatial = v / channel_vectors;

                    // Odd width, height, and depth coordinates contain data.
                    if ((spatial & 4161u) == 4161u) {
                        const uint32_t compact_spatial =
                            ((spatial >> 1) & 31u) |
                            ((spatial >> 2) & 992u) |
                            ((spatial >> 3) & ~1023u);
                        const uint32_t channel = v % channel_vectors;
                        value = __ldg(
                            input + compact_spatial * channel_vectors + channel);
                    }
                }

                streaming_store(output + v, value);
            }
        }
    }
}

std::vector<torch::Tensor> prepare_transpose3d(
    torch::Tensor input, torch::Tensor weight)
{
    const c10::cuda::CUDAGuard guard(input.device());

    TORCH_CHECK(input.is_cuda() && weight.is_cuda());
    TORCH_CHECK(input.device() == weight.device());
    TORCH_CHECK(input.scalar_type() == at::kFloat);
    TORCH_CHECK(weight.scalar_type() == at::kFloat);
    TORCH_CHECK(input.is_contiguous() && weight.is_contiguous());
    TORCH_CHECK(
        input.dim() == 5 && input.size(2) == 16 &&
        input.size(3) == 32 && input.size(4) == 32);
    TORCH_CHECK(
        weight.dim() == 5 && weight.size(0) == input.size(1) &&
        weight.size(2) == 3 && weight.size(3) == 3 && weight.size(4) == 3);

    const int64_t batch = input.size(0);
    const int ic = input.size(1);
    const int oc = weight.size(1);

    auto input_storage = torch::empty(
        {batch, 16, 32, 32, ic}, input.options());
    auto weight_storage = torch::empty(
        {oc, 3, 3, 3, ic}, weight.options());

    const int channel_tiles = (ic + 31) / 32;
    const int64_t input_blocks64 = batch * 512 * channel_tiles;
    const int64_t weight_blocks = (weight.numel() + 255) / 256;
    TORCH_CHECK(input_blocks64 + weight_blocks < INT32_MAX);

    const int input_blocks = static_cast<int>(input_blocks64);
    const int blocks = static_cast<int>(input_blocks64 + weight_blocks);
    const auto stream = c10::cuda::getCurrentCUDAStream();

    prepare_kernel<<<blocks, 256, 0, stream.stream()>>>(
        input.data_ptr<float>(),
        weight.data_ptr<float>(),
        input_storage.data_ptr<float>(),
        weight_storage.data_ptr<float>(),
        ic, oc, channel_tiles, input_blocks, weight.numel());

    C10_CUDA_KERNEL_LAUNCH_CHECK();

    return {
        input_storage.permute({0, 4, 1, 2, 3}),
        weight_storage.permute({0, 4, 1, 2, 3})
    };
}

torch::Tensor expand_transpose3d(torch::Tensor input)
{
    const c10::cuda::CUDAGuard guard(input.device());

    TORCH_CHECK(input.is_cuda());
    TORCH_CHECK(input.scalar_type() == at::kFloat);
    TORCH_CHECK(input.is_contiguous(at::MemoryFormat::ChannelsLast3d));
    TORCH_CHECK(
        input.dim() == 5 && input.size(2) == 16 &&
        input.size(3) == 32 && input.size(4) == 32);

    const int64_t batch = input.size(0);
    const int64_t channels = input.size(1);
    TORCH_CHECK(channels % 4 == 0);

    constexpr int64_t spatial_stride = 32 * 64 * 64;
    const int64_t main_count = batch * channels * spatial_stride;
    const int64_t total_count = main_count + channels * (4096 + 64 + 1);

    TORCH_CHECK(total_count / 4 < UINT32_MAX - 4194304u);

    auto storage = torch::empty({total_count}, input.options());

    const uint32_t main_vectors = main_count / 4;
    const uint32_t total_vectors = total_count / 4;
    const uint32_t channel_vectors = channels / 4;
    const int blocks = static_cast<int>(
        std::min<uint32_t>((total_vectors + 1023) / 1024, 4096));

    const auto stream = c10::cuda::getCurrentCUDAStream();
    const float4* source =
        reinterpret_cast<const float4*>(input.data_ptr<float>());
    float4* destination =
        reinterpret_cast<float4*>(storage.data_ptr<float>());

    if (channel_vectors == 16) {
        expand_kernel<16><<<blocks, 256, 0, stream.stream()>>>(
            source, destination, main_vectors, total_vectors, channel_vectors);
    } else {
        expand_kernel<0><<<blocks, 256, 0, stream.stream()>>>(
            source, destination, main_vectors, total_vectors, channel_vectors);
    }

    C10_CUDA_KERNEL_LAUNCH_CHECK();

    // Boundary rows and planes overlap only locations containing zero.
    return storage.as_strided(
        {batch, channels, 33, 65, 65},
        {
            channels * spatial_stride,
            1,
            channels * 4096,
            channels * 64,
            channels
        });
}
"""

_ops = load_inline(
    name="sparse_transpose3d_channels_last_v5",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["prepare_transpose3d", "expand_transpose3d"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
    verbose=False,
)


class ModelNew(nn.Module):
    def __init__(
        self,
        in_channels: int,
        out_channels: int,
        kernel_size: int,
        stride: int = 1,
        padding: int = 0,
        dilation: int = 1,
        bias: bool = False,
    ):
        super().__init__()
        self.conv_transpose3d = nn.ConvTranspose3d(
            in_channels,
            out_channels,
            kernel_size=(kernel_size, kernel_size, kernel_size),
            stride=stride,
            padding=padding,
            dilation=dilation,
            bias=bias,
        )

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        conv = self.conv_transpose3d
        supported = (
            x.is_cuda
            and x.ndim == 5
            and x.dtype == torch.float32
            and conv.weight.dtype == torch.float32
            and x.device == conv.weight.device
            and x.shape[0] > 0
            and x.shape[1] == conv.in_channels
            and x.shape[2:] == (16, 32, 32)
            and conv.out_channels % 4 == 0
            and conv.kernel_size == (3, 3, 3)
            and conv.stride == (2, 2, 2)
            and conv.padding == (1, 1, 1)
            and conv.dilation == (2, 2, 2)
            and conv.output_padding == (0, 0, 0)
            and conv.groups == 1
            and conv.bias is None
            and not torch.is_autocast_enabled()
            and x.shape[0] * conv.out_channels < 131000
        )
        if not supported:
            return conv(x)

        prepared_input, prepared_weight = _ops.prepare_transpose3d(
            x.contiguous(), conv.weight.contiguous()
        )
        compact = F.conv3d(
            prepared_input,
            prepared_weight,
            bias=None,
            stride=1,
            padding=1,
            dilation=1,
        )
        compact = compact.contiguous(memory_format=torch.channels_last_3d)
        return _ops.expand_transpose3d(compact)
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 1.07 ms; the PyTorch reference (with TF32 enabled) takes 1.94 ms; speedup 1.813x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| void expand_kernel<16>(float4 const*, float4*, unsigned int, unsigned int, unsigned int) | 516.4 | 1 |
| sm80_xmma_fprop_implicit_gemm_indexed_tf32f32_tf32f32_f32_nhwckrsc_nchw_tilesize256x64x32_ | 306.4 | 1 |
| void cudnn::engines_precompiled::nchwToNhwcKernel<float, float, float, false, true, (cudnn | 104.6 | 1 |
| void cudnn::engines_precompiled::convertTensor_kernel<float, float, float, (cudnnKernelDat | 60.4 | 2 |
| prepare_kernel(float const*, float const*, float*, float*, int, int, int, int, long) | 52.7 | 1 |

The fastest correct solution so far (2.034x speedup) is:
```python
import torch
import torch.nn as nn
import torch.nn.functional as F
from torch.utils.cpp_extension import load_inline


_cpp_source = r"""
#include <torch/extension.h>

torch::Tensor reverse_transpose_weight(torch::Tensor weight);
torch::Tensor expand_sparse_volume(torch::Tensor input);
"""

_cuda_source = r"""
#include <torch/extension.h>
#include <c10/cuda/CUDAGuard.h>
#include <c10/cuda/CUDAStream.h>
#include <c10/cuda/CUDAException.h>
#include <cuda_runtime.h>
#include <algorithm>
#include <cstdint>

__global__ void reverse_transpose_weight_kernel(
    const float* __restrict__ input,
    float* __restrict__ output,
    int in_channels,
    int out_channels,
    int64_t count)
{
    for (int64_t i = int64_t(blockIdx.x) * blockDim.x + threadIdx.x;
         i < count;
         i += int64_t(blockDim.x) * gridDim.x) {
        const int k = i % 27;
        const int ic = (i / 27) % in_channels;
        const int oc = i / (27LL * in_channels);
        output[i] =
            input[(int64_t(ic) * out_channels + oc) * 27 + 26 - k];
    }
}

/*
Physical storage uses a 32 x 64 x 64 volume per channel.

The logical 33 x 65 x 65 volume also includes its zero boundary
planes. Those planes can share storage with subsequent zero rows
and planes. Extra initialized storage covers the final boundary.

Four output values are written per thread:
    [0, input[2*w], 0, input[2*w+1]]
on odd depth/height coordinates, and zeros everywhere else.
*/
__global__ void expand_sparse_volume_kernel(
    const float2* __restrict__ input,
    float4* __restrict__ output,
    uint32_t main_vectors,
    uint32_t total_vectors)
{
    const uint32_t step = blockDim.x * gridDim.x;

    for (uint32_t v = blockIdx.x * blockDim.x + threadIdx.x;
         v < total_vectors;
         v += step) {
        float4 value = make_float4(0.f, 0.f, 0.f, 0.f);

        // In vector coordinates, height parity is bit 4 and depth
        // parity is bit 10. Removing those bits gives the compact
        // input's float2 index.
        if (v < main_vectors && (v & 1040u) == 1040u) {
            const uint32_t source =
                (v & 15u) |
                ((v >> 1) & 496u) |
                ((v >> 2) & ~511u);

            const float2 pair = input[source];
            value.y = pair.x;
            value.w = pair.y;
        }

        output[v] = value;
    }
}

torch::Tensor reverse_transpose_weight(torch::Tensor weight)
{
    const c10::cuda::CUDAGuard guard(weight.device());
    TORCH_CHECK(weight.scalar_type() == at::kFloat);
    TORCH_CHECK(weight.is_contiguous());

    const int ic = weight.size(0);
    const int oc = weight.size(1);
    auto output = torch::empty({oc, ic, 3, 3, 3}, weight.options());

    constexpr int threads = 256;
    const int blocks = static_cast<int>(
        std::min<int64_t>((output.numel() + threads - 1) / threads, 4096));
    const auto stream = c10::cuda::getCurrentCUDAStream();

    reverse_transpose_weight_kernel<<<blocks, threads, 0, stream.stream()>>>(
        weight.data_ptr<float>(),
        output.data_ptr<float>(),
        ic, oc, output.numel());

    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return output;
}

torch::Tensor expand_sparse_volume(torch::Tensor input)
{
    const c10::cuda::CUDAGuard guard(input.device());
    TORCH_CHECK(input.scalar_type() == at::kFloat);
    TORCH_CHECK(input.is_contiguous());
    TORCH_CHECK(
        input.dim() == 5 &&
        input.size(2) == 16 &&
        input.size(3) == 32 &&
        input.size(4) == 32);

    const int64_t batch = input.size(0);
    const int64_t channels = input.size(1);

    constexpr int64_t row_stride = 64;
    constexpr int64_t plane_stride = 64 * 64;
    constexpr int64_t channel_stride = 32 * 64 * 64;

    const int64_t main_count = batch * channels * channel_stride;

    // The largest logical offset is main_count + 4160.
    // Round the allocation up to a complete float4.
    constexpr int64_t tail_count = plane_stride + row_stride + 4;
    const int64_t total_count = main_count + tail_count;

    TORCH_CHECK(total_count / 4 <= UINT32_MAX);

    auto storage = torch::empty({total_count}, input.options());

    const uint32_t main_vectors = static_cast<uint32_t>(main_count / 4);
    const uint32_t total_vectors = static_cast<uint32_t>(total_count / 4);

    constexpr int threads = 256;
    const int blocks = static_cast<int>(
        std::min<uint32_t>(
            (total_vectors + threads - 1) / threads, 4096));
    const auto stream = c10::cuda::getCurrentCUDAStream();

    expand_sparse_volume_kernel<<<blocks, threads, 0, stream.stream()>>>(
        reinterpret_cast<const float2*>(input.data_ptr<float>()),
        reinterpret_cast<float4*>(storage.data_ptr<float>()),
        main_vectors,
        total_vectors);

    C10_CUDA_KERNEL_LAUNCH_CHECK();

    return storage.as_strided(
        {batch, channels, 33, 65, 65},
        {
            channels * channel_stride,
            channel_stride,
            plane_stride,
            row_stride,
            1
        });
}
"""

_ops = load_inline(
    name="sparse_transpose3d_compact_boundaries_v4",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["reverse_transpose_weight", "expand_sparse_volume"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
    verbose=False,
)


class ModelNew(nn.Module):
    def __init__(
        self,
        in_channels: int,
        out_channels: int,
        kernel_size: int,
        stride: int = 1,
        padding: int = 0,
        dilation: int = 1,
        bias: bool = False,
    ):
        super().__init__()
        self.conv_transpose3d = nn.ConvTranspose3d(
            in_channels,
            out_channels,
            kernel_size=(kernel_size, kernel_size, kernel_size),
            stride=stride,
            padding=padding,
            dilation=dilation,
            bias=bias,
        )

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        conv = self.conv_transpose3d
        supported = (
            x.is_cuda
            and x.ndim == 5
            and x.dtype == torch.float32
            and conv.weight.dtype == torch.float32
            and x.device == conv.weight.device
            and x.shape[0] > 0
            and x.shape[1] == conv.in_channels
            and x.shape[2:] == (16, 32, 32)
            and conv.kernel_size == (3, 3, 3)
            and conv.stride == (2, 2, 2)
            and conv.padding == (1, 1, 1)
            and conv.dilation == (2, 2, 2)
            and conv.output_padding == (0, 0, 0)
            and conv.groups == 1
            and conv.bias is None
            and not torch.is_autocast_enabled()
            and x.shape[0] * conv.out_channels < 131071
        )
        if not supported:
            return conv(x)

        weight = _ops.reverse_transpose_weight(conv.weight.contiguous())
        compact = F.conv3d(
            x.contiguous(),
            weight,
            bias=None,
            stride=1,
            padding=1,
            dilation=1,
        ).contiguous()

        return _ops.expand_sparse_volume(compact)
```

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.
