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):
    """
    Model that performs a transposed convolution, global average pooling, adds a bias, applies log-sum-exp, sum, and multiplication.
    """
    def __init__(self, in_channels, out_channels, kernel_size, bias_shape):
        super(Model, self).__init__()
        self.conv_transpose = nn.ConvTranspose2d(in_channels, out_channels, kernel_size)
        self.bias = nn.Parameter(torch.randn(bias_shape))

    def forward(self, x):
        x = self.conv_transpose(x)
        x = torch.mean(x, dim=(2, 3), keepdim=True)  # Global average pooling
        x = x + self.bias
        x = torch.logsumexp(x, dim=1, keepdim=True)  # Log-sum-exp
        x = torch.sum(x, dim=(2, 3))  # Sum
        x = x * 10.0  # Multiplication
        return x

batch_size = 16
in_channels = 64
out_channels = 128
height = width = 512
kernel_size = 3
bias_shape = (out_channels, 1, 1)

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

def get_init_inputs():
    return [in_channels, out_channels, kernel_size, bias_shape]
    ```
    
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 3 of 6 for this task. Your previous solution (turn 2) was:
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline


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

constexpr int THREADS = 256;
constexpr int TILE = 4096;

__device__ __forceinline__ float warp_sum(float v) {
    #pragma unroll
    for (int d = 16; d > 0; d >>= 1)
        v += __shfl_down_sync(0xffffffff, v, d);
    return v;
}

__device__ __forceinline__ float warp_max(float v) {
    #pragma unroll
    for (int d = 16; d > 0; d >>= 1)
        v = fmaxf(v, __shfl_down_sync(0xffffffff, v, d));
    return v;
}

template<bool VECTORIZED>
__global__ void spatial_sum_kernel(
    const float* __restrict__ x,
    float* __restrict__ partials,
    int64_t spatial,
    int parts,
    int64_t total_blocks
) {
    __shared__ float warp_sums[8];

    for (int64_t block = blockIdx.x;
         block < total_blocks;
         block += gridDim.x) {
        const int64_t plane = block / parts;
        const int part = block % parts;
        const int64_t start = int64_t(part) * TILE;
        float value = 0.0f;

        if constexpr (VECTORIZED) {
            const float4* input =
                reinterpret_cast<const float4*>(x + plane * spatial);
            float a = 0.0f, b = 0.0f, c = 0.0f, d = 0.0f;

            #pragma unroll
            for (int j = 0; j < TILE / (4 * THREADS); ++j) {
                const int64_t position =
                    start / 4 + threadIdx.x + j * THREADS;
                if (position < spatial / 4) {
                    const float4 v = input[position];
                    a += v.x;
                    b += v.y;
                    c += v.z;
                    d += v.w;
                }
            }
            value = (a + b) + (c + d);
        } else {
            #pragma unroll
            for (int j = 0; j < TILE / THREADS; ++j) {
                const int64_t position =
                    start + threadIdx.x + j * THREADS;
                if (position < spatial)
                    value += x[plane * spatial + position];
            }
        }

        value = warp_sum(value);
        if ((threadIdx.x & 31) == 0)
            warp_sums[threadIdx.x >> 5] = value;
        __syncthreads();

        if (threadIdx.x < 32) {
            value = threadIdx.x < 8 ? warp_sums[threadIdx.x] : 0.0f;
            value = warp_sum(value);
            if (threadIdx.x == 0)
                partials[block] = value;
        }
        __syncthreads();
    }
}

__global__ void finish_kernel(
    const float* __restrict__ partials,
    const float* __restrict__ weight,
    const float* __restrict__ conv_bias,
    const float* __restrict__ extra_bias,
    float* __restrict__ output,
    int in_channels,
    int out_channels,
    int kernel_elements,
    int parts,
    float inverse_output_area
) {
    extern __shared__ float storage[];
    float* input_sums = storage;
    float* logits = input_sums + in_channels;
    float* reduction = logits + out_channels;

    const int batch = blockIdx.x;
    const int tid = threadIdx.x;
    const int lane = tid & 31;
    const int warp = tid >> 5;
    const float negative_infinity = __int_as_float(0xff800000);

    for (int i = tid; i < in_channels; i += THREADS) {
        const int64_t base = (int64_t(batch) * in_channels + i) * parts;
        float value = 0.0f;
        for (int p = 0; p < parts; ++p)
            value += partials[base + p];
        input_sums[i] = value * inverse_output_area;
    }
    __syncthreads();

    // The full spatial sum factorizes for an unpadded transposed convolution.
    float local_max = negative_infinity;
    for (int c = tid; c < out_channels; c += THREADS) {
        float value = 0.0f;
        for (int i = 0; i < in_channels; ++i) {
            const int64_t base =
                (int64_t(i) * out_channels + c) * kernel_elements;
            float weight_sum = 0.0f;
            for (int k = 0; k < kernel_elements; ++k)
                weight_sum += weight[base + k];
            value = fmaf(input_sums[i], weight_sum, value);
        }
        value = (value + conv_bias[c]) + extra_bias[c];
        logits[c] = value;
        local_max = fmaxf(local_max, value);
    }

    local_max = warp_max(local_max);
    if (lane == 0)
        reduction[warp] = local_max;
    __syncthreads();

    if (warp == 0) {
        float value = lane < 8 ? reduction[lane] : negative_infinity;
        value = warp_max(value);
        if (lane == 0)
            reduction[8] = value;
    }
    __syncthreads();

    const float maximum = reduction[8];
    float local_sum = 0.0f;
    for (int c = tid; c < out_channels; c += THREADS)
        local_sum += expf(logits[c] - maximum);

    local_sum = warp_sum(local_sum);
    if (lane == 0)
        reduction[warp] = local_sum;
    __syncthreads();

    if (warp == 0) {
        float value = lane < 8 ? reduction[lane] : 0.0f;
        value = warp_sum(value);
        if (lane == 0)
            output[batch] = 10.0f * (maximum + logf(value));
    }
}

torch::Tensor pooled_transpose_cuda(
    torch::Tensor x,
    torch::Tensor weight,
    torch::Tensor conv_bias,
    torch::Tensor extra_bias
) {
    const c10::cuda::CUDAGuard device_guard(x.device());

    const int batch = x.size(0);
    const int in_channels = x.size(1);
    const int out_channels = weight.size(1);
    const int kh = weight.size(2);
    const int kw = weight.size(3);
    const int64_t spatial = x.size(2) * x.size(3);
    const int parts = (spatial + TILE - 1) / TILE;

    auto output = torch::empty({batch, 1}, x.options());
    if (batch == 0)
        return output;

    auto partials = torch::empty({batch, in_channels, parts}, x.options());
    const int64_t total_blocks = int64_t(batch) * in_channels * parts;
    const int blocks = total_blocks < 65535 ? int(total_blocks) : 65535;
    const auto stream = at::cuda::getCurrentCUDAStream();

    if (spatial % 4 == 0) {
        spatial_sum_kernel<true><<<blocks, THREADS, 0, stream>>>(
            x.data_ptr<float>(), partials.data_ptr<float>(),
            spatial, parts, total_blocks
        );
    } else {
        spatial_sum_kernel<false><<<blocks, THREADS, 0, stream>>>(
            x.data_ptr<float>(), partials.data_ptr<float>(),
            spatial, parts, total_blocks
        );
    }
    C10_CUDA_KERNEL_LAUNCH_CHECK();

    const int64_t output_area =
        (x.size(2) + kh - 1) * (x.size(3) + kw - 1);
    const size_t shared_bytes =
        size_t(in_channels + out_channels + 9) * sizeof(float);

    finish_kernel<<<batch, THREADS, shared_bytes, stream>>>(
        partials.data_ptr<float>(),
        weight.data_ptr<float>(),
        conv_bias.data_ptr<float>(),
        extra_bias.data_ptr<float>(),
        output.data_ptr<float>(),
        in_channels,
        out_channels,
        kh * kw,
        parts,
        1.0f / float(output_area)
    );
    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return output;
}
"""

_extension = load_inline(
    name="fused_transpose_global_pool_lse_v2",
    cpp_sources=r"""
#include <torch/extension.h>
torch::Tensor pooled_transpose_cuda(
    torch::Tensor x,
    torch::Tensor weight,
    torch::Tensor conv_bias,
    torch::Tensor extra_bias
);
""",
    cuda_sources=_cuda_source,
    functions=["pooled_transpose_cuda"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
    verbose=False,
)


class ModelNew(nn.Module):
    def __init__(self, in_channels, out_channels, kernel_size, bias_shape):
        super().__init__()
        self.conv_transpose = nn.ConvTranspose2d(
            in_channels, out_channels, kernel_size
        )
        self.bias = nn.Parameter(torch.randn(bias_shape))

    def forward(self, x):
        conv = self.conv_transpose
        use_custom = (
            x.is_cuda
            and x.dtype == torch.float32
            and conv.weight.dtype == torch.float32
            and conv.bias.dtype == torch.float32
            and self.bias.dtype == torch.float32
            and x.device == conv.weight.device == conv.bias.device == self.bias.device
            and x.ndim == 4
            and x.shape[1] == conv.in_channels
            and x.shape[2] > 0
            and x.shape[3] > 0
            and tuple(self.bias.shape) == (conv.out_channels, 1, 1)
            and conv.in_channels + conv.out_channels + 9 <= 12000
            and not torch.is_autocast_enabled()
        )

        if use_custom:
            return _extension.pooled_transpose_cuda(
                x.contiguous(),
                conv.weight.contiguous(),
                conv.bias.contiguous(),
                self.bias.contiguous(),
            )

        x = conv(x)
        x = x.mean(dim=(2, 3), keepdim=True)
        x = x + self.bias
        x = torch.logsumexp(x, dim=1, keepdim=True)
        return x.sum(dim=(2, 3)) * 10.0
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 0.797 ms; the PyTorch reference (with TF32 enabled) takes 14.8 ms; speedup 18.570x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| void spatial_sum_kernel<true>(float const*, float*, long, int, long) | 724.5 | 1 |
| finish_kernel(float const*, float const*, float const*, float const*, float*, int, int, in | 50.9 | 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.
