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 5 of 6 for this task. Your previous solution (turn 4) was:
```python
# 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
from torch.utils.cpp_extension import load_inline

# Mathematical shortcut (valid for ConvTranspose2d with padding == 0 and
# groups == 1): every input pixel's kernel "stamp" lies fully inside the
# transposed-convolution output, so the global average of its output is
#   mean(out)[b,o] = ( sum_i sum(x[b,i]) * sum(W[i,o]) ) / (Ho*Wo) + cbias[o]
# The whole model therefore reduces to:
#   1) per-(batch,channel) spatial sums of the input (one memory-bound pass
#      over 1 GiB that never materializes the transposed-conv output)
#   2) a tiny fused epilogue: dot product with cached kernel-weight sums +
#      biases + numerically stable log-sum-exp + scaling.
# Both steps run in ONE kernel launch: the last block of each batch (detected
# with a thread-fence/atomic counter) performs the epilogue for that batch.
# Speed tricks vs. the previous version:
#   * weight sums (ksum) are precomputed once in Python and cached
#   * the __threadfence is executed by thread 0 only (the only writer)
#   * 8x-unrolled float4 streaming loads with 32-bit indexing
#   * grid ordered so concurrently-resident blocks sweep memory linearly
_cuda_source = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <cuda_runtime.h>
#include <algorithm>

#define T1 256
#define NWARP (T1 / 32)

__device__ __forceinline__ double warp_sum(double v) {
    for (int off = 16; off > 0; off >>= 1)
        v += __shfl_down_sync(0xffffffffu, v, off);
    return v;
}

// Sum of one chunk of a contiguous row of hw floats (32-bit indexing,
// 8x unrolled streaming loads).
template <bool VEC>
__device__ __forceinline__ double chunk_sum(const float* __restrict__ row,
                                            int hw, int chunk, int chunks) {
    if (VEC) {
        const int n4 = hw >> 2;
        const int per = (n4 + chunks - 1) / chunks;
        const int start = per * chunk;
        int end = start + per;
        if (end > n4) end = n4;
        const float4* p4 = reinterpret_cast<const float4*>(row) + start;
        const int cnt = end - start;
        float4 aA = make_float4(0.0f, 0.0f, 0.0f, 0.0f);
        float4 aB = make_float4(0.0f, 0.0f, 0.0f, 0.0f);
        int i = threadIdx.x;
        for (; i + 7 * T1 < cnt; i += 8 * T1) {
            const float4 v0 = __ldcs(p4 + i);
            const float4 v1 = __ldcs(p4 + i + T1);
            const float4 v2 = __ldcs(p4 + i + 2 * T1);
            const float4 v3 = __ldcs(p4 + i + 3 * T1);
            const float4 v4 = __ldcs(p4 + i + 4 * T1);
            const float4 v5 = __ldcs(p4 + i + 5 * T1);
            const float4 v6 = __ldcs(p4 + i + 6 * T1);
            const float4 v7 = __ldcs(p4 + i + 7 * T1);
            aA.x += (v0.x + v1.x) + (v2.x + v3.x);
            aA.y += (v0.y + v1.y) + (v2.y + v3.y);
            aA.z += (v0.z + v1.z) + (v2.z + v3.z);
            aA.w += (v0.w + v1.w) + (v2.w + v3.w);
            aB.x += (v4.x + v5.x) + (v6.x + v7.x);
            aB.y += (v4.y + v5.y) + (v6.y + v7.y);
            aB.z += (v4.z + v5.z) + (v6.z + v7.z);
            aB.w += (v4.w + v5.w) + (v6.w + v7.w);
        }
        for (; i + 3 * T1 < cnt; i += 4 * T1) {
            const float4 v0 = __ldcs(p4 + i);
            const float4 v1 = __ldcs(p4 + i + T1);
            const float4 v2 = __ldcs(p4 + i + 2 * T1);
            const float4 v3 = __ldcs(p4 + i + 3 * T1);
            aA.x += (v0.x + v1.x) + (v2.x + v3.x);
            aA.y += (v0.y + v1.y) + (v2.y + v3.y);
            aA.z += (v0.z + v1.z) + (v2.z + v3.z);
            aA.w += (v0.w + v1.w) + (v2.w + v3.w);
        }
        for (; i < cnt; i += T1) {
            const float4 v0 = __ldcs(p4 + i);
            aA.x += v0.x; aA.y += v0.y; aA.z += v0.z; aA.w += v0.w;
        }
        return (double)(((aA.x + aB.x) + (aA.y + aB.y)) +
                        ((aA.z + aB.z) + (aA.w + aB.w)));
    } else {
        const int per = (hw + chunks - 1) / chunks;
        const int start = per * chunk;
        int end = start + per;
        if (end > hw) end = hw;
        const float* p = row + start;
        const int cnt = end - start;
        float aA = 0.0f, aB = 0.0f;
        int i = threadIdx.x;
        for (; i + 7 * T1 < cnt; i += 8 * T1) {
            aA += (p[i] + p[i + T1]) + (p[i + 2 * T1] + p[i + 3 * T1]);
            aB += (p[i + 4 * T1] + p[i + 5 * T1]) + (p[i + 6 * T1] + p[i + 7 * T1]);
        }
        for (; i + 3 * T1 < cnt; i += 4 * T1)
            aA += (p[i] + p[i + T1]) + (p[i + 2 * T1] + p[i + 3 * T1]);
        for (; i < cnt; i += T1) aA += p[i];
        return (double)(aA + aB);
    }
}

// Grid: blockIdx.x = chunk index, blockIdx.y = flattened (batch * Cin).
// Consecutive blockIdx.x covers adjacent memory, so the resident blocks sweep
// the tensor linearly (good DRAM locality). The last block of each batch
// fuses the whole epilogue (dot with cached weight sums, biases, log-sum-exp).
template <bool VEC>
__global__ void __launch_bounds__(T1, 4)
fused_convt_kernel(const float* __restrict__ x,
                   const float* __restrict__ ksum,   // (Cin, Cout)
                   const float* __restrict__ cbias,  // (Cout) or nullptr
                   const float* __restrict__ bias,   // (Cout)
                   double* __restrict__ partials,    // (B*Cin, chunks)
                   int*   __restrict__ counters,     // (B), zeroed on entry/exit
                   float* __restrict__ out,          // (B)
                   int hw, int chunks, int Cin, int Cout,
                   double inv_hw, double scale) {
    const int bc    = (int)blockIdx.y;
    const int chunk = (int)blockIdx.x;
    const int b     = bc / Cin;
    const int lane  = threadIdx.x & 31;
    const int warp  = threadIdx.x >> 5;

    // ---- 1) block-wide partial spatial sum of x[b, c] chunk ----
    const double v = chunk_sum<VEC>(x + (size_t)bc * (size_t)hw, hw, chunk, chunks);
    __shared__ double sh[NWARP];
    double r = warp_sum(v);
    if (lane == 0) sh[warp] = r;
    __syncthreads();
    if (warp == 0) {
        double rr = (lane < NWARP) ? sh[lane] : 0.0;
        rr = warp_sum(rr);
        if (lane == 0) partials[(size_t)bc * chunks + chunk] = rr;
    }

    // ---- 2) last-block-of-batch detection (fence only in the writer) ----
    __shared__ bool is_last;
    if (threadIdx.x == 0) {
        __threadfence();
        const int old = atomicAdd(&counters[b], 1);
        is_last = (old == Cin * chunks - 1);
        if (is_last) atomicExch(&counters[b], 0);  // reset for the next call
    }
    __syncthreads();
    if (!is_last) return;
    __threadfence();

    // ---- 3) fused epilogue for batch b (single block) ----
    extern __shared__ double smem[];
    double* xs  = smem;        // Cin doubles
    double* vv  = smem + Cin;  // Cout doubles
    double* red = vv + Cout;   // 32 doubles

    const int t = threadIdx.x;
    const double* pb = partials + (size_t)b * (size_t)Cin * chunks;
    for (int i = t; i < Cin; i += T1) {
        const double* p = pb + (size_t)i * chunks;
        double s = 0.0;
        for (int c = 0; c < chunks; ++c) s += p[c];
        xs[i] = s;
    }
    __syncthreads();

    // vv[o] = ( sum_i xs[i] * ksum[i,o] ) * inv_hw + bias[o] + cbias[o]
    // Two threads per output channel, each covering half of the inputs.
    const int pairs = T1 / 2;
    for (int o0 = 0; o0 < Cout; o0 += pairs) {
        const int o = o0 + (t >> 1);
        const bool active = (o < Cout);
        double acc = 0.0;
        if (active) {
            const float* kp = ksum + o;
            for (int i = (t & 1); i < Cin; i += 2)
                acc += xs[i] * (double)kp[(size_t)i * Cout];
        }
        acc += __shfl_xor_sync(0xffffffffu, acc, 1);
        if (active && ((t & 1) == 0)) {
            double val = acc * inv_hw + (double)bias[o];
            if (cbias != nullptr) val += (double)cbias[o];
            vv[o] = val;
        }
    }
    __syncthreads();

    // ---- numerically stable log-sum-exp over channels ----
    double m = -1.0e308;
    for (int o = t; o < Cout; o += T1) m = fmax(m, vv[o]);
    for (int off = 16; off > 0; off >>= 1)
        m = fmax(m, __shfl_down_sync(0xffffffffu, m, off));
    if (lane == 0) red[warp] = m;
    __syncthreads();
    if (warp == 0) {
        double mm = (lane < NWARP) ? red[lane] : -1.0e308;
        for (int off = 16; off > 0; off >>= 1)
            mm = fmax(mm, __shfl_down_sync(0xffffffffu, mm, off));
        if (lane == 0) red[0] = mm;
    }
    __syncthreads();
    const double mx = red[0];
    __syncthreads();

    double s = 0.0;
    for (int o = t; o < Cout; o += T1) s += exp(vv[o] - mx);
    s = warp_sum(s);
    if (lane == 0) red[warp] = s;
    __syncthreads();
    if (warp == 0) {
        double ss = (lane < NWARP) ? red[lane] : 0.0;
        ss = warp_sum(ss);
        if (lane == 0) out[b] = (float)(scale * (mx + log(ss)));
    }
}

torch::Tensor fused_forward(torch::Tensor x, torch::Tensor ksum,
                            torch::Tensor cbias, torch::Tensor bias,
                            torch::Tensor counters,
                            double inv_hw, double scale) {
    const long B    = x.size(0);
    const long Cin  = x.size(1);
    const long hw   = x.size(2) * x.size(3);
    const long Cout = ksum.size(1);
    const long bc   = B * Cin;

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

    const bool vec     = ((hw & 3L) == 0L);
    const long work    = vec ? (hw >> 2) : hw;
    const long per_thr = 16;
    long chunks_l = (work + (long)T1 * per_thr - 1) / ((long)T1 * per_thr);
    chunks_l      = std::max<long>(1L, std::min<long>(chunks_l, 65535L));
    const int chunks = (int)chunks_l;

    auto partials = torch::empty(
        {bc, (long)chunks},
        torch::TensorOptions().dtype(torch::kFloat64).device(x.device()));
    auto out = torch::empty({B, 1}, x.options());

    const float* cb_ptr =
        (cbias.defined() && cbias.numel() > 0) ? cbias.data_ptr<float>() : nullptr;
    const size_t smem = (size_t)(Cin + Cout + 32) * sizeof(double);

    dim3 grid((unsigned)chunks, (unsigned)bc, 1);
    if (vec) {
        fused_convt_kernel<true><<<grid, T1, smem, stream>>>(
            x.data_ptr<float>(), ksum.data_ptr<float>(), cb_ptr,
            bias.data_ptr<float>(), partials.data_ptr<double>(),
            counters.data_ptr<int>(), out.data_ptr<float>(),
            (int)hw, chunks, (int)Cin, (int)Cout, inv_hw, scale);
    } else {
        fused_convt_kernel<false><<<grid, T1, smem, stream>>>(
            x.data_ptr<float>(), ksum.data_ptr<float>(), cb_ptr,
            bias.data_ptr<float>(), partials.data_ptr<double>(),
            counters.data_ptr<int>(), out.data_ptr<float>(),
            (int)hw, chunks, (int)Cin, (int)Cout, inv_hw, scale);
    }
    return out;
}
"""

_cpp_source = (
    "torch::Tensor fused_forward(torch::Tensor x, torch::Tensor ksum, "
    "torch::Tensor cbias, torch::Tensor bias, torch::Tensor counters, "
    "double inv_hw, double scale);"
)

_fused_op = load_inline(
    name="fused_convt_gap_bias_lse_v4",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["fused_forward"],
    verbose=False,
    extra_cuda_cflags=["-O3"],
)


class ModelNew(nn.Module):
    """
    Optimized version of Model. Replaces the expensive transposed convolution
    + global-average-pooling chain with a mathematically equivalent single
    fused custom CUDA kernel (valid for padding == 0 and groups == 1, which is
    the configuration used by this model). Falls back to the original PyTorch
    ops for any other configuration.
    """

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

        ct = self.conv_transpose
        self._static_ok = (
            ct.groups == 1
            and tuple(ct.padding) == (0, 0)
            and ct.weight.dtype == torch.float32
            and self.bias.dtype == torch.float32
            and self.bias.numel() == ct.out_channels
            and (ct.in_channels + ct.out_channels + 32) * 8 <= 32 * 1024
        )
        self._sH, self._sW = ct.stride
        self._dH, self._dW = ct.dilation
        self._kH, self._kW = ct.kernel_size
        self._opH, self._opW = ct.output_padding
        self._ksum = None
        self._ksum_key = None
        self._counters = None
        self._empty_bias = None

    def forward(self, x):
        if (
            self._static_ok
            and x.dim() == 4
            and x.is_cuda
            and x.dtype == torch.float32
        ):
            ct = self.conv_transpose
            dev = x.device
            B = int(x.shape[0])
            C = int(x.shape[1])
            H = int(x.shape[2])
            W = int(x.shape[3])

            if B * C <= 65535 and H * W < 2147483647:
                # Cache the per-(in,out) kernel-weight sums; they only change
                # when the weights are updated.
                w = ct.weight
                key = (w.data_ptr(), w._version)
                if key != self._ksum_key:
                    self._ksum = w.detach().sum(dim=(2, 3)).contiguous()
                    self._ksum_key = key

                ctr = self._counters
                if ctr is None or ctr.device != dev or ctr.numel() < B:
                    ctr = torch.zeros(max(B, 64), device=dev, dtype=torch.int32)
                    self._counters = ctr

                cb = ct.bias
                if cb is None:
                    eb = self._empty_bias
                    if eb is None or eb.device != dev:
                        eb = torch.empty(0, device=dev, dtype=torch.float32)
                        self._empty_bias = eb
                    cb = eb
                else:
                    cb = cb.contiguous()

                # Output spatial size of the transposed convolution.
                Ho = (H - 1) * self._sH + self._dH * (self._kH - 1) + self._opH + 1
                Wo = (W - 1) * self._sW + self._dW * (self._kW - 1) + self._opW + 1
                inv_hw = 1.0 / float(Ho * Wo)

                return self._fwd(
                    x.contiguous(),
                    self._ksum,
                    cb,
                    self.bias.contiguous().view(-1),
                    ctr,
                    inv_hw,
                    10.0,
                )

        # Generic fallback: original computation.
        x = self.conv_transpose(x)
        x = torch.mean(x, dim=(2, 3), keepdim=True)
        x = x + self.bias
        x = torch.logsumexp(x, dim=1, keepdim=True)
        x = torch.sum(x, dim=(2, 3))
        x = x * 10.0
        return x
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 0.77 ms; the PyTorch reference (with TF32 enabled) takes 14.8 ms; speedup 19.221x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| void fused_convt_kernel<true>(float const*, float const*, float const*, float const*, doub | 735.6 | 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.
