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 convolution, applies Group Normalization, Tanh, HardSwish, 
    Residual Addition, and LogSumExp.
    """
    def __init__(self, in_channels, out_channels, kernel_size, groups, eps=1e-5):
        super(Model, self).__init__()
        self.conv = nn.Conv2d(in_channels, out_channels, kernel_size)
        self.group_norm = nn.GroupNorm(groups, out_channels, eps=eps)
        self.tanh = nn.Tanh()
        self.hard_swish = nn.Hardswish()

    def forward(self, x):
        # Convolution
        x_conv = self.conv(x)
        # Group Normalization
        x_norm = self.group_norm(x_conv)
        # Tanh
        x_tanh = self.tanh(x_norm)
        # HardSwish
        x_hard_swish = self.hard_swish(x_tanh)
        # Residual Addition
        x_res = x_conv + x_hard_swish
        # LogSumExp
        x_logsumexp = torch.logsumexp(x_res, dim=1, keepdim=True)
        return x_logsumexp

batch_size = 128
in_channels = 8
out_channels = 64
height, width = 128, 128
kernel_size = 3
groups = 16

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

def get_init_inputs():
    return [in_channels, out_channels, kernel_size, groups]
    ```
    
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
# 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

torch.backends.cudnn.benchmark = True

# Custom CUDA kernels:
#  1) group_stats_kernel: computes per-(batch, group) mean / rstd of the conv output.
#  2) fused_act_lse_*_kernel: fuses GroupNorm affine + Tanh + HardSwish + residual add
#     + LogSumExp over channels into a single pass over the data (vectorized float4 path
#     plus a scalar fallback for odd spatial sizes).
_cuda_src = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <cuda_runtime.h>

#define THREADS 256
#define NEG_INF __int_as_float(0xff800000)

__device__ __forceinline__ float fast_tanh(float y) {
    // tanh(y) = sign(y) * (1 - e^{-2|y|}) / (1 + e^{-2|y|}); no overflow possible.
    float e = __expf(-2.0f * fabsf(y));
    float t = __fdividef(1.0f - e, 1.0f + e);
    return copysignf(t, y);
}

__device__ __forceinline__ float act_res(float v, float a, float b) {
    // groupnorm affine: y = v * a + b, then tanh, then hardswish, then residual add.
    float y = v * a + b;
    float t = fast_tanh(y);
    float hs = t * fminf(fmaxf(t + 3.0f, 0.0f), 6.0f) * (1.0f / 6.0f);
    return v + hs;
}

__device__ __forceinline__ void lse_update(float r, float &m, float &s) {
    // online logsumexp update
    if (r > m) {
        s = s * __expf(m - r) + 1.0f;
        m = r;
    } else {
        s += __expf(r - m);
    }
}

// Kernel 1: per (n, group) mean and rstd of the conv output.
__global__ void group_stats_kernel(const float *__restrict__ x,
                                   float *__restrict__ mean,
                                   float *__restrict__ rstd,
                                   int C, int HW, int G, int CperG, float eps) {
    const int ng = blockIdx.x;
    const int n = ng / G;
    const int g = ng - n * G;
    const int c0 = g * CperG;
    const float *base = x + (size_t)n * (size_t)C * (size_t)HW;
    float sum = 0.0f, sumsq = 0.0f;
    if ((HW & 3) == 0) {
        const int HW4 = HW >> 2;
        for (int c = c0; c < c0 + CperG; ++c) {
            const float4 *p = reinterpret_cast<const float4 *>(base + (size_t)c * (size_t)HW);
            for (int i = threadIdx.x; i < HW4; i += blockDim.x) {
                float4 v = p[i];
                sum += (v.x + v.y) + (v.z + v.w);
                sumsq += (v.x * v.x + v.y * v.y) + (v.z * v.z + v.w * v.w);
            }
        }
    } else {
        for (int c = c0; c < c0 + CperG; ++c) {
            const float *p = base + (size_t)c * (size_t)HW;
            for (int i = threadIdx.x; i < HW; i += blockDim.x) {
                float v = p[i];
                sum += v;
                sumsq += v * v;
            }
        }
    }
    __shared__ float ssum[THREADS / 32];
    __shared__ float sssq[THREADS / 32];
    const int lane = threadIdx.x & 31;
    const int wid = threadIdx.x >> 5;
    #pragma unroll
    for (int o = 16; o > 0; o >>= 1) {
        sum += __shfl_down_sync(0xffffffffu, sum, o);
        sumsq += __shfl_down_sync(0xffffffffu, sumsq, o);
    }
    if (lane == 0) {
        ssum[wid] = sum;
        sssq[wid] = sumsq;
    }
    __syncthreads();
    if (wid == 0) {
        const int nw = THREADS / 32;
        sum = (lane < nw) ? ssum[lane] : 0.0f;
        sumsq = (lane < nw) ? sssq[lane] : 0.0f;
        #pragma unroll
        for (int o = 16; o > 0; o >>= 1) {
            sum += __shfl_down_sync(0xffffffffu, sum, o);
            sumsq += __shfl_down_sync(0xffffffffu, sumsq, o);
        }
        if (lane == 0) {
            const float cnt = (float)((long long)CperG * (long long)HW);
            const float m = sum / cnt;
            float var = sumsq / cnt - m * m;
            var = fmaxf(var, 0.0f);
            mean[ng] = m;
            rstd[ng] = rsqrtf(var + eps);
        }
    }
}

// Kernel 2 (float4 vectorized): fused GroupNorm + Tanh + HardSwish + residual + LogSumExp.
__global__ void fused_act_lse_vec4_kernel(const float *__restrict__ x,
                                          const float *__restrict__ w,
                                          const float *__restrict__ b,
                                          const float *__restrict__ mean,
                                          const float *__restrict__ rstd,
                                          float *__restrict__ out,
                                          int C, int HW, int G, int CperG) {
    const int n = blockIdx.y;
    const int HW4 = HW >> 2;
    extern __shared__ float sm[];
    float *sa = sm;        // per-channel scale  = gamma * rstd(group)
    float *sb = sm + C;    // per-channel offset = beta - mean(group) * scale
    for (int c = threadIdx.x; c < C; c += blockDim.x) {
        const int ng = n * G + c / CperG;
        const float a = w[c] * rstd[ng];
        sa[c] = a;
        sb[c] = b[c] - mean[ng] * a;
    }
    __syncthreads();
    const int hw4 = blockIdx.x * blockDim.x + threadIdx.x;
    if (hw4 >= HW4) return;
    const float4 *xp = reinterpret_cast<const float4 *>(x + (size_t)n * (size_t)C * (size_t)HW) + hw4;
    float m0 = NEG_INF, m1 = NEG_INF, m2 = NEG_INF, m3 = NEG_INF;
    float s0 = 0.0f, s1 = 0.0f, s2 = 0.0f, s3 = 0.0f;
    const size_t stride = (size_t)HW4;
    for (int c = 0; c < C; ++c) {
        const float4 v = xp[c * stride];
        const float a = sa[c], bb = sb[c];
        lse_update(act_res(v.x, a, bb), m0, s0);
        lse_update(act_res(v.y, a, bb), m1, s1);
        lse_update(act_res(v.z, a, bb), m2, s2);
        lse_update(act_res(v.w, a, bb), m3, s3);
    }
    float4 o;
    o.x = m0 + logf(s0);
    o.y = m1 + logf(s1);
    o.z = m2 + logf(s2);
    o.w = m3 + logf(s3);
    reinterpret_cast<float4 *>(out + (size_t)n * (size_t)HW)[hw4] = o;
}

// Kernel 2 (scalar fallback for spatial size not divisible by 4).
__global__ void fused_act_lse_kernel(const float *__restrict__ x,
                                     const float *__restrict__ w,
                                     const float *__restrict__ b,
                                     const float *__restrict__ mean,
                                     const float *__restrict__ rstd,
                                     float *__restrict__ out,
                                     int C, int HW, int G, int CperG) {
    const int n = blockIdx.y;
    extern __shared__ float sm[];
    float *sa = sm;
    float *sb = sm + C;
    for (int c = threadIdx.x; c < C; c += blockDim.x) {
        const int ng = n * G + c / CperG;
        const float a = w[c] * rstd[ng];
        sa[c] = a;
        sb[c] = b[c] - mean[ng] * a;
    }
    __syncthreads();
    const int hw = blockIdx.x * blockDim.x + threadIdx.x;
    if (hw >= HW) return;
    const float *xp = x + (size_t)n * (size_t)C * (size_t)HW + hw;
    float m = NEG_INF, s = 0.0f;
    for (int c = 0; c < C; ++c) {
        const float v = xp[(size_t)c * (size_t)HW];
        lse_update(act_res(v, sa[c], sb[c]), m, s);
    }
    out[(size_t)n * (size_t)HW + hw] = m + logf(s);
}

torch::Tensor fused_gn_act_lse(torch::Tensor x, torch::Tensor w, torch::Tensor b,
                               int64_t groups, double eps) {
    TORCH_CHECK(x.is_cuda() && x.dim() == 4 && x.scalar_type() == at::kFloat,
                "x must be a CUDA float32 4D tensor");
    auto xc = x.contiguous();
    auto wc = w.contiguous();
    auto bc = b.contiguous();
    const int N = xc.size(0), C = xc.size(1), H = xc.size(2), W = xc.size(3);
    const int HW = H * W;
    const int G = (int)groups;
    TORCH_CHECK(C % G == 0, "channels must be divisible by groups");
    const int CperG = C / G;
    auto opts = xc.options();
    auto mean = at::empty({N * G}, opts);
    auto rstd = at::empty({N * G}, opts);
    auto out = at::empty({N, 1, H, W}, opts);
    auto stream = at::cuda::getCurrentCUDAStream();
    group_stats_kernel<<<N * G, THREADS, 0, stream>>>(
        xc.data_ptr<float>(), mean.data_ptr<float>(), rstd.data_ptr<float>(),
        C, HW, G, CperG, (float)eps);
    const int smem = 2 * C * (int)sizeof(float);
    if ((HW & 3) == 0) {
        dim3 grid((HW / 4 + THREADS - 1) / THREADS, N);
        fused_act_lse_vec4_kernel<<<grid, THREADS, smem, stream>>>(
            xc.data_ptr<float>(), wc.data_ptr<float>(), bc.data_ptr<float>(),
            mean.data_ptr<float>(), rstd.data_ptr<float>(),
            out.data_ptr<float>(), C, HW, G, CperG);
    } else {
        dim3 grid((HW + THREADS - 1) / THREADS, N);
        fused_act_lse_kernel<<<grid, THREADS, smem, stream>>>(
            xc.data_ptr<float>(), wc.data_ptr<float>(), bc.data_ptr<float>(),
            mean.data_ptr<float>(), rstd.data_ptr<float>(),
            out.data_ptr<float>(), C, HW, G, CperG);
    }
    return out;
}
"""

_cpp_src = "torch::Tensor fused_gn_act_lse(torch::Tensor x, torch::Tensor w, torch::Tensor b, int64_t groups, double eps);"

_fused = load_inline(
    name="fused_gn_tanh_hswish_res_lse",
    cpp_sources=_cpp_src,
    cuda_sources=_cuda_src,
    functions=["fused_gn_act_lse"],
    verbose=False,
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
)


class ModelNew(nn.Module):
    """
    Optimized model: keeps the cuDNN convolution, and replaces
    GroupNorm + Tanh + HardSwish + Residual Add + LogSumExp with a pair of
    fused custom CUDA kernels (stats kernel + fused elementwise/reduction kernel).
    """
    def __init__(self, in_channels, out_channels, kernel_size, groups, eps=1e-5):
        super(ModelNew, self).__init__()
        self.conv = nn.Conv2d(in_channels, out_channels, kernel_size)
        self.group_norm = nn.GroupNorm(groups, out_channels, eps=eps)
        self.groups = groups
        self.eps = float(eps)

    def forward(self, x):
        # Convolution (cuDNN)
        x_conv = self.conv(x)
        if x_conv.is_cuda and x_conv.dtype == torch.float32:
            # Fused GroupNorm + Tanh + HardSwish + Residual + LogSumExp
            return _fused.fused_gn_act_lse(
                x_conv,
                self.group_norm.weight,
                self.group_norm.bias,
                self.groups,
                self.eps,
            )
        # Fallback (e.g. CPU tensors): original PyTorch operator sequence
        x_norm = self.group_norm(x_conv)
        x_tanh = torch.tanh(x_norm)
        x_hard_swish = F.hardswish(x_tanh)
        x_res = x_conv + x_hard_swish
        return torch.logsumexp(x_res, dim=1, keepdim=True)
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 2.79 ms; the PyTorch reference (with TF32 enabled) takes 9.69 ms; speedup 3.473x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| sm80_xmma_fprop_implicit_gemm_tf32f32_tf32f32_f32_nhwckrsc_nchw_tilesize128x128x16_stage4_ | 1027.2 | 1 |
| void at::native::elementwise_kernel<128, 2, at::native::gpu_kernel_impl_nocast<at::native: | 777.9 | 1 |
| fused_act_lse_vec4_kernel(float const*, float const*, float const*, float const*, float co | 410.6 | 1 |
| group_stats_kernel(float const*, float*, float*, int, int, int, int, float) | 385.0 | 1 |
| void cudnn::engines_precompiled::nchwToNhwcKernel<float, float, float, false, true, (cudnn | 147.7 | 2 |

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.
