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
import torch.nn.functional as F
import math

# From https://github.com/karpathy/minGPT/blob/master/mingpt/model.py

class Model(nn.Module):
    """
    Implementation of the GELU activation function currently in Google BERT repo (identical to OpenAI GPT).
    Reference: Gaussian Error Linear Units (GELU) paper: https://arxiv.org/abs/1606.08415
    """
    def __init__(self):
        super(Model, self).__init__()
    
    def forward(self, x):
        return 0.5 * x * (1.0 + torch.tanh(math.sqrt(2.0 / math.pi) * (x + 0.044715 * torch.pow(x, 3.0))))

batch_size = 8192
dim = 8192

def get_inputs():
    return [torch.rand(batch_size, dim)]

def get_init_inputs():
    return []
    ```
    
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 4 of 6 for this task. Your previous solution (turn 3) 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 <algorithm>
#include <cstdint>

__device__ __forceinline__ float gelu(float x) {
    const float u = 0.7978845608028654f *
                    fmaf(0.044715f * x, x * x, x);
    float t;
#if defined(__CUDA_ARCH__) && __CUDA_ARCH__ >= 750
    asm("tanh.approx.f32 %0, %1;" : "=f"(t) : "f"(u));
#else
    t = tanhf(u);
#endif
    return (0.5f * x) * (1.0f + t);
}

__device__ __forceinline__ float4 gelu4(float4 v) {
    v.x = gelu(v.x);
    v.y = gelu(v.y);
    v.z = gelu(v.z);
    v.w = gelu(v.w);
    return v;
}

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

template <typename Index, bool FullTiles>
__global__ __launch_bounds__(256)
void gelu_vector_kernel(
    const float* __restrict__ x,
    float* __restrict__ y,
    Index n
) {
    constexpr int threads = 256;
    constexpr int vectors_per_tile = threads * 4;
    const Index nv = n / 4;
    const float4* __restrict__ xv =
        reinterpret_cast<const float4*>(x);
    float4* __restrict__ yv = reinterpret_cast<float4*>(y);

    const Index tile_stride =
        static_cast<Index>(gridDim.x) * vectors_per_tile;

    for (Index tile = static_cast<Index>(blockIdx.x) * vectors_per_tile;
         tile < nv;
         tile += tile_stride) {
        const Index base = tile + threadIdx.x;
        float4 values[4];

        #pragma unroll
        for (int j = 0; j < 4; ++j) {
            const Index i = base + j * threads;
            if (FullTiles || i < nv)
                values[j] = xv[i];
        }

        #pragma unroll
        for (int j = 0; j < 4; ++j) {
            const Index i = base + j * threads;
            if (FullTiles || i < nv)
                values[j] = gelu4(values[j]);
        }

        #pragma unroll
        for (int j = 0; j < 4; ++j) {
            const Index i = base + j * threads;
            if (FullTiles || i < nv)
                streaming_store(yv + i, values[j]);
        }
    }

    if (!FullTiles && blockIdx.x == 0) {
        const Index i = nv * 4 + threadIdx.x;
        if (i < n)
            y[i] = gelu(x[i]);
    }
}

template <typename Index>
__global__ void gelu_scalar_kernel(
    const float* __restrict__ x,
    float* __restrict__ y,
    Index n
) {
    const Index stride = static_cast<Index>(blockDim.x) * gridDim.x;
    for (Index i = static_cast<Index>(blockIdx.x) * blockDim.x
                   + threadIdx.x;
         i < n;
         i += stride) {
        y[i] = gelu(x[i]);
    }
}

template <typename Index>
void launch_gelu(
    const float* x, float* y, int64_t n, cudaStream_t stream
) {
    constexpr int threads = 256;
    if ((reinterpret_cast<uintptr_t>(x) & 15) == 0) {
        constexpr int elements_per_tile = threads * 16;
        const int blocks = static_cast<int>(
            std::min<int64_t>(
                (n + elements_per_tile - 1) / elements_per_tile, 65535));

        if (n % elements_per_tile == 0) {
            gelu_vector_kernel<Index, true>
                <<<blocks, threads, 0, stream>>>(x, y, static_cast<Index>(n));
        } else {
            gelu_vector_kernel<Index, false>
                <<<blocks, threads, 0, stream>>>(x, y, static_cast<Index>(n));
        }
    } else {
        const int blocks = static_cast<int>(
            std::min<int64_t>((n + threads - 1) / threads, 4096));
        gelu_scalar_kernel<Index>
            <<<blocks, threads, 0, stream>>>(x, y, static_cast<Index>(n));
    }
}

torch::Tensor gelu_cuda(torch::Tensor input) {
    TORCH_CHECK(input.is_cuda(), "Expected a CUDA tensor");
    TORCH_CHECK(input.scalar_type() == at::kFloat, "Expected float32");

    const c10::cuda::CUDAGuard guard(input.device());
    auto x = input.contiguous();
    auto y = torch::empty_like(x);
    const int64_t n = x.numel();
    if (n == 0)
        return y;

    const cudaStream_t stream = at::cuda::getCurrentCUDAStream();
    if (n < (int64_t(1) << 30)) {
        launch_gelu<int>(
            x.data_ptr<float>(), y.data_ptr<float>(), n, stream);
    } else {
        launch_gelu<int64_t>(
            x.data_ptr<float>(), y.data_ptr<float>(), n, stream);
    }
    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return y;
}
"""

_gelu_extension = load_inline(
    name="fused_gelu_streaming_v3",
    cpp_sources="torch::Tensor gelu_cuda(torch::Tensor input);",
    cuda_sources=_cuda_source,
    functions=["gelu_cuda"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3", "--use_fast_math"],
    verbose=False,
)


class ModelNew(nn.Module):
    def __init__(self):
        super().__init__()

    def forward(self, x):
        return _gelu_extension.gelu_cuda(x)
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 0.408 ms; the PyTorch reference (with TF32 enabled) takes 3.56 ms; speedup 8.725x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| void gelu_vector_kernel<int, true>(float const*, float*, int) | 392.8 | 1 |

The fastest correct solution so far (8.725x speedup) is:
```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 <algorithm>
#include <cstdint>

__device__ __forceinline__ float gelu(float x) {
    const float x3 = x * x * x;
    return (0.5f * x) *
           (1.0f + tanhf(0.7978845608028654f * (x + 0.044715f * x3)));
}

__global__ void gelu_kernel(
    const float* __restrict__ x,
    float* __restrict__ y,
    int64_t n
) {
    const int64_t tid =
        static_cast<int64_t>(blockIdx.x) * blockDim.x + threadIdx.x;
    const int64_t stride =
        static_cast<int64_t>(blockDim.x) * gridDim.x;
    const int64_t vectors = n / 4;

    const float4* xv = reinterpret_cast<const float4*>(x);
    float4* yv = reinterpret_cast<float4*>(y);

    for (int64_t i = tid; i < vectors; i += stride) {
        float4 v = xv[i];
        v.x = gelu(v.x);
        v.y = gelu(v.y);
        v.z = gelu(v.z);
        v.w = gelu(v.w);
        yv[i] = v;
    }

    const int64_t tail = vectors * 4 + tid;
    if (tail < n) {
        y[tail] = gelu(x[tail]);
    }
}

__global__ void gelu_scalar_kernel(
    const float* __restrict__ x,
    float* __restrict__ y,
    int64_t n
) {
    const int64_t tid =
        static_cast<int64_t>(blockIdx.x) * blockDim.x + threadIdx.x;
    const int64_t stride =
        static_cast<int64_t>(blockDim.x) * gridDim.x;

    for (int64_t i = tid; i < n; i += stride) {
        y[i] = gelu(x[i]);
    }
}

torch::Tensor gelu_cuda(torch::Tensor input) {
    TORCH_CHECK(input.is_cuda(), "Expected a CUDA tensor");
    TORCH_CHECK(input.scalar_type() == at::kFloat, "Expected float32");

    const c10::cuda::CUDAGuard guard(input.device());
    auto x = input.contiguous();
    auto y = torch::empty_like(x);
    const int64_t n = x.numel();
    if (n == 0) return y;

    constexpr int threads = 256;
    const auto stream = at::cuda::getCurrentCUDAStream();

    if ((reinterpret_cast<uintptr_t>(x.data_ptr<float>()) & 15) == 0) {
        const int64_t work = (n + 3) / 4;
        const int blocks = static_cast<int>(
            std::min<int64_t>((work + threads - 1) / threads, 4096));
        gelu_kernel<<<blocks, threads, 0, stream>>>(
            x.data_ptr<float>(), y.data_ptr<float>(), n);
    } else {
        const int blocks = static_cast<int>(
            std::min<int64_t>((n + threads - 1) / threads, 4096));
        gelu_scalar_kernel<<<blocks, threads, 0, stream>>>(
            x.data_ptr<float>(), y.data_ptr<float>(), n);
    }

    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return y;
}
"""

_gelu_extension = load_inline(
    name="fused_vectorized_tanh_gelu",
    cpp_sources="torch::Tensor gelu_cuda(torch::Tensor input);",
    cuda_sources=_cuda_source,
    functions=["gelu_cuda"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3", "--use_fast_math"],
    verbose=False,
)


class ModelNew(nn.Module):
    def __init__(self):
        super().__init__()

    def forward(self, x):
        return _gelu_extension.gelu_cuda(x)
```

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.
