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):
    """
    Simple model that performs Layer Normalization.
    """
    def __init__(self, normalized_shape: tuple):
        """
        Initializes the LayerNorm layer.

        Args:
            normalized_shape (tuple): Shape of the input tensor to be normalized.
        """
        super(Model, self).__init__()
        self.ln = nn.LayerNorm(normalized_shape=normalized_shape)

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        """
        Applies Layer Normalization to the input tensor.

        Args:
            x (torch.Tensor): Input tensor of shape (*, normalized_shape).

        Returns:
            torch.Tensor: Output tensor with Layer Normalization applied, same shape as input.
        """
        return self.ln(x)

batch_size = 16
features = 64
dim1 = 256
dim2 = 256

def get_inputs():
    x = torch.rand(batch_size, features, dim1, dim2)
    return [x]

def get_init_inputs():
    return [(features, dim1, dim2)]
    ```
    
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>

constexpr int THREADS = 256;
constexpr int CHUNK = 4096;
constexpr int ROW_TILE = 8;

struct Moments {
    float mean;
    float m2;
    float count;
};

__device__ __forceinline__ Moments merge(Moments a, Moments b) {
    if (a.count == 0.f) return b;
    if (b.count == 0.f) return a;
    float count = a.count + b.count;
    float delta = b.mean - a.mean;
    float ratio = b.count / count;
    return {
        a.mean + delta * ratio,
        a.m2 + b.m2 + delta * delta * a.count * ratio,
        count
    };
}

__device__ __forceinline__ Moments warp_reduce(Moments v) {
    #pragma unroll
    for (int offset = 16; offset > 0; offset >>= 1) {
        Moments other;
        other.mean = __shfl_down_sync(0xffffffff, v.mean, offset);
        other.m2 = __shfl_down_sync(0xffffffff, v.m2, offset);
        other.count = __shfl_down_sync(0xffffffff, v.count, offset);
        if ((threadIdx.x & 31) + offset < 32)
            v = merge(v, other);
    }
    return v;
}

__device__ __forceinline__ Moments block_reduce(Moments v) {
    __shared__ Moments shared[THREADS / 32];
    v = warp_reduce(v);
    if ((threadIdx.x & 31) == 0)
        shared[threadIdx.x >> 5] = v;
    __syncthreads();

    if (threadIdx.x < 32) {
        v = threadIdx.x < THREADS / 32
            ? shared[threadIdx.x]
            : Moments{0.f, 0.f, 0.f};
        v = warp_reduce(v);
    }
    return v;
}

__device__ __forceinline__ float2 merge_equal(
    float2 a, float2 b, float count)
{
    float delta = b.x - a.x;
    return make_float2(
        a.x + delta * 0.5f,
        a.y + b.y + delta * delta * (count * 0.5f));
}

__global__ void partial_moments(
    const float* __restrict__ x,
    float2* __restrict__ partial,
    int64_t columns,
    int chunks)
{
    int chunk = blockIdx.x;
    int row = blockIdx.y;
    int64_t base = static_cast<int64_t>(row) * columns
                 + static_cast<int64_t>(chunk) * CHUNK;
    const float4* input = reinterpret_cast<const float4*>(x + base);

    float4 mean = make_float4(0.f, 0.f, 0.f, 0.f);
    float4 m2 = make_float4(0.f, 0.f, 0.f, 0.f);

    #pragma unroll
    for (int k = 0; k < CHUNK / (THREADS * 4); ++k) {
        float4 v = input[threadIdx.x + k * THREADS];
        float reciprocal = 1.f / static_cast<float>(k + 1);

        float dx = v.x - mean.x;
        float dy = v.y - mean.y;
        float dz = v.z - mean.z;
        float dw = v.w - mean.w;

        mean.x += dx * reciprocal;
        mean.y += dy * reciprocal;
        mean.z += dz * reciprocal;
        mean.w += dw * reciprocal;

        m2.x += dx * (v.x - mean.x);
        m2.y += dy * (v.y - mean.y);
        m2.z += dz * (v.z - mean.z);
        m2.w += dw * (v.w - mean.w);
    }

    constexpr float component_count = CHUNK / (THREADS * 4);
    float2 a = merge_equal(
        make_float2(mean.x, m2.x),
        make_float2(mean.y, m2.y),
        component_count);
    float2 b = merge_equal(
        make_float2(mean.z, m2.z),
        make_float2(mean.w, m2.w),
        component_count);
    float2 total = merge_equal(a, b, component_count * 2.f);

    float count = CHUNK / THREADS;
    #pragma unroll
    for (int offset = 16; offset > 0; offset >>= 1) {
        float2 other = make_float2(
            __shfl_down_sync(0xffffffff, total.x, offset),
            __shfl_down_sync(0xffffffff, total.y, offset));
        total = merge_equal(total, other, count);
        count *= 2.f;
    }

    __shared__ float2 warp_results[THREADS / 32];
    if ((threadIdx.x & 31) == 0)
        warp_results[threadIdx.x >> 5] = total;
    __syncthreads();

    if (threadIdx.x < 32) {
        total = threadIdx.x < THREADS / 32
            ? warp_results[threadIdx.x]
            : make_float2(0.f, 0.f);

        count = CHUNK / (THREADS / 32);
        #pragma unroll
        for (int offset = THREADS / 64; offset > 0; offset >>= 1) {
            float2 other = make_float2(
                __shfl_down_sync(0xffffffff, total.x, offset),
                __shfl_down_sync(0xffffffff, total.y, offset));
            total = merge_equal(total, other, count);
            count *= 2.f;
        }

        if (threadIdx.x == 0)
            partial[static_cast<int64_t>(row) * chunks + chunk] = total;
    }
}

__global__ void finish_moments(
    const float2* __restrict__ partial,
    float2* __restrict__ statistics,
    int chunks,
    int64_t columns,
    float epsilon)
{
    int row = blockIdx.x;
    Moments value{0.f, 0.f, 0.f};

    for (int i = threadIdx.x; i < chunks; i += THREADS) {
        float2 p = partial[static_cast<int64_t>(row) * chunks + i];
        value = merge(value, Moments{p.x, p.y, static_cast<float>(CHUNK)});
    }

    value = block_reduce(value);

    if (threadIdx.x == 0) {
        float variance = fmaxf(value.m2 / static_cast<float>(columns), 0.f);
        statistics[row] = make_float2(
            value.mean, rsqrtf(variance + epsilon));
    }
}

template<bool FULL_TILE>
__global__ void normalize_affine_tiled(
    const float4* __restrict__ x,
    const float4* __restrict__ weight,
    const float4* __restrict__ bias,
    const float2* __restrict__ statistics,
    float4* __restrict__ output,
    int64_t vectors_per_row,
    int rows)
{
    int64_t column =
        static_cast<int64_t>(blockIdx.x) * THREADS + threadIdx.x;
    if (column >= vectors_per_row) return;

    // Reuse the affine parameters across eight independent rows.
    float4 w = weight[column];
    float4 b = bias[column];
    int first_row = blockIdx.y * ROW_TILE;

    #pragma unroll
    for (int r = 0; r < ROW_TILE; ++r) {
        int row = first_row + r;
        if (FULL_TILE || row < rows) {
            int64_t index =
                static_cast<int64_t>(row) * vectors_per_row + column;
            float4 v = x[index];
            float2 stats = statistics[row];

            float4 result;
            result.x = ((v.x - stats.x) * stats.y) * w.x + b.x;
            result.y = ((v.y - stats.x) * stats.y) * w.y + b.y;
            result.z = ((v.z - stats.x) * stats.y) * w.z + b.z;
            result.w = ((v.w - stats.x) * stats.y) * w.w + b.w;
            output[index] = result;
        }
    }
}

torch::Tensor large_layer_norm_cuda(
    torch::Tensor x,
    torch::Tensor weight,
    torch::Tensor bias,
    double epsilon)
{
    c10::cuda::CUDAGuard guard(x.device());

    int64_t columns = weight.numel();
    int rows = static_cast<int>(x.numel() / columns);
    int chunks = static_cast<int>(columns / CHUNK);

    auto output = torch::empty_like(x);
    auto partial = torch::empty({rows, chunks, 2}, x.options());
    auto statistics = torch::empty({rows, 2}, x.options());

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

    partial_moments<<<dim3(chunks, rows), THREADS, 0, stream>>>(
        x.data_ptr<float>(),
        reinterpret_cast<float2*>(partial.data_ptr<float>()),
        columns,
        chunks);
    C10_CUDA_KERNEL_LAUNCH_CHECK();

    finish_moments<<<rows, THREADS, 0, stream>>>(
        reinterpret_cast<const float2*>(partial.data_ptr<float>()),
        reinterpret_cast<float2*>(statistics.data_ptr<float>()),
        chunks,
        columns,
        static_cast<float>(epsilon));
    C10_CUDA_KERNEL_LAUNCH_CHECK();

    int64_t vectors_per_row = columns / 4;
    dim3 grid(
        (vectors_per_row + THREADS - 1) / THREADS,
        (rows + ROW_TILE - 1) / ROW_TILE);

    const float4* input_ptr =
        reinterpret_cast<const float4*>(x.data_ptr<float>());
    const float4* weight_ptr =
        reinterpret_cast<const float4*>(weight.data_ptr<float>());
    const float4* bias_ptr =
        reinterpret_cast<const float4*>(bias.data_ptr<float>());
    const float2* stats_ptr =
        reinterpret_cast<const float2*>(statistics.data_ptr<float>());
    float4* output_ptr =
        reinterpret_cast<float4*>(output.data_ptr<float>());

    if (rows % ROW_TILE == 0) {
        normalize_affine_tiled<true><<<grid, THREADS, 0, stream>>>(
            input_ptr, weight_ptr, bias_ptr, stats_ptr,
            output_ptr, vectors_per_row, rows);
    } else {
        normalize_affine_tiled<false><<<grid, THREADS, 0, stream>>>(
            input_ptr, weight_ptr, bias_ptr, stats_ptr,
            output_ptr, vectors_per_row, rows);
    }
    C10_CUDA_KERNEL_LAUNCH_CHECK();

    return output;
}
"""

_extension = load_inline(
    name="large_layer_norm_row_tiled_v2",
    cpp_sources=r"""
        #include <torch/extension.h>
        torch::Tensor large_layer_norm_cuda(
            torch::Tensor x,
            torch::Tensor weight,
            torch::Tensor bias,
            double epsilon);
    """,
    cuda_sources=_cuda_source,
    functions=["large_layer_norm_cuda"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
    verbose=False,
)


class _LayerNormFunction(torch.autograd.Function):
    @staticmethod
    def forward(ctx, x, weight, bias, normalized_shape, epsilon):
        ctx.normalized_shape = normalized_shape
        ctx.epsilon = epsilon
        ctx.save_for_backward(x, weight, bias)
        return _extension.large_layer_norm_cuda(x, weight, bias, epsilon)

    @staticmethod
    def backward(ctx, grad_output):
        x, weight, bias = ctx.saved_tensors
        _, mean, rstd = torch.ops.aten.native_layer_norm.default(
            x, ctx.normalized_shape, weight, bias, ctx.epsilon
        )
        dx, dw, db = torch.ops.aten.native_layer_norm_backward.default(
            grad_output.contiguous(),
            x,
            ctx.normalized_shape,
            mean,
            rstd,
            weight,
            bias,
            list(ctx.needs_input_grad[:3]),
        )
        return dx, dw, db, None, None


class ModelNew(nn.Module):
    def __init__(self, normalized_shape: tuple):
        super().__init__()
        self.ln = nn.LayerNorm(normalized_shape=normalized_shape)
        self.normalized_shape = tuple(self.ln.normalized_shape)
        self.columns = self.ln.weight.numel()

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        weight, bias = self.ln.weight, self.ln.bias
        supported = (
            x.is_cuda
            and x.dtype == torch.float32
            and x.is_contiguous()
            and weight.dtype == torch.float32
            and bias.dtype == torch.float32
            and weight.device == x.device
            and bias.device == x.device
            and weight.is_contiguous()
            and bias.is_contiguous()
            and tuple(x.shape[-len(self.normalized_shape):])
                == self.normalized_shape
            and self.columns >= 4096
            and self.columns % 4096 == 0
            and 0 < x.numel() // self.columns <= 65535
            and x.data_ptr() % 16 == 0
            and weight.data_ptr() % 16 == 0
            and bias.data_ptr() % 16 == 0
        )
        if not supported:
            return self.ln(x)

        return _LayerNormFunction.apply(
            x, weight, bias, self.normalized_shape, self.ln.eps
        )
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 0.672 ms; the PyTorch reference (with TF32 enabled) takes 10.1 ms; speedup 15.030x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| void normalize_affine_tiled<true>(float4 const*, float4 const*, float4 const*, float2 cons | 422.1 | 1 |
| partial_moments(float const*, float2*, long, int) | 202.8 | 1 |
| finish_moments(float2 const*, float2*, int, long, float) | 5.8 | 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.
