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 6 of 6 for this task. Your previous solution (turn 5) 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 = 16384;
constexpr int VECTORS_PER_THREAD = CHUNK / (4 * THREADS);
constexpr int ROW_TILE = 16;
constexpr int COLUMN_THREADS = 128;
constexpr int ROW_LANES = THREADS / COLUMN_THREADS;

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

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;
    const float count = a.count + b.count;
    const float delta = b.mean - a.mean;
    const 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 value)
{
    #pragma unroll
    for (int offset = 16; offset > 0; offset >>= 1) {
        Moments other{
            __shfl_down_sync(0xffffffff, value.mean, offset),
            __shfl_down_sync(0xffffffff, value.m2, offset),
            __shfl_down_sync(0xffffffff, value.count, offset)
        };
        if ((threadIdx.x & 31) + offset < 32)
            value = merge(value, other);
    }
    return value;
}

__global__ void partial_moments(
    const float4* __restrict__ x,
    float2* __restrict__ partial,
    int64_t vectors_per_row,
    int chunks)
{
    const int64_t base =
        static_cast<int64_t>(blockIdx.y) * vectors_per_row
        + static_cast<int64_t>(blockIdx.x) * (CHUNK / 4)
        + threadIdx.x;

    const float4 first = x[base];
    const float anchor = first.x;
    const float dy = first.y - anchor;
    const float dz = first.z - anchor;
    const float dw = first.w - anchor;

    float4 sum = make_float4(0.f, dy, dz, dw);
    float4 square = make_float4(0.f, dy * dy, dz * dz, dw * dw);

    #pragma unroll
    for (int k = 1; k < VECTORS_PER_THREAD; ++k) {
        const float4 v = x[base + k * THREADS];
        const float a = v.x - anchor;
        const float b = v.y - anchor;
        const float c = v.z - anchor;
        const float d = v.w - anchor;

        sum.x += a;
        sum.y += b;
        sum.z += c;
        sum.w += d;

        square.x = fmaf(a, a, square.x);
        square.y = fmaf(b, b, square.y);
        square.z = fmaf(c, c, square.z);
        square.w = fmaf(d, d, square.w);
    }

    constexpr float reciprocal = 1.f / (4 * VECTORS_PER_THREAD);
    const float s = (sum.x + sum.y) + (sum.z + sum.w);
    const float q = (square.x + square.y) + (square.z + square.w);

    float2 total = make_float2(
        anchor + s * reciprocal,
        fmaxf(fmaf(-s, s * reciprocal, q), 0.f));

    float count = CHUNK / THREADS;
    #pragma unroll
    for (int offset = 16; offset > 0; offset >>= 1) {
        const 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 shared[THREADS / 32];
    if ((threadIdx.x & 31) == 0)
        shared[threadIdx.x >> 5] = total;
    __syncthreads();

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

        #pragma unroll
        for (int offset = THREADS / 64; offset > 0; offset >>= 1) {
            const 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>(blockIdx.y) * chunks
                    + blockIdx.x] = total;
        }
    }
}

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

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

    value = warp_reduce(value);

    __shared__ Moments shared[THREADS / 32];
    if ((threadIdx.x & 31) == 0)
        shared[threadIdx.x >> 5] = value;
    __syncthreads();

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

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

template<bool FULL_TILE>
__global__ void normalize_affine(
    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)
{
    __shared__ float2 row_stats[ROW_TILE];

    const int first_row = blockIdx.y * ROW_TILE;
    if (threadIdx.x < ROW_TILE) {
        const int row = first_row + threadIdx.x;
        row_stats[threadIdx.x] = (FULL_TILE || row < rows)
            ? statistics[row]
            : make_float2(0.f, 0.f);
    }
    __syncthreads();

    const int column_lane = threadIdx.x % COLUMN_THREADS;
    const int row_lane = threadIdx.x / COLUMN_THREADS;
    const int64_t column =
        static_cast<int64_t>(blockIdx.x) * COLUMN_THREADS + column_lane;

    if (column >= vectors_per_row) return;

    const float4 w = weight[column];
    const float4 b = bias[column];

    #pragma unroll 4
    for (int r = 0; r < ROW_TILE / ROW_LANES; ++r) {
        const int local_row = r * ROW_LANES + row_lane;
        const int row = first_row + local_row;

        if (FULL_TILE || row < rows) {
            const int64_t index =
                static_cast<int64_t>(row) * vectors_per_row + column;
            const float4 v = x[index];
            const float2 s = row_stats[local_row];

            float4 result;
            result.x = ((v.x - s.x) * s.y) * w.x + b.x;
            result.y = ((v.y - s.x) * s.y) * w.y + b.y;
            result.z = ((v.z - s.x) * s.y) * w.z + b.z;
            result.w = ((v.w - s.x) * s.y) * w.w + b.w;

            __stcs(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());

    const int64_t columns = weight.numel();
    const int rows = static_cast<int>(x.numel() / columns);
    const int chunks = static_cast<int>(columns / CHUNK);
    const int64_t vectors_per_row = columns / 4;

    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();

    const float4* xp =
        reinterpret_cast<const float4*>(x.data_ptr<float>());
    const float4* wp =
        reinterpret_cast<const float4*>(weight.data_ptr<float>());
    const float4* bp =
        reinterpret_cast<const float4*>(bias.data_ptr<float>());
    float4* op =
        reinterpret_cast<float4*>(output.data_ptr<float>());
    float2* pp =
        reinterpret_cast<float2*>(partial.data_ptr<float>());
    float2* sp =
        reinterpret_cast<float2*>(statistics.data_ptr<float>());

    partial_moments<<<dim3(chunks, rows), THREADS, 0, stream>>>(
        xp, pp, vectors_per_row, chunks);
    C10_CUDA_KERNEL_LAUNCH_CHECK();

    finish_moments<<<rows, THREADS, 0, stream>>>(
        pp, sp, chunks, columns, static_cast<float>(epsilon));
    C10_CUDA_KERNEL_LAUNCH_CHECK();

    const dim3 grid(
        static_cast<unsigned>(
            (vectors_per_row + COLUMN_THREADS - 1) / COLUMN_THREADS),
        (rows + ROW_TILE - 1) / ROW_TILE);

    if (rows % ROW_TILE == 0) {
        normalize_affine<true><<<grid, THREADS, 0, stream>>>(
            xp, wp, bp, sp, op, vectors_per_row, rows);
    } else {
        normalize_affine<false><<<grid, THREADS, 0, stream>>>(
            xp, wp, bp, sp, op, vectors_per_row, rows);
    }
    C10_CUDA_KERNEL_LAUNCH_CHECK();

    return output;
}
"""

_extension = load_inline(
    name="large_layer_norm_centered_interleaved_v5",
    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 >= 16384
            and self.columns % 16384 == 0
            and self.columns // 16384 <= 2147483647
            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)

        if torch.is_grad_enabled() and (
            x.requires_grad or weight.requires_grad or bias.requires_grad
        ):
            return _LayerNormFunction.apply(
                x, weight, bias, self.normalized_shape, self.ln.eps
            )

        return _extension.large_layer_norm_cuda(
            x, weight, bias, self.ln.eps
        )
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 0.654 ms; the PyTorch reference (with TF32 enabled) takes 10.1 ms; speedup 15.443x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| void normalize_affine<true>(float4 const*, float4 const*, float4 const*, float2 const*, fl | 405.4 | 1 |
| partial_moments(float4 const*, float2*, long, int) | 202.5 | 1 |
| finish_moments(float2 const*, float2*, int, long, float) | 4.8 | 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.
