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):
    """
    A model that performs an exclusive cumulative sum (does not include the current element).

    Parameters:
        dim (int): The dimension along which to perform the exclusive cumulative sum.
    """

    def __init__(self, dim):
        super(Model, self).__init__()
        self.dim = dim

    def forward(self, x):
        exclusive_cumsum = torch.cat((torch.zeros_like(x.select(self.dim, 0).unsqueeze(self.dim)), x), dim=self.dim)[:-1]
        return torch.cumsum(exclusive_cumsum, dim=self.dim)

batch_size = 32768
input_shape = (32768,)
dim = 1

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

def get_init_inputs():
    return [dim]

    ```
    
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


_cpp_source = r"""
#include <torch/extension.h>
torch::Tensor exclusive_scan_cuda(torch::Tensor x);
"""

_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 = 1024;
constexpr int WARPS = THREADS / 32;
constexpr unsigned MASK = 0xffffffffu;

__device__ __forceinline__ float warp_scan(float value) {
    #pragma unroll
    for (int offset = 1; offset < 32; offset <<= 1) {
        asm volatile(
            "{\n"
            "  .reg .pred p;\n"
            "  .reg .f32 other;\n"
            "  shfl.sync.bfly.b32 other, %0, 0, 31, %2;\n"
            "  shfl.sync.up.b32 other|p, %0, %1, 0, %2;\n"
            "  @p add.f32 %0, %0, other;\n"
            "}\n"
            : "+f"(value)
            : "r"(offset), "r"(MASK));
    }
    return value;
}

template<int ROUNDS>
__global__ __launch_bounds__(THREADS, 1)
void row_scan_kernel(
    const float* __restrict__ input,
    float* __restrict__ output)
{
    constexpr int COLUMNS = THREADS * ROUNDS * 4;
    constexpr int WARP_COLUMNS = 32 * ROUNDS * 4;
    constexpr int OUTPUT_STRIDE = COLUMNS + 4;

    const int lane = threadIdx.x & 31;
    const int warp = threadIdx.x >> 5;
    const int64_t row = blockIdx.x;

    const float4* src = reinterpret_cast<const float4*>(
        input + row * COLUMNS + warp * WARP_COLUMNS);

    float4 prefixes[ROUNDS];
    float warp_total = 0.0f;

    #pragma unroll
    for (int i = 0; i < ROUNDS; ++i) {
        const float4 v = src[i * 32 + lane];
        const float xy = v.x + v.y;
        const float xyz = xy + v.z;
        const float inclusive = warp_scan(xyz + v.w);
        const float previous = __shfl_up_sync(MASK, inclusive, 1);
        const float base = warp_total + (lane == 0 ? 0.0f : previous);

        prefixes[i] = make_float4(
            base,
            base + v.x,
            base + xy,
            base + xyz);

        warp_total += __shfl_sync(MASK, inclusive, 31);
    }

    __shared__ float warp_offsets[WARPS];
    if (lane == 31)
        warp_offsets[warp] = warp_total;
    __syncthreads();

    float* output_row = output + row * OUTPUT_STRIDE;

    if (warp == 0) {
        const float inclusive = warp_scan(warp_offsets[lane]);
        const float previous = __shfl_up_sync(MASK, inclusive, 1);
        warp_offsets[lane] = lane == 0 ? 0.0f : previous;

        if (lane == 31)
            output_row[COLUMNS] = inclusive;
    }
    __syncthreads();

    const float offset = warp_offsets[warp];
    float4* dst = reinterpret_cast<float4*>(
        output_row + warp * WARP_COLUMNS);

    #pragma unroll
    for (int i = 0; i < ROUNDS; ++i) {
        const float4 p = prefixes[i];
        dst[i * 32 + lane] = make_float4(
            offset + p.x,
            offset + p.y,
            offset + p.z,
            offset + p.w);
    }
}

torch::Tensor exclusive_scan_cuda(torch::Tensor x) {
    const c10::cuda::CUDAGuard guard(x.device());
    const int64_t rows = x.size(0) - 1;
    const int64_t columns = x.size(1);

    auto output = torch::empty_strided(
        {rows, columns + 1},
        {columns + 4, 1},
        x.options());

    if (rows == 0)
        return output;

    const auto stream = at::cuda::getCurrentCUDAStream();
    const dim3 grid(static_cast<unsigned int>(rows));

    switch (columns) {
        case 4096:
            row_scan_kernel<1><<<grid, THREADS, 0, stream>>>(
                x.data_ptr<float>(), output.data_ptr<float>());
            break;
        case 8192:
            row_scan_kernel<2><<<grid, THREADS, 0, stream>>>(
                x.data_ptr<float>(), output.data_ptr<float>());
            break;
        case 16384:
            row_scan_kernel<4><<<grid, THREADS, 0, stream>>>(
                x.data_ptr<float>(), output.data_ptr<float>());
            break;
        case 32768:
            row_scan_kernel<8><<<grid, THREADS, 0, stream>>>(
                x.data_ptr<float>(), output.data_ptr<float>());
            break;
        default:
            TORCH_CHECK(false, "Unsupported row length");
    }

    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return output;
}
"""

_extension = load_inline(
    name="exclusive_cumsum_prescan_v5",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["exclusive_scan_cuda"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
    verbose=False,
)


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

    def forward(self, x):
        if (
            x.is_cuda
            and x.dtype == torch.float32
            and x.ndim == 2
            and self.dim in (1, -1)
            and x.is_contiguous()
            and x.data_ptr() % 16 == 0
            and 0 < x.size(0) <= 2147483647
            and x.size(1) in (4096, 8192, 16384, 32768)
            and not (torch.is_grad_enabled() and x.requires_grad)
        ):
            return _extension.exclusive_scan_cuda(x)

        shifted = torch.cat(
            (
                torch.zeros_like(x.select(self.dim, 0).unsqueeze(self.dim)),
                x,
            ),
            dim=self.dim,
        )[:-1]
        return torch.cumsum(shifted, dim=self.dim)
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 6.32 ms; the PyTorch reference (with TF32 enabled) takes 16.2 ms; speedup 2.563x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| void row_scan_kernel<8>(float const*, float*) | 6320.0 | 1 |

The fastest correct solution so far (2.576x speedup) is:
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline


_cpp_source = r"""
#include <torch/extension.h>
torch::Tensor exclusive_scan_cuda(torch::Tensor x);
"""

_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 = 1024;
constexpr int WARPS = THREADS / 32;
constexpr unsigned MASK = 0xffffffffu;

__device__ __forceinline__ float warp_inclusive_sum(float value) {
    const int lane = threadIdx.x & 31;
    #pragma unroll
    for (int delta = 1; delta < 32; delta <<= 1) {
        float other = __shfl_up_sync(MASK, value, delta);
        if (lane >= delta)
            value += other;
    }
    return value;
}

template<int ROUNDS>
__global__ __launch_bounds__(THREADS, 1)
void row_scan_kernel(
    const float* __restrict__ input,
    float* __restrict__ output)
{
    constexpr int COLUMNS = THREADS * ROUNDS * 4;
    constexpr int WARP_COLUMNS = 32 * ROUNDS * 4;
    constexpr int OUTPUT_STRIDE = COLUMNS + 4;

    const int row = blockIdx.x;
    const int lane = threadIdx.x & 31;
    const int warp = threadIdx.x >> 5;

    const float4* input_vectors = reinterpret_cast<const float4*>(
        input + static_cast<int64_t>(row) * COLUMNS
              + warp * WARP_COLUMNS);

    float4 values[ROUNDS];
    float thread_total = 0.0f;

    #pragma unroll
    for (int i = 0; i < ROUNDS; ++i) {
        const float4 v = input_vectors[i * 32 + lane];
        values[i] = v;
        thread_total += ((v.x + v.y) + v.z) + v.w;
    }

    #pragma unroll
    for (int delta = 16; delta > 0; delta >>= 1)
        thread_total += __shfl_down_sync(MASK, thread_total, delta);

    __shared__ float warp_offsets[WARPS];
    if (lane == 0)
        warp_offsets[warp] = thread_total;
    __syncthreads();

    float* output_row =
        output + static_cast<int64_t>(row) * OUTPUT_STRIDE;

    if (warp == 0) {
        const float inclusive = warp_inclusive_sum(warp_offsets[lane]);
        const float previous = __shfl_up_sync(MASK, inclusive, 1);
        warp_offsets[lane] = lane == 0 ? 0.0f : previous;
        if (lane == 31)
            output_row[COLUMNS] = inclusive;
    }
    __syncthreads();

    float carry = warp_offsets[warp];
    float4* output_vectors = reinterpret_cast<float4*>(
        output_row + warp * WARP_COLUMNS);

    #pragma unroll
    for (int i = 0; i < ROUNDS; ++i) {
        const float4 v = values[i];
        const float total = ((v.x + v.y) + v.z) + v.w;
        const float inclusive = warp_inclusive_sum(total);
        const float previous = __shfl_up_sync(MASK, inclusive, 1);
        const float prefix = carry + (lane == 0 ? 0.0f : previous);

        float4 result;
        result.x = prefix;
        result.y = prefix + v.x;
        result.z = prefix + (v.x + v.y);
        result.w = prefix + ((v.x + v.y) + v.z);
        output_vectors[i * 32 + lane] = result;

        carry += __shfl_sync(MASK, inclusive, 31);
    }
}

torch::Tensor exclusive_scan_cuda(torch::Tensor x) {
    const c10::cuda::CUDAGuard guard(x.device());

    const int64_t rows = x.size(0) - 1;
    const int64_t columns = x.size(1);

    // Padding keeps every row aligned for vectorized exclusive-scan stores.
    auto output = torch::empty_strided(
        {rows, columns + 1}, {columns + 4, 1}, x.options());

    if (rows == 0)
        return output;

    const auto stream = at::cuda::getCurrentCUDAStream();
    const dim3 grid(static_cast<unsigned int>(rows));

    switch (columns) {
        case 4096:
            row_scan_kernel<1><<<grid, THREADS, 0, stream>>>(
                x.data_ptr<float>(), output.data_ptr<float>());
            break;
        case 8192:
            row_scan_kernel<2><<<grid, THREADS, 0, stream>>>(
                x.data_ptr<float>(), output.data_ptr<float>());
            break;
        case 16384:
            row_scan_kernel<4><<<grid, THREADS, 0, stream>>>(
                x.data_ptr<float>(), output.data_ptr<float>());
            break;
        case 32768:
            row_scan_kernel<8><<<grid, THREADS, 0, stream>>>(
                x.data_ptr<float>(), output.data_ptr<float>());
            break;
        default:
            TORCH_CHECK(false, "Unsupported row length");
    }

    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return output;
}
"""

_extension = load_inline(
    name="exclusive_cumsum_single_pass_v2",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["exclusive_scan_cuda"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
    verbose=False,
)


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

    def forward(self, x):
        if (
            x.is_cuda
            and x.dtype == torch.float32
            and x.ndim == 2
            and self.dim in (1, -1)
            and x.is_contiguous()
            and 0 < x.size(0) <= 2147483647
            and x.size(1) in (4096, 8192, 16384, 32768)
            and not (torch.is_grad_enabled() and x.requires_grad)
        ):
            # The original [:-1] removes the last row, not the last column.
            return _extension.exclusive_scan_cuda(x)

        shifted = torch.cat(
            (
                torch.zeros_like(x.select(self.dim, 0).unsqueeze(self.dim)),
                x,
            ),
            dim=self.dim,
        )[:-1]
        return torch.cumsum(shifted, dim=self.dim)
```

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.
