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
# 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
from torch.utils.cpp_extension import load_inline

# Fused exclusive-cumsum kernel (improved over the previous best):
#   * One thread block per row; each thread owns TWO float4 groups per tile
#     (tile = 4096 elements) -> 8 tiles per 32768-element row, so few
#     barriers per row.
#   * Register double-buffered prefetch of the next tile with streaming
#     cache hints (__ldcs/__stcs): the data is touched exactly once.
#   * Block-wide scan in double precision via warp shuffles + one shared
#     array of per-warp totals (both groups' warp scans are issued before a
#     single barrier to minimize sync count: 3 barriers per tile).
#   * Results are staged in shared memory so the global stores are fully
#     coalesced even though output rows are only 4-byte aligned
#     (row length N+1 is odd).
#   * __launch_bounds__(512, 3) keeps 3 resident blocks per SM so barriers
#     and memory latency overlap across blocks.
exclusive_cumsum_cuda_source = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <cuda_runtime.h>

#define NT 512
#define NWARP (NT / 32)

__device__ __forceinline__ double warp_incl_scan_d(double v, const int lane) {
    #pragma unroll
    for (int o = 1; o < 32; o <<= 1) {
        double y = __shfl_up_sync(0xffffffffu, v, o);
        if (lane >= o) v += y;
    }
    return v;
}

__global__ void __launch_bounds__(NT, 3)
exclusive_cumsum_rows_kernel(const float* __restrict__ x,
                             float* __restrict__ out,
                             const int n4,   // N / 4
                             const int N) {  // input row length
    __shared__ double wsum[2][NWARP];
    __shared__ float stage[NT * 8];

    const long row = blockIdx.x;
    const float4* __restrict__ in_row = reinterpret_cast<const float4*>(x + row * (long)N);
    float* __restrict__ out_row = out + row * (long)(N + 1);

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

    const int g4 = n4 >> 9;        // full 1024-float4 (4096-element) tiles
    const int r4 = n4 & 1023;      // leftover float4 count (< 1024)

    double carry = 0.0;

    float4 a = make_float4(0.f, 0.f, 0.f, 0.f);
    float4 b = make_float4(0.f, 0.f, 0.f, 0.f);
    if (g4 > 0) {
        a = __ldcs(in_row + tid);
        b = __ldcs(in_row + NT + tid);
    }

    for (int t = 0; t < g4; ++t) {
        const float4 va = a;
        const float4 vb = b;
        const int nt = t + 1;
        if (nt < g4) {
            a = __ldcs(in_row + (nt << 10) + tid);
            b = __ldcs(in_row + (nt << 10) + NT + tid);
        }

        const double sa = (double)va.x + (double)va.y + (double)va.z + (double)va.w;
        const double sb = (double)vb.x + (double)vb.y + (double)vb.z + (double)vb.w;
        const double inclA = warp_incl_scan_d(sa, lane);
        const double inclB = warp_incl_scan_d(sb, lane);
        if (lane == 31) {
            wsum[0][warp] = inclA;
            wsum[1][warp] = inclB;
        }
        __syncthreads();
        if (warp == 0) {
            double wa = (lane < NWARP) ? wsum[0][lane] : 0.0;
            double wb = (lane < NWARP) ? wsum[1][lane] : 0.0;
            wa = warp_incl_scan_d(wa, lane);
            wb = warp_incl_scan_d(wb, lane);
            if (lane < NWARP) {
                wsum[0][lane] = wa;
                wsum[1][lane] = wb;
            }
        }
        __syncthreads();

        const double preA   = (warp == 0) ? 0.0 : wsum[0][warp - 1];
        const double totalA = wsum[0][NWARP - 1];
        const double preB   = (warp == 0) ? 0.0 : wsum[1][warp - 1];
        const double totalB = wsum[1][NWARP - 1];

        double run = carry + preA + (inclA - sa);
        stage[tid * 4 + 0] = (float)run;  run += (double)va.x;
        stage[tid * 4 + 1] = (float)run;  run += (double)va.y;
        stage[tid * 4 + 2] = (float)run;  run += (double)va.z;
        stage[tid * 4 + 3] = (float)run;

        run = carry + totalA + preB + (inclB - sb);
        stage[NT * 4 + tid * 4 + 0] = (float)run;  run += (double)vb.x;
        stage[NT * 4 + tid * 4 + 1] = (float)run;  run += (double)vb.y;
        stage[NT * 4 + tid * 4 + 2] = (float)run;  run += (double)vb.z;
        stage[NT * 4 + tid * 4 + 3] = (float)run;

        carry += totalA + totalB;
        __syncthreads();  // staging complete

        const int ob = t << 12;
        #pragma unroll
        for (int j = 0; j < 8; ++j) {
            __stcs(out_row + ob + j * NT + tid, stage[j * NT + tid]);
        }
        // No trailing barrier: the next tile's staging writes happen only
        // after its two scan barriers, by which time every thread has
        // finished the writeback above (program order).
    }

    if (r4 > 0) {
        const int base4 = g4 << 10;
        float4 va = make_float4(0.f, 0.f, 0.f, 0.f);
        float4 vb = make_float4(0.f, 0.f, 0.f, 0.f);
        if (tid < r4)          va = __ldcs(in_row + base4 + tid);
        if (NT + tid < r4)     vb = __ldcs(in_row + base4 + NT + tid);

        const double sa = (double)va.x + (double)va.y + (double)va.z + (double)va.w;
        const double sb = (double)vb.x + (double)vb.y + (double)vb.z + (double)vb.w;
        const double inclA = warp_incl_scan_d(sa, lane);
        const double inclB = warp_incl_scan_d(sb, lane);
        if (lane == 31) {
            wsum[0][warp] = inclA;
            wsum[1][warp] = inclB;
        }
        __syncthreads();
        if (warp == 0) {
            double wa = (lane < NWARP) ? wsum[0][lane] : 0.0;
            double wb = (lane < NWARP) ? wsum[1][lane] : 0.0;
            wa = warp_incl_scan_d(wa, lane);
            wb = warp_incl_scan_d(wb, lane);
            if (lane < NWARP) {
                wsum[0][lane] = wa;
                wsum[1][lane] = wb;
            }
        }
        __syncthreads();

        const double preA   = (warp == 0) ? 0.0 : wsum[0][warp - 1];
        const double totalA = wsum[0][NWARP - 1];
        const double preB   = (warp == 0) ? 0.0 : wsum[1][warp - 1];
        const double totalB = wsum[1][NWARP - 1];

        double run = carry + preA + (inclA - sa);
        stage[tid * 4 + 0] = (float)run;  run += (double)va.x;
        stage[tid * 4 + 1] = (float)run;  run += (double)va.y;
        stage[tid * 4 + 2] = (float)run;  run += (double)va.z;
        stage[tid * 4 + 3] = (float)run;

        run = carry + totalA + preB + (inclB - sb);
        stage[NT * 4 + tid * 4 + 0] = (float)run;  run += (double)vb.x;
        stage[NT * 4 + tid * 4 + 1] = (float)run;  run += (double)vb.y;
        stage[NT * 4 + tid * 4 + 2] = (float)run;  run += (double)vb.z;
        stage[NT * 4 + tid * 4 + 3] = (float)run;

        carry += totalA + totalB;
        __syncthreads();  // staging complete

        const int ob = g4 << 12;
        #pragma unroll
        for (int j = 0; j < 8; ++j) {
            const int k = ob + j * NT + tid;
            if (k < N) __stcs(out_row + k, stage[j * NT + tid]);
        }
    }

    // Append the row total at position N.
    if (tid == 0) __stcs(out_row + N, (float)carry);
}

torch::Tensor exclusive_cumsum_rows(torch::Tensor x, int64_t out_rows) {
    TORCH_CHECK(x.is_cuda(), "x must be a CUDA tensor");
    TORCH_CHECK(x.scalar_type() == at::kFloat, "x must be float32");
    TORCH_CHECK(x.is_contiguous(), "x must be contiguous");
    const int64_t N = x.size(-1);
    TORCH_CHECK(N > 0 && (N % 4) == 0, "last dim must be a positive multiple of 4");
    TORCH_CHECK(N < (1LL << 30), "last dim too large for this kernel");

    auto out = at::empty({out_rows, N + 1}, x.options());
    if (out_rows <= 0) return out;

    auto stream = at::cuda::getCurrentCUDAStream();
    exclusive_cumsum_rows_kernel<<<(int)out_rows, NT, 0, stream>>>(
        x.data_ptr<float>(), out.data_ptr<float>(), (int)(N >> 2), (int)N);
    return out;
}
"""

exclusive_cumsum_cpp_source = (
    "torch::Tensor exclusive_cumsum_rows(torch::Tensor x, int64_t out_rows);"
)

exclusive_cumsum_ext = load_inline(
    name="exclusive_cumsum_rows_ext_v2",
    cpp_sources=exclusive_cumsum_cpp_source,
    cuda_sources=exclusive_cumsum_cuda_source,
    functions=["exclusive_cumsum_rows"],
    verbose=False,
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
)


class ModelNew(nn.Module):
    """
    Optimized exclusive cumulative sum.

    For a CUDA float32 contiguous tensor with `dim` equal to the last
    dimension (the benchmark case), a single fused CUDA kernel computes the
    per-row exclusive cumsum (with the row total appended, last slice of
    dim-0 dropped) directly, reproducing the reference semantics without
    materializing the concatenated intermediate tensor. All other cases fall
    back to the exact reference implementation.
    """

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

    def forward(self, x):
        d = self.dim if self.dim >= 0 else self.dim + x.dim()
        if (
            x.is_cuda
            and x.dtype == torch.float32
            and x.is_contiguous()
            and x.dim() >= 2
            and d == x.dim() - 1
            and x.size(0) >= 1
            and x.size(-1) > 0
            and (x.size(-1) % 4) == 0
        ):
            n = x.size(-1)
            total_rows = x.numel() // n
            # Number of last-dim rows removed by the reference's `[:-1]`
            # slice on dimension 0.
            tail_rows = x.numel() // (x.size(0) * n)
            out_rows = total_rows - tail_rows
            out = exclusive_cumsum_ext.exclusive_cumsum_rows(
                x.view(total_rows, n), out_rows
            )
            out_shape = list(x.shape)
            out_shape[0] -= 1
            out_shape[-1] = n + 1
            return out.view(out_shape)

        # Exact reference fallback for all other configurations.
        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)
```
Result of evaluating it:
The solution compiled but failed at run time. Error output:
```
torch.AcceleratorError: CUDA error: an illegal memory access was encountered
Search for `cudaErrorIllegalAddress' in https://docs.nvidia.com/cuda/cuda-runtime-api/group__CUDART__TYPES.html for more information.
CUDA kernel errors might be asynchronously reported at some other API call, so the stacktrace below might be incorrect.
For debugging consider passing CUDA_LAUNCH_BLOCKING=1
The CUDA driver logged these messages, which may provide useful details:
Sticky error detected
Returning 700 (CUDA_ERROR_ILLEGAL_ADDRESS) from cuMemFree_v2
```

The fastest correct solution so far (2.527x speedup) is:
```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
from torch.utils.cpp_extension import load_inline

# Custom CUDA kernel: for each input row of length N it produces a row of
# length N+1 containing the exclusive cumulative sum with the row total
# appended (i.e. out[0]=0, out[k]=sum(x[0..k-1]), out[N]=sum(x[0..N-1])),
# processing only the first `out_rows` rows. This reproduces exactly the
# semantics of the reference (cat-with-zero + slice + cumsum) for the case
# dim == last dimension, while fusing everything into a single kernel and
# avoiding the huge intermediate concatenated tensor.
exclusive_cumsum_cuda_source = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <cuda_runtime.h>

#define NT 1024
#define NWARP (NT / 32)

__device__ __forceinline__ double warp_inclusive_scan_d(double v) {
    #pragma unroll
    for (int offset = 1; offset < 32; offset <<= 1) {
        double y = __shfl_up_sync(0xffffffffu, v, offset);
        if ((threadIdx.x & 31) >= offset) v += y;
    }
    return v;
}

// One thread block per row. Each thread handles one float4 per tile.
__global__ void __launch_bounds__(NT)
exclusive_cumsum_rows_kernel(const float* __restrict__ x,
                             float* __restrict__ out,
                             const long n4,   // N / 4
                             const long N) {  // row length of input
    __shared__ double warp_sums[NWARP];
    __shared__ float stage[NT * 4];

    const long row = blockIdx.x;
    const float4* __restrict__ in_row = reinterpret_cast<const float4*>(x + row * N);
    float* __restrict__ out_row = out + row * (N + 1);

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

    const long num_tiles = (n4 + NT - 1) / NT;
    double carry = 0.0;

    for (long t = 0; t < num_tiles; ++t) {
        const long idx = t * NT + tid;
        const bool valid = idx < n4;

        float4 v = make_float4(0.f, 0.f, 0.f, 0.f);
        if (valid) v = __ldg(in_row + idx);

        const double sum = (double)v.x + (double)v.y + (double)v.z + (double)v.w;

        // Block-wide exclusive scan of the per-thread sums (in double precision).
        double incl = warp_inclusive_scan_d(sum);
        if (lane == 31) warp_sums[warp] = incl;
        __syncthreads();
        if (warp == 0) {
            double w = (lane < NWARP) ? warp_sums[lane] : 0.0;
            w = warp_inclusive_scan_d(w);
            if (lane < NWARP) warp_sums[lane] = w;
        }
        __syncthreads();

        const double warp_prefix = (warp == 0) ? 0.0 : warp_sums[warp - 1];
        const double tile_total  = warp_sums[NWARP - 1];
        const double base        = carry + warp_prefix + (incl - sum);

        // Stage the four results so the global stores below are fully coalesced.
        stage[tid * 4 + 0] = (float)(base);
        stage[tid * 4 + 1] = (float)(base + (double)v.x);
        stage[tid * 4 + 2] = (float)(base + (double)v.x + (double)v.y);
        stage[tid * 4 + 3] = (float)(base + (double)v.x + (double)v.y + (double)v.z);

        // The very last thread appends the row total at position N.
        if (idx == n4 - 1) {
            out_row[N] = (float)(base + sum);
        }

        carry += tile_total;
        __syncthreads();

        const long tile_base = t * (long)(NT * 4);
        #pragma unroll
        for (int j = 0; j < 4; ++j) {
            const long k = tile_base + j * NT + tid;
            if (k < N) out_row[k] = stage[j * NT + tid];
        }
        __syncthreads();
    }
}

torch::Tensor exclusive_cumsum_rows(torch::Tensor x, int64_t out_rows) {
    TORCH_CHECK(x.is_cuda(), "x must be a CUDA tensor");
    TORCH_CHECK(x.scalar_type() == at::kFloat, "x must be float32");
    TORCH_CHECK(x.is_contiguous(), "x must be contiguous");
    const int64_t N = x.size(-1);
    TORCH_CHECK(N > 0 && (N % 4) == 0, "last dim must be a positive multiple of 4");

    auto out = at::empty({out_rows, N + 1}, x.options());
    if (out_rows <= 0) return out;

    const long n4 = (long)(N / 4);
    auto stream = at::cuda::getCurrentCUDAStream();
    exclusive_cumsum_rows_kernel<<<(int)out_rows, NT, 0, stream>>>(
        x.data_ptr<float>(), out.data_ptr<float>(), n4, (long)N);
    return out;
}
"""

exclusive_cumsum_cpp_source = (
    "torch::Tensor exclusive_cumsum_rows(torch::Tensor x, int64_t out_rows);"
)

exclusive_cumsum_ext = load_inline(
    name="exclusive_cumsum_rows_ext",
    cpp_sources=exclusive_cumsum_cpp_source,
    cuda_sources=exclusive_cumsum_cuda_source,
    functions=["exclusive_cumsum_rows"],
    verbose=False,
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
)


class ModelNew(nn.Module):
    """
    Optimized exclusive cumulative sum.

    For a CUDA float32 contiguous tensor with `dim` equal to the last
    dimension (the benchmark case), a single fused CUDA kernel computes the
    per-row exclusive cumsum (with the row total appended, last slice of
    dim-0 dropped) directly, reproducing the reference semantics without
    materializing the concatenated intermediate tensor. All other cases fall
    back to the exact reference implementation.
    """

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

    def forward(self, x):
        d = self.dim if self.dim >= 0 else self.dim + x.dim()
        if (
            x.is_cuda
            and x.dtype == torch.float32
            and x.is_contiguous()
            and x.dim() >= 2
            and d == x.dim() - 1
            and x.size(0) >= 1
            and x.size(-1) > 0
            and (x.size(-1) % 4) == 0
        ):
            n = x.size(-1)
            total_rows = x.numel() // n
            # Number of last-dim rows removed by the reference's `[:-1]`
            # slice on dimension 0.
            tail_rows = x.numel() // (x.size(0) * n)
            out_rows = total_rows - tail_rows
            out = exclusive_cumsum_ext.exclusive_cumsum_rows(
                x.view(total_rows, n), out_rows
            )
            out_shape = list(x.shape)
            out_shape[0] -= 1
            out_shape[-1] = n + 1
            return out.view(out_shape)

        # Exact reference fallback for all other configurations.
        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)
```

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.
