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 5 of 6 for this task. Your previous solution (turn 4) 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

# Warp-per-row exclusive cumsum (corrected and improved).
#
# For each input row of length N the kernel writes a row of length N+1:
#   out[0] = 0, out[k] = sum(x[0..k-1]), out[N] = row total.
# This matches the reference (cat a zero slab along dim, slice [:-1] on dim 0,
# cumsum along dim) when dim is the last dimension, while fusing everything
# into a single kernel with no concatenated intermediate tensor.
#
# Design notes:
#   * Each warp owns one row -> no __syncthreads anywhere; the cross-tile
#     carry stays in a register and all scans are warp shuffles.
#   * Per 256-element tile each lane loads two float4: "a" (elements
#     4*lane..4*lane+3) and "b" (elements 128+4*lane..131+4*lane). Two
#     SEPARATE warp scans (one over the a-sums, one over the b-sums) keep the
#     element ordering correct: the b-part base is carry + totalA + exclB.
#     (Scanning the combined a+b sums is wrong because the b elements lie
#     after every a element in row order.)
#   * Per-lane results are staged in shared memory with a conflict-free
#     padded layout (slot(e) = e + e/32, STAGE = 264) so the global stores
#     are fully coalesced despite each lane owning strided elements.
#   * One-tile software prefetch keeps global loads in flight, and streaming
#     cache hints (__ldcs/__stcs) are used since the data is touched once.
exclusive_cumsum_cuda_source = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <cuda_runtime.h>

#define NT 256
#define NWARP (NT / 32)
#define STAGE 264  // 256 floats + 8 floats of padding (1 per 32 elements)

__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, 6)
exclusive_cumsum_warp_row_kernel(const float* __restrict__ x,
                                 float* __restrict__ out,
                                 const long n4,   // N / 4
                                 const long N,    // input row length
                                 const long rows) {
    __shared__ float stage[NWARP][STAGE];

    const int lane = threadIdx.x & 31;
    const int warp = threadIdx.x >> 5;
    const long row = (long)blockIdx.x * NWARP + warp;
    if (row >= rows) return;

    const float4* __restrict__ in_row = reinterpret_cast<const float4*>(x + row * N);
    float* __restrict__ out_row = out + row * (N + 1);
    float* __restrict__ st = stage[warp];

    const long P  = n4 >> 6;         // full 64-float4 (256-element) tiles
    const int  r4 = (int)(n4 & 63);  // leftover float4 count (< 64)

    // Padded staging slots for this lane's two 4-element groups.
    const int s0 = 4 * lane + (lane >> 3);
    const int s1 = 132 + s0;

    double carry = 0.0;

    float4 na = make_float4(0.f, 0.f, 0.f, 0.f);
    float4 nb = make_float4(0.f, 0.f, 0.f, 0.f);
    if (P > 0) {
        na = __ldcs(in_row + lane);
        nb = __ldcs(in_row + 32 + lane);
    }

    for (long t = 0; t < P; ++t) {
        const float4 a = na;
        const float4 b = nb;
        const long nt = t + 1;
        if (nt < P) {
            na = __ldcs(in_row + (nt << 6) + lane);
            nb = __ldcs(in_row + (nt << 6) + 32 + lane);
        }

        // Scan the a-group (elements 4*lane..4*lane+3 of this tile).
        const double sa = (double)a.x + (double)a.y + (double)a.z + (double)a.w;
        const double inclA = warp_incl_scan_d(sa, lane);
        const double totalA = __shfl_sync(0xffffffffu, inclA, 31);

        double run = carry + (inclA - sa);
        st[s0 + 0] = (float)run;  run += (double)a.x;
        st[s0 + 1] = (float)run;  run += (double)a.y;
        st[s0 + 2] = (float)run;  run += (double)a.z;
        st[s0 + 3] = (float)run;  run += (double)a.w;

        // Scan the b-group (elements 128+4*lane..128+4*lane+3 of this tile).
        // All a elements precede all b elements, hence the totalA offset.
        const double sb = (double)b.x + (double)b.y + (double)b.z + (double)b.w;
        const double inclB = warp_incl_scan_d(sb, lane);
        const double totalB = __shfl_sync(0xffffffffu, inclB, 31);

        run = carry + totalA + (inclB - sb);
        st[s1 + 0] = (float)run;  run += (double)b.x;
        st[s1 + 1] = (float)run;  run += (double)b.y;
        st[s1 + 2] = (float)run;  run += (double)b.z;
        st[s1 + 3] = (float)run;

        carry += totalA + totalB;
        __syncwarp(0xffffffffu);

        // Coalesced writeback of the whole 256-element tile from staging.
        const long ob = t << 8;
        #pragma unroll
        for (int i = 0; i < 8; ++i) {
            __stcs(out_row + ob + i * 32 + lane, st[i * 33 + lane]);
        }
        __syncwarp(0xffffffffu);
    }

    if (r4 > 0) {
        const long base4 = P << 6;
        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 (lane < r4)      a = __ldcs(in_row + base4 + lane);
        if (32 + lane < r4) b = __ldcs(in_row + base4 + 32 + lane);

        const double sa = (double)a.x + (double)a.y + (double)a.z + (double)a.w;
        const double inclA = warp_incl_scan_d(sa, lane);
        const double totalA = __shfl_sync(0xffffffffu, inclA, 31);

        double run = carry + (inclA - sa);
        st[s0 + 0] = (float)run;  run += (double)a.x;
        st[s0 + 1] = (float)run;  run += (double)a.y;
        st[s0 + 2] = (float)run;  run += (double)a.z;
        st[s0 + 3] = (float)run;  run += (double)a.w;

        const double sb = (double)b.x + (double)b.y + (double)b.z + (double)b.w;
        const double inclB = warp_incl_scan_d(sb, lane);
        const double totalB = __shfl_sync(0xffffffffu, inclB, 31);

        run = carry + totalA + (inclB - sb);
        st[s1 + 0] = (float)run;  run += (double)b.x;
        st[s1 + 1] = (float)run;  run += (double)b.y;
        st[s1 + 2] = (float)run;  run += (double)b.z;
        st[s1 + 3] = (float)run;

        carry += totalA + totalB;
        __syncwarp(0xffffffffu);

        const long ob = P << 8;
        #pragma unroll
        for (int i = 0; i < 8; ++i) {
            const long e = ob + i * 32 + lane;
            if (e < N) __stcs(out_row + e, st[i * 33 + lane]);
        }
        __syncwarp(0xffffffffu);
    }

    // Append the row total at position N.
    if (lane == 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");

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

    const long n4 = (long)(N >> 2);
    const int blocks = (int)((out_rows + NWARP - 1) / NWARP);
    auto stream = at::cuda::getCurrentCUDAStream();
    exclusive_cumsum_warp_row_kernel<<<blocks, NT, 0, stream>>>(
        x.data_ptr<float>(), out.data_ptr<float>(), n4, (long)N, (long)out_rows);
    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_warp_row_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 warp-per-row 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 is correct on the test inputs. Mean runtime 6.94 ms; the PyTorch reference (with TF32 enabled) takes 16.2 ms; speedup 2.334x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| exclusive_cumsum_warp_row_kernel(float const*, float*, long, long, long) | 6908.9 | 1 |

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.
