```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_cumsum_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 <cub/cub.cuh>
#include <cuda_runtime.h>
#include <cstdint>

constexpr int THREADS = 256;
constexpr int ITEMS = 16;
constexpr int TILE = THREADS * ITEMS;

__global__ void tile_sums_kernel(
    const float* __restrict__ x,
    float* __restrict__ sums,
    int64_t columns,
    int tiles)
{
    using Reduce = cub::BlockReduce<float, THREADS>;
    __shared__ typename Reduce::TempStorage storage;

    const int preceding_tiles = tiles - 1;
    const int64_t row = int64_t(blockIdx.x) / preceding_tiles;
    const int tile = blockIdx.x % preceding_tiles;
    const int64_t base = row * columns + int64_t(tile) * TILE;

    float total = 0.0f;
    #pragma unroll
    for (int i = 0; i < ITEMS; ++i) {
        total += x[base + threadIdx.x + i * THREADS];
    }

    const float aggregate = Reduce(storage).Sum(total);
    if (threadIdx.x == 0) {
        sums[row * preceding_tiles + tile] = aggregate;
    }
}

__global__ void scan_tiles_kernel(
    const float* __restrict__ x,
    const float* __restrict__ sums,
    float* __restrict__ output,
    int64_t columns,
    int tiles)
{
    using Load = cub::BlockLoad<
        float, THREADS, ITEMS, cub::BLOCK_LOAD_WARP_TRANSPOSE>;
    using Scan = cub::BlockScan<float, THREADS>;
    using Store = cub::BlockStore<
        float, THREADS, ITEMS, cub::BLOCK_STORE_WARP_TRANSPOSE>;

    __shared__ union {
        typename Load::TempStorage load;
        typename Scan::TempStorage scan;
        typename Store::TempStorage store;
    } storage;
    __shared__ float prefix;

    const int64_t row = int64_t(blockIdx.x) / tiles;
    const int tile = blockIdx.x % tiles;
    const int64_t start = int64_t(tile) * TILE;
    const int valid = static_cast<int>(
        columns - start < TILE ? columns - start : TILE);

    if (threadIdx.x == 0) {
        double offset = 0.0;
        for (int i = 0; i < tile; ++i) {
            offset += sums[row * (tiles - 1) + i];
        }
        prefix = static_cast<float>(offset);
        if (tile == 0) {
            output[row * (columns + 1)] = 0.0f;
        }
    }

    float values[ITEMS];
    Load(storage.load).Load(
        x + row * columns + start, values, valid, 0.0f);
    __syncthreads();

    Scan(storage.scan).InclusiveSum(values, values);
    __syncthreads();

    #pragma unroll
    for (int i = 0; i < ITEMS; ++i) {
        values[i] += prefix;
    }

    Store(storage.store).Store(
        output + row * (columns + 1) + start + 1,
        values,
        valid);
}

torch::Tensor exclusive_cumsum_cuda(torch::Tensor x)
{
    TORCH_CHECK(x.is_cuda(), "Expected a CUDA tensor");
    TORCH_CHECK(x.scalar_type() == at::kFloat, "Expected float32");
    TORCH_CHECK(x.dim() == 2 && x.is_contiguous(),
                "Expected a contiguous matrix");
    TORCH_CHECK(x.size(1) > 0, "The scan dimension must be nonempty");

    const c10::cuda::CUDAGuard guard(x.device());
    const int64_t rows = x.size(0) > 0 ? x.size(0) - 1 : 0;
    const int64_t columns = x.size(1);
    const int tiles = static_cast<int>((columns + TILE - 1) / TILE);

    // Preserve the reference's [:-1], which slices dimension zero.
    auto output = torch::empty({rows, columns + 1}, x.options());
    if (rows == 0) {
        return output;
    }

    auto sums = torch::empty({rows, int64_t(tiles - 1)}, x.options());
    cudaStream_t stream = at::cuda::getCurrentCUDAStream();

    if (tiles > 1) {
        tile_sums_kernel<<<
            static_cast<unsigned int>(rows * (tiles - 1)),
            THREADS, 0, stream>>>(
                x.data_ptr<float>(),
                sums.data_ptr<float>(),
                columns,
                tiles);
        C10_CUDA_KERNEL_LAUNCH_CHECK();
    }

    scan_tiles_kernel<<<
        static_cast<unsigned int>(rows * tiles),
        THREADS, 0, stream>>>(
            x.data_ptr<float>(),
            sums.data_ptr<float>(),
            output.data_ptr<float>(),
            columns,
            tiles);
    C10_CUDA_KERNEL_LAUNCH_CHECK();

    return output;
}
"""

_extension = load_inline(
    name="exclusive_cumsum_tiled_cuda",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["exclusive_cumsum_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.shape[1] <= 65536
            and not x.requires_grad
        ):
            return _extension.exclusive_cumsum_cuda(x)

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