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 computes Hinge Loss for binary classification tasks.

    Parameters:
        None
    """
    def __init__(self):
        super(Model, self).__init__()

    def forward(self, predictions, targets):
        """
        Computes mean hinge loss.

        Args:
            predictions (torch.Tensor): Predictions tensor of shape (batch_size, num_classes), float32.
            targets (torch.Tensor): Targets tensor of shape (batch_size,), float32.

        Returns:
            torch.Tensor: Scalar mean hinge loss.
        """
        return torch.mean(torch.clamp(1 - predictions * targets, min=0))
    ```
    
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
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline

_cpp_source = r"""
#include <torch/extension.h>
torch::Tensor hinge_loss_cuda(torch::Tensor predictions, torch::Tensor targets);
"""

_cuda_source = r"""
#include <torch/extension.h>
#include <ATen/ExpandUtils.h>
#include <ATen/cuda/CUDAContext.h>
#include <c10/cuda/CUDAGuard.h>
#include <c10/cuda/CUDAException.h>
#include <cuda_runtime.h>
#include <algorithm>
#include <cstdint>
#include <limits>

struct BroadcastInfo {
    int ndim;
    int64_t sizes[64];
    int64_t ps[64];
    int64_t ts[64];
};

__device__ __forceinline__ float hinge(float p, float t) {
    const float value = __fsub_rn(1.0f, __fmul_rn(p, t));
    return value < 0.0f ? 0.0f : value;
}

__device__ __forceinline__ float block_sum(float value) {
    __shared__ float scratch[8];
    const int lane = threadIdx.x & 31;
    const int warp = threadIdx.x >> 5;

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

    if (lane == 0)
        scratch[warp] = value;
    __syncthreads();

    value = threadIdx.x < 8 ? scratch[lane] : 0.0f;
    if (warp == 0) {
        #pragma unroll
        for (int delta = 16; delta > 0; delta >>= 1)
            value += __shfl_down_sync(0xffffffff, value, delta);
    }
    return value;
}

__device__ __forceinline__ void write_partial(
    float value, float* output, int64_t count
) {
    value = block_sum(value);
    if (threadIdx.x == 0) {
        const int index = blockIdx.y * gridDim.x + blockIdx.x;
        output[index] = gridDim.x * gridDim.y == 1
            ? (count ? value / static_cast<float>(count) : nanf(""))
            : value;
    }
}

template<bool Scalar>
__device__ __forceinline__ float4 read4(
    const float* pointer, int64_t offset, float scalar
) {
    if (Scalar)
        return make_float4(scalar, scalar, scalar, scalar);
    return *reinterpret_cast<const float4*>(pointer + offset);
}

// The two-dimensional grid removes integer division from the input loop.
// Each iteration handles 4096 elements with independent accumulators.
template<bool PScalar, bool TScalar, bool Vectorized>
__global__ void rectangular_kernel(
    const float* __restrict__ predictions,
    const float* __restrict__ targets,
    float* __restrict__ output,
    int64_t rows,
    int64_t columns,
    int64_t prediction_row_stride,
    int64_t target_row_stride,
    int64_t count
) {
    float s0 = 0.0f;
    float s1 = 0.0f;
    float s2 = 0.0f;
    float s3 = 0.0f;

    constexpr int tile_size = Vectorized ? 4096 : 1024;
    const int64_t step = static_cast<int64_t>(gridDim.x) * tile_size;

    for (int64_t row = blockIdx.y; row < rows; row += gridDim.y) {
        const float* p = predictions + row * prediction_row_stride;
        const float* t = targets + row * target_row_stride;
        const float pv_scalar = PScalar ? p[0] : 0.0f;
        const float tv_scalar = TScalar ? t[0] : 0.0f;

        if (Vectorized) {
            for (int64_t base = static_cast<int64_t>(blockIdx.x) * 4096
                                + threadIdx.x * 4;
                 base < columns; base += step) {
                #pragma unroll
                for (int j = 0; j < 4; ++j) {
                    const int64_t col = base + j * 1024;
                    if (col < columns) {
                        const float4 p4 = read4<PScalar>(p, col, pv_scalar);
                        const float4 t4 = read4<TScalar>(t, col, tv_scalar);
                        s0 += hinge(p4.x, t4.x);
                        s1 += hinge(p4.y, t4.y);
                        s2 += hinge(p4.z, t4.z);
                        s3 += hinge(p4.w, t4.w);
                    }
                }
            }
        } else {
            for (int64_t base = static_cast<int64_t>(blockIdx.x) * 1024
                                + threadIdx.x;
                 base < columns; base += step) {
                if (base < columns)
                    s0 += hinge(PScalar ? pv_scalar : p[base],
                                TScalar ? tv_scalar : t[base]);
                if (base + 256 < columns)
                    s1 += hinge(PScalar ? pv_scalar : p[base + 256],
                                TScalar ? tv_scalar : t[base + 256]);
                if (base + 512 < columns)
                    s2 += hinge(PScalar ? pv_scalar : p[base + 512],
                                TScalar ? tv_scalar : t[base + 512]);
                if (base + 768 < columns)
                    s3 += hinge(PScalar ? pv_scalar : p[base + 768],
                                TScalar ? tv_scalar : t[base + 768]);
            }
        }
    }

    write_partial((s0 + s1) + (s2 + s3), output, count);
}

__global__ void outer_kernel(
    const float* __restrict__ rows,
    const float* __restrict__ columns,
    float* __restrict__ output,
    int64_t m,
    int64_t n,
    int64_t row_stride,
    int64_t column_stride
) {
    float sums[4] = {0.0f, 0.0f, 0.0f, 0.0f};

    for (int64_t rb = static_cast<int64_t>(blockIdx.y) * 8;
         rb < m; rb += static_cast<int64_t>(gridDim.y) * 8) {
        for (int64_t cb = static_cast<int64_t>(blockIdx.x) * 1024
                          + threadIdx.x;
             cb < n; cb += static_cast<int64_t>(gridDim.x) * 1024) {
            float tv[4];
            #pragma unroll
            for (int j = 0; j < 4; ++j) {
                const int64_t col = cb + j * 256;
                tv[j] = col < n ? columns[col * column_stride] : 0.0f;
            }

            #pragma unroll
            for (int r = 0; r < 8; ++r) {
                if (rb + r < m) {
                    const float pv = rows[(rb + r) * row_stride];
                    #pragma unroll
                    for (int j = 0; j < 4; ++j) {
                        if (cb + j * 256 < n)
                            sums[j] += hinge(pv, tv[j]);
                    }
                }
            }
        }
    }

    write_partial(
        (sums[0] + sums[1]) + (sums[2] + sums[3]), output, m * n);
}

template<typename Index>
__global__ void broadcast_kernel(
    const float* __restrict__ predictions,
    const float* __restrict__ targets,
    float* __restrict__ output,
    int64_t count,
    BroadcastInfo info
) {
    float sum = 0.0f;
    const int64_t step = static_cast<int64_t>(blockDim.x) * gridDim.x;

    for (int64_t i = static_cast<int64_t>(blockIdx.x) * blockDim.x
                     + threadIdx.x;
         i < count; i += step) {
        Index remaining = static_cast<Index>(i);
        int64_t po = 0;
        int64_t to = 0;
        for (int d = info.ndim - 1; d >= 0; --d) {
            const Index size = static_cast<Index>(info.sizes[d]);
            const Index coordinate = remaining % size;
            remaining /= size;
            po += static_cast<int64_t>(coordinate) * info.ps[d];
            to += static_cast<int64_t>(coordinate) * info.ts[d];
        }
        sum += hinge(predictions[po], targets[to]);
    }

    write_partial(sum, output, count);
}

__global__ void finish_kernel(
    const float* __restrict__ partials,
    float* __restrict__ output,
    int blocks,
    int64_t count
) {
    float sum = 0.0f;
    for (int i = threadIdx.x; i < blocks; i += 256)
        sum += partials[i];

    sum = block_sum(sum);
    if (threadIdx.x == 0)
        output[0] = sum / static_cast<float>(count);
}

template<bool Vectorized>
void launch_rectangular(
    const float* p,
    const float* t,
    float* output,
    int64_t rows,
    int64_t columns,
    int64_t prs,
    int64_t trs,
    int64_t count,
    bool pscalar,
    bool tscalar,
    dim3 grid,
    cudaStream_t stream
) {
    if (pscalar && tscalar) {
        rectangular_kernel<true, true, Vectorized>
            <<<grid, 256, 0, stream>>>(
                p, t, output, rows, columns, prs, trs, count);
    } else if (pscalar) {
        rectangular_kernel<true, false, Vectorized>
            <<<grid, 256, 0, stream>>>(
                p, t, output, rows, columns, prs, trs, count);
    } else if (tscalar) {
        rectangular_kernel<false, true, Vectorized>
            <<<grid, 256, 0, stream>>>(
                p, t, output, rows, columns, prs, trs, count);
    } else {
        rectangular_kernel<false, false, Vectorized>
            <<<grid, 256, 0, stream>>>(
                p, t, output, rows, columns, prs, trs, count);
    }
}

torch::Tensor hinge_loss_cuda(
    torch::Tensor predictions, torch::Tensor targets
) {
    TORCH_CHECK(predictions.is_cuda() && targets.is_cuda(),
                "Inputs must be CUDA tensors");
    TORCH_CHECK(predictions.device() == targets.device(),
                "Inputs must be on the same device");
    TORCH_CHECK(predictions.scalar_type() == at::kFloat &&
                targets.scalar_type() == at::kFloat,
                "Inputs must have dtype float32");

    const c10::cuda::CUDAGuard guard(predictions.device());
    auto expanded = at::broadcast_tensors({predictions, targets});
    const auto& p = expanded[0];
    const auto& t = expanded[1];
    TORCH_CHECK(p.dim() <= 64, "At most 64 dimensions are supported");

    const int64_t count = p.numel();
    BroadcastInfo info{};
    for (int d = 0; d < p.dim(); ++d) {
        if (p.size(d) == 1)
            continue;
        const int previous = info.ndim - 1;
        if (previous >= 0 &&
            info.ps[previous] == p.stride(d) * p.size(d) &&
            info.ts[previous] == t.stride(d) * p.size(d)) {
            info.sizes[previous] *= p.size(d);
            info.ps[previous] = p.stride(d);
            info.ts[previous] = t.stride(d);
        } else {
            const int index = info.ndim++;
            info.sizes[index] = p.size(d);
            info.ps[index] = p.stride(d);
            info.ts[index] = t.stride(d);
        }
    }

    const bool pscalar = predictions.numel() == 1;
    const bool tscalar = targets.numel() == 1;
    const bool linear = (p.is_contiguous() || pscalar) &&
                        (t.is_contiguous() || tscalar);
    const bool outer_pt = info.ndim == 2 &&
                          info.ps[1] == 0 && info.ts[0] == 0;
    const bool outer_tp = info.ndim == 2 &&
                          info.ts[1] == 0 && info.ps[0] == 0;
    const bool outer = count > 0 && (outer_pt || outer_tp);

    const bool rect2d = count > 0 && info.ndim == 2 &&
                       info.sizes[1] >= 128 &&
                       (info.ps[1] == 0 || info.ps[1] == 1) &&
                       (info.ts[1] == 0 || info.ts[1] == 1);
    const bool rectangular = count > 0 && (linear || rect2d);

    int64_t rows = 1;
    int64_t columns = count;
    int64_t prs = 0;
    int64_t trs = 0;
    bool pcs = pscalar;
    bool tcs = tscalar;

    if (!linear && rect2d) {
        rows = info.sizes[0];
        columns = info.sizes[1];
        prs = info.ps[0];
        trs = info.ts[0];
        pcs = info.ps[1] == 0;
        tcs = info.ts[1] == 0;
    }

    const float* pp = p.data_ptr<float>();
    const float* tp = t.data_ptr<float>();
    const bool vectorized = rectangular && columns % 4 == 0 &&
        (pcs || (reinterpret_cast<uintptr_t>(pp) % 16 == 0 &&
                 prs % 4 == 0)) &&
        (tcs || (reinterpret_cast<uintptr_t>(tp) % 16 == 0 &&
                 trs % 4 == 0));

    dim3 grid(1, 1);
    if (outer) {
        grid.x = static_cast<unsigned int>(std::min<int64_t>(
            2048, (info.sizes[1] + 1023) / 1024));
        grid.y = static_cast<unsigned int>(std::min<int64_t>(
            (info.sizes[0] + 7) / 8,
            std::max<int64_t>(1, 2048 / grid.x)));
    } else if (rectangular) {
        const int tile = vectorized ? 4096 : 1024;
        grid.x = static_cast<unsigned int>(std::min<int64_t>(
            2048, (columns + tile - 1) / tile));
        grid.y = static_cast<unsigned int>(std::min<int64_t>(
            rows, std::max<int64_t>(1, 2048 / grid.x)));
    } else {
        grid.x = static_cast<unsigned int>(std::max<int64_t>(
            1, std::min<int64_t>(2048, (count + 1023) / 1024)));
    }

    const int blocks = grid.x * grid.y;
    auto output = torch::empty({}, predictions.options());
    auto partials = blocks == 1
        ? output : torch::empty({blocks}, predictions.options());
    float* destination = partials.data_ptr<float>();
    const cudaStream_t stream = at::cuda::getCurrentCUDAStream();

    if (outer) {
        outer_kernel<<<grid, 256, 0, stream>>>(
            outer_pt ? pp : tp,
            outer_pt ? tp : pp,
            destination,
            info.sizes[0],
            info.sizes[1],
            outer_pt ? info.ps[0] : info.ts[0],
            outer_pt ? info.ts[1] : info.ps[1]);
    } else if (rectangular) {
        if (vectorized) {
            launch_rectangular<true>(
                pp, tp, destination, rows, columns, prs, trs,
                count, pcs, tcs, grid, stream);
        } else {
            launch_rectangular<false>(
                pp, tp, destination, rows, columns, prs, trs,
                count, pcs, tcs, grid, stream);
        }
    } else if (count <= std::numeric_limits<int32_t>::max()) {
        broadcast_kernel<int32_t><<<grid, 256, 0, stream>>>(
            pp, tp, destination, count, info);
    } else {
        broadcast_kernel<int64_t><<<grid, 256, 0, stream>>>(
            pp, tp, destination, count, info);
    }
    C10_CUDA_KERNEL_LAUNCH_CHECK();

    if (blocks > 1) {
        finish_kernel<<<1, 256, 0, stream>>>(
            destination, output.data_ptr<float>(), blocks, count);
        C10_CUDA_KERNEL_LAUNCH_CHECK();
    }

    return output;
}
"""

_hinge_extension = load_inline(
    name="hinge_loss_grid2d_v4",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["hinge_loss_cuda"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3", "--fmad=false"],
    verbose=False,
)


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

    def forward(self, predictions, targets):
        return _hinge_extension.hinge_loss_cuda(predictions, targets)
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 3 ms; the PyTorch reference (with TF32 enabled) takes 21.7 ms; speedup 7.233x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| void rectangular_kernel<false, false, true>(float const*, float const*, float*, long, long | 2987.5 | 1 |
| finish_kernel(float const*, float*, int, long) | 2.9 | 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.
