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):
    """
    Simple model that performs a matrix multiplication, applies minimum, and subtracts a constant.
    """
    def __init__(self, in_features, out_features, constant):
        super(Model, self).__init__()
        self.linear = nn.Linear(in_features, out_features)
        self.constant = nn.Parameter(torch.tensor(constant))

    def forward(self, x):
        x = self.linear(x)
        x = torch.min(x, self.constant)
        x = x - self.constant
        return x

batch_size = 128
in_features = 16384
out_features = 16384
constant = 2.0

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

def get_init_inputs():
    return [in_features, out_features, constant]
    ```
    
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 3 of 6 for this task. Your previous solution (turn 2) 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 linear_min_sub_cuda(
    torch::Tensor x,
    torch::Tensor weight,
    torch::Tensor bias,
    torch::Tensor constant);
"""

_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>
#include <cublas_v2.h>
#include <climits>

__device__ __forceinline__ float epilogue(float x, float b, float c) {
    float y = x + b;
    return (isnan(y) ? y : (isnan(c) ? c : fminf(y, c))) - c;
}

__global__ void finish_vectorized(
    float4* __restrict__ output,
    const float4* __restrict__ bias,
    const float* __restrict__ constant,
    int64_t count,
    int64_t width)
{
    const float c = constant[0];
    for (int64_t i = int64_t(blockIdx.x) * blockDim.x + threadIdx.x;
         i < count;
         i += int64_t(blockDim.x) * gridDim.x) {
        float4 v = output[i];
        const float4 b = bias[i % width];
        v.x = epilogue(v.x, b.x, c);
        v.y = epilogue(v.y, b.y, c);
        v.z = epilogue(v.z, b.z, c);
        v.w = epilogue(v.w, b.w, c);
        output[i] = v;
    }
}

__global__ void finish_scalar(
    float* __restrict__ output,
    const float* __restrict__ bias,
    const float* __restrict__ constant,
    int64_t count,
    int64_t width)
{
    const float c = constant[0];
    for (int64_t i = int64_t(blockIdx.x) * blockDim.x + threadIdx.x;
         i < count;
         i += int64_t(blockDim.x) * gridDim.x) {
        output[i] = epilogue(output[i], bias[i % width], c);
    }
}

torch::Tensor linear_min_sub_cuda(
    torch::Tensor x,
    torch::Tensor weight,
    torch::Tensor bias,
    torch::Tensor constant)
{
    TORCH_CHECK(x.is_cuda() && weight.is_cuda() &&
                bias.is_cuda() && constant.is_cuda(),
                "Expected CUDA tensors");
    TORCH_CHECK(x.scalar_type() == at::kFloat &&
                weight.scalar_type() == at::kFloat &&
                bias.scalar_type() == at::kFloat &&
                constant.scalar_type() == at::kFloat,
                "Expected float32 tensors");
    TORCH_CHECK(x.device() == weight.device() &&
                x.device() == bias.device() &&
                x.device() == constant.device(),
                "Tensors must share a device");
    TORCH_CHECK(x.dim() >= 1 && weight.dim() == 2 &&
                bias.dim() == 1 && constant.numel() == 1,
                "Invalid tensor dimensions");
    TORCH_CHECK(x.size(-1) == weight.size(1) &&
                bias.numel() == weight.size(0),
                "Incompatible tensor shapes");

    const c10::cuda::CUDAGuard guard(x.device());
    x = x.contiguous();
    weight = weight.contiguous();
    bias = bias.contiguous();
    constant = constant.contiguous();

    const int64_t k64 = weight.size(1);
    const int64_t n64 = weight.size(0);
    TORCH_CHECK(k64 > 0, "Input width must be positive");
    const int64_t m64 = x.numel() / k64;
    TORCH_CHECK(m64 <= INT_MAX && n64 <= INT_MAX && k64 <= INT_MAX,
                "Matrix dimensions exceed cuBLAS limits");

    auto shape = x.sizes().vec();
    shape.back() = n64;
    auto output = torch::empty(shape, x.options());
    if (output.numel() == 0) return output;

    const int m = static_cast<int>(m64);
    const int n = static_cast<int>(n64);
    const int k = static_cast<int>(k64);

    cudaStream_t stream = at::cuda::getCurrentCUDAStream();
    cublasHandle_t handle = at::cuda::getCurrentCUDABlasHandle();

    cublasPointerMode_t previous_pointer_mode;
    TORCH_CHECK(
        cublasGetPointerMode(handle, &previous_pointer_mode)
            == CUBLAS_STATUS_SUCCESS,
        "Could not query cuBLAS pointer mode");
    TORCH_CHECK(
        cublasSetPointerMode(handle, CUBLAS_POINTER_MODE_HOST)
            == CUBLAS_STATUS_SUCCESS,
        "Could not set cuBLAS pointer mode");

    const float alpha = 1.0f;
    const float beta = 0.0f;

    // Row-major Y = X W^T, expressed as column-major Y^T = W X^T.
    // Explicitly enable TF32 tensor cores for this GEMM.
    const cublasStatus_t status = cublasGemmEx(
        handle,
        CUBLAS_OP_T,
        CUBLAS_OP_N,
        n, m, k,
        &alpha,
        weight.data_ptr<float>(), CUDA_R_32F, k,
        x.data_ptr<float>(), CUDA_R_32F, k,
        &beta,
        output.data_ptr<float>(), CUDA_R_32F, n,
        CUBLAS_COMPUTE_32F_FAST_TF32,
        CUBLAS_GEMM_DEFAULT_TENSOR_OP);

    const cublasStatus_t restore_status =
        cublasSetPointerMode(handle, previous_pointer_mode);
    TORCH_CHECK(status == CUBLAS_STATUS_SUCCESS,
                "cuBLAS GEMM failed: ", static_cast<int>(status));
    TORCH_CHECK(restore_status == CUBLAS_STATUS_SUCCESS,
                "Could not restore cuBLAS pointer mode");

    constexpr int threads = 256;
    const bool vectorized =
        n % 4 == 0 &&
        reinterpret_cast<uintptr_t>(bias.data_ptr<float>()) % 16 == 0;
    const int64_t count = output.numel();
    const int64_t work = vectorized ? count / 4 : count;
    const int blocks = static_cast<int>(
        std::min<int64_t>((work + threads - 1) / threads, 4096));

    if (vectorized) {
        finish_vectorized<<<blocks, threads, 0, stream>>>(
            reinterpret_cast<float4*>(output.data_ptr<float>()),
            reinterpret_cast<const float4*>(bias.data_ptr<float>()),
            constant.data_ptr<float>(),
            count / 4,
            n / 4);
    } else {
        finish_scalar<<<blocks, threads, 0, stream>>>(
            output.data_ptr<float>(),
            bias.data_ptr<float>(),
            constant.data_ptr<float>(),
            count,
            n);
    }
    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return output;
}
"""

_ops = load_inline(
    name="linear_tf32_min_sub_v2",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["linear_min_sub_cuda"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
    extra_ldflags=["-lcublas"],
    verbose=False,
)


class _LinearMinSub(torch.autograd.Function):
    @staticmethod
    def forward(ctx, x, weight, bias, constant):
        ctx.save_for_backward(x, weight, bias, constant)
        return _ops.linear_min_sub_cuda(x, weight, bias, constant)

    @staticmethod
    def backward(ctx, grad_output):
        x, weight, bias, constant = ctx.saved_tensors
        y = torch.nn.functional.linear(x, weight, bias)
        less = y < constant
        equal = y == constant
        nan = torch.isnan(y) | torch.isnan(constant)
        dy_factor = less.to(y.dtype) + equal.to(y.dtype) * 0.5
        dc_factor = (y > constant).to(y.dtype) + equal.to(y.dtype) * 0.5
        dy_factor = torch.where(nan, torch.ones_like(dy_factor), dy_factor)
        dc_factor = torch.where(nan, torch.ones_like(dc_factor), dc_factor)

        grad_y = grad_output * dy_factor
        grad_x = grad_weight = grad_bias = grad_constant = None

        if ctx.needs_input_grad[0]:
            grad_x = grad_y.matmul(weight)
        if ctx.needs_input_grad[1]:
            grad_weight = (
                grad_y.reshape(-1, weight.shape[0]).t()
                @ x.reshape(-1, weight.shape[1])
            )
        if ctx.needs_input_grad[2]:
            grad_bias = grad_y.reshape(-1, weight.shape[0]).sum(0)
        if ctx.needs_input_grad[3]:
            grad_constant = (
                grad_output * dc_factor - grad_output
            ).sum_to_size(constant.shape)
        return grad_x, grad_weight, grad_bias, grad_constant


class ModelNew(nn.Module):
    def __init__(self, in_features, out_features, constant):
        super().__init__()
        self.linear = nn.Linear(in_features, out_features)
        self.constant = nn.Parameter(torch.tensor(constant))

    def forward(self, x):
        weight = self.linear.weight
        bias = self.linear.bias
        constant = self.constant

        if (
            x.is_cuda
            and x.dtype == torch.float32
            and weight.dtype == torch.float32
            and bias.dtype == torch.float32
            and constant.dtype == torch.float32
            and x.shape[-1] > 0
            and not torch.is_autocast_enabled()
        ):
            if torch.is_grad_enabled() and any(
                t.requires_grad for t in (x, weight, bias, constant)
            ):
                return _LinearMinSub.apply(x, weight, bias, constant)
            return _ops.linear_min_sub_cuda(x, weight, bias, constant)

        y = self.linear(x)
        return torch.minimum(y, constant) - constant
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 0.961 ms; the PyTorch reference (with TF32 enabled) takes 0.918 ms; speedup 0.955x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| void cutlass::Kernel2<cutlass_80_tensorop_s1688gemm_256x128_32x3_tn_align4>(cutlass_80_ten | 907.2 | 1 |
| finish_vectorized(float4*, float4 const*, float const*, long, long) | 9.0 | 1 |
| Memset (Device) | 1.1 | 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.
