```python
import torch
import torch.nn as nn
import torch.nn.functional as F
from torch.utils.cpp_extension import load_inline

_cpp_source = """
#include <torch/extension.h>
torch::Tensor min_sub_cuda(torch::Tensor x, torch::Tensor constant);
"""

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

template <typename scalar_t>
__global__ void min_sub_kernel(
    const scalar_t* __restrict__ x,
    const scalar_t* __restrict__ constant,
    scalar_t* __restrict__ output,
    int64_t n)
{
    using acc_t = at::acc_type<scalar_t, true>;
    const acc_t c = static_cast<acc_t>(constant[0]);

    for (int64_t i = static_cast<int64_t>(blockIdx.x) * blockDim.x
                     + threadIdx.x;
         i < n;
         i += static_cast<int64_t>(blockDim.x) * gridDim.x) {
        const acc_t v = static_cast<acc_t>(x[i]);
        const acc_t m = isnan(v) ? v : (isnan(c) ? c : (v < c ? v : c));
        output[i] = static_cast<scalar_t>(m - c);
    }
}

torch::Tensor min_sub_cuda(torch::Tensor x, torch::Tensor constant)
{
    TORCH_CHECK(x.is_cuda() && constant.is_cuda(), "Expected CUDA tensors");
    TORCH_CHECK(x.device() == constant.device(), "Device mismatch");
    TORCH_CHECK(x.scalar_type() == constant.scalar_type(), "Dtype mismatch");
    TORCH_CHECK(x.is_contiguous(), "Expected contiguous input");
    TORCH_CHECK(constant.numel() == 1, "Expected a scalar constant");

    const c10::cuda::CUDAGuard guard(x.device());
    auto output = torch::empty_like(x);
    const int64_t n = x.numel();
    if (n == 0) {
        return output;
    }

    constexpr int threads = 256;
    const int blocks = static_cast<int>(
        std::min<int64_t>((n + threads - 1) / threads, 4096));
    const auto stream = at::cuda::getCurrentCUDAStream();

    AT_DISPATCH_FLOATING_TYPES_AND2(
        at::ScalarType::Half, at::ScalarType::BFloat16,
        x.scalar_type(), "min_sub_cuda", [&] {
            min_sub_kernel<scalar_t><<<blocks, threads, 0, stream>>>(
                x.data_ptr<scalar_t>(),
                constant.data_ptr<scalar_t>(),
                output.data_ptr<scalar_t>(),
                n);
        });

    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return output;
}
"""

_fused_ops = load_inline(
    name="linear_min_sub_fused",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["min_sub_cuda"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
    verbose=False,
)


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):
        y = F.linear(x, self.linear.weight, self.linear.bias)
        if (
            not y.is_cuda
            or y.dtype != self.constant.dtype
            or (torch.is_grad_enabled() and
                (y.requires_grad or self.constant.requires_grad))
        ):
            return torch.minimum(y, self.constant) - self.constant
        return _fused_ops.min_sub_cuda(y, self.constant)
```