Write a custom CUDA kernel to optimize `PATS` activation function.

Formula: f(x) = x * arctan(k * PI * sigmoid(x))

Problem Analysis:
1. Computationally Intensive & Memory Bound: The operation is element-wise but involves a chain of two transcendental functions (sigmoid contains exp, arctan).
2. Operator Chaining: A standard PyTorch implementation creates intermediate tensors for sigmoid and arctan, wasting memory bandwidth.

Optimization Strategy: Fused Element-wise Kernel with Vectorization

1. One-Thread-per-Element: Map each element to a CUDA thread.

2. Vectorized Loads (float4): Use `float4` to process 128 bits per memory transaction.

3. Fused In-Register Math:
   - For each element `x`:
     `sig_val = 1.0f / (1.0f + __expf(-x))`
     `atan_arg = k * PI * sig_val`
     `atan_val = atanf(atan_arg)`
     `result = x * atan_val`
   - All computations are fused in registers.

4. One-Pass: Fuse all steps into a single read-compute-write kernel.
  
Here's an example to show you the syntax of inline embedding custom CUDA operators in torch: The example given architecture is:   
  
```python
import torch
import torch.nn as nn
import math

BATCH_SIZE = 4096
HIDDEN_DIM = 4096
SHAPE = (BATCH_SIZE, HIDDEN_DIM)

K_VALUE = 0.5

class PATS(nn.Module):
    """
    "PATS: a new neural network activation function with parameter" (ICCCS 2020)
    https://ieeexplore.ieee.org/document/9118471
    Formula: f(x) = x * arctan(k * PI * sigmoid(x))
    """
    def __init__(self, k=0.5):
        super(PATS, self).__init__()
        self.k = k
        self.pi = math.pi

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        sig = torch.sigmoid(x)
        return x * torch.arctan(self.k * self.pi * sig)

class Model(nn.Module):
    def __init__(self, k=0.5):
        super(Model, self).__init__()
        self.act = PATS(k=k)
    
    def forward(self, x):
        return self.act(x)

def get_inputs():
    input_tensor = torch.randn(SHAPE, dtype=torch.float32) * 5.0
    return [input_tensor.contiguous()]

def get_init_inputs():
    return [K_VALUE]