Write a custom CUDA kernel to optimize the ASU activation function as defined in the provided table.

The mathematical definition is:
f(x) = x * sin(x)

Problem Analysis:
1. Memory Bandwidth: The operation is element-wise and strictly memory-bound. The arithmetic intensity is low (one sin, one mul). Standard PyTorch implementation executes `sin(x)` followed by `x * result`, involving intermediate memory traffic.
2. Precision: Trigonometric functions are sensitive to precision. Double precision (float64) is required for strict accuracy alignment with the reference.

Optimization Strategy: Fused Vectorized Kernel in Double Precision

1. Data Type: Use `double` for all computations to guarantee numerical stability and accuracy.

2. Vectorized Memory Access: Use `double2` types to load/store 128 bits (2 doubles) per instruction. This is the optimal transaction size for float64 data on GPUs, significantly reducing instruction overhead and maximizing bandwidth.

3. Fused Computation: Compute `val * sin(val)` entirely in registers. This fuses the two element-wise operations into a single kernel pass (1 read, 1 write).

4. Grid-Stride Loop: Implement a robust grid-stride loop to handle arbitrary input tensor sizes efficiently.
  
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

BATCH_SIZE = 4096
DIM = 4096
SHAPE = (BATCH_SIZE, DIM)

DTYPE = torch.float64

class ASU(nn.Module):
    """
    Amplifying Sine Unit: An Oscillatory Activation Function for Deep Neural Networks to Recover Nonlinear Oscillations Efficiently 
    https://arxiv.org/pdf/2304.09759
    Formula: f(x) = x * sin(x)
    """
    def __init__(self):
        super(ASU, self).__init__()

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        return x * torch.sin(x)

class Model(nn.Module):
    def __init__(self):
        super(Model, self).__init__()
        self.act = ASU()
    
    def forward(self, x: torch.Tensor) -> torch.Tensor:
        return self.act(x)

def get_inputs():
    x = torch.randn(SHAPE, dtype=DTYPE)
    return [x.contiguous()]

def get_init_inputs():
    return []