You write custom CUDA kernels to replace the pytorch operators in the given GeGLU 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 chunk+gelu+elementwise_mul), or algorithmic changes (such as optimized memory access patterns). You are only limited by your imagination.

# Technologies Used in This Code

## Core Libraries
- **PyTorch**: Deep learning framework
- **CUDA**: NVIDIA GPU parallel computing
- **C++**: Kernel implementation

## CUDA Components
- **CUDA kernel**: `complex_ops_kernel`
- **CUDA math functions**: `expf()`, `logf()`, `sinf()`, `cosf()`, `atan2f()`
- **Device functions**: `__device__` helper functions for complex operations
- **Complex arithmetic**: Three custom complex number operations

## Complex Number Operations
1. **Complex exponential**: exp(z) = exp(real)·(cos(imag) + i·sin(imag))
2. **Complex logarithm**: log(z) = log|z| + i·arg(z)
3. **Complex power**: wᵖ = exp(p·log(w))
- **Composition**: Computes exp(log(exp(z))ᵖ) = exp(z)ᵖ (mathematically)

## Mathematical Implementation
- **Complex exponential**: Euler's formula implementation
- **Complex logarithm**: Polar form using atan2 for angle
- **Complex power**: Via logarithm and exponential (zᵖ = exp(p·log(z)))
- **Parameterized power**: User-defined complex exponent (p_re + i·p_im)

## Architecture
- **Device functions**: Reusable complex operation helpers
- **Element-wise parallelism**: One thread per complex number
- **Three-step pipeline**: exp → log → pow composition
- **Batch processing**: Handles multiple complex numbers

## CUDA Optimizations
- **Modular design**: Separate device functions for each operation
- **Mathematical identities**: Leverages exp(log(exp(z))) = exp(z)
- **Efficient operations**: Optimized complex arithmetic
- **Single kernel**: Fused three operations

## Performance Features
- **GPU acceleration**: Parallel complex operations
- **Reusable functions**: Modular device function design
- **Numerical precision**: Proper complex number handling
- **Parameterized**: User-defined complex exponent

## Numerical Considerations
- **Branch cuts**: Complex logarithm has branch cut on negative real axis
- **Overflow**: exp() can overflow for large real parts
- **Domain issues**: log(0) undefined
- **Multiple values**: Complex power may have multiple values

## Mathematical Properties
- **Identity relation**: exp(log(exp(z))) = exp(z) exactly
- **Complex power**: Generalization of real exponentiation
- **Analytic functions**: exp and log are analytic (except branch cuts)
- **Composition**: exp ∘ log ∘ exp = exp (mathematically)

## Use Case Applications
- **Complex analysis**: Advanced complex number manipulations
- **Signal processing**: Complex exponent operations
- **Physics**: Quantum mechanics wave functions
- **Mathematics**: Complex function evaluation

## Implementation Details
- **Tensor shape**: Expects [batch_size, 2] for complex numbers
- **Device functions**: `__device__` for GPU-only reusable code
- **Complex exponent**: User provides p_re and p_im parameters
- **Output format**: Same interleaved complex format as input

## Unique Aspects
- **Three-operation chain**: Unique exp-log-pow composition
- **Parameterized power**: Complex-valued exponent
- **Mathematical identity**: Should compute exp(z)ᵖ (within numerical error)
- **Device function library**: Reusable complex arithmetic functions



Here's an example to show you the syntax of inline embedding custom CUDA operators in torch: The example given architecture is:
import torch
import torch.nn as nn


class Model(nn.Module):
    def __init__(self, p_re, p_im):
        super(Model, self).__init__()
        self.p_c = torch.complex(torch.tensor(p_re), torch.tensor(p_im))

    def forward(self, z_f):
        z_c = torch.complex(z_f[..., 0], z_f[..., 1])

        y_c = torch.exp(z_c)
        w_c = torch.log(y_c)
        out_c = torch.pow(w_c, self.p_c)

        return torch.stack([out_c.real, out_c.imag], dim=-1)


batch_size = 1024
dim = 2


def get_inputs():
    z_f = torch.randn(batch_size, dim)
    return [z_f]


def get_init_inputs():
    return [0.5, 0.5]