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**: `zscore_sigmoid_denormalize_kernel`
- **CUDA math functions**: `expf()`, `fmaf()` (fused multiply-add)
- **Element-wise parallelism**: One thread per element
- **FMA optimization**: Using fused multiply-add for better precision

## Mathematical Operations Pipeline
1. **Z-score normalization**: `(x - mean) / std`
2. **Sigmoid activation**: `1 / (1 + exp(-y))`
3. **Inverse Z-score**: `z * std + mean`
- **Fused operations**: Three-step transformation in single kernel

## Architecture
- **Standard 1D grid**: Simple block/grid configuration
- **Element-wise computation**: Independent processing per element
- **Memory pattern**: Coalesced memory access

## CUDA Math Optimizations
- **expf()**: Single-precision exponential
- **fmaf()**: Fused multiply-add for denormalization (z*std + mean)
- **Efficient computation**: Minimizes rounding errors

## Mathematical Properties
- **Reversible transformation**: Sigmoid on normalized data
- **Range preservation**: Output remains in similar range as input
- **Non-linear transformation**: Sigmoid introduces non-linearity
- **Parameterized**: User-provided mean and std parameters

## Performance Features
- **GPU acceleration**: Parallel computation across all elements
- **Fused kernel**: Three operations combined for efficiency
- **Numerical stability**: FMA reduces rounding errors
- **Simple operations**: Moderate computational cost

## Numerical Considerations
- **Std requirement**: std ≠ 0 (no protection in code)
- **Numerical range**: Sigmoid output in (0,1)
- **Denormalization**: Maps sigmoid output back to original scale
- **Potential overflow**: exp(-y) could underflow for large y

## Use Case Applications
- **Normalized activation**: Sigmoid on standardized data
- **Range transformation**: Map data through normalized sigmoid
- **Custom scaling**: User-defined mean/std for specific ranges
- **Element-wise processing**: Independent transformation per element



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, mean, std):
        super(Model, self).__init__()
        self.mean = mean
        self.std = std

    def forward(self, x):
        y = (x - self.mean) / self.std
        z = torch.sigmoid(y)

        return z * self.std + self.mean


batch_size = 1024
dim = 1024


def get_inputs():
    x = torch.randn(batch_size, dim) * 5.0 + 10.0
    return [x]


def get_init_inputs():
    return [10.0, 5.0]