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**: `quantile_normalize_kernel`
- **Element-wise parallelism**: One thread per element
- **2D indexing**: Combines batch and dimension indices
- **Nested loops**: Inner loop for ranking computation

## Statistical Operations
- **Quantile ranking**: Compute rank of each element within its row
- **Normalization**: Convert rank to uniform [0,1] distribution
- **Range expansion**: Map from [0,1] to [min_val, max_val]
- **Percentile transformation**: Convert values to percentile scores

## Algorithm Characteristics
- **Rank computation**: O(n²) pairwise comparisons per row
- **Batch independence**: Each row processed independently
- **Tie handling**: Stable ranking with `(x_ptr[j] == val && j < i)`
- **Uniform distribution**: Output follows uniform distribution

## Architecture
- **Flattened parallelism**: Total threads = batch_size × dim
- **Batch-dimension mapping**: idx → (batch_idx, element_idx)
- **Memory access**: Coalesced within each row
- **Computational intensity**: High due to inner ranking loop

## Performance Considerations
- **Computational cost**: O(batch×dim²) comparisons - potentially expensive
- **Parallel efficiency**: Each thread does full ranking of its row
- **Memory pattern**: Sequential access within rows
- **Scalability**: Performance degrades with larger dimensions

## Numerical Considerations
- **Tie-breaking**: Stable ranking for equal values
- **Division**: Uses `(dim - 1)` for proper [0,1] range
- **Float conversion**: `(float)rank` for floating-point division
- **Range calculation**: `max_val - min_val` computed once per thread

## Unique Aspects
- **Quantile transformation**: Maps distribution to uniform
- **Non-parametric**: Makes no distributional assumptions
- **Rank-based**: Only relative ordering matters, not absolute values
- **Batch-wise operation**: Each sample normalized independently
- **Output range**: User-defined min_val to max_val

## Use Case Applications
- **Distribution normalization**: Make different distributions comparable
- **Non-parametric scaling**: Useful for unknown/irregular distributions
- **Data preprocessing**: Prepare data for algorithms requiring uniform inputs
- **Statistical testing**: Quantile-based transformations

## Limitations
- **Quadratic complexity**: Not suitable for very large dimensions
- **No optimization**: Simple brute-force ranking algorithm
- **Batch memory**: Each thread accesses entire row multiple times
- **No early exit**: Always completes full dim comparisons




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, min_val, max_val):
        super(Model, self).__init__()
        self.min_val = min_val
        self.max_val = max_val
        self.range = max_val - min_val

    def forward(self, x):
        ranks = x.argsort(dim=-1).argsort(dim=-1).float()

        # Quantile Normalization to [0, 1]
        norm = ranks / (x.size(-1) - 1)

        # Range Expand
        return norm * self.range + self.min_val


batch_size = 256
dim = 256


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


def get_init_inputs():
    return [-1.0, 1.0]