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.

This code implements Fisher-Rao distance + RMSNorm (Root Mean Square Normalization) with CUDA optimizations:

Element-wise parallelism - Each thread computes distance between x[i] and y[i] independently.

Fisher-Rao metric - Computes |log(x) - log(y)| as distance on probability simplex.

Numerical stability - Adds ε=1e-6 to absolute values to avoid log(0).

Fused distance calculation - Computes absolute log difference in single kernel.

Memory coalescing - Contiguous memory access patterns.

CUDA math functions - Uses fabsf() and logf() for hardware acceleration.

Simple grid-stride mapping - Standard 1D grid/block for element-wise operations.

Custom RMSNorm implementation - PyTorch module for root mean square normalization.

Post-processing - Applies RMSNorm to Fisher-Rao distance elements.

Batch processing - Handles all elements in parallel regardless of shape.





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 RMSNorm(nn.Module):
    def __init__(self, dim, eps=1e-6):
        super().__init__()
        self.scale = nn.Parameter(torch.ones(dim))
        self.eps = eps

    def forward(self, x):
        var = x.pow(2).mean(-1, keepdim=True)
        norm_x = x * torch.rsqrt(var + self.eps)
        return norm_x * self.scale

class Model(nn.Module):
    def __init__(self, dim):
        super(Model, self).__init__()
        self.rmsnorm = RMSNorm(dim)

    def forward(self, x, y):
        eps = 1e-6
        val_x = torch.abs(x) + eps
        val_y = torch.abs(y) + eps
        dist = torch.abs(torch.log(val_x) - torch.log(val_y))
        out = self.rmsnorm(dist)
        return out.mean()

batch_size = 16
input_dim = 1024

def get_inputs():
    x = torch.randn(batch_size, input_dim)
    y = torch.randn(batch_size, input_dim)
    return [x, y]

def get_init_inputs():
    return [input_dim]