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 Jaccard similarity + Legendre polynomial with CUDA optimizations:

Parallel reduction - Warp shuffle for three simultaneous sums: dot product, x², t².

Separate shared memory arrays - Three dedicated shared buffers to avoid bank conflicts.

Fused operations - Computes Jaccard coefficient and 3rd-order Legendre polynomial in one kernel.

Grid-stride loop - Threads process multiple elements for load balancing.

Numerical stability - Adds 1e-6 to denominator to prevent division by zero.

Memory coalescing - Contiguous tensor access patterns.

Batch parallelism - One CUDA block per input row for batch processing.



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, target):
        super(Model, self).__init__()
        self.target = nn.Parameter(target)

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        dot = torch.sum(x * self.target, dim=-1)
        norm_x_sq = torch.sum(x * x, dim=-1)
        norm_target_sq = torch.sum(self.target * self.target, dim=-1)

        jaccard = dot / (norm_x_sq + norm_target_sq - dot + 1e-6)

        j2 = jaccard * jaccard
        j3 = j2 * jaccard
        return 0.5 * (5.0 * j3 - 3.0 * jaccard)


batch_size = 128
input_dim = 1024


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


def get_init_inputs():
    target = torch.randn(input_dim)
    return [target]