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 Jaro-Winkler similarity with softmax normalization using CUDA optimizations:

Dual reduction operations - Implements both warp reduction sum (warp_reduce_sum) and max (warp_reduce_max).

Softmax normalization - Computes softmax for both x and target in shared memory for reuse.

Shared memory caching - Stores normalized target probabilities in shared memory to avoid recomputation.

Numerical stability - Uses max subtraction (exp(val - max)) for stable softmax computation.

Jaro-Winkler approximation - Computes similarity using min values and prefix matches (first 4 elements).

Complex multi-stage reduction - Multiple reduction phases: max, sum, min matching, prefix sum.

Shared memory reuse - Reuses shared memory buffers for different reduction stages.

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

Batch parallelism - One CUDA block per input row with dynamic shared memory allocation.

Fused kernel - Combines softmax normalization, matching computation, and Jaro-Winkler calculation in single kernel.



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:
        x_prob = torch.softmax(x, dim=-1)
        t_prob = torch.softmax(self.target, dim=-1)

        min_vals = torch.min(x_prob, t_prob)
        m = torch.sum(min_vals, dim=-1)

        sim_j = (2.0 * m + 1.0) / 3.0

        prefix = torch.sum(min_vals[:, :4], dim=-1)

        return sim_j + 0.1 * prefix * (1.0 - sim_j)


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]