Symbolic Density Models of One-in-a-Billion Statistical Tails via Importance Sampling and Genetic Programming

Abstract

This paper explores the application of symbolic regression for building models of probability distributions in which the accuracy at the distributions’ tails is critical. The problem is of importance to cutting-edge industrial integrated circuit design, such as designing SRAM memory components (bitcells, sense amps) where each component has extremely low probability of failure. A naive approach is infeasible because it would require billions of Monte Carlo circuit simulations. This paper demonstrates a flow that efficiently generates samples at the tails using importance sampling, then builds genetic programming symbolic regression models in a space that captures the tails – the normal quantile space. These symbolic density models allow the circuit designers to analyze the tradeoff between high-sigma yields and circuit performance. The flow is validated on two modern industrial problems: a bitcell circuit on a 45nm TSMC process, and a sense amp circuit on a 28nm TSMC process.