MACACO: Modeling and analysis of circuits for approximate computing

Approximate computing, which refers to a class of techniques that relax the requirement of exact equivalence between the specification and implementation of a computing system, has attracted significant interest in recent years. We propose a systematic methodology, called MACACO, for the Modeling and Analysis of Circuits for Approximate Computing. The proposed methodology can be utilized to analyze how an approximate circuit behaves with reference to a conventional correct implementation, by computing metrics such as worst-case error, average-case error, error probability, and error distribution. The methodology applies to both timing-induced approximations such as voltage over-scaling or over-clocking, and functional approximations based on logic complexity reduction. The first step in MACACO is the construction of an equivalent untimed circuit that represents the behavior of the approximate circuit at a given voltage and clock period. Next, we construct a virtual error circuit that represents the error in the approximate circuit's output for any given input or input sequence. Finally, we apply conventional Boolean analysis techniques (SAT solvers, BDDs) and statistical techniques (Monte-Carlo simulation) in order to compute the various metrics of interest. We have applied the proposed methodology to analyze a range of approximate designs for datapath building blocks. Our results show that MACACO can help a designer to systematically evaluate the impact of approximate circuits, and to choose between different approximate implementations, thereby facilitating the adoption of such circuits for approximate computing.