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Approximate circuits that trade the chip area or power consumption for the precision of the computation play a key role in development of energy-aware systems. Designing complex approximate circuits is, however, very difficult, especially, when a given approximation error has to be guaranteed. Evolutionary search algorithms together with SAT-based error evaluation currently represent one of the most successful approaches for automated circuit approximation. In this paper, we apply satisfiability solving not only for circuit evaluation but also for its minimisation. We consider and evaluate several approaches to this task, both inspired by existing works as well as novel ones. Our experiments show that a combined strategy, integrating evolutionary search and SMT-based sub-circuit minimisation (using quantified theory of arrays) that we propose, is able to find complex approximate circuits (e.g. 16-bit multipliers) with considerably better trade-offs between the circuit precision and size than existing approaches.

Satisfiability Solving Meets Evolutionary Optimisation in Designing Approximate Circuits