cecApprox: Enabling Automated Combinational Equivalence Checking for Approximate Circuits

Approximate circuits have become ubiquitous in error-resilient applications. Given their widespread use, formal verification of these approximate designs is essential. Recently, there have been attempts that use formal verification techniques based on Boolean Satisfiability (SAT) Methods and Binary Decisions Diagrams (BDDs) to perform formal error analysis. These methods guarantee that the designed approximate circuits satisfy the error metrics, e.g., Mean Square Error (MSE), etc. However, in certain approximate circuits tailored for a particular input data distribution, only performing error analysis is not enough. In addition to error metrics, the functional specifications in these approximate circuits should also be satisfied. Otherwise, it can lead to a larger-than-expected deterioration in the output quality. In this work, for the first time, we alleviate this issue by proposing an automated formal verification methodology, cecApprox, that guarantees the approximate circuit matches the functional specification. This is crucial for circuits that are tailored for a particular input distribution. We show that in the designs where the structure is preserved, the corrected circuit can be generated from the approximate circuit. This corrected circuit can then be formally verified by performing combinational equivalence checking with the exact circuit as the golden reference. If the corrected circuit is equivalent to the exact circuit, it guarantees that the approximate circuit matches the given functional specification in addition to error metrics. Hence, the cecApprox methodology complements the formal error analysis method. We show the efficacy of cecApprox by generating and formally verifying millions of different 8-bit and 16-bit structural preserved approximate adders and multipliers using combinational equivalence checking.