Analysis of Approximate adders with single functional error

Approximate computing has been extensively employed in arithmetic circuits such as a ripple carry adder (RCA); in an approximate adder, errors are introduced by design. VLSI circuits are also prone to errors caused by external and physical phenomena (such as cosmic rays or a stuck-at); hereafter, these errors are referred to as functional. This paper investigates the combined effects of single functional error (SFE) in an approximate cell and an RCA. The exact and approximate cell designs are considered using a state transition diagram-based analysis to identify relationships between the types of functional error in a cell against the expected behavior for all possible cases. A probabilistic analysis for an exact RCA is proposed; it shows excellent agreement with the simulation results for different metrics such as the error rate (ER). Next, an error analysis is performed on the RCA by considering the number of approximate cells as well as the location of the erroneous cell; the simulation results and modeling analysis of the exact RCA show and prove that the ER and MED (Mean Error Distance) of odd-numbered cases is higher than for evennumbered cases. Moreover, the MED for an approximate RCA in the presence of an SFE is higher than for the exact RCA. Simulation and analytical results show that an approximate RCA in the presence of an SFE can have a serious degradation in accuracy performance if the approximate cell type is not properly selected. In addition, the binary tree-based mathematical analysis and pseudocode are provided to support the exhaustive simulation results for the ER of the RCA using different approximate cells.