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Triple modular redundancy (TMR) is one of the most common techniques in fault-tolerant systems, in which the output is determined by a majority voter. However, the design diversity of replicated modules and/or soft errors that are more likely to happen in the nanoscale era may affect the majority voting scheme. Besides, the significant overheads of the TMR scheme may limit its usage in energy consumption and area-constrained critical systems. However, for most inherently error-resilient applications such as image processing and vision deployed in critical systems (such as autonomous vehicles and robotics), achieving a given level of reliability has more priority than precise results. Therefore, these applications can benefit from the approximate computing paradigm to achieve higher energy efficiency and a lower area. This article proposes an energy-efficient approximate reliability (X-Rel) framework to overcome the aforementioned challenges of the TMR systems and get the full potential of approximate computing without sacrificing the desired reliability constraint and output quality. The X-Rel framework relies on relaxing the precision of the voter based on a systematical error bounding method that leverages user-defined quality and reliability constraints. Afterward, the size of the achieved voter is used to approximate the TMR modules such that the overall area and energy consumption are minimized. The effectiveness of employing the proposed X-Rel technique in a TMR structure, for different quality constraints as well as with various reliability bounds, is evaluated in a 15-nm FinFET technology. The results of the X-Rel voter show delay, area, and energy consumption reductions of up to 86%, 87%, and 98%, respectively, when compared to those of the state-of-the-art approximate TMR voters. Also, the effectiveness of the proposed X-Rel-based TMR structure is assessed in four benchmark applications from different domains. For these benchmarks, the results show 1.59   $\times$   , 2.35   $\times$   , and 3.39   $\times$    energy-delay-area-product (EDAP) reduction for less than 1%, 5%, and 10% output quality degradations, respectively. Finally, an image processing application is benchmarked to evaluate the X-Rel framework efficacy in the presence of errors, where the results show up to a 4.78   $\times$    higher output image quality in comparison with the typical TMR voters.

X-Rel: Energy-Efficient and Low-Overhead Approximate Reliability Framework for Error-Tolerant Applications Deployed in Critical Systems