Approximate computing strives to achieve the highest performance-, area-, and power-efficiency for a given quality constraint and vice versa. Conventional approximate design methodology restricts the introduction of errors to avoid a high loss in quality. However, this limits the computing efficiency and the number of pareto-optimal design alternatives for a quality-efficiency tradeoff. This paper presents a novel self-healing (SH) methodology for an approximate square-accumulate (SAC) architecture. SAC refers to a hardware architecture that computes the inner product of a vector with itself. SH exploits the algorithmic error resilience of the SAC structure to ensure an effective quality-efficiency tradeoff, wherein the squarer is regarded as an approximation stage, and the accumulator as a healing stage. We propose to deploy an approximate squarer mirror pair , such that the error introduced by one approximate squarer mirrors the error introduced by the other, i.e., the errors generated by the approximate squarers are approximately the additive inverse of each other. This helps the healing stage (accumulator) to automatically average out the error originated in the approximation stage, and thereby to minimize the quality loss. For random input vectors, SH demonstrates up to 25% and 18.6% better area and power efficiency, respectively, with a better quality output than the conventional approximate computing methodology. As a case study, SH is applied to one of the computationally expensive components (SAC) of the radio astronomy calibration application, where it shows up to 46.7% better quality for equivalent computing efficiency as that of conventional methodology.