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In numerous regular statistical models, median bias reduction (Kenne Pagui et al., 2017) has proven to be a noteworthy improvement over maximum likelihood, alternative to mean bias reduction. The estimator is obtained as solution to a modified score equation ensuring smaller asymptotic median bias than the maximum likelihood estimator. This paper provides a simplified algebraic form of the adjustment term for general regular models. With the new formula, the estimation procedure benefits from a considerable computational gain by avoiding multiple summations and thus allows an efficient implementation. More importantly, the new formulation allows to highlight how the median bias reduction adjustment can be obtained by adding an extra term to the mean bias reduction adjustment. Illustrations are provided through new applications of median bias reduction to two regression models not belonging to the generalized linear models class, extended beta regression and beta-binomial regression. Mean bias reduction is also provided here for the latter model. Simulation studies show remarkable componentwise median centering of the median bias reduced estimator, while variability and coverage of related confidence intervals are comparable with those of mean bias reduction. Moreover, empirical results for the beta-binomial model show that the method is successful in solving maximum likelihood boundary estimate problem.