The combination of successive substitution and the Newton method provides a robust and efficient solution to the non-linear iso-fugacity and mass balance equations for two-phase split computations in compositional simulations. In the successive substitution, the non-linear Rachford-Rice equation introduces some complexity, but the Newton method in combination with the bisection method provides a robust solution. For three-phase split calculations, the literature indicates that the computed three-phase region is smaller than the measured data. Some of the computed results also show some inconsistency in the phase diagram when there is three-phase region. We have found that the solution to the two Rachford-Rice equations in three phases can solve the problem. Our proposal is deceptively simple. We use a two-dimensional bisection method to provide good enough initial guesses for the Newton method in solving the two Rachford-Rice equations. In this work, we provide examples of various degree of complexity to demonstrate how powerful the combined bisection-Newton method is in three-phase split calculations. The proposed method is both infallible and efficient. To the best of our knowledge, the use of the bisection method in two variables has not been attempted in the past.