The length values selection for a determined type of linkage to achieve the necessary task, dimensional synthesis, is classified into three classes based on the mechanism’s task: function generation, path generation, and motion generation. The case considered in this study, Function generation synthesis, aims to create a relation between the angular motions of the input and output links of the mechanism. For this problem, a semi-analytical method called the Newton-HCM is used for numerical solutions, which combines Newton’s method with the semi-analytical Homotopy Continuation Method (HCM). Function generation synthesis of a planar four-bar linkage for four and five precision points is the main challenge of the current study, which is highly nonlinear and complicated to solve. Numerical examples of the function generation problem for a four-bar linkage with four and five precision points are presented and authenticate the excellent performance of the proposed algorithm.