A continuum robot is a unique type of robots which move because of the elastic deformation of their bodies. The kinematics of such robots is typically described using constant curvature assumption. Such an assumption, however, does not completely describe the kinematics of a reallife continuum robot. As a result, variable curvature assumptions describe the kinematics of the continuum robot better, however, they are more complicated to formulate and work with. In particular, the existing methods of solving the inverse kinematics problem of multisection continuum robots with variable curvature suffer from a variety of deficiencies. Those deficiencies include complex matrix calculations, singularity problems, unscalability, and inability to find a numeric solution in some cases. In this work, we present FABRIKx: fast and reliable algorithm to solve the problem of inverse kinematics of the multisection continuum robot with variable curvature. In particular, to describe the variable curvature, we utilize a piecewise constant curvature assumption. The proposed algorithm combines both tangent and chord approaches to solve the inverse kinematics problem. The inverse kinematics of a single bending section of piecewise constant curvature is also described. To evaluate FABRIKx effectiveness, we compare it with the Jacobian-based and FABRIKc-based algorithms via simulation studies for different robots. The obtained results show that FABRIKx demonstrates a higher success rate and a lower solution time.