The moment-of-fluid method (MOF) is a second-order accurate interface reconstruction method which can be seen as an extension of the volume-of-fluid method with piecewise linear interface construction (VOF-PLIC). MOF involves a computationally intensive minimization problem that needs to be solved on every cell containing several materials. We propose a new fast and robust reconstruction algorithm to tackle this problem on rectangular hexahedral cells. Our approach uses explicit analytic formulas of the objective function that does not use any geometric computations such as half-space–polyhedron intersections. The numerical results show that the proposed method is more robust and more than 200 times faster than the original approach. Additionally, we propose a faster reconstruction algorithm on convex polyhedral cells. All the methods presented in this article have been implemented and verified on the open-source code Notus