This paper introduces effective numerical methods for the planning and fast replanning of safe motions to ensure the safety, balance, and integrity of humanoid robots over the whole motion duration. Our safe methods do not depend on, nor are connected to, any type of modeling or constraints. To plan safe motions, certain constraints have to be satisfied over a continuous interval of time. Classical methods revert to time-grid discretization, which can be risky for the robot. We introduce a hybrid method to plan safe motions, which combines a classical unsafe method with a verification step that checks constraint violation and computes excess by the usage of interval analysis. When the robot meets unexpected situations, it has to replan a new motion, which is often too time consuming. Hence, we introduce a new method to rapidly replan safe motions, i.e., in less than 2 s CPU time. It computes offline feasible subsets in the vicinity of safe motions and finds online a solution in these subsets without actually recomputing the nonlinear constraints. Our methods are validated by the use the HOAP-3 robot, where the motions are run with no balance controller.