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 nor are connected to any type of modelling 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 for planning safe motions, which combines a classical unsafe method with a verification step that checks constraint violation and computes excess using 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 for rapidly replanning safe motions, i.e., in less than 2s CPU time. It computes off-line feasible subsets in the vicinity of safe motions and finds on-line a solution in these subsets without actually computing again the nonlinear constraints. Our methods are validated using the HOAP-3 robot, where the motions are run without any balance controller.