The class of feasible directions methods is a powerful tool for solving constrained minimization problems, min-max problems, and unconstrained minimization problems in the absence of continuity of the gradient. The different versions proposed either involve all the gradients and do not require "antizigzagging precautions" or involve only the constraints in a neighborhood of the current feasible point and do require "antizigzagging precautions" [i-i0]. This paper considers the class of feasible directions methods which do require "antizigzagging precautions". Such methods require that at each iteration the gradient of the cost function, the gradients of the active constraints, and the gradients of the "almost" active constraints be evaluated. The method we propose only requires that at each iteration, the gradient of the cost function and the gradient of the active constraints, be recomputed, and we use the previously computed gradients of the almost active constraints, provided that these gradients are not too "old". This results in a considerable saving of computational time when the constraint functions are complicated.