We discuss two closely related interval arithmetic systems: i) the of directed (generalized) intervals studied by E. Kaucher, and ii) the syste intervals together with the outer and inner interval operations. A relation two systems becomes feasible due to introduction of special notations and a normal form of directed intervals. As an application, it has been shown that interval systems can be used for the computation of tight inner and outer in of ranges of functions and consequently for the development of software for computation of ranges of functions.