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