The C++ class library CoStLy for the rigorous computation of complex function values or ranges is presented. Rectangular complex interval arithmetic is used for the computations. In the CoStLy procedures, all truncation and roundoff errors are calculated during the course of the floating-point computation and enclosed into the result. The library contains procedures for root and power functions, the exponential, trigonometric and hyperbolic functions, their inverse functions, and some auxiliary functions, such as the absolute value or the argument functio