A universal algorithm for the calculation of the density-of-states, partition, and other thermodynamic functions of internal one-dimensional nonharmonic degrees of freedom is developed and analyzed. The algorithm is exact in the classical limit and results in analytical formulas for many types of potentials. Quantum effects are taken into account via the inverse Laplace transform of the classical partition function corrected for quantum effects by the method of Fitter and Gwinn. Explicit formulas are reported for a variety of selected potential types (double-well inversion, quartic oscillator, etc.). An extension of the classical treatment to multidimensional potentials is described.