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SZpack is a numerical library which allows fast and precise computation of the Sunyaev-Zeldovich (SZ) signal for hot, moving clusters of galaxies. Both explicit numerical integration as well as approximate representation of the SZ signals can be obtained. Variations of the electron temperature and bulk velocity along the line-of-sight can be included. SZpack allows very fast and precise (<~0.001% at frequencies h nu <~ 30kT_g and electron temperature kTe ~ 75 keV) computation and its accuracy practically eliminates uncertainties related to more expensive numerical evaluation of the Boltzmann collision term. It furthermore cleanly separates kinematic corrections from scattering physics, effects that previously have not been clarified.