Binary population synthesis calculations and associated predictions,especially event rates, are known to depend on a significant number of inputmodel parameters with different degrees of sensitivity. At the same time, forsystems with relatively low formation rates, such simulations are heavilycomputationally demanding and therefore the needed explorations of thehigh-dimensional parameter space require major -- often prohibitive --computational resources. In the present study, to better understand several keyevent rates involving binary evolution and binaries with two compact objects inMilky Way-like galaxies and to provide ways of reducing the computational costsof complete parameter space explorations: (i) we perform a methodical parameterstudy of the \emph{StarTrack} population synthesis code ; and (ii) we develop aformalism and methodology for the derivation of {\em multi-dimensional fits}for event rates. We significantly generalize our earlier study, and we focus onways of thoroughly assessing the accuracy of the fits. We anticipate that theefficient tools developed here can be applied in lieu of large-scale populationcalculations and will facilitate the exploration of the dependence of ratepredictions on a wide range binary evolution parameters. Such explorations canthen allow the derivation of constraints on these parameters, given empiricalrate constraints and accounting for fitting errors. Here we describe in detailthe principles and practice behind constructing these fits, estimating theiraccuracy, and comparing them with observations in a manner that accounts fortheir errors.
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