Background: COSMIC Function Points and traditional Function Points (i.e., IFPUG Function Points and more recent variation of Function Points, such as NESMA and FISMA) are probably the best known and most widely used Functional Size Measurement methods. The relationship between the two kinds of Function Points still needs to be investigated. If traditional Function Points could be accurately converted into COSMIC Function Points and vice versa, then, by measuring one kind of Function Points, one would be able to obtain the other kind of Function Points, and one might measure one or the other kind interchangeably. Several studies have been performed to evaluate whether a correlation or a conversion function between the two measures exists. Specifically, it has been suggested that the relationship between traditional Function Points and COSMIC Function Points may not be linear, i.e., the value of COSMIC Function Points seems to increase more than proportionally to an increase of traditional Function Points. Objective: This paper aims at verifying this hypothesis using available datasets that collect both FP and CFP size measures. Method: Rigorous statistical analysis techniques are used, specifically Piecewise Linear Regression, whose applicability conditions are systematically checked. The Piecewise Linear Regression curve is a series of interconnected segments. In this paper, we focused on Piecewise Linear Regression curves composed of two segments. We also used Linear and Parabolic Regressions, to check if and to what extent Piecewise Linear Regression may provide an advantage over other regression techniques. We used two categories of regression techniques: Ordinary Least Squares regression is based on the usual minimization of the sum of squares of the residuals, or, equivalently, on the minimization of the average squared residual; Least Median of Squares regression is a robust regression technique that is based on the minimization of the median squared residual. Using a robust regression technique helps filter out the excessive influence of outliers. Results: It appears that the analysis of the relationship between traditional Function Points and COSMIC Function Points based on the aforementioned data analysis techniques yields valid significant models. However, different results for the various available datasets are achieved. In practice, we obtained statistically valid linear, piecewise linear, and non-linear conversion formulas for several datasets. In general, none of these is better than the others in a statistically significant manner. Conclusions: Practitioners interested in the conversion of FP measures into CFP (or vice versa) cannot just pick a conversion model and be sure that it will yield the best results. All the regression models we tested provide good results with some datasets. In practice, all the models described in the paper - in particular, both linear and non-linear ones - should be evaluated in order to identify the ones that are best suited for the specific dataset at hand.