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Measuring software functional size via standard Function Points Analysis (FPA) requires the availability of fully specified requirements and specific competencies. Most of the time, the need to measure software functional size occurs well in advance with respect to these ideal conditions, under the lack of complete information or skilled experts. To work around the constraints of the official measurement process, several estimation methods for FPA have been proposed and are commonly used. Among these, the International Function Points User Group (IFPUG) has adopted the "High-level FPA" method (also known as the NESMA method). This method avoids weighting each data and transaction function by using fixed weights instead. Applying High-level FPA, or similar estimation methods, is faster and easier than carrying out the official measurement process but inevitably yields an approximation in the measures. In this article, we contribute to the problem of estimating software functional size measures by using machine learning. To the best of our knowledge, machine learning methods were never applied to the early estimation of software functional size. Our goal is to understand whether machine learning techniques yield estimates of FPA measures that are more accurate than those obtained with High-level FPA or similar methods. An empirical study on a large dataset of functional size predictors was carried out to train and test three of the most popular and robust machine learning methods, namely Random Forests, Support Vector Regression, and Neural Networks. A systematic experimental phase, with cycles of dataset filtering and splitting, parameter tuning, and model training and validation, is presented. The estimation accuracy of the obtained models was then evaluated and compared to that of fixed-weight models (e.g., High-level FPA) and linear regression models, also using a second dataset as the test set. We found that Support Vector Regression yields quite accurate estimation models. However, the obtained level of accuracy does not appear significantly better with respect to High-level FPA or to models built via ordinary least squares regression. Noticeably, fairly good accuracy levels were obtained by models that do not even require discerning among different types of transactions and data. [ABSTRACT FROM AUTHOR]