Modern biomedical research is increasingly data-driven. To create the required big datasets, health data needs to be shared or reused, which often leads to privacy challenges. Data anonymization is an important protection method where data is transformed such that privacy guarantees can be provided according to formal models. For applications in practice, anonymization methods need to be integrated into scalable and reliable tools. In this work, we tackle the problem of achieving reliability. Privacy models often involve mathematical definitions using real numbers which are typically approximated using floating-point numbers when implemented as software. We study the effect on the privacy guarantees provided and present a reliable computing framework based on fractional and interval arithmetic for improving the reliability of implementations. Extensive evaluations demonstrate that reliable data anonymization is practical and that it can be achieved with minor impacts on executions times and data utility.