Differential privacy is among the most prominent techniques for preserving privacy of sensitive data, owing to its robust mathematical guarantees and general applicability to a vast array of computations on data, including statistical analysis and machine learning. Previous work demonstrated that concrete implementations of differential privacy mechanisms are vulnerable to statistical attacks. This vulnerability is caused by the approximation of real values to floating point numbers. This paper presents a practical solution to the finite-precision floating point vulnerability, where the inverse transform sampling of the Laplace distribution can itself be inverted, thus enabling an attack where the original value can be retrieved with non-negligible advantage. The proposed solution has the advantages of being (i) mathematically sound, (ii) generalisable to any infinitely divisible probability distribution, and (iii) of simple implementation in modern architectures. Finally, the solution has been designed to make side channel attack in-feasible, because of inherently exponential, in the size of the domain, brute force attacks.