Modern computing has adopted the floating point type as a default way to describe computations with real numbers. Thanks to dedicated hardware support, such computations are efficient on modern architectures. However, rigorous reasoning about the resulting programs remains difficult, because of a large gap between the finite floating point representation and the infinite-precision real-number semantics that serves as the mental model for the developers. Because programming languages do not provide support for estimating errors, some computations in practice are performed more and some less precisely than needed. We present a library solution for rigorous arithmetic computation. Our numerical data type library tracks a (double) floating point value, but also a guaranteed upper bound on the error between this value and the ideal value that would be computed in the real-value semantics. Our implementation involves a set of linear approximations based on an extension of affine arithmetic. The derived approximations cover most of the standard mathematical operations including trigonometric functions, and are more comprehensive than any publicly available ones. Moreover, while interval arithmetic rapidly yields overly pessimistic estimates, our approach remains precise for a range of computational tasks of interest. We evaluate the library on a number of examples from numerical analysis and physical simulations. We found it to be a useful tool for gaining confidence in the correctness of the computation.