Algorithm 1029: Encapsulated Error, a Direct Approach to Evaluate Floating-Point Accuracy

Abstract

Floating-point numbers represent only a subset of real numbers. As such, floating-point arithmetic introduces approximations that can compound and have a significant impact on numerical simulations. We introduce encapsulated error, a new way to estimate the numerical error of an application and provide a reference implementation, the Shaman library. Our method uses dedicated arithmetic over a type that encapsulates both the result the user would have had with the original computation and an approximation of its numerical error. We thus can measure the number of significant digits of any result or intermediate result in a simulation. We show that this approach, although simple, gives results competitive with state-of-the-art methods. It has a smaller overhead, and it is compatible with parallelism, making it suitable for the study of large-scale applications.