A Reduced Product of Absolute and Relative Error Bounds for Floating-Point Analysis

Rigorous estimation of bounds on errors in finite precision computation has become a key point of many formal verification tools. The primary interest of the use of such tools is generally to obtain worst-case bounds on the absolute errors. However, the natural bound on the elementary error committed by each floating-point arithmetic operation is a bound on the relative error, which suggests that relative error bounds could also play a role in the process of computing tight error estimations. In this work, we introduce a very simple interval-based abstraction, combining absolute and relative error propagations. We demonstrate with a prototype implementation how this simple product allows us in many cases to improve absolute error bounds, and even to often favorably compare with state-of-the art tools, that rely on much more costly relational abstractions or optimization-based estimations.