Stupid is as Stupid Does: Taking the Square Root of the Square of a Floating-Point Number

Sylvie Boldo (1, 2)

1 TOCCATA - Formally Verified Programs, Certified Tools and Numerical Computations
2 LRI - Laboratoire de Recherche en Informatique

Floating-point experts know that mathematical formulas may fail or give imprecise results when implemented in floating-point arithmetic. This article describes an example where, surprisingly, it is absolutely not the case. Indeed, using radix 2 and an unbounded exponent range, the computation of the square root of the square of a floating-point number a is exactly |a|. A consequence is the fact that the floating-point computation of a/ sqrt (a^2 + b^2) is always in the interval [−1, 1]. This removes the need for a test when calling an arccos or an arcsin on this value. For more guarantees, this property was formally checked using the Coq proof assistant and the Flocq library. The conclusion will give hints on what happens without assumptions and in other radices, where the behavior is very different.

Mots clés: Floating-point; formal proof; square; square root; radix; Coq

Domaines: Logique en informatique [cs.LO]

Sylvie Boldo. Stupid is as Stupid Does: Taking the Square Root of the Square of a Floating-Point Number. Seventh and Eighth International Workshop on Numerical Software Verification, Apr 2015, Seattle, WA, United States. pp.50--55. ⟨hal-01148409⟩