Augmented precision square roots, 2-D norms, and discussion on correctly rounding {x^2+y^2}

Nicolas Brisebarre (1) , Mioara Maria Joldes (1) , Peter Kornerup (2) , Érik Martin-Dorel (1) , Jean-Michel Muller (1)

1 ARENAIRE - Computer arithmetic
2 IMADA - Department of Mathematics and Computer Science [Odense]

Define an "augmented precision" algorithm as an algorithm that returns, in precision-p floating-point arithmetic, its result as the unevaluated sum of two floating-point numbers, with a relative error of the order of 2^(−2p). Assuming an FMA instruction is available, we perform a tight error analysis of an augmented precision algorithm for the square root, and introduce two slightly different augmented precision algorithms for the 2D-norm sqrt(x^2 + y^2). Then we give tight lower bounds on the minimum distance (in ulps) between sqrt(x^2 + y^2) and a midpoint when sqrt(x^2 + y^2) is not itself a midpoint. This allows us to determine cases when our algorithms make it possible to return correctly-rounded 2D-norms.

Mots clés: Floating-point arithmetic; compensated algorithms; square-root; Correct rounding; 2D-norms; accurate computations

Domaines: Autre [cs.OH]

Nicolas Brisebarre, Mioara Maria Joldes, Peter Kornerup, Érik Martin-Dorel, Jean-Michel Muller. Augmented precision square roots, 2-D norms, and discussion on correctly rounding {x^2+y^2}. 20th IEEE Symposium on Computer Arithmetic (ARITH-20), Jul 2011, Tübingen, Germany. pp.23-30, ⟨10.1109/ARITH.2011.13⟩. ⟨ensl-00545591v2⟩