Augmented precision square roots , 2 - D norms , and discussion on correctly rounding √ x 2 + y 2 .

Define an “augmented precision” algorithm as an algorithm that returns, in precision-p floating-point arithmetic, its result as the unevaluated sum of two floatingpoint 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 p x2 + y2. Then we give tight lower bounds on the minimum distance (in ulps) between p x2 + y2 and a midpoint when p x2 + y2 is not itself a midpoint. This allows us to determine cases when our algorithms make it possible to return correctlyrounded 2D-norms.