Augmented Precision Square Roots and 2-D Norms, and Discussion on Correctly Rounding sqrt(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 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 √x
²
+y
²
. Then we give tight lower bounds on the minimum distance (in ulps) between √x
²
+y
²
and a midpoint when √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.