Augmented Precision Square Roots and 2-D Norms, and Discussion on Correctly Rounding sqrt(x^2+y^2)

2011 IEEE 20th Symposium on Computer Arithmetic Pub Date : 2011-07-25 DOI:10.1109/ARITH.2011.13

N. Brisebarre, Mioara Joldes, Peter Kornerup, Érik Martin-Dorel, J. Muller

引用次数: 1

Abstract

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.

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增广精度平方根与二维范数，以及对平方根(x^2+y^2)正确舍入的讨论

将“增广精度”算法定义为这样一种算法:在精度为p的浮点运算中，其结果返回为两个浮点数的未求值和，相对误差为2^(-2p)。假设FMA指令可用，我们对平方根的增广精度算法进行了严格的误差分析，并为2d范数根号(x^2+y^2)引入了两种略有不同的增广精度算法。然后，当根号(x²+y²)本身不是中点时，我们给出根号(x²+y²)与中点之间的最小距离(以ulps为单位)的严格下界。这使我们能够确定当我们的算法能够返回正确舍入的2d规范时的情况。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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2011 IEEE 20th Symposium on Computer Arithmetic

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