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Mathematics of Computation

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Sharp ULP rounding error bound
for the hypotenuse function

Author: Abraham Ziv
Journal: Math. Comp. 68 (1999), 1143-1148
MSC (1991): Primary 65G05; Secondary 65D20
Published electronically: February 13, 1999
MathSciNet review: 1648423
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Abstract: The hypotenuse function, $z=\sqrt{x^2+y^2}$, is sometimes included in math library packages. Assuming that it is being computed by a straightforward algorithm, in a binary floating point environment, with round to nearest rounding mode, a sharp roundoff error bound is derived, for arbitrary precision. For IEEE single precision, or higher, the bound implies that $|\overline z-z|<1.222\, ulp(z)$ and $|\overline z-z|<1.222\, ulp(\overline z)$. Numerical experiments indicate that this bound is sharp and cannot be improved.

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Additional Information

Abraham Ziv
Affiliation: IBM Israel, Science and Technology, Matam–Advanced Technology Center, Haifa 31905, Israel

Keywords: Rounding error, error analysis, relative error, error bound, floating point, ULP, hypotenuse function, math library
Received by editor(s): December 1, 1997
Published electronically: February 13, 1999
Article copyright: © Copyright 1999 American Mathematical Society