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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
DOI: https://doi.org/10.1090/S0025-5718-99-01103-5
Published electronically: February 13, 1999
MathSciNet review: 1648423
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Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. Nicholas J. Higham, Accuracy and stability of numerical algorithms, SIAM, Philadelphia, PA, 1996. MR 97a:65047
  • 2. IEEE standard for binary floating point arithmetic. An American national standard, ANSI/IEEE Std 754-1985.
  • 3. Pat H. Sterbenz, Floating-point computation, Prentice-Hall, Englewood Cliffs, NJ, 1974. MR 50:1556
  • 4. J. H. Wilkinson, Rounding errors in algebraic processes, Prentice-Hall, Englewood Cliffs, NJ, 1963. MR 28:4661
  • 5. Abraham Ziv, Converting approximate error bounds into exact ones, Math. Comp. 64 (1995), 265-277. MR 95c:65074

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Additional Information

Abraham Ziv
Affiliation: IBM Israel, Science and Technology, Matam–Advanced Technology Center, Haifa 31905, Israel
Email: ziv@haifasc3.vnet.ibm.com

DOI: https://doi.org/10.1090/S0025-5718-99-01103-5
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

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