Sharp ULP rounding error bound for the hypotenuse function
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- by Abraham Ziv PDF
- Math. Comp. 68 (1999), 1143-1148 Request permission
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
- Nicholas J. Higham, Accuracy and stability of numerical algorithms, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. MR 1368629
- IEEE standard for binary floating point arithmetic. An American national standard, ANSI/IEEE Std 754-1985.
- Pat H. Sterbenz, Floating-point computation, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR 0349062
- J. H. Wilkinson, Rounding errors in algebraic processes, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0161456
- Abraham Ziv, Converting approximate error bounds into exact ones, Math. Comp. 64 (1995), no. 209, 265–277. MR 1260129, DOI 10.1090/S0025-5718-1995-1260129-1
Additional Information
- Abraham Ziv
- Affiliation: IBM Israel, Science and Technology, Matam–Advanced Technology Center, Haifa 31905, Israel
- Email: ziv@haifasc3.vnet.ibm.com
- Received by editor(s): December 1, 1997
- Published electronically: February 13, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Math. Comp. 68 (1999), 1143-1148
- MSC (1991): Primary 65G05; Secondary 65D20
- DOI: https://doi.org/10.1090/S0025-5718-99-01103-5
- MathSciNet review: 1648423