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Number-Theoretic Test Generation for Directed Rounding
July 2000 (vol. 49 no. 7)
pp. 651-658

Abstract—We present methods to systematically generate the hardest test cases for multiplication, division, and square root subject to directed rounding, essentially extending previous work on number-theoretic floating-point testing to rounding modes other than to-nearest. The algorithms focus upon the rounding boundaries of the modes truncate, to-minus-infinity, and to-infinity, and programs based on them require little beyond exact arithmetic in the working precision to create billions of edge cases. We will show that the amount of work required to calculate trial multiplicands pays off in the form of free extra tests due to an interconnection among the operations considered herein. Although these tests do not replace proofs of correctness, they can be used to gain a high degree of confidence that the accuracy requirements mandated by IEEE Standard 754 have been satisfied.

[1] J.T. Coonen, “Contributions to a Proposed Standard for Binary Floating-Point Arithmetic,” PhD thesis, Univ. of California at Berkeley, 1984.
[2] IEEE 754-1985 Standard for Binary Floating-Point Arithmetic, New York: IEEE, 1985.
[3] W. Kahan, “A Test for Correctly Rounded SQRT,” http://www.cs.berkeley.edu/~wkahanSQRTest.ps ( 31 May 1996).
[4] W. Kahan, “Checking whether Floating-Point Division Is Correctly Rounded,” manuscript, Apr. 1987.
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[6] I. Koren, Computer Arithmetic Algorithms.Englewood Cliffs, N.J.: Prentice Hall, 1993.
[7] P.T.P. Tang, “Testing Computer Arithmetic by Elementary Number Theory,” preprint MCS-P84-0889, Math. and Computer Science Division, Argonne Nat'l Laboratory, Aug. 1989.
[8] The U.C. Berkeley Test Suite, available from Netlib,http://www.netlib.org/fpucbtest.tgz

Index Terms:
Arithmetic testing, IEEE Standard 754, rounding functions, Hensel lifting, hyperSPARC.
Citation:
Michael Parks, "Number-Theoretic Test Generation for Directed Rounding," IEEE Transactions on Computers, vol. 49, no. 7, pp. 651-658, July 2000, doi:10.1109/12.863034
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