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A Continued-Fraction Analysis Of Trigonometric Argument Reduction
November 1995 (vol. 44 no. 11)
pp. 1348-1351

Abstract—The calculation of a trigonometric function of a large argument x is effectively carried out by finding the integer N and 0 ≤α < 1 such that $x = (N + \alpha){\pi\over 4}$. This reduction modulo ${\pi\over 4}$ makes it possible to calculate a trigonometric function of a reduced argument, either $\alpha{\pi\over 4}$ or $(1 - \alpha){\pi\over 4}$, which lies in the interval $(0,{\pi\over 4})$. Payne and Hanek [1] described an efficient algorithm for computing α to a predetermined level of accuracy. They noted that if x differs only slightly from an integral multiple of ${\pi\over 2}$, the reduction must be carried out quite accurately to avoid a large loss of significance in the reduced argument. We present a simple method using continued fractions for determining, for all numbers x represented in an IEEE floating-point format, the specific x for which the greatest number of insignificant leading bits occur. Applications are made to IEEE single-precision and double-precision formats and two extended-precision formats.

[1] M. Payne and R. Hanek,“Radian reduction for trigonometric functions,” SIGNUM Newsletter, vol. 18, pp. 19-24, 1983.
[2] ANSI/IEEE Std. 754-1985, Binary Floating-Point Arithmetic, IEEE Press, Piscataway, N.J., 1985 (also called ISO/IEC 559).
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[4] P.T.P. Tang,“Some software implementations of the functions sine and cosine,” ANL 90/3, pp. 1-27, 1990.
[5] W.J. Cody, Jr., and W. Waite,Software Manual for the Elementary Functions, p. 125,Englewood Cliffs: Prentice-Hall, Inc., 1980.
[6] A.Y. Khinchin,Continued Fractions, p. 24,Chicago: The University of Chicago Press, 1964.
[7] Tim Peters,communication to nceg, 1991.
[8] V. Paxson,A Program for Testing IEEE Decimal-Binary Conversion, available as testbase.ps.Z from ftp.ee.lbl.gov., pp. 1-40.
[9] MC88110 Second Generation RISC Microprocessor User’s Manual, pp. 4-3 Motorola, Inc., 1991.
[10] R.P. Brent,“Fast multiple-precision evaluation of elementary functions,” J. ACM, vol. 23, pp. 242, 1976.

Index Terms:
Argument reduction, computer arithmetic, continued fractions, non-linear optimization, trigonometric functions.
Citation:
Roger Alan Smith, "A Continued-Fraction Analysis Of Trigonometric Argument Reduction," IEEE Transactions on Computers, vol. 44, no. 11, pp. 1348-1351, Nov. 1995, doi:10.1109/12.475133
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