Publication 1995 Issue No. 11 - November Abstract - A Continued-Fraction Analysis Of Trigonometric Argument Reduction
A Continued-Fraction Analysis Of Trigonometric Argument Reduction
November 1995 (vol. 44 no. 11)
pp. 1348-1351
 ASCII Text x Roger Alan Smith, "A Continued-Fraction Analysis Of Trigonometric Argument Reduction," IEEE Transactions on Computers, vol. 44, no. 11, pp. 1348-1351, November, 1995.
 BibTex x @article{ 10.1109/12.475133,author = {Roger Alan Smith},title = {A Continued-Fraction Analysis Of Trigonometric Argument Reduction},journal ={IEEE Transactions on Computers},volume = {44},number = {11},issn = {0018-9340},year = {1995},pages = {1348-1351},doi = {http://doi.ieeecomputersociety.org/10.1109/12.475133},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on ComputersTI - A Continued-Fraction Analysis Of Trigonometric Argument ReductionIS - 11SN - 0018-9340SP1348EP1351EPD - 1348-1351A1 - Roger Alan Smith, PY - 1995KW - Argument reductionKW - computer arithmeticKW - continued fractionsKW - non-linear optimizationKW - trigonometric functions.VL - 44JA - IEEE Transactions on ComputersER -

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 $x = \left(N + \alpha\right)\left\{\pi\over 4\right\}$. This reduction modulo $\left\{\pi\over 4\right\}$ makes it possible to calculate a trigonometric function of a reduced argument, either $\alpha\left\{\pi\over 4\right\}$ or $\left(1 - \alpha\right)\left\{\pi\over 4\right\}$, which lies in the interval $\left(0,\left\{\pi\over 4\right\}\right)$. 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 $\left\{\pi\over 2\right\}$, 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.

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[2] ANSI/IEEE Std. 754-1985, Binary Floating-Point Arithmetic, IEEE Press, Piscataway, N.J., 1985 (also called ISO/IEC 559).
[3] S. Gal and B. Bachelis, "An Accurate Elementary Mathematical Library for the IEEE Floating Point Standard," ACM Trans. Math. Software, vol. 17, pp. 26-45, Mar. 1991.
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[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