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<p><it>Abstract</it>—The calculation of a trigonometric function of a large argument <it>x</it> is effectively carried out by finding the integer <it>N</it> and 0 ≤α < 1 such that <math><tmath>$x = (N + \alpha){\pi\over 4}$</tmath></math>. This reduction modulo <math><tmath>${\pi\over 4}$</tmath></math> makes it possible to calculate a trigonometric function of a reduced argument, either <math><tmath>$\alpha{\pi\over 4}$</tmath></math> or <math><tmath>$(1 - \alpha){\pi\over 4}$</tmath></math>, which lies in the interval <math><tmath>$(0,{\pi\over 4})$</tmath></math>. Payne and Hanek [<ref rid="BIBC13481" type="bib">1</ref>] described an efficient algorithm for computing α to a predetermined level of accuracy. They noted that if <it>x</it> differs only slightly from an integral multiple of <math><tmath>${\pi\over 2}$</tmath></math>, 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 <it>x</it> represented in an IEEE floating-point format, the specific <it>x</it> 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.</p>
Argument reduction, computer arithmetic, continued fractions, non-linear optimization, trigonometric functions.
Roger Alan Smith, "A Continued-Fraction Analysis Of Trigonometric Argument Reduction", IEEE Transactions on Computers, vol. 44, no. , pp. 1348-1351, November 1995, doi:10.1109/12.475133
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