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Computation of Arctan N for −∞ < N < +∞

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Rational (R) and polynomial (P) approximations to Arctan N are studied with the aim of computing this function, to any prescribed accuracy and without unduly increasing the number PC of stored constants, in a minimum number M of multiplications (and divisions for R approximations). The number Dg of first correct significant digits in principle is not bounded. The results corresponding to the values 8, 10, 18 and 20 of this number are as follows; (Table of results is given for M ranging in value from 4 to 10.) If M is increased, subroutines with smaller PC are easily deduced from our general results. Thus, for instance, rational approximations with Dg = 6 can be obtained in three multiplications only, if PC = 19 (combination m* = 3, q = 10); but the same accuracy Dg = 6 characterizes also the cases M = 4 with PC = 11 and M = 5 with PC = 7 (combinations m* = 4, q = 6 and m = 5, q = 4). If polynomial approximations are used, Dg = 6 is obtained for M = 5, PC = 7, but also for M = 4 and PC = 11. No subroutines with a stored table of values of Arctan x are considered.

Note: The Institute of Electrical and Electronics Engineers, Incorporated is distributing this Article with permission of the International Business Machines Corporation (IBM) who is the exclusive owner. The recipient of this Article may not assign, sublicense, lease, rent or otherwise transfer, reproduce, prepare derivative works, publicly display or perform, or distribute the Article.  

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IBM Journal of Research and Development  (Volume:2 ,  Issue: 1 )