Radian reduction for trigonometric functions

ABSTRACT

An accurate reduction poses little difficulty for arguments of a few radians. However for, say, a CRAY1, H format on the VAX, or double extended in the proposed IEEE standard, the maximum argument which might be presented for reduction is of the order of 2^16000 radians. Accurate reduction of such an argument would require storage of π (or its reciprocal) to over 16,000 bits. Direct reduction by division (or multiplication) then requires generation of a somewhat larger number of bits in the result in order to guarantee the accuracy of the reduction. Of these bits only the low few bits of the integer part of the quotient (product) and enough bits to correctly round the remainder are relevant; the rest will be discarded.

REFERENCES

• 1: William Gosper, Acceleration of Series, Massachusetts Institute of Technology, Cambridge, MA, 1974
• 2: R. W. Gosper, "A Calculus of Series Rearrangements," pp. 121--151 in "Algorithms and Complexity: New Directions and Recent Results," edited by J. F. Traub, Academic Press, 1976.