A fixed-point iteration computes arctan(t) to arbitrary odd convergence order 2P+1 by subtracting a partial arctan series applied to the tangent of the current error.
BRENT: Fast Algorithms for High-Precision Computat ion of Elementary Functions, Australian National University Canberra, ACT 0200, Austra lia 2006
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A fixed point iteration method for the arctangent with any odd order of convergence based on sine and cosine
A fixed-point iteration computes arctan(t) to arbitrary odd convergence order 2P+1 by subtracting a partial arctan series applied to the tangent of the current error.