FLINT3.7.0-dev
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arb_atan_newton, arb_atan_arf_newton

Synopsis#include <flint/arb.h>
void arb_atan_newton(arb_t res, const arb_t x, slong prec)source
void arb_atan_arf_newton(arb_t res, const arf_t x, slong prec)source

Compute atan⁡x\operatorname{atan} x from a lower-precision value tt by atan⁡x=t+atan⁡((xcos⁡t−sin⁡t)/(cos⁡t+xsin⁡t))\operatorname{atan} x = t + \operatorname{atan}((x \cos t - \sin t)/(\cos t + x \sin t)), with the correction evaluated by a short Taylor series, so that the precision multiplies by 5 to 11 per step. Arguments with ∣x∣>16|x| > 16 use atan⁡x=±π/2−atan⁡(1/x)\operatorname{atan} x = \pm\pi/2 - \operatorname{atan}(1/x). Used for arb_atan_arf from 3400 bits.

Cite this page

The FLINT team, “arb_atan_newton, arb_atan_arf_newton (arb.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/arb/arb_atan_newton.html (generated 2026-11-02).