`arb_sin_cos_arf_atan_reduction`

Synopsis#include <flint/arb.h>

void arb_sin_cos_arf_atan_reduction(arb_t res1, arb_t res2, const arf_t x, slong prec)source

Computes $\sin x$ and $\cos x$ by reduction with the arguments of small Gaussian primes [HJ2024]: $x - \sum c_i \theta_i$ is made small with $\theta_i = \arg(a_i + b_i i)$ , and the result is multiplied by the Gaussian integer $\prod (a_i + b_i i)^{c_i}$ normalised to modulus 1. Used for arb_sin_cos from 2600 bits to 4 million bits.

References#

[HJ2024] J. van der Hoeven and F. Johansson, "Fast multiple precision exp(x) with precomputations", 2024 IEEE 31st Symposium on Computer Arithmetic (ARITH) (2024), https://hal.science/hal-04454093

Cite this page

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

@misc{flint-doc-arb-arb-sin-cos-arf-atan-reduction,
  author       = {{The FLINT team}},
  title        = {{arb\_sin\_cos\_arf\_atan\_reduction (arb.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/arb/arb_sin_cos_arf_atan_reduction.html},
  note         = {Generated 2026-11-02}
}

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