FLINT3.7.0-dev
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DFLOAT_LOG_EPS_1, DFLOAT_LOG_EPS_2, DFLOAT_LOG_EPS_3, DFLOAT_LOG_EPS_4, DFLOAT_LOG_EPS, DFLOAT_ATAN_EPS_1, DFLOAT_ATAN_EPS_2, DFLOAT_ATAN_EPS_3, DFLOAT_ATAN_EPS_4, DFLOAT_ATAN_EPS

Synopsis#include <flint/dfloat.h>
DFLOAT_LOG_EPS_1source
DFLOAT_LOG_EPS_2source
DFLOAT_LOG_EPS_3source
DFLOAT_LOG_EPS_4source
DFLOAT_LOG_EPS(n)source
DFLOAT_ATAN_EPS_1source
DFLOAT_ATAN_EPS_2source
DFLOAT_ATAN_EPS_3source
DFLOAT_ATAN_EPS_4source
DFLOAT_ATAN_EPS(n)source

Relative error bounds of the logarithm and arctangent kernels: ∣dN_log(x)−log⁡x∣≤ε∣log⁡x∣|\mathtt{dN\_log}(x) - \log x| \le \varepsilon |\log x| for 2−1022≤x<210222^{-1022} \le x < 2^{1022}, with ε\varepsilon about 2−49.32^{-49.3}, 2−99.32^{-99.3}, 2−150.22^{-150.2} and 2−200.82^{-200.8}, and ∣dN_atan(x)−atan⁡x∣≤ε∣atan⁡x∣|\mathtt{dN\_atan}(x) - \operatorname{atan} x| \le \varepsilon |\operatorname{atan} x| for all finite xx, with ε\varepsilon about 2−49.82^{-49.8}, 2−99.62^{-99.6}, 2−150.52^{-150.5} and 2−201.32^{-201.3}. Both bounds are relative everywhere, including near x=1x = 1 for the logarithm and near 0 for the arctangent.

Cite this page

The FLINT team, “DFLOAT_LOG_EPS_1, DFLOAT_LOG_EPS_2, DFLOAT_LOG_EPS_3, DFLOAT_LOG_EPS_4, DFLOAT_LOG_EPS, DFLOAT_ATAN_EPS_1, DFLOAT_ATAN_EPS_2, DFLOAT_ATAN_EPS_3, DFLOAT_ATAN_EPS_4, DFLOAT_ATAN_EPS (dfloat.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/dfloat/DFLOAT_LOG_EPS_1.html (generated 2026-11-02).