FLINT3.7.0-dev
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_mp_real_neglog_newton, _mp_real_neglog_newton_tune, _mp_real_atan_newton, _mp_real_atan_newton_tune

Synopsis#include <flint/mp_real.h>
void _mp_real_neglog_newton(nn_ptr y, ulong * err, nn_srcptr x, slong n)source
void _mp_real_neglog_newton_tune(nn_ptr y, ulong * err, nn_srcptr x, slong n, int forward, slong N)source
void _mp_real_atan_newton(nn_ptr y, ulong * err, nn_srcptr x, slong n)source
void _mp_real_atan_newton_tune(nn_ptr y, ulong * err, nn_srcptr x, slong n, int forward, slong N)source

Set (y,n)(y, n) to −log⁡x-\log x for (x,n)(x, n) in [1/2,1)[1/2, 1), respectively to atan⁡x\operatorname{atan} x for (x,n)(x, n) in [0,1)[0, 1), by one Newton–Taylor step over the forward function: −log⁡x=t−log⁡(1+w)-\log x = t - \log(1 + w) with w=xet−1w = x e^t - 1, resp. atan⁡x=t+atan⁡w\operatorname{atan} x = t + \operatorname{atan} w with w=(xcos⁡t−sin⁡t)/(xsin⁡t+cos⁡t)w = (x\cos t - \sin t)/(x \sin t + \cos t), from a starting value tt correct to a fraction of the precision. err is computed at run time and at most 2.

The forward functions are the diophantine exponential and sine and cosine at n+1n + 1 limbs. The _tune versions take the forward algorithm (0 diophantine, 1 bitwise, 2 table-free) and the number NN of terms of the series in ww (up to 16, 0 for the default); the starting value is computed at about 1/(N+1)1/(N + 1) resp. 1/(2N)1/(2N) of the precision, by the bitwise kernels up to 512 limbs and recursively above. The whole costs one forward evaluation plus about a tenth of one. These are the kernels of mp_real_log_bits and mp_real_atan_bits above 600 limbs.

Cite this page

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