pqc/external/flint-2.4.3/double_extras/lambertw.c

169 lines
5.4 KiB
C
Raw Normal View History

2014-05-18 22:03:37 +00:00
/*=============================================================================
This file is part of FLINT.
FLINT is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
FLINT is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with FLINT; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
=============================================================================*/
/******************************************************************************
Copyright (C) 2012 Fredrik Johansson
******************************************************************************/
#include "double_extras.h"
#define POLY(p, x) d_polyval((p), sizeof(p) / sizeof(double), (x))
static const double pol1[4] = {
0.2278634396856248853716, 0.6685854654191353381433,
0.4670475452404395343887, 0.061184972065242761167 };
static const double pol2[5] = {
0.2278636537503804204913, 0.8964421845409468074626,
1.0217927151592500702475, 0.34513102625055769873401,
0.020801230123523916719604 };
static const double pol3[6] = {
0.00005767860320327097931, 0.029896654795890461899563,
0.0378739044968912982585405, 0.00971957088414193124615358,
0.000488576886695502361566636, 1.150549466178344373015667e-6 };
static const double pol4[5] = {
0.030306172539339585635388, 0.066596680780796068408204,
0.035483738872057375987452, 0.00506436278851840340711316,
0.0001465263028844943142786722 };
static const double pol5[6] = {
0.00048233868073637531461, 0.004268700087824343609188,
0.00127714949974214706149789, 0.0000799706171559085390983949,
1.186347211803672341928371e-6, 2.943454067276155504308283e-9 };
static const double pol6[6] = {
0.00553288881087242781512, 0.0043904877060733941697614,
0.00069354549834088964895342, 0.0000288257440032545960408328,
3.01054066921000066105342e-7, 4.94316029290773314755549e-10 };
static const double pol7[4] = {
-0.93011683587619427070, -2.9702322028603227386,
-2.0759083419960793148, -0.042485660005713612806 };
static const double pol8[4] = {
0.93011683587619458392, 4.3654074566738568022,
6.1437079650412473506, 2.4613195056093927345 };
static const double pol9[11] = {
-1.0000000000000000000, 2.3316439815971242034,
-1.8121878856393634902, 1.9366311144923597554,
-2.3535512018816145168, 3.0668589010506319129,
-4.1753356002581771389, 5.8580237298747741488,
-8.4010322175239773710, 12.250753501314460424,
-18.100697012472442755 };
static const double pol10[6] = {
-5.1972986075163593071, -37.478686466672907613,
-96.155193004929291698, -102.23856988136744607,
-37.181958033133170210, -0.48504976999675644134 };
static const double pol11[6] = {
5.1972986074950082685, 45.274634378414741754, 150.20768172029114131,
233.88699813222871981, 167.13313463159765859, 42.171248374042409414 };
/* avoid overflows in the formula when x is close to 2^EMAX */
#define RESCALE 1.1102230246251565404e-16
static double
halley(double x, double w)
{
double t, u, v;
/* exp() does not overflow, since w is an underestimate
when the asymptotic series is used */
t = exp(w) * RESCALE;
u = 2*w + 2;
v = w*t - x * RESCALE;
t = w - u*v / (u*t*(w+1) - (w+2)*v);
return t;
}
/* this should be exactly 6627126856707895 * 2^(-54) ~=
0.36787944117144228, which
is the most negative double in the domain */
#define ONE_OVER_E ldexp(6627126856707895.0, -54)
/* difference from -1/e */
#define CORRECTION 4.3082397558469466e-17
double
d_lambertw(double x)
{
double t, u, w;
if (x == 0.0 || x != x || x == D_INF)
return x;
if (x < 0.0)
{
/* complex result */
if (x < -ONE_OVER_E)
return D_NAN;
/* close to zero */
else if (x > -1e-9)
return x - x * x;
/* close to the singularity at -1/e */
else if (x + ONE_OVER_E < 0.0003)
return POLY(pol9, sqrt((x + ONE_OVER_E) + CORRECTION));
/* otherwise get initial value for Halley iteration */
if (x + ONE_OVER_E < 0.04)
w = POLY(pol9, sqrt((x + ONE_OVER_E) + CORRECTION));
else
w = x * (1.0 + x * POLY(pol10, x) / POLY(pol11, x));
}
else
{
/* close to zero */
if (x <= 0.03125)
{
if (x < 1e-9)
return x - x * x;
else
return x * (1.0 + x * POLY(pol7, x) / POLY(pol8, x));
}
/* get initial value for Halley iteration */
if (x <= 1.0)
w = x * POLY(pol1, x) / POLY(pol2, x);
else if (x <= 6.0)
w = POLY(pol3, x) / POLY(pol4, x);
else if (x <= 40.0)
w = POLY(pol5, x) / POLY(pol6, x);
else
{
/* asymptotic series */
t = log(x);
u = log(t);
w = (2*t*t*t - 2*(1+(t-1)*t)*u + u*u)/(2*t*t);
/* one extra refinement */
if (x < 1e15)
w = halley(x, w);
}
}
return halley(x, w);
}