-/* Copyright (C) 2011-2025 Free Software Foundation, Inc.
- This file is part of the GNU C Library.
+/* Correctly-rounded inverse hyperbolic tangent function. Binary64 version.
- The GNU C Library is free software; you can redistribute it and/or
- modify it under the terms of the GNU Lesser General Public
- License as published by the Free Software Foundation; either
- version 2.1 of the License, or (at your option) any later version.
+Copyright (c) 2023-2025 Alexei Sibidanov.
- The GNU C Library 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
- Lesser General Public License for more details.
+The original version of this file was copied from the CORE-MATH
+project (file src/binary64/atanh/atanh.c, revision 4da7f241).
- You should have received a copy of the GNU Lesser General Public
- License along with the GNU C Library; if not, see
- <https://www.gnu.org/licenses/>. */
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
-/* __ieee754_atanh(x)
- Method :
- 1.Reduced x to positive by atanh(-x) = -atanh(x)
- 2.For x>=0.5
- 1 2x x
- atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
- 2 1 - x 1 - x
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE. */
- For x<0.5
- atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
+#include <array_length.h>
+#include <stdint.h>
+#include <math.h>
+#include <libm-alias-finite.h>
+#include "math_config.h"
- Special cases:
- atanh(x) is NaN if |x| > 1 with signal;
- atanh(NaN) is that NaN with no signal;
- atanh(+-1) is +-INF with signal.
+static inline double
+fasttwosum (double x, double y, double *e)
+{
+ double s = x + y, z = s - x;
+ *e = y - z;
+ return s;
+}
- */
+static inline double
+fasttwosub (double x, double y, double *e)
+{
+ double s = x - y, z = x - s;
+ *e = z - y;
+ return s;
+}
-#include <float.h>
-#include <inttypes.h>
-#include <math.h>
-#include <math-barriers.h>
-#include <math_private.h>
-#include <math-underflow.h>
-#include <libm-alias-finite.h>
+static inline double
+adddd (double xh, double xl, double ch, double cl, double *l)
+{
+ double s = xh + ch, d = s - xh;
+ *l = ((ch - d) + (xh + (d - s))) + (xl + cl);
+ return s;
+}
+
+static inline double
+muldd (double xh, double xl, double ch, double cl, double *l)
+{
+ double ahlh = ch * xl, alhh = cl * xh, ahhh = ch * xh,
+ ahhl = fma (ch, xh, -ahhh);
+ ahhl += alhh + ahlh;
+ ch = ahhh + ahhl;
+ *l = (ahhh - ch) + ahhl;
+ return ch;
+}
-static const double huge = 1e300;
+static inline double
+mulddd (double xh, double xl, double ch, double *l)
+{
+ double ahlh = ch * xl, ahhh = ch * xh, ahhl = fma (ch, xh, -ahhh);
+ ahhl += ahlh;
+ ch = ahhh + ahhl;
+ *l = (ahhh - ch) + ahhl;
+ return ch;
+}
+
+static inline double
+polydd (double xh, double xl, int n, const double c[][2], double *l)
+{
+ int i = n - 1;
+ double ch = c[i][0] + *l, cl = ((c[i][0] - ch) + *l) + c[i][1];
+ while (--i >= 0)
+ {
+ ch = muldd (xh, xl, ch, cl, &cl);
+ double th = ch + c[i][0], tl = (c[i][0] - th) + ch;
+ ch = th;
+ cl += tl + c[i][1];
+ }
+ *l = cl;
+ return ch;
+}
+
+static double __attribute__ ((noinline)) as_atanh_refine (double, double,
+ double, double);
+static double __attribute__ ((noinline))
+as_atanh_zero (double x)
+{
+ static const double ch[][2]
+ = { { 0x1.5555555555555p-2, 0x1.5555555555555p-56 },
+ { 0x1.999999999999ap-3, -0x1.999999999611cp-57 },
+ { 0x1.2492492492492p-3, 0x1.2492490f76b25p-57 },
+ { 0x1.c71c71c71c71cp-4, 0x1.c71cd5c38a112p-58 },
+ { 0x1.745d1745d1746p-4, -0x1.7556c4165f4cap-59 },
+ { 0x1.3b13b13b13b14p-4, -0x1.b893c3b36052ep-59 },
+ { 0x1.1111111111105p-4, 0x1.4e1afd723ed1fp-59 },
+ { 0x1.e1e1e1e1e2678p-5, -0x1.f86ea96fb1435p-59 },
+ { 0x1.af286bc9f90ccp-5, 0x1.1e51a6e54fde9p-60 },
+ { 0x1.8618618c779b6p-5, -0x1.ab913de95c3bfp-61 },
+ { 0x1.642c84aa383ebp-5, 0x1.632e747641b12p-59 },
+ { 0x1.47ae2d205013cp-5, -0x1.0c9617e7bcff2p-60 },
+ { 0x1.2f664d60473f9p-5, 0x1.3adb3e2b7f35ep-61 } };
+
+ static const double cl[]
+ = { 0x1.1a9a91fd692afp-5, 0x1.06dfbb35e7f44p-5, 0x1.037bed4d7588fp-5,
+ 0x1.5aca6d6d720d6p-6, 0x1.99ea5700d53a5p-5 };
+
+ double x2 = x * x, x2l = fma (x, x, -x2);
+ double y2
+ = x2
+ * (cl[0] + x2 * (cl[1] + x2 * (cl[2] + x2 * (cl[3] + x2 * (cl[4])))));
+ double y1 = polydd (x2, x2l, 13, ch, &y2);
+ y1 = mulddd (y1, y2, x, &y2);
+ y1 = muldd (y1, y2, x2, x2l, &y2);
+ double y0 = fasttwosum (x, y1, &y1);
+ y1 = fasttwosum (y1, y2, &y2);
+ uint64_t t = asuint64 (y1);
+ if (__glibc_unlikely (!(t & (~UINT64_C(0) >> 12))))
+ {
+ uint64_t w = asuint64 (y2);
+ if ((w ^ t) >> 63)
+ t--;
+ else
+ t++;
+ y1 = asdouble (t);
+ }
+ return y0 + y1;
+}
#ifndef SECTION
-# define SECTION
+# define SECTION
#endif
+
SECTION
double
__ieee754_atanh (double x)
{
- double xa = fabs (x);
- double t;
- if (isless (xa, 0.5))
- {
- if (__glibc_unlikely (xa < 0x1.0p-28))
+ double ax = fabs (x);
+ uint64_t aix = asuint64 (ax);
+ if (__glibc_unlikely (aix >= UINT64_C(0x3ff0000000000000)))
+ { // |x| >= 1
+ if (aix == UINT64_C(0x3ff0000000000000))
+ // |x| = 1
+ return __math_divzero (x < 0.0);
+ if (aix > UINT64_C(0x7ff0000000000000))
+ return x + x; // nan
+ return __math_invalid (x);
+ }
+
+ if (__glibc_unlikely (aix < UINT64_C(0x3fd0000000000000)))
+ { // |x| < 0x1p-2
+ // atanh(x) rounds to x to nearest for |x| < 0x1.d12ed0af1a27fp-27
+ if (__glibc_unlikely (aix < UINT64_C(0x3e4d12ed0af1a27f)))
{
- math_force_eval (huge + x);
- math_check_force_underflow (x);
- return x;
+ // |x| < 0x1.d12ed0af1a27fp-27
+ /* We have underflow exactly when 0 < |x| < 2^-1022:
+ for RNDU, atanh(2^-1022-2^-1074) would round to 2^-1022-2^-1075
+ with unbounded exponent range */
+ return __math_check_uflow_zero_lt (x, 0x1p-1022,
+ fma (x, 0x1p-55, x));
}
-
- t = xa + xa;
- t = 0.5 * __log1p (t + t * xa / (1.0 - xa));
+ double x2 = x * x;
+ static const double c[] = { 0x1.999999999999ap-3, 0x1.2492492492244p-3,
+ 0x1.c71c71c79715fp-4, 0x1.745d16f777723p-4,
+ 0x1.3b13ca4174634p-4, 0x1.110c9724989bdp-4,
+ 0x1.e2d17608a5b2ep-5, 0x1.a0b56308cba0bp-5,
+ 0x1.fb6341208ad2ep-5 };
+ double dx2 = fma (x, x, -x2);
+ double x4 = x2 * x2, x3 = x2 * x, x8 = x4 * x4;
+ double dx3 = fma (x2, x, -x3) + dx2 * x;
+ double p
+ = (c[0] + x2 * c[1]) + x4 * (c[2] + x2 * c[3])
+ + x8 * ((c[4] + x2 * c[5]) + x4 * (c[6] + x2 * c[7]) + x8 * c[8]);
+ double t = 0x1.5555555555555p-56 + x2 * p;
+ double pl, ph = fasttwosum (0x1.5555555555555p-2, t, &pl);
+ ph = muldd (ph, pl, x3, dx3, &pl);
+ double tl;
+ ph = fasttwosum (x, ph, &tl);
+ pl += tl;
+ double eps = x * (x4 * 0x1.dp-53 + 0x1p-103);
+ double lb = ph + (pl - eps), ub = ph + (pl + eps);
+ if (__glibc_likely (lb == ub))
+ return lb;
+ return as_atanh_zero (x);
}
- else if (__glibc_likely (isless (xa, 1.0)))
- t = 0.5 * __log1p ((xa + xa) / (1.0 - xa));
- else
- {
- if (isgreater (xa, 1.0))
- return (x - x) / (x - x);
- return x / 0.0;
- }
+ double pl, ph = fasttwosum (1, ax, &pl), ql, qh = fasttwosub (1, ax, &ql),
+ iqh = 1 / qh, th = ph * iqh,
+ tl = fma (ph, iqh, -th)
+ + (pl + ph * (fma (-qh, iqh, 1) - ql * iqh)) * iqh;
- return copysign (t, x);
-}
+ static const struct
+ {
+ uint16_t c0;
+ short c1;
+ } B[] = {
+ { 301, 27565 }, { 7189, 24786 }, { 13383, 22167 }, { 18923, 19696 },
+ { 23845, 17361 }, { 28184, 15150 }, { 31969, 13054 }, { 35231, 11064 },
+ { 37996, 9173 }, { 40288, 7372 }, { 42129, 5657 }, { 43542, 4020 },
+ { 44546, 2457 }, { 45160, 962 }, { 45399, -468 }, { 45281, -1838 },
+ { 44821, -3151 }, { 44032, -4412 }, { 42929, -5622 }, { 41522, -6786 },
+ { 39825, -7905 }, { 37848, -8982 }, { 35602, -10020 }, { 33097, -11020 },
+ { 30341, -11985 }, { 27345, -12916 }, { 24115, -13816 }, { 20661, -14685 },
+ { 16989, -15526 }, { 13107, -16339 }, { 9022, -17126 }, { 4740, -17889 }
+ };
+ static const double r1[]
+ = { 0x1p+0, 0x1.f5076p-1, 0x1.ea4bp-1, 0x1.dfc98p-1, 0x1.d5818p-1,
+ 0x1.cb72p-1, 0x1.c199cp-1, 0x1.b7f76p-1, 0x1.ae8ap-1, 0x1.a5504p-1,
+ 0x1.9c492p-1, 0x1.93738p-1, 0x1.8ace6p-1, 0x1.8258ap-1, 0x1.7a114p-1,
+ 0x1.71f76p-1, 0x1.6a09ep-1, 0x1.6247ep-1, 0x1.5ab08p-1, 0x1.5342cp-1,
+ 0x1.4bfdap-1, 0x1.44e08p-1, 0x1.3dea6p-1, 0x1.371a8p-1, 0x1.306fep-1,
+ 0x1.29e9ep-1, 0x1.2387ap-1, 0x1.1d488p-1, 0x1.172b8p-1, 0x1.11302p-1,
+ 0x1.0b558p-1, 0x1.059bp-1, 0x1p-1 };
+ static const double r2[]
+ = { 0x1p+0, 0x1.ffa74p-1, 0x1.ff4eap-1, 0x1.fef62p-1, 0x1.fe9dap-1,
+ 0x1.fe452p-1, 0x1.fdeccp-1, 0x1.fd946p-1, 0x1.fd3c2p-1, 0x1.fce3ep-1,
+ 0x1.fc8bcp-1, 0x1.fc33ap-1, 0x1.fbdbap-1, 0x1.fb83ap-1, 0x1.fb2bcp-1,
+ 0x1.fad3ep-1, 0x1.fa7c2p-1, 0x1.fa246p-1, 0x1.f9ccap-1, 0x1.f975p-1,
+ 0x1.f91d8p-1, 0x1.f8c6p-1, 0x1.f86e8p-1, 0x1.f8172p-1, 0x1.f7bfep-1,
+ 0x1.f768ap-1, 0x1.f7116p-1, 0x1.f6ba4p-1, 0x1.f6632p-1, 0x1.f60c2p-1,
+ 0x1.f5b52p-1, 0x1.f55e4p-1, 0x1.f5076p-1 };
+ static const double l1[][2] = { { 0x0p+0, 0x0p+0 },
+ { -0x1.532c1269e2038p-27, 0x1.62e5p-7 },
+ { 0x1.ce42d81b54e84p-27, 0x1.62e3cp-6 },
+ { -0x1.25826f815ec3dp-26, 0x1.0a2acp-5 },
+ { 0x1.0db1b1e7cee11p-26, 0x1.62e4ap-5 },
+ { -0x1.1f3a8c6c95003p-26, 0x1.bb9dcp-5 },
+ { -0x1.774cd4fb8c30dp-26, 0x1.0a2b2p-4 },
+ { 0x1.452e56c030a0ap-29, 0x1.3687fp-4 },
+ { 0x1.6b63c4966a79ap-28, 0x1.62e41p-4 },
+ { -0x1.b20a21ccb525ep-28, 0x1.8f40ap-4 },
+ { 0x1.4006cfb3d8f85p-26, 0x1.bb9d1p-4 },
+ { -0x1.cdb026b310c41p-26, 0x1.e7f9bp-4 },
+ { -0x1.69124fdc0f16dp-26, 0x1.0a2b08p-3 },
+ { -0x1.084656cdc2727p-26, 0x1.205958p-3 },
+ { -0x1.376fa8b0357fdp-26, 0x1.3687cp-3 },
+ { 0x1.e56ae55a47b4ap-28, 0x1.4cb5e8p-3 },
+ { 0x1.070ff8834eeb4p-26, 0x1.62e44p-3 },
+ { 0x1.623516109f4fep-26, 0x1.79129p-3 },
+ { -0x1.ec656b95fbdacp-29, 0x1.8f40bp-3 },
+ { 0x1.f0ca2e729f51p-28, 0x1.a56ed8p-3 },
+ { -0x1.7d260a858354ap-26, 0x1.bb9d68p-3 },
+ { 0x1.e7279075503d3p-27, 0x1.d1cb9p-3 },
+ { 0x1.39e1a0a503873p-27, 0x1.e7f9dp-3 },
+ { 0x1.cd86d7b87c3d6p-26, 0x1.fe27d8p-3 },
+ { 0x1.060ab88de341ep-26, 0x1.0a2b24p-2 },
+ { 0x1.20a860d3f939p-28, 0x1.154244p-2 },
+ { -0x1.dacee95fc2f1p-27, 0x1.205974p-2 },
+ { 0x1.45de3a86e0acap-26, 0x1.2b707p-2 },
+ { 0x1.c164cbfb991afp-27, 0x1.3687bp-2 },
+ { 0x1.d3f66b24225efp-26, 0x1.419ec4p-2 },
+ { 0x1.fc023efa144bap-26, 0x1.4cb5f8p-2 },
+ { 0x1.086a8af6f26cp-28, 0x1.57cd28p-2 },
+ { -0x1.05c610ca86c39p-30, 0x1.62e43p-2 } };
+ static const double l2[][2] = { { 0x0p+0, 0x0p+0 },
+ { -0x1.37e152a129e4ep-28, 0x1.632p-12 },
+ { -0x1.3f6c916b8be9cp-26, 0x1.63p-11 },
+ { 0x1.20505936739d5p-26, 0x1.0a24p-10 },
+ { -0x1.23e2e8cb541bap-26, 0x1.62dcp-10 },
+ { -0x1.acb7983ac4f5ep-32, 0x1.bbap-10 },
+ { 0x1.6f7c7689c63aep-28, 0x1.0a2ap-9 },
+ { 0x1.f5ca695b4c58bp-30, 0x1.368cp-9 },
+ { -0x1.c6c18bd953226p-27, 0x1.62e6p-9 },
+ { 0x1.7a516c34846bdp-26, 0x1.8f46p-9 },
+ { -0x1.f3b83dd8b853p-27, 0x1.bbap-9 },
+ { -0x1.c3459046e4e57p-31, 0x1.e8p-9 },
+ { 0x1.b5c7e34cb79f6p-38, 0x1.0a2cp-8 },
+ { -0x1.2487e9af9a692p-27, 0x1.205cp-8 },
+ { 0x1.f21bbc4ad79cep-26, 0x1.3687p-8 },
+ { -0x1.550ffc857b731p-29, 0x1.4cb7p-8 },
+ { 0x1.87458ec1b7b34p-27, 0x1.62e2p-8 },
+ { 0x1.103d4fe83ee81p-26, 0x1.7911p-8 },
+ { 0x1.810483d3b398cp-27, 0x1.8f44p-8 },
+ { -0x1.2085cb340608ep-27, 0x1.a573p-8 },
+ { 0x1.12698a119c42fp-26, 0x1.bb9dp-8 },
+ { -0x1.edb8c172b4c33p-26, 0x1.d1ccp-8 },
+ { -0x1.8b55b87a5e238p-26, 0x1.e7fep-8 },
+ { 0x1.be5e17763f78ap-26, 0x1.fe2bp-8 },
+ { -0x1.c2d496790073ep-30, 0x1.0a2a8p-7 },
+ { 0x1.6542f523abeecp-26, 0x1.1541p-7 },
+ { -0x1.b7fdbe5b193f8p-26, 0x1.205ap-7 },
+ { 0x1.fa4d42fe30c7cp-26, 0x1.2b7p-7 },
+ { 0x1.0d46ad04adc86p-26, 0x1.36888p-7 },
+ { -0x1.1c22d02d17c4cp-26, 0x1.419fp-7 },
+ { 0x1.a7d1e330dcccep-30, 0x1.4cb7p-7 },
+ { 0x1.187025e656ba3p-31, 0x1.57cdp-7 },
+ { -0x1.532c1269e2038p-27, 0x1.62e5p-7 } };
+ static const double c[]
+ = { -0x1p+0, 0x1.555555555553p+0, -0x1.fffffffffffap+0,
+ 0x1.99999e33a6366p+1, -0x1.555559ef9525fp+2 };
+ uint64_t t = asuint64 (th);
+ int ex = t >> 52, e = ex - 0x3ff;
+ t &= ~UINT64_C(0) >> 12;
+ double ed = e;
+ uint64_t i = t >> (52 - 5);
+ int64_t d = t & (~UINT64_C(0) >> 17);
+ uint64_t j = (t + ((uint64_t) B[i].c0 << 33)
+ + ((int64_t) B[i].c1 * (d >> 16))) >> (52 - 10);
+ t |= UINT64_C(0x3ff) << 52;
+ int i1 = j >> 5, i2 = j & 0x1f;
+ double tf = asdouble (t);
+ double r = (0.5 * r1[i1]) * r2[i2], dx = fma (r, tf, -0.5),
+ dx2 = dx * dx, rx = r * tf, dxl = fma (r, tf, -rx);
+ double f
+ = dx2 * ((c[0] + dx * c[1]) + dx2 * (c[2] + dx * c[3] + dx2 * c[4]));
+ const double l2h = 0x1.62e42fefa3ap-2, l2l = -0x1.0ca86c3898dp-50;
+ double lh = (l1[i1][1] + l2[i2][1]) + l2h * ed, ll;
+ lh = fasttwosum (lh, rx - 0.5, &ll);
+ ll += l2l * ed + (l1[i1][0] + l2[i2][0]) + dxl + 0.5 * tl / th;
+ ll += f;
+ lh *= copysign (1, x);
+ ll *= copysign (1, x);
+ double eps = 31e-24 + dx2 * 0x1p-49;
+ double lb = lh + (ll - eps), ub = lh + (ll + eps);
+ if (__glibc_likely (lb == ub))
+ return lb;
+ th = fasttwosum (th, tl, &tl);
+ return as_atanh_refine (x, th, tl,
+ 0x1.71547652b82fep+1 * fabs (lh + ll));
+}
#ifndef __ieee754_atanh
libm_alias_finite (__ieee754_atanh, __atanh)
#endif
+
+static __attribute__ ((noinline)) double
+as_atanh_database (double x, double f)
+{
+ static const double db[][3] = {
+ { 0x1.2dbb7b1c91363p-2, 0x1.36f33d51c264dp-2, 0x1p-56 },
+ { 0x1.c493dc899e4a5p-2, 0x1.e611aa58ab608p-2, -0x1p-56 },
+ };
+ double ax = fabs (x), sgn = copysign (1, x);
+ int a = 0, b = array_length (db) - 1, m = (a + b) / 2;
+ while (a <= b)
+ { // binary search
+ if (db[m][0] < ax)
+ a = m + 1;
+ else if (db[m][0] == ax)
+ {
+ f = sgn * db[m][1] + sgn * db[m][2];
+ break;
+ }
+ else
+ b = m - 1;
+ m = (a + b) / 2;
+ }
+ return f;
+}
+
+static double
+as_atanh_refine (double x, double zh, double zl, double a)
+{
+ static const double t1[]
+ = { 0x1p+0, 0x1.ea4afap-1, 0x1.d5818ep-1, 0x1.c199bep-1,
+ 0x1.ae89f98p-1, 0x1.9c4918p-1, 0x1.8ace54p-1, 0x1.7a1147p-1,
+ 0x1.6a09e68p-1, 0x1.5ab07ep-1, 0x1.4bfdad8p-1, 0x1.3dea65p-1,
+ 0x1.306fe08p-1, 0x1.2387a7p-1, 0x1.172b84p-1, 0x1.0b5587p-1,
+ 0x1p-1 };
+ static const double t2[]
+ = { 0x1p+0, 0x1.fe9d968p-1, 0x1.fd3c228p-1, 0x1.fbdba38p-1,
+ 0x1.fa7c18p-1, 0x1.f91d8p-1, 0x1.f7bfdbp-1, 0x1.f663278p-1,
+ 0x1.f507658p-1, 0x1.f3ac948p-1, 0x1.f252b38p-1, 0x1.f0f9c2p-1,
+ 0x1.efa1bfp-1, 0x1.ee4aaap-1, 0x1.ecf483p-1, 0x1.eb9f488p-1 };
+ static const double t3[]
+ = { 0x1p+0, 0x1.ffe9d2p-1, 0x1.ffd3a58p-1, 0x1.ffbd798p-1,
+ 0x1.ffa74e8p-1, 0x1.ff91248p-1, 0x1.ff7afb8p-1, 0x1.ff64d38p-1,
+ 0x1.ff4eac8p-1, 0x1.ff38868p-1, 0x1.ff22618p-1, 0x1.ff0c3dp-1,
+ 0x1.fef61ap-1, 0x1.fedff78p-1, 0x1.fec9d68p-1, 0x1.feb3b6p-1 };
+ static const double t4[]
+ = { 0x1p+0, 0x1.fffe9dp-1, 0x1.fffd3ap-1, 0x1.fffbd78p-1,
+ 0x1.fffa748p-1, 0x1.fff9118p-1, 0x1.fff7ae8p-1, 0x1.fff64cp-1,
+ 0x1.fff4e9p-1, 0x1.fff386p-1, 0x1.fff2238p-1, 0x1.fff0c08p-1,
+ 0x1.ffef5d8p-1, 0x1.ffedfa8p-1, 0x1.ffec98p-1, 0x1.ffeb35p-1 };
+ static const double LL[4][17][3] = {
+ {
+ { 0x0p+0, 0x0p+0, 0x0p+0 },
+ { 0x1.62e432b24p-6, -0x1.745af34bb54b8p-42, -0x1.17e3ec05cde7p-97 },
+ { 0x1.62e42e4a8p-5, 0x1.111a4eadf312p-44, 0x1.cff3027abb119p-93 },
+ { 0x1.0a2b233f1p-4, -0x1.88ac4ec78af8p-42, 0x1.4fa087ca75dfdp-93 },
+ { 0x1.62e43056cp-4, 0x1.6bd65e8b0b7p-46, -0x1.b18e160362c24p-95 },
+ { 0x1.bb9d3cbd6p-4, 0x1.de14aa55ec2bp-42, -0x1.c6ac3f1862a6bp-94 },
+ { 0x1.0a2b244dap-3, 0x1.94def487fea7p-42, -0x1.dead1a4581acfp-94 },
+ { 0x1.3687aa9b78p-3, 0x1.9cec9a50db22p-43, 0x1.34a70684f8e0ep-93 },
+ { 0x1.62e42fabap-3, -0x1.d69047a3aebp-44, -0x1.4e061f79144e2p-95 },
+ { 0x1.8f40b56d28p-3, 0x1.de7d755fd2e2p-42, 0x1.bdc7ecf001489p-94 },
+ { 0x1.bb9d3b61fp-3, 0x1.c14f1445b12p-46, 0x1.a1d78cbdc5b58p-93 },
+ { 0x1.e7f9c11f08p-3, -0x1.6e3e0000dae7p-43, 0x1.6a4559fadde98p-94 },
+ { 0x1.0a2b242ec4p-2, 0x1.bb7cf852a5fe8p-42, 0x1.a6aef11ee43bdp-93 },
+ { 0x1.205966c764p-2, 0x1.ad3a5f214294p-45, 0x1.5cc344fa10652p-93 },
+ { 0x1.3687a98aacp-2, 0x1.1623671842fp-45, -0x1.0b428fe1f9e43p-94 },
+ { 0x1.4cb5ec93f4p-2, 0x1.3d50980ea513p-42, 0x1.67f0ea083b1c4p-93 },
+ { 0x1.62e42fefa4p-2, -0x1.8432a1b0e264p-44, 0x1.803f2f6af40f3p-93 },
+ },
+ {
+ { 0x0p+0, 0x0p+0, 0x0p+0 },
+ { 0x1.62e462b4p-10, 0x1.061d003b97318p-42, 0x1.d7faee66a2e1ep-93 },
+ { 0x1.62e44c92p-9, 0x1.95a7bff5e239p-42, -0x1.f7e788a87135p-95 },
+ { 0x1.0a2b1e33p-8, 0x1.2a3a1a65aa3ap-43, -0x1.54599c9605442p-93 },
+ { 0x1.62e4367cp-8, -0x1.4a995b6d9ddcp-45, -0x1.56bb79b254f33p-100 },
+ { 0x1.bb9d449ap-8, 0x1.8a119c42e9bcp-42, -0x1.8ecf7d8d661f1p-93 },
+ { 0x1.0a2b1f19p-7, 0x1.8863771bd10a8p-42, 0x1.e9731de7f0155p-94 },
+ { 0x1.3687ad11p-7, 0x1.e026a347ca1c8p-42, 0x1.fadc62522444dp-97 },
+ { 0x1.62e436f28p-7, 0x1.25b84f71b70b8p-42, -0x1.fcb3f98612d27p-96 },
+ { 0x1.8f40b7b38p-7, -0x1.62a0a4fd4758p-43, 0x1.3cb3c35d9f6a1p-93 },
+ { 0x1.bb9d3abbp-7, -0x1.0ec48f94d786p-42, -0x1.6b47d410e4cc7p-93 },
+ { 0x1.e7f9bb23p-7, 0x1.e4415cbc97ap-43, -0x1.3729fdb677231p-93 },
+ { 0x1.0a2b22478p-6, -0x1.cb73f4505b03p-42, -0x1.1b3b3a3bc370ap-93 },
+ { 0x1.2059691e8p-6, -0x1.abcc3412f264p-43, -0x1.fe6e998e48673p-95 },
+ { 0x1.3687a768p-6, -0x1.43901e5c97a9p-42, 0x1.b54cdd52a5d88p-96 },
+ { 0x1.4cb5eb5d8p-6, -0x1.8f106f00f13b8p-42, -0x1.8f793f5fce148p-93 },
+ { 0x1.62e432b24p-6, -0x1.745af34bb54b8p-42, -0x1.17e3ec05cde7p-97 },
+ },
+ {
+ { 0x0p+0, 0x0p+0, 0x0p+0 },
+ { 0x1.62e7bp-14, -0x1.868625640a68p-44, -0x1.34bf0db910f65p-93 },
+ { 0x1.62e35f6p-13, -0x1.2ee3d96b696ap-43, 0x1.a2948cd558655p-94 },
+ { 0x1.0a2b4b2p-12, 0x1.53edbcf1165p-47, -0x1.cfc26ccf6d0e4p-97 },
+ { 0x1.62e4be1p-12, 0x1.783e334614p-52, -0x1.04b96da30e63ap-93 },
+ { 0x1.bb9e085p-12, -0x1.60785f20acb2p-43, -0x1.f33369bf7dff1p-96 },
+ { 0x1.0a2b94dp-11, 0x1.fd4b3a273353p-42, -0x1.685a35575eff1p-96 },
+ { 0x1.368810f8p-11, 0x1.7ded26dc813p-47, -0x1.4c4d1abca79bfp-96 },
+ { 0x1.62e47878p-11, 0x1.7d2bee9a1f63p-42, 0x1.860233b7ad13p-93 },
+ { 0x1.8f40cb48p-11, -0x1.af034eaf471cp-42, 0x1.ae748822d57b7p-94 },
+ { 0x1.bb9d094p-11, -0x1.7a223013a20fp-42, -0x1.1e499087075b6p-93 },
+ { 0x1.e7fa32c8p-11, -0x1.b2e67b1b59bdp-43, -0x1.54a41eda30fa6p-93 },
+ { 0x1.0a2b237p-10, -0x1.7ad97ff4ac7ap-44, 0x1.f932da91371ddp-93 },
+ { 0x1.2059a338p-10, -0x1.96422d90df4p-44, -0x1.90800fbbf2ed3p-94 },
+ { 0x1.36879824p-10, 0x1.0f9054001812p-44, 0x1.9567e01e48f9ap-93 },
+ { 0x1.4cb602cp-10, -0x1.0d709a5ec0b5p-43, 0x1.253dfd44635d2p-94 },
+ { 0x1.62e462b4p-10, 0x1.061d003b97318p-42, 0x1.d7faee66a2e1ep-93 },
+ },
+ {
+ { 0x0p+0, 0x0p+0, 0x0p+0 },
+ { 0x1.63007cp-18, -0x1.db0e38e5aaaap-43, 0x1.259a7b94815b9p-93 },
+ { 0x1.6300f6p-17, 0x1.2b1c75580438p-44, 0x1.78cabba01e3e4p-93 },
+ { 0x1.0a2115p-16, -0x1.5ff223730759p-42, 0x1.8074feacfe49dp-95 },
+ { 0x1.62e1ecp-16, -0x1.85d6f6487ce4p-45, 0x1.05485074b9276p-93 },
+ { 0x1.bba301p-16, -0x1.af5d58a7c921p-43, -0x1.30a8c0fd2ff5fp-93 },
+ { 0x1.0a32298p-15, 0x1.590faa0883bdp-43, 0x1.95e9bda999947p-93 },
+ { 0x1.3682f1p-15, 0x1.f0224376efaf8p-42, -0x1.5843c0db50d1p-93 },
+ { 0x1.62e3d8p-15, -0x1.142c13daed4ap-43, 0x1.c68a61183ce87p-93 },
+ { 0x1.8f44dd8p-15, -0x1.aa489f399931p-43, 0x1.11c5c376854eap-94 },
+ { 0x1.bb9601p-15, 0x1.9904d8b6a3638p-42, 0x1.8c89554493c8fp-93 },
+ { 0x1.e7f744p-15, 0x1.5785ddbe7cba8p-42, 0x1.e7ff3cde7d70cp-94 },
+ { 0x1.0a2c53p-14, -0x1.6d9e8780d0d5p-43, 0x1.ad9c178106693p-94 },
+ { 0x1.205d134p-14, -0x1.214a2e893fccp-43, 0x1.548a9500c9822p-93 },
+ { 0x1.3685e28p-14, 0x1.e23588646103p-43, 0x1.2a97b26da2d88p-94 },
+ { 0x1.4cb6c18p-14, 0x1.2b7cfcea9e0d8p-42, -0x1.5095048a6b824p-93 },
+ { 0x1.62e7bp-14, -0x1.868625640a68p-44, -0x1.34bf0db910f65p-93 },
+ },
+ };
+ static const double ch[][2] = {
+ { 0x1p-1, 0x1.24b67ee516e3bp-111 },
+ { -0x1p-2, -0x1.932ce43199a8dp-110 },
+ { 0x1.5555555555555p-3, 0x1.55540c15cf91fp-57 },
+ };
+ static const double cl[3]
+ = { -0x1p-3, 0x1.9999999a0754fp-4, -0x1.55555555c3157p-4 };
+ uint64_t t = asuint64 (zh);
+ int ex = t >> 52, e = ex - 0x3ff;
+ t &= ~UINT64_C (0) >> 12;
+ t |= UINT64_C (0x3ff) << 52;
+ double ed = e;
+ uint64_t v = asuint64 (a - ed + 0x1.00008p+0);
+ uint64_t i = (v - (UINT64_C(0x3ff) << 52)) >> (52 - 16);
+ int i1 = i >> 12, i2 = (i >> 8) & 0xf, i3 = (i >> 4) & 0xf, i4 = i & 0xf;
+ const double l20 = 0x1.62e42fefa3ap-2, l21 = -0x1.0ca86c3898dp-50,
+ l22 = 0x1.f97b57a079ap-104;
+ double el2 = l22 * ed, el1 = l21 * ed, el0 = l20 * ed;
+ double L[3];
+ L[0] = LL[0][i1][0] + LL[1][i2][0] + (LL[2][i3][0] + LL[3][i4][0]) + el0;
+ L[1] = LL[0][i1][1] + LL[1][i2][1] + (LL[2][i3][1] + LL[3][i4][1]);
+ L[2] = LL[0][i1][2] + LL[1][i2][2] + (LL[2][i3][2] + LL[3][i4][2]);
+ double t12 = t1[i1] * t2[i2], t34 = t3[i3] * t4[i4];
+ double th = t12 * t34, tl = fma (t12, t34, -th);
+ double tf = asdouble (t);
+ double dh = th * tf, dl = fma (th, tf, -dh);
+ double sh = tl * tf, sl = fma (tl, tf, -sh);
+ double xl, xh = fasttwosum (dh - 1, dl, &xl);
+ t = asuint64 (zl);
+ t -= (int64_t) e << 52;
+ xl += th * asdouble (t);
+ xh = adddd (xh, xl, sh, sl, &xl);
+ sl = xh * (cl[0] + xh * (cl[1] + xh * cl[2]));
+ sh = polydd (xh, xl, 3, ch, &sl);
+ sh = muldd (xh, xl, sh, sl, &sl);
+ sh = adddd (sh, sl, el1, el2, &sl);
+ sh = adddd (sh, sl, L[1], L[2], &sl);
+ double v2, v0 = fasttwosum (L[0], sh, &v2), v1 = fasttwosum (v2, sl, &v2);
+ t = asuint64 (v1);
+ if (__glibc_unlikely (!(t & (~UINT64_C(0) >> 12))))
+ {
+ uint64_t w = asuint64 (v2);
+ if ((w ^ t) >> 63)
+ t--;
+ else
+ t++;
+ v1 = asdouble (t);
+ }
+ uint64_t t0 = asuint64 (v0);
+ uint64_t er = ((t + 1) & (~UINT64_C(0) >> 12)),
+ de = ((t0 >> 52) & 0x7ff) - ((t >> 52) & 0x7ff);
+ v0 *= copysign (1, x);
+ v1 *= copysign (1, x);
+ double res = v0 + v1;
+ if (__glibc_unlikely (de > 104 || er < 3))
+ return as_atanh_database (x, res);
+ return res;
+}
projects / thirdparty / glibc.git / commitdiff
