math/k_casinh.c

   1 /* Return arc hyperbole sine for double value, with the imaginary part
   2    of the result possibly adjusted for use in computing other
   3    functions.
   4    Copyright (C) 1997-2014 Free Software Foundation, Inc.
   5    This file is part of the GNU C Library.
   6
   7    The GNU C Library is free software; you can redistribute it and/or
   8    modify it under the terms of the GNU Lesser General Public
   9    License as published by the Free Software Foundation; either
  10    version 2.1 of the License, or (at your option) any later version.
  11
  12    The GNU C Library is distributed in the hope that it will be useful,
  13    but WITHOUT ANY WARRANTY; without even the implied warranty of
  14    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  15    Lesser General Public License for more details.
  16
  17    You should have received a copy of the GNU Lesser General Public
  18    License along with the GNU C Library; if not, see
  19    <http://www.gnu.org/licenses/>.  */
  20
  21 #include <complex.h>
  22 #include <math.h>
  23 #include <math_private.h>
  24 #include <float.h>
  25
  26 /* Return the complex inverse hyperbolic sine of finite nonzero Z,
  27    with the imaginary part of the result subtracted from pi/2 if ADJ
  28    is nonzero.  */
  29
  30 __complex__ double
  31 __kernel_casinh (__complex__ double x, int adj)
  32 {
  33   __complex__ double res;
  34   double rx, ix;
  35   __complex__ double y;
  36
  37   /* Avoid cancellation by reducing to the first quadrant.  */
  38   rx = fabs (__real__ x);
  39   ix = fabs (__imag__ x);
  40
  41   if (rx >= 1.0 / DBL_EPSILON || ix >= 1.0 / DBL_EPSILON)
  42     {
  43       /* For large x in the first quadrant, x + csqrt (1 + x * x)
  44          is sufficiently close to 2 * x to make no significant
  45          difference to the result; avoid possible overflow from
  46          the squaring and addition.  */
  47       __real__ y = rx;
  48       __imag__ y = ix;
  49
  50       if (adj)
  51         {
  52           double t = __real__ y;
  53           __real__ y = __copysign (__imag__ y, __imag__ x);
  54           __imag__ y = t;
  55         }
  56
  57       res = __clog (y);
  58       __real__ res += M_LN2;
  59     }
  60   else if (rx >= 0.5 && ix < DBL_EPSILON / 8.0)
  61     {
  62       double s = __ieee754_hypot (1.0, rx);
  63
  64       __real__ res = __ieee754_log (rx + s);
  65       if (adj)
  66         __imag__ res = __ieee754_atan2 (s, __imag__ x);
  67       else
  68         __imag__ res = __ieee754_atan2 (ix, s);
  69     }
  70   else if (rx < DBL_EPSILON / 8.0 && ix >= 1.5)
  71     {
  72       double s = __ieee754_sqrt ((ix + 1.0) * (ix - 1.0));
  73
  74       __real__ res = __ieee754_log (ix + s);
  75       if (adj)
  76         __imag__ res = __ieee754_atan2 (rx, __copysign (s, __imag__ x));
  77       else
  78         __imag__ res = __ieee754_atan2 (s, rx);
  79     }
  80   else if (ix > 1.0 && ix < 1.5 && rx < 0.5)
  81     {
  82       if (rx < DBL_EPSILON * DBL_EPSILON)
  83         {
  84           double ix2m1 = (ix + 1.0) * (ix - 1.0);
  85           double s = __ieee754_sqrt (ix2m1);
  86
  87           __real__ res = __log1p (2.0 * (ix2m1 + ix * s)) / 2.0;
  88           if (adj)
  89             __imag__ res = __ieee754_atan2 (rx, __copysign (s, __imag__ x));
  90           else
  91             __imag__ res = __ieee754_atan2 (s, rx);
  92         }
  93       else
  94         {
  95           double ix2m1 = (ix + 1.0) * (ix - 1.0);
  96           double rx2 = rx * rx;
  97           double f = rx2 * (2.0 + rx2 + 2.0 * ix * ix);
  98           double d = __ieee754_sqrt (ix2m1 * ix2m1 + f);
  99           double dp = d + ix2m1;
 100           double dm = f / dp;
 101           double r1 = __ieee754_sqrt ((dm + rx2) / 2.0);
 102           double r2 = rx * ix / r1;
 103
 104           __real__ res = __log1p (rx2 + dp + 2.0 * (rx * r1 + ix * r2)) / 2.0;
 105           if (adj)
 106             __imag__ res = __ieee754_atan2 (rx + r1, __copysign (ix + r2,
 107                                                                  __imag__ x));
 108           else
 109             __imag__ res = __ieee754_atan2 (ix + r2, rx + r1);
 110         }
 111     }
 112   else if (ix == 1.0 && rx < 0.5)
 113     {
 114       if (rx < DBL_EPSILON / 8.0)
 115         {
 116           __real__ res = __log1p (2.0 * (rx + __ieee754_sqrt (rx))) / 2.0;
 117           if (adj)
 118             __imag__ res = __ieee754_atan2 (__ieee754_sqrt (rx),
 119                                             __copysign (1.0, __imag__ x));
 120           else
 121             __imag__ res = __ieee754_atan2 (1.0, __ieee754_sqrt (rx));
 122         }
 123       else
 124         {
 125           double d = rx * __ieee754_sqrt (4.0 + rx * rx);
 126           double s1 = __ieee754_sqrt ((d + rx * rx) / 2.0);
 127           double s2 = __ieee754_sqrt ((d - rx * rx) / 2.0);
 128
 129           __real__ res = __log1p (rx * rx + d + 2.0 * (rx * s1 + s2)) / 2.0;
 130           if (adj)
 131             __imag__ res = __ieee754_atan2 (rx + s1, __copysign (1.0 + s2,
 132                                                                  __imag__ x));
 133           else
 134             __imag__ res = __ieee754_atan2 (1.0 + s2, rx + s1);
 135         }
 136     }
 137   else if (ix < 1.0 && rx < 0.5)
 138     {
 139       if (ix >= DBL_EPSILON)
 140         {
 141           if (rx < DBL_EPSILON * DBL_EPSILON)
 142             {
 143               double onemix2 = (1.0 + ix) * (1.0 - ix);
 144               double s = __ieee754_sqrt (onemix2);
 145
 146               __real__ res = __log1p (2.0 * rx / s) / 2.0;
 147               if (adj)
 148                 __imag__ res = __ieee754_atan2 (s, __imag__ x);
 149               else
 150                 __imag__ res = __ieee754_atan2 (ix, s);
 151             }
 152           else
 153             {
 154               double onemix2 = (1.0 + ix) * (1.0 - ix);
 155               double rx2 = rx * rx;
 156               double f = rx2 * (2.0 + rx2 + 2.0 * ix * ix);
 157               double d = __ieee754_sqrt (onemix2 * onemix2 + f);
 158               double dp = d + onemix2;
 159               double dm = f / dp;
 160               double r1 = __ieee754_sqrt ((dp + rx2) / 2.0);
 161               double r2 = rx * ix / r1;
 162
 163               __real__ res
 164                 = __log1p (rx2 + dm + 2.0 * (rx * r1 + ix * r2)) / 2.0;
 165               if (adj)
 166                 __imag__ res = __ieee754_atan2 (rx + r1,
 167                                                 __copysign (ix + r2,
 168                                                             __imag__ x));
 169               else
 170                 __imag__ res = __ieee754_atan2 (ix + r2, rx + r1);
 171             }
 172         }
 173       else
 174         {
 175           double s = __ieee754_hypot (1.0, rx);
 176
 177           __real__ res = __log1p (2.0 * rx * (rx + s)) / 2.0;
 178           if (adj)
 179             __imag__ res = __ieee754_atan2 (s, __imag__ x);
 180           else
 181             __imag__ res = __ieee754_atan2 (ix, s);
 182         }
 183       if (__real__ res < DBL_MIN)
 184         {
 185           volatile double force_underflow = __real__ res * __real__ res;
 186           (void) force_underflow;
 187         }
 188     }
 189   else
 190     {
 191       __real__ y = (rx - ix) * (rx + ix) + 1.0;
 192       __imag__ y = 2.0 * rx * ix;
 193
 194       y = __csqrt (y);
 195
 196       __real__ y += rx;
 197       __imag__ y += ix;
 198
 199       if (adj)
 200         {
 201           double t = __real__ y;
 202           __real__ y = copysign (__imag__ y, __imag__ x);
 203           __imag__ y = t;
 204         }
 205
 206       res = __clog (y);
 207     }
 208
 209   /* Give results the correct sign for the original argument.  */
 210   __real__ res = __copysign (__real__ res, __real__ x);
 211   __imag__ res = __copysign (__imag__ res, (adj ? 1.0 : __imag__ x));
 212
 213   return res;
 214 }