math/k_casinh_template.c

   1 /* Return arc hyperbolic sine for a complex float type, with the
   2    imaginary part of the result possibly adjusted for use in
   3    computing other functions.
   4    Copyright (C) 1997-2017 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 CFLOAT
  31 M_DECL_FUNC (__kernel_casinh) (CFLOAT x, int adj)
  32 {
  33   CFLOAT res;
  34   FLOAT rx, ix;
  35   CFLOAT y;
  36
  37   /* Avoid cancellation by reducing to the first quadrant.  */
  38   rx = M_FABS (__real__ x);
  39   ix = M_FABS (__imag__ x);
  40
  41   if (rx >= 1 / M_EPSILON || ix >= 1 / M_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           FLOAT t = __real__ y;
  53           __real__ y = M_COPYSIGN (__imag__ y, __imag__ x);
  54           __imag__ y = t;
  55         }
  56
  57       res = M_SUF (__clog) (y);
  58       __real__ res += (FLOAT) M_MLIT (M_LN2);
  59     }
  60   else if (rx >= M_LIT (0.5) && ix < M_EPSILON / 8)
  61     {
  62       FLOAT s = M_HYPOT (1, rx);
  63
  64       __real__ res = M_LOG (rx + s);
  65       if (adj)
  66         __imag__ res = M_ATAN2 (s, __imag__ x);
  67       else
  68         __imag__ res = M_ATAN2 (ix, s);
  69     }
  70   else if (rx < M_EPSILON / 8 && ix >= M_LIT (1.5))
  71     {
  72       FLOAT s = M_SQRT ((ix + 1) * (ix - 1));
  73
  74       __real__ res = M_LOG (ix + s);
  75       if (adj)
  76         __imag__ res = M_ATAN2 (rx, M_COPYSIGN (s, __imag__ x));
  77       else
  78         __imag__ res = M_ATAN2 (s, rx);
  79     }
  80   else if (ix > 1 && ix < M_LIT (1.5) && rx < M_LIT (0.5))
  81     {
  82       if (rx < M_EPSILON * M_EPSILON)
  83         {
  84           FLOAT ix2m1 = (ix + 1) * (ix - 1);
  85           FLOAT s = M_SQRT (ix2m1);
  86
  87           __real__ res = M_LOG1P (2 * (ix2m1 + ix * s)) / 2;
  88           if (adj)
  89             __imag__ res = M_ATAN2 (rx, M_COPYSIGN (s, __imag__ x));
  90           else
  91             __imag__ res = M_ATAN2 (s, rx);
  92         }
  93       else
  94         {
  95           FLOAT ix2m1 = (ix + 1) * (ix - 1);
  96           FLOAT rx2 = rx * rx;
  97           FLOAT f = rx2 * (2 + rx2 + 2 * ix * ix);
  98           FLOAT d = M_SQRT (ix2m1 * ix2m1 + f);
  99           FLOAT dp = d + ix2m1;
 100           FLOAT dm = f / dp;
 101           FLOAT r1 = M_SQRT ((dm + rx2) / 2);
 102           FLOAT r2 = rx * ix / r1;
 103
 104           __real__ res = M_LOG1P (rx2 + dp + 2 * (rx * r1 + ix * r2)) / 2;
 105           if (adj)
 106             __imag__ res = M_ATAN2 (rx + r1, M_COPYSIGN (ix + r2, __imag__ x));
 107           else
 108             __imag__ res = M_ATAN2 (ix + r2, rx + r1);
 109         }
 110     }
 111   else if (ix == 1 && rx < M_LIT (0.5))
 112     {
 113       if (rx < M_EPSILON / 8)
 114         {
 115           __real__ res = M_LOG1P (2 * (rx + M_SQRT (rx))) / 2;
 116           if (adj)
 117             __imag__ res = M_ATAN2 (M_SQRT (rx), M_COPYSIGN (1, __imag__ x));
 118           else
 119             __imag__ res = M_ATAN2 (1, M_SQRT (rx));
 120         }
 121       else
 122         {
 123           FLOAT d = rx * M_SQRT (4 + rx * rx);
 124           FLOAT s1 = M_SQRT ((d + rx * rx) / 2);
 125           FLOAT s2 = M_SQRT ((d - rx * rx) / 2);
 126
 127           __real__ res = M_LOG1P (rx * rx + d + 2 * (rx * s1 + s2)) / 2;
 128           if (adj)
 129             __imag__ res = M_ATAN2 (rx + s1, M_COPYSIGN (1 + s2, __imag__ x));
 130           else
 131             __imag__ res = M_ATAN2 (1 + s2, rx + s1);
 132         }
 133     }
 134   else if (ix < 1 && rx < M_LIT (0.5))
 135     {
 136       if (ix >= M_EPSILON)
 137         {
 138           if (rx < M_EPSILON * M_EPSILON)
 139             {
 140               FLOAT onemix2 = (1 + ix) * (1 - ix);
 141               FLOAT s = M_SQRT (onemix2);
 142
 143               __real__ res = M_LOG1P (2 * rx / s) / 2;
 144               if (adj)
 145                 __imag__ res = M_ATAN2 (s, __imag__ x);
 146               else
 147                 __imag__ res = M_ATAN2 (ix, s);
 148             }
 149           else
 150             {
 151               FLOAT onemix2 = (1 + ix) * (1 - ix);
 152               FLOAT rx2 = rx * rx;
 153               FLOAT f = rx2 * (2 + rx2 + 2 * ix * ix);
 154               FLOAT d = M_SQRT (onemix2 * onemix2 + f);
 155               FLOAT dp = d + onemix2;
 156               FLOAT dm = f / dp;
 157               FLOAT r1 = M_SQRT ((dp + rx2) / 2);
 158               FLOAT r2 = rx * ix / r1;
 159
 160               __real__ res = M_LOG1P (rx2 + dm + 2 * (rx * r1 + ix * r2)) / 2;
 161               if (adj)
 162                 __imag__ res = M_ATAN2 (rx + r1, M_COPYSIGN (ix + r2,
 163                                                              __imag__ x));
 164               else
 165                 __imag__ res = M_ATAN2 (ix + r2, rx + r1);
 166             }
 167         }
 168       else
 169         {
 170           FLOAT s = M_HYPOT (1, rx);
 171
 172           __real__ res = M_LOG1P (2 * rx * (rx + s)) / 2;
 173           if (adj)
 174             __imag__ res = M_ATAN2 (s, __imag__ x);
 175           else
 176             __imag__ res = M_ATAN2 (ix, s);
 177         }
 178       math_check_force_underflow_nonneg (__real__ res);
 179     }
 180   else
 181     {
 182       __real__ y = (rx - ix) * (rx + ix) + 1;
 183       __imag__ y = 2 * rx * ix;
 184
 185       y = M_SUF (__csqrt) (y);
 186
 187       __real__ y += rx;
 188       __imag__ y += ix;
 189
 190       if (adj)
 191         {
 192           FLOAT t = __real__ y;
 193           __real__ y = M_COPYSIGN (__imag__ y, __imag__ x);
 194           __imag__ y = t;
 195         }
 196
 197       res = M_SUF (__clog) (y);
 198     }
 199
 200   /* Give results the correct sign for the original argument.  */
 201   __real__ res = M_COPYSIGN (__real__ res, __real__ x);
 202   __imag__ res = M_COPYSIGN (__imag__ res, (adj ? 1 : __imag__ x));
 203
 204   return res;
 205 }