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) 1997-2023 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    <https://www.gnu.org/licenses/>.  */
  20
  21 #include <complex.h>
  22 #include <math.h>
  23 #include <math_private.h>
  24 #include <math-underflow.h>
  25 #include <float.h>
  26
  27 /* Return the complex inverse hyperbolic sine of finite nonzero Z,
  28    with the imaginary part of the result subtracted from pi/2 if ADJ
  29    is nonzero.  */
  30
  31 CFLOAT
  32 M_DECL_FUNC (__kernel_casinh) (CFLOAT x, int adj)
  33 {
  34   CFLOAT res;
  35   FLOAT rx, ix;
  36   CFLOAT y;
  37
  38   /* Avoid cancellation by reducing to the first quadrant.  */
  39   rx = M_FABS (__real__ x);
  40   ix = M_FABS (__imag__ x);
  41
  42   if (rx >= 1 / M_EPSILON || ix >= 1 / M_EPSILON)
  43     {
  44       /* For large x in the first quadrant, x + csqrt (1 + x * x)
  45          is sufficiently close to 2 * x to make no significant
  46          difference to the result; avoid possible overflow from
  47          the squaring and addition.  */
  48       __real__ y = rx;
  49       __imag__ y = ix;
  50
  51       if (adj)
  52         {
  53           FLOAT t = __real__ y;
  54           __real__ y = M_COPYSIGN (__imag__ y, __imag__ x);
  55           __imag__ y = t;
  56         }
  57
  58       res = M_SUF (__clog) (y);
  59       __real__ res += M_MLIT (M_LN2);
  60     }
  61   else if (rx >= M_LIT (0.5) && ix < M_EPSILON / 8)
  62     {
  63       FLOAT s = M_HYPOT (1, rx);
  64
  65       __real__ res = M_LOG (rx + s);
  66       if (adj)
  67         __imag__ res = M_ATAN2 (s, __imag__ x);
  68       else
  69         __imag__ res = M_ATAN2 (ix, s);
  70     }
  71   else if (rx < M_EPSILON / 8 && ix >= M_LIT (1.5))
  72     {
  73       FLOAT s = M_SQRT ((ix + 1) * (ix - 1));
  74
  75       __real__ res = M_LOG (ix + s);
  76       if (adj)
  77         __imag__ res = M_ATAN2 (rx, M_COPYSIGN (s, __imag__ x));
  78       else
  79         __imag__ res = M_ATAN2 (s, rx);
  80     }
  81   else if (ix > 1 && ix < M_LIT (1.5) && rx < M_LIT (0.5))
  82     {
  83       if (rx < M_EPSILON * M_EPSILON)
  84         {
  85           FLOAT ix2m1 = (ix + 1) * (ix - 1);
  86           FLOAT s = M_SQRT (ix2m1);
  87
  88           __real__ res = M_LOG1P (2 * (ix2m1 + ix * s)) / 2;
  89           if (adj)
  90             __imag__ res = M_ATAN2 (rx, M_COPYSIGN (s, __imag__ x));
  91           else
  92             __imag__ res = M_ATAN2 (s, rx);
  93         }
  94       else
  95         {
  96           FLOAT ix2m1 = (ix + 1) * (ix - 1);
  97           FLOAT rx2 = rx * rx;
  98           FLOAT f = rx2 * (2 + rx2 + 2 * ix * ix);
  99           FLOAT d = M_SQRT (ix2m1 * ix2m1 + f);
 100           FLOAT dp = d + ix2m1;
 101           FLOAT dm = f / dp;
 102           FLOAT r1 = M_SQRT ((dm + rx2) / 2);
 103           FLOAT r2 = rx * ix / r1;
 104
 105           __real__ res = M_LOG1P (rx2 + dp + 2 * (rx * r1 + ix * r2)) / 2;
 106           if (adj)
 107             __imag__ res = M_ATAN2 (rx + r1, M_COPYSIGN (ix + r2, __imag__ x));
 108           else
 109             __imag__ res = M_ATAN2 (ix + r2, rx + r1);
 110         }
 111     }
 112   else if (ix == 1 && rx < M_LIT (0.5))
 113     {
 114       if (rx < M_EPSILON / 8)
 115         {
 116           __real__ res = M_LOG1P (2 * (rx + M_SQRT (rx))) / 2;
 117           if (adj)
 118             __imag__ res = M_ATAN2 (M_SQRT (rx), M_COPYSIGN (1, __imag__ x));
 119           else
 120             __imag__ res = M_ATAN2 (1, M_SQRT (rx));
 121         }
 122       else
 123         {
 124           FLOAT d = rx * M_SQRT (4 + rx * rx);
 125           FLOAT s1 = M_SQRT ((d + rx * rx) / 2);
 126           FLOAT s2 = M_SQRT ((d - rx * rx) / 2);
 127
 128           __real__ res = M_LOG1P (rx * rx + d + 2 * (rx * s1 + s2)) / 2;
 129           if (adj)
 130             __imag__ res = M_ATAN2 (rx + s1, M_COPYSIGN (1 + s2, __imag__ x));
 131           else
 132             __imag__ res = M_ATAN2 (1 + s2, rx + s1);
 133         }
 134     }
 135   else if (ix < 1 && rx < M_LIT (0.5))
 136     {
 137       if (ix >= M_EPSILON)
 138         {
 139           if (rx < M_EPSILON * M_EPSILON)
 140             {
 141               FLOAT onemix2 = (1 + ix) * (1 - ix);
 142               FLOAT s = M_SQRT (onemix2);
 143
 144               __real__ res = M_LOG1P (2 * rx / s) / 2;
 145               if (adj)
 146                 __imag__ res = M_ATAN2 (s, __imag__ x);
 147               else
 148                 __imag__ res = M_ATAN2 (ix, s);
 149             }
 150           else
 151             {
 152               FLOAT onemix2 = (1 + ix) * (1 - ix);
 153               FLOAT rx2 = rx * rx;
 154               FLOAT f = rx2 * (2 + rx2 + 2 * ix * ix);
 155               FLOAT d = M_SQRT (onemix2 * onemix2 + f);
 156               FLOAT dp = d + onemix2;
 157               FLOAT dm = f / dp;
 158               FLOAT r1 = M_SQRT ((dp + rx2) / 2);
 159               FLOAT r2 = rx * ix / r1;
 160
 161               __real__ res = M_LOG1P (rx2 + dm + 2 * (rx * r1 + ix * r2)) / 2;
 162               if (adj)
 163                 __imag__ res = M_ATAN2 (rx + r1, M_COPYSIGN (ix + r2,
 164                                                              __imag__ x));
 165               else
 166                 __imag__ res = M_ATAN2 (ix + r2, rx + r1);
 167             }
 168         }
 169       else
 170         {
 171           FLOAT s = M_HYPOT (1, rx);
 172
 173           __real__ res = M_LOG1P (2 * rx * (rx + s)) / 2;
 174           if (adj)
 175             __imag__ res = M_ATAN2 (s, __imag__ x);
 176           else
 177             __imag__ res = M_ATAN2 (ix, s);
 178         }
 179       math_check_force_underflow_nonneg (__real__ res);
 180     }
 181   else
 182     {
 183       __real__ y = (rx - ix) * (rx + ix) + 1;
 184       __imag__ y = 2 * rx * ix;
 185
 186       y = M_SUF (__csqrt) (y);
 187
 188       __real__ y += rx;
 189       __imag__ y += ix;
 190
 191       if (adj)
 192         {
 193           FLOAT t = __real__ y;
 194           __real__ y = M_COPYSIGN (__imag__ y, __imag__ x);
 195           __imag__ y = t;
 196         }
 197
 198       res = M_SUF (__clog) (y);
 199     }
 200
 201   /* Give results the correct sign for the original argument.  */
 202   __real__ res = M_COPYSIGN (__real__ res, __real__ x);
 203   __imag__ res = M_COPYSIGN (__imag__ res, (adj ? 1 : __imag__ x));
 204
 205   return res;
 206 }