libquadmath/math/casinhq_kernel.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) 1997-2018 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 "quadmath-imp.h"
  22
  23 /* Return the complex inverse hyperbolic sine of finite nonzero Z,
  24    with the imaginary part of the result subtracted from pi/2 if ADJ
  25    is nonzero.  */
  26
  27 __complex128
  28 __quadmath_kernel_casinhq (__complex128 x, int adj)
  29 {
  30   __complex128 res;
  31   __float128 rx, ix;
  32   __complex128 y;
  33
  34   /* Avoid cancellation by reducing to the first quadrant.  */
  35   rx = fabsq (__real__ x);
  36   ix = fabsq (__imag__ x);
  37
  38   if (rx >= 1 / FLT128_EPSILON || ix >= 1 / FLT128_EPSILON)
  39     {
  40       /* For large x in the first quadrant, x + csqrt (1 + x * x)
  41          is sufficiently close to 2 * x to make no significant
  42          difference to the result; avoid possible overflow from
  43          the squaring and addition.  */
  44       __real__ y = rx;
  45       __imag__ y = ix;
  46
  47       if (adj)
  48         {
  49           __float128 t = __real__ y;
  50           __real__ y = copysignq (__imag__ y, __imag__ x);
  51           __imag__ y = t;
  52         }
  53
  54       res = clogq (y);
  55       __real__ res += (__float128) M_LN2q;
  56     }
  57   else if (rx >= 0.5Q && ix < FLT128_EPSILON / 8)
  58     {
  59       __float128 s = hypotq (1, rx);
  60
  61       __real__ res = logq (rx + s);
  62       if (adj)
  63         __imag__ res = atan2q (s, __imag__ x);
  64       else
  65         __imag__ res = atan2q (ix, s);
  66     }
  67   else if (rx < FLT128_EPSILON / 8 && ix >= 1.5Q)
  68     {
  69       __float128 s = sqrtq ((ix + 1) * (ix - 1));
  70
  71       __real__ res = logq (ix + s);
  72       if (adj)
  73         __imag__ res = atan2q (rx, copysignq (s, __imag__ x));
  74       else
  75         __imag__ res = atan2q (s, rx);
  76     }
  77   else if (ix > 1 && ix < 1.5Q && rx < 0.5Q)
  78     {
  79       if (rx < FLT128_EPSILON * FLT128_EPSILON)
  80         {
  81           __float128 ix2m1 = (ix + 1) * (ix - 1);
  82           __float128 s = sqrtq (ix2m1);
  83
  84           __real__ res = log1pq (2 * (ix2m1 + ix * s)) / 2;
  85           if (adj)
  86             __imag__ res = atan2q (rx, copysignq (s, __imag__ x));
  87           else
  88             __imag__ res = atan2q (s, rx);
  89         }
  90       else
  91         {
  92           __float128 ix2m1 = (ix + 1) * (ix - 1);
  93           __float128 rx2 = rx * rx;
  94           __float128 f = rx2 * (2 + rx2 + 2 * ix * ix);
  95           __float128 d = sqrtq (ix2m1 * ix2m1 + f);
  96           __float128 dp = d + ix2m1;
  97           __float128 dm = f / dp;
  98           __float128 r1 = sqrtq ((dm + rx2) / 2);
  99           __float128 r2 = rx * ix / r1;
 100
 101           __real__ res = log1pq (rx2 + dp + 2 * (rx * r1 + ix * r2)) / 2;
 102           if (adj)
 103             __imag__ res = atan2q (rx + r1, copysignq (ix + r2, __imag__ x));
 104           else
 105             __imag__ res = atan2q (ix + r2, rx + r1);
 106         }
 107     }
 108   else if (ix == 1 && rx < 0.5Q)
 109     {
 110       if (rx < FLT128_EPSILON / 8)
 111         {
 112           __real__ res = log1pq (2 * (rx + sqrtq (rx))) / 2;
 113           if (adj)
 114             __imag__ res = atan2q (sqrtq (rx), copysignq (1, __imag__ x));
 115           else
 116             __imag__ res = atan2q (1, sqrtq (rx));
 117         }
 118       else
 119         {
 120           __float128 d = rx * sqrtq (4 + rx * rx);
 121           __float128 s1 = sqrtq ((d + rx * rx) / 2);
 122           __float128 s2 = sqrtq ((d - rx * rx) / 2);
 123
 124           __real__ res = log1pq (rx * rx + d + 2 * (rx * s1 + s2)) / 2;
 125           if (adj)
 126             __imag__ res = atan2q (rx + s1, copysignq (1 + s2, __imag__ x));
 127           else
 128             __imag__ res = atan2q (1 + s2, rx + s1);
 129         }
 130     }
 131   else if (ix < 1 && rx < 0.5Q)
 132     {
 133       if (ix >= FLT128_EPSILON)
 134         {
 135           if (rx < FLT128_EPSILON * FLT128_EPSILON)
 136             {
 137               __float128 onemix2 = (1 + ix) * (1 - ix);
 138               __float128 s = sqrtq (onemix2);
 139
 140               __real__ res = log1pq (2 * rx / s) / 2;
 141               if (adj)
 142                 __imag__ res = atan2q (s, __imag__ x);
 143               else
 144                 __imag__ res = atan2q (ix, s);
 145             }
 146           else
 147             {
 148               __float128 onemix2 = (1 + ix) * (1 - ix);
 149               __float128 rx2 = rx * rx;
 150               __float128 f = rx2 * (2 + rx2 + 2 * ix * ix);
 151               __float128 d = sqrtq (onemix2 * onemix2 + f);
 152               __float128 dp = d + onemix2;
 153               __float128 dm = f / dp;
 154               __float128 r1 = sqrtq ((dp + rx2) / 2);
 155               __float128 r2 = rx * ix / r1;
 156
 157               __real__ res = log1pq (rx2 + dm + 2 * (rx * r1 + ix * r2)) / 2;
 158               if (adj)
 159                 __imag__ res = atan2q (rx + r1, copysignq (ix + r2,
 160                                                              __imag__ x));
 161               else
 162                 __imag__ res = atan2q (ix + r2, rx + r1);
 163             }
 164         }
 165       else
 166         {
 167           __float128 s = hypotq (1, rx);
 168
 169           __real__ res = log1pq (2 * rx * (rx + s)) / 2;
 170           if (adj)
 171             __imag__ res = atan2q (s, __imag__ x);
 172           else
 173             __imag__ res = atan2q (ix, s);
 174         }
 175       math_check_force_underflow_nonneg (__real__ res);
 176     }
 177   else
 178     {
 179       __real__ y = (rx - ix) * (rx + ix) + 1;
 180       __imag__ y = 2 * rx * ix;
 181
 182       y = csqrtq (y);
 183
 184       __real__ y += rx;
 185       __imag__ y += ix;
 186
 187       if (adj)
 188         {
 189           __float128 t = __real__ y;
 190           __real__ y = copysignq (__imag__ y, __imag__ x);
 191           __imag__ y = t;
 192         }
 193
 194       res = clogq (y);
 195     }
 196
 197   /* Give results the correct sign for the original argument.  */
 198   __real__ res = copysignq (__real__ res, __real__ x);
 199   __imag__ res = copysignq (__imag__ res, (adj ? 1 : __imag__ x));
 200
 201   return res;
 202 }