1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223
|
/* Return arc hyperbole sine for long double value, with the imaginary
part of the result possibly adjusted for use in computing other
functions.
Copyright (C) 1997-2014 Free Software Foundation, Inc.
This file is part of the GNU C Library.
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.
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.
You should have received a copy of the GNU Lesser General Public
License along with the GNU C Library; if not, see
<http://www.gnu.org/licenses/>. */
#include <complex.h>
#include <math.h>
#include <math_private.h>
#include <float.h>
/* To avoid spurious overflows, use this definition to treat IBM long
double as approximating an IEEE-style format. */
#if LDBL_MANT_DIG == 106
# undef LDBL_EPSILON
# define LDBL_EPSILON 0x1p-106L
#endif
/* Return the complex inverse hyperbolic sine of finite nonzero Z,
with the imaginary part of the result subtracted from pi/2 if ADJ
is nonzero. */
__complex__ long double
__kernel_casinhl (__complex__ long double x, int adj)
{
__complex__ long double res;
long double rx, ix;
__complex__ long double y;
/* Avoid cancellation by reducing to the first quadrant. */
rx = fabsl (__real__ x);
ix = fabsl (__imag__ x);
if (rx >= 1.0L / LDBL_EPSILON || ix >= 1.0L / LDBL_EPSILON)
{
/* For large x in the first quadrant, x + csqrt (1 + x * x)
is sufficiently close to 2 * x to make no significant
difference to the result; avoid possible overflow from
the squaring and addition. */
__real__ y = rx;
__imag__ y = ix;
if (adj)
{
long double t = __real__ y;
__real__ y = __copysignl (__imag__ y, __imag__ x);
__imag__ y = t;
}
res = __clogl (y);
__real__ res += M_LN2l;
}
else if (rx >= 0.5L && ix < LDBL_EPSILON / 8.0L)
{
long double s = __ieee754_hypotl (1.0L, rx);
__real__ res = __ieee754_logl (rx + s);
if (adj)
__imag__ res = __ieee754_atan2l (s, __imag__ x);
else
__imag__ res = __ieee754_atan2l (ix, s);
}
else if (rx < LDBL_EPSILON / 8.0L && ix >= 1.5L)
{
long double s = __ieee754_sqrtl ((ix + 1.0L) * (ix - 1.0L));
__real__ res = __ieee754_logl (ix + s);
if (adj)
__imag__ res = __ieee754_atan2l (rx, __copysignl (s, __imag__ x));
else
__imag__ res = __ieee754_atan2l (s, rx);
}
else if (ix > 1.0L && ix < 1.5L && rx < 0.5L)
{
if (rx < LDBL_EPSILON * LDBL_EPSILON)
{
long double ix2m1 = (ix + 1.0L) * (ix - 1.0L);
long double s = __ieee754_sqrtl (ix2m1);
__real__ res = __log1pl (2.0L * (ix2m1 + ix * s)) / 2.0L;
if (adj)
__imag__ res = __ieee754_atan2l (rx, __copysignl (s, __imag__ x));
else
__imag__ res = __ieee754_atan2l (s, rx);
}
else
{
long double ix2m1 = (ix + 1.0L) * (ix - 1.0L);
long double rx2 = rx * rx;
long double f = rx2 * (2.0L + rx2 + 2.0L * ix * ix);
long double d = __ieee754_sqrtl (ix2m1 * ix2m1 + f);
long double dp = d + ix2m1;
long double dm = f / dp;
long double r1 = __ieee754_sqrtl ((dm + rx2) / 2.0L);
long double r2 = rx * ix / r1;
__real__ res
= __log1pl (rx2 + dp + 2.0L * (rx * r1 + ix * r2)) / 2.0L;
if (adj)
__imag__ res = __ieee754_atan2l (rx + r1, __copysignl (ix + r2,
__imag__ x));
else
__imag__ res = __ieee754_atan2l (ix + r2, rx + r1);
}
}
else if (ix == 1.0L && rx < 0.5L)
{
if (rx < LDBL_EPSILON / 8.0L)
{
__real__ res = __log1pl (2.0L * (rx + __ieee754_sqrtl (rx))) / 2.0L;
if (adj)
__imag__ res = __ieee754_atan2l (__ieee754_sqrtl (rx),
__copysignl (1.0L, __imag__ x));
else
__imag__ res = __ieee754_atan2l (1.0L, __ieee754_sqrtl (rx));
}
else
{
long double d = rx * __ieee754_sqrtl (4.0L + rx * rx);
long double s1 = __ieee754_sqrtl ((d + rx * rx) / 2.0L);
long double s2 = __ieee754_sqrtl ((d - rx * rx) / 2.0L);
__real__ res = __log1pl (rx * rx + d + 2.0L * (rx * s1 + s2)) / 2.0L;
if (adj)
__imag__ res = __ieee754_atan2l (rx + s1,
__copysignl (1.0L + s2,
__imag__ x));
else
__imag__ res = __ieee754_atan2l (1.0L + s2, rx + s1);
}
}
else if (ix < 1.0L && rx < 0.5L)
{
if (ix >= LDBL_EPSILON)
{
if (rx < LDBL_EPSILON * LDBL_EPSILON)
{
long double onemix2 = (1.0L + ix) * (1.0L - ix);
long double s = __ieee754_sqrtl (onemix2);
__real__ res = __log1pl (2.0L * rx / s) / 2.0L;
if (adj)
__imag__ res = __ieee754_atan2l (s, __imag__ x);
else
__imag__ res = __ieee754_atan2l (ix, s);
}
else
{
long double onemix2 = (1.0L + ix) * (1.0L - ix);
long double rx2 = rx * rx;
long double f = rx2 * (2.0L + rx2 + 2.0L * ix * ix);
long double d = __ieee754_sqrtl (onemix2 * onemix2 + f);
long double dp = d + onemix2;
long double dm = f / dp;
long double r1 = __ieee754_sqrtl ((dp + rx2) / 2.0L);
long double r2 = rx * ix / r1;
__real__ res
= __log1pl (rx2 + dm + 2.0L * (rx * r1 + ix * r2)) / 2.0L;
if (adj)
__imag__ res = __ieee754_atan2l (rx + r1,
__copysignl (ix + r2,
__imag__ x));
else
__imag__ res = __ieee754_atan2l (ix + r2, rx + r1);
}
}
else
{
long double s = __ieee754_hypotl (1.0L, rx);
__real__ res = __log1pl (2.0L * rx * (rx + s)) / 2.0L;
if (adj)
__imag__ res = __ieee754_atan2l (s, __imag__ x);
else
__imag__ res = __ieee754_atan2l (ix, s);
}
if (__real__ res < LDBL_MIN)
{
volatile long double force_underflow = __real__ res * __real__ res;
(void) force_underflow;
}
}
else
{
__real__ y = (rx - ix) * (rx + ix) + 1.0L;
__imag__ y = 2.0L * rx * ix;
y = __csqrtl (y);
__real__ y += rx;
__imag__ y += ix;
if (adj)
{
long double t = __real__ y;
__real__ y = __copysignl (__imag__ y, __imag__ x);
__imag__ y = t;
}
res = __clogl (y);
}
/* Give results the correct sign for the original argument. */
__real__ res = __copysignl (__real__ res, __real__ x);
__imag__ res = __copysignl (__imag__ res, (adj ? 1.0L : __imag__ x));
return res;
}
|