/* Return arc hyperbole sine for 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> /* 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__ double __kernel_casinh (__complex__ double x, int adj) { __complex__ double res; double rx, ix; __complex__ double y; /* Avoid cancellation by reducing to the first quadrant. */ rx = fabs (__real__ x); ix = fabs (__imag__ x); if (rx >= 1.0 / DBL_EPSILON || ix >= 1.0 / DBL_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) { double t = __real__ y; __real__ y = __copysign (__imag__ y, __imag__ x); __imag__ y = t; } res = __clog (y); __real__ res += M_LN2; } else if (rx >= 0.5 && ix < DBL_EPSILON / 8.0) { double s = __ieee754_hypot (1.0, rx); __real__ res = __ieee754_log (rx + s); if (adj) __imag__ res = __ieee754_atan2 (s, __imag__ x); else __imag__ res = __ieee754_atan2 (ix, s); } else if (rx < DBL_EPSILON / 8.0 && ix >= 1.5) { double s = __ieee754_sqrt ((ix + 1.0) * (ix - 1.0)); __real__ res = __ieee754_log (ix + s); if (adj) __imag__ res = __ieee754_atan2 (rx, __copysign (s, __imag__ x)); else __imag__ res = __ieee754_atan2 (s, rx); } else if (ix > 1.0 && ix < 1.5 && rx < 0.5) { if (rx < DBL_EPSILON * DBL_EPSILON) { double ix2m1 = (ix + 1.0) * (ix - 1.0); double s = __ieee754_sqrt (ix2m1); __real__ res = __log1p (2.0 * (ix2m1 + ix * s)) / 2.0; if (adj) __imag__ res = __ieee754_atan2 (rx, __copysign (s, __imag__ x)); else __imag__ res = __ieee754_atan2 (s, rx); } else { double ix2m1 = (ix + 1.0) * (ix - 1.0); double rx2 = rx * rx; double f = rx2 * (2.0 + rx2 + 2.0 * ix * ix); double d = __ieee754_sqrt (ix2m1 * ix2m1 + f); double dp = d + ix2m1; double dm = f / dp; double r1 = __ieee754_sqrt ((dm + rx2) / 2.0); double r2 = rx * ix / r1; __real__ res = __log1p (rx2 + dp + 2.0 * (rx * r1 + ix * r2)) / 2.0; if (adj) __imag__ res = __ieee754_atan2 (rx + r1, __copysign (ix + r2, __imag__ x)); else __imag__ res = __ieee754_atan2 (ix + r2, rx + r1); } } else if (ix == 1.0 && rx < 0.5) { if (rx < DBL_EPSILON / 8.0) { __real__ res = __log1p (2.0 * (rx + __ieee754_sqrt (rx))) / 2.0; if (adj) __imag__ res = __ieee754_atan2 (__ieee754_sqrt (rx), __copysign (1.0, __imag__ x)); else __imag__ res = __ieee754_atan2 (1.0, __ieee754_sqrt (rx)); } else { double d = rx * __ieee754_sqrt (4.0 + rx * rx); double s1 = __ieee754_sqrt ((d + rx * rx) / 2.0); double s2 = __ieee754_sqrt ((d - rx * rx) / 2.0); __real__ res = __log1p (rx * rx + d + 2.0 * (rx * s1 + s2)) / 2.0; if (adj) __imag__ res = __ieee754_atan2 (rx + s1, __copysign (1.0 + s2, __imag__ x)); else __imag__ res = __ieee754_atan2 (1.0 + s2, rx + s1); } } else if (ix < 1.0 && rx < 0.5) { if (ix >= DBL_EPSILON) { if (rx < DBL_EPSILON * DBL_EPSILON) { double onemix2 = (1.0 + ix) * (1.0 - ix); double s = __ieee754_sqrt (onemix2); __real__ res = __log1p (2.0 * rx / s) / 2.0; if (adj) __imag__ res = __ieee754_atan2 (s, __imag__ x); else __imag__ res = __ieee754_atan2 (ix, s); } else { double onemix2 = (1.0 + ix) * (1.0 - ix); double rx2 = rx * rx; double f = rx2 * (2.0 + rx2 + 2.0 * ix * ix); double d = __ieee754_sqrt (onemix2 * onemix2 + f); double dp = d + onemix2; double dm = f / dp; double r1 = __ieee754_sqrt ((dp + rx2) / 2.0); double r2 = rx * ix / r1; __real__ res = __log1p (rx2 + dm + 2.0 * (rx * r1 + ix * r2)) / 2.0; if (adj) __imag__ res = __ieee754_atan2 (rx + r1, __copysign (ix + r2, __imag__ x)); else __imag__ res = __ieee754_atan2 (ix + r2, rx + r1); } } else { double s = __ieee754_hypot (1.0, rx); __real__ res = __log1p (2.0 * rx * (rx + s)) / 2.0; if (adj) __imag__ res = __ieee754_atan2 (s, __imag__ x); else __imag__ res = __ieee754_atan2 (ix, s); } if (__real__ res < DBL_MIN) { volatile double force_underflow = __real__ res * __real__ res; (void) force_underflow; } } else { __real__ y = (rx - ix) * (rx + ix) + 1.0; __imag__ y = 2.0 * rx * ix; y = __csqrt (y); __real__ y += rx; __imag__ y += ix; if (adj) { double t = __real__ y; __real__ y = copysign (__imag__ y, __imag__ x); __imag__ y = t; } res = __clog (y); } /* Give results the correct sign for the original argument. */ __real__ res = __copysign (__real__ res, __real__ x); __imag__ res = __copysign (__imag__ res, (adj ? 1.0 : __imag__ x)); return res; }