/* mpfr_coth - Hyperbolic cotangent function. Copyright 2005-2023 Free Software Foundation, Inc. Contributed by the AriC and Caramba projects, INRIA. This file is part of the GNU MPFR Library. The GNU MPFR 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 3 of the License, or (at your option) any later version. The GNU MPFR 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 MPFR Library; see the file COPYING.LESSER. If not, see https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ /* the hyperbolic cotangent is defined by coth(x) = 1/tanh(x) coth (NaN) = NaN. coth (+Inf) = 1 coth (-Inf) = -1 coth (+0) = +Inf. coth (-0) = -Inf. */ #define FUNCTION mpfr_coth #define INVERSE mpfr_tanh #define ACTION_NAN(y) do { MPFR_SET_NAN(y); MPFR_RET_NAN; } while (1) #define ACTION_INF(y) return mpfr_set_si (y, MPFR_IS_POS(x) ? 1 : -1, rnd_mode) #define ACTION_ZERO(y,x) do { MPFR_SET_SAME_SIGN(y,x); MPFR_SET_INF(y); \ MPFR_SET_DIVBY0 (); MPFR_RET(0); } while (1) /* We know |coth(x)| > 1, thus if the approximation z is such that 1 <= z <= 1 + 2^(-p) where p is the target precision, then the result is either 1 or nextabove(1) = 1 + 2^(1-p). */ #define ACTION_SPECIAL \ if (MPFR_GET_EXP(z) == 1) /* 1 <= |z| < 2 */ \ { \ /* the following is exact by Sterbenz theorem */ \ mpfr_sub_si (z, z, MPFR_SIGN (z), MPFR_RNDN); \ if (MPFR_IS_ZERO(z) || MPFR_GET_EXP(z) <= - (mpfr_exp_t) precy) \ { \ mpfr_add_si (z, z, MPFR_SIGN (z), MPFR_RNDN); \ break; \ } \ } /* The analysis is adapted from that for mpfr_csc: near x=0, coth(x) = 1/x + x/3 + ..., more precisely we have |coth(x) - 1/x| <= 0.32 for |x| <= 1. Like for csc, the error term has the same sign as 1/x, thus |coth(x)| >= |1/x|. Then: (i) either x is a power of two, then 1/x is exactly representable, and as long as 1/2*ulp(1/x) > 0.32, we can conclude; (ii) otherwise assume x has <= n bits, and y has <= n+1 bits, then |y - 1/x| >= 2^(-2n) ufp(y), where ufp means unit in first place. Since |coth(x) - 1/x| <= 0.32, if 2^(-2n) ufp(y) >= 0.64, then |y - coth(x)| >= 2^(-2n-1) ufp(y), and rounding 1/x gives the correct result. If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1). A sufficient condition is thus EXP(x) + 1 <= -2 MAX(PREC(x),PREC(Y)). */ #define ACTION_TINY(y,x,r) \ if (MPFR_EXP(x) + 1 <= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(y))) \ { \ int signx = MPFR_SIGN(x); \ inexact = mpfr_ui_div (y, 1, x, r); \ if (inexact == 0) /* x is a power of two */ \ { /* result always 1/x, except when rounding away from zero */ \ if (rnd_mode == MPFR_RNDA) \ rnd_mode = (signx > 0) ? MPFR_RNDU : MPFR_RNDD; \ if (rnd_mode == MPFR_RNDU) \ { \ if (signx > 0) \ mpfr_nextabove (y); /* 2^k + epsilon */ \ inexact = 1; \ } \ else if (rnd_mode == MPFR_RNDD) \ { \ if (signx < 0) \ mpfr_nextbelow (y); /* -2^k - epsilon */ \ inexact = -1; \ } \ else /* round to zero, or nearest */ \ inexact = -signx; \ } \ MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); \ goto end; \ } #include "gen_inverse.h" |