| /* mpfr_coth - Hyperbolic cotangent function. |
| |
| Copyright 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc. |
| Contributed by the Arenaire and Cacao 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 2.1 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.LIB. If not, 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) = +0. |
| coth (-0) = -0. |
| */ |
| |
| #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_ZERO(y); \ |
| 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) > 0 ? 1 : -1, GMP_RNDN); \ |
| if (MPFR_IS_ZERO(z) || MPFR_GET_EXP(z) <= - (mp_exp_t) precy) \ |
| { \ |
| mpfr_add_si (z, z, MPFR_SIGN(z) > 0 ? 1 : -1, GMP_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 * (mp_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 == GMP_RNDU) \ |
| { \ |
| if (signx > 0) \ |
| mpfr_nextabove (y); /* 2^k + epsilon */ \ |
| inexact = 1; \ |
| } \ |
| else if (rnd_mode == GMP_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" |
