1 /* mpfr_coth - Hyperbolic cotangent function. 2 3 Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc. 4 Contributed by the Arenaire and Caramel projects, INRIA. 5 6 This file is part of the GNU MPFR Library. 7 8 The GNU MPFR Library is free software; you can redistribute it and/or modify 9 it under the terms of the GNU Lesser General Public License as published by 10 the Free Software Foundation; either version 3 of the License, or (at your 11 option) any later version. 12 13 The GNU MPFR Library is distributed in the hope that it will be useful, but 14 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 15 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public 16 License for more details. 17 18 You should have received a copy of the GNU Lesser General Public License 19 along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see 20 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., 21 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ 22 23 /* the hyperbolic cotangent is defined by coth(x) = 1/tanh(x) 24 coth (NaN) = NaN. 25 coth (+Inf) = 1 26 coth (-Inf) = -1 27 coth (+0) = +Inf. 28 coth (-0) = -Inf. 29 */ 30 31 #define FUNCTION mpfr_coth 32 #define INVERSE mpfr_tanh 33 #define ACTION_NAN(y) do { MPFR_SET_NAN(y); MPFR_RET_NAN; } while (1) 34 #define ACTION_INF(y) return mpfr_set_si (y, MPFR_IS_POS(x) ? 1 : -1, rnd_mode) 35 #define ACTION_ZERO(y,x) do { MPFR_SET_SAME_SIGN(y,x); MPFR_SET_INF(y); \ 36 mpfr_set_divby0 (); MPFR_RET(0); } while (1) 37 38 /* We know |coth(x)| > 1, thus if the approximation z is such that 39 1 <= z <= 1 + 2^(-p) where p is the target precision, then the 40 result is either 1 or nextabove(1) = 1 + 2^(1-p). */ 41 #define ACTION_SPECIAL \ 42 if (MPFR_GET_EXP(z) == 1) /* 1 <= |z| < 2 */ \ 43 { \ 44 /* the following is exact by Sterbenz theorem */ \ 45 mpfr_sub_si (z, z, MPFR_SIGN(z) > 0 ? 1 : -1, MPFR_RNDN); \ 46 if (MPFR_IS_ZERO(z) || MPFR_GET_EXP(z) <= - (mpfr_exp_t) precy) \ 47 { \ 48 mpfr_add_si (z, z, MPFR_SIGN(z) > 0 ? 1 : -1, MPFR_RNDN); \ 49 break; \ 50 } \ 51 } 52 53 /* The analysis is adapted from that for mpfr_csc: 54 near x=0, coth(x) = 1/x + x/3 + ..., more precisely we have 55 |coth(x) - 1/x| <= 0.32 for |x| <= 1. Like for csc, the error term has 56 the same sign as 1/x, thus |coth(x)| >= |1/x|. Then: 57 (i) either x is a power of two, then 1/x is exactly representable, and 58 as long as 1/2*ulp(1/x) > 0.32, we can conclude; 59 (ii) otherwise assume x has <= n bits, and y has <= n+1 bits, then 60 |y - 1/x| >= 2^(-2n) ufp(y), where ufp means unit in first place. 61 Since |coth(x) - 1/x| <= 0.32, if 2^(-2n) ufp(y) >= 0.64, then 62 |y - coth(x)| >= 2^(-2n-1) ufp(y), and rounding 1/x gives the correct 63 result. If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1). 64 A sufficient condition is thus EXP(x) + 1 <= -2 MAX(PREC(x),PREC(Y)). */ 65 #define ACTION_TINY(y,x,r) \ 66 if (MPFR_EXP(x) + 1 <= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(y))) \ 67 { \ 68 int signx = MPFR_SIGN(x); \ 69 inexact = mpfr_ui_div (y, 1, x, r); \ 70 if (inexact == 0) /* x is a power of two */ \ 71 { /* result always 1/x, except when rounding away from zero */ \ 72 if (rnd_mode == MPFR_RNDA) \ 73 rnd_mode = (signx > 0) ? MPFR_RNDU : MPFR_RNDD; \ 74 if (rnd_mode == MPFR_RNDU) \ 75 { \ 76 if (signx > 0) \ 77 mpfr_nextabove (y); /* 2^k + epsilon */ \ 78 inexact = 1; \ 79 } \ 80 else if (rnd_mode == MPFR_RNDD) \ 81 { \ 82 if (signx < 0) \ 83 mpfr_nextbelow (y); /* -2^k - epsilon */ \ 84 inexact = -1; \ 85 } \ 86 else /* round to zero, or nearest */ \ 87 inexact = -signx; \ 88 } \ 89 MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); \ 90 goto end; \ 91 } 92 93 #include "gen_inverse.h" 94