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Diffstat (limited to 'src/coth.c')
| -rw-r--r-- | src/coth.c | 93 |
1 files changed, 93 insertions, 0 deletions
diff --git a/src/coth.c b/src/coth.c new file mode 100644 index 0000000..5cc093b --- /dev/null +++ b/src/coth.c @@ -0,0 +1,93 @@ +/* mpfr_coth - Hyperbolic cotangent function. + +Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. +Contributed by the AriC and Caramel 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 +http://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) > 0 ? 1 : -1, MPFR_RNDN); \ + if (MPFR_IS_ZERO(z) || MPFR_GET_EXP(z) <= - (mpfr_exp_t) precy) \ + { \ + mpfr_add_si (z, z, MPFR_SIGN(z) > 0 ? 1 : -1, 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" |