source/libs/mpfr/mpfr-src/src/coth.c

   1 /* mpfr_coth - Hyperbolic cotangent function.
   2
   3 Copyright 2005-2016 Free Software Foundation, Inc.
   4 Contributed by the AriC and Caramba 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"