contrib/mpfr/src/csc.c

   1 /* mpfr_csc - cosecant function.
   2
   3 Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
   4 Contributed by the AriC 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 cosecant is defined by csc(x) = 1/sin(x).
  24    csc (NaN) = NaN.
  25    csc (+Inf) = csc (-Inf) = NaN.
  26    csc (+0) = +Inf.
  27    csc (-0) = -Inf.
  28 */
  29
  30 #define FUNCTION mpfr_csc
  31 #define INVERSE  mpfr_sin
  32 #define ACTION_NAN(y) do { MPFR_SET_NAN(y); MPFR_RET_NAN; } while (1)
  33 #define ACTION_INF(y) do { MPFR_SET_NAN(y); MPFR_RET_NAN; } while (1)
  34 #define ACTION_ZERO(y,x) do { MPFR_SET_SAME_SIGN(y,x); MPFR_SET_INF(y); \
  35                               mpfr_set_divby0 (); MPFR_RET(0); } while (1)
  36 /* near x=0, we have csc(x) = 1/x + x/6 + ..., more precisely we have
  37    |csc(x) - 1/x| <= 0.2 for |x| <= 1. The analysis is similar to that for
  38    gamma(x) near x=0 (see gamma.c), except here the error term has the same
  39    sign as 1/x, thus |csc(x)| >= |1/x|. Then:
  40    (i) either x is a power of two, then 1/x is exactly representable, and
  41        as long as 1/2*ulp(1/x) > 0.2, we can conclude;
  42    (ii) otherwise assume x has <= n bits, and y has <= n+1 bits, then
  43    |y - 1/x| >= 2^(-2n) ufp(y), where ufp means unit in first place.
  44    Since |csc(x) - 1/x| <= 0.2, if 2^(-2n) ufp(y) >= 0.4, then
  45    |y - csc(x)| >= 2^(-2n-1) ufp(y), and rounding 1/x gives the correct result.
  46    If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1).
  47    A sufficient condition is thus EXP(x) <= -2 MAX(PREC(x),PREC(Y)). */
  48 #define ACTION_TINY(y,x,r) \
  49   if (MPFR_EXP(x) <= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(y))) \
  50     {                                                                   \
  51       int signx = MPFR_SIGN(x);                                         \
  52       inexact = mpfr_ui_div (y, 1, x, r);                               \
  53       if (inexact == 0) /* x is a power of two */                       \
  54         { /* result always 1/x, except when rounding away from zero */  \
  55           if (rnd_mode == MPFR_RNDA)                                    \
  56             rnd_mode = (signx > 0) ? MPFR_RNDU : MPFR_RNDD;             \
  57           if (rnd_mode == MPFR_RNDU)                                    \
  58             {                                                           \
  59               if (signx > 0)                                            \
  60                 mpfr_nextabove (y); /* 2^k + epsilon */                 \
  61               inexact = 1;                                              \
  62             }                                                           \
  63           else if (rnd_mode == MPFR_RNDD)                               \
  64             {                                                           \
  65               if (signx < 0)                                            \
  66                 mpfr_nextbelow (y); /* -2^k - epsilon */                \
  67               inexact = -1;                                             \
  68             }                                                           \
  69           else /* round to zero, or nearest */                          \
  70             inexact = -signx;                                           \
  71         }                                                               \
  72       MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);                \
  73       goto end;                                                         \
  74     }
  75
  76 #include "gen_inverse.h"