1  /* mpfr_csc - cosecant function.
       2  
       3  Copyright 2005-2023 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  https://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"