1/* mpfr_csc - cosecant function.
2
3Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
4Contributed by the AriC and Caramel projects, INRIA.
5
6This file is part of the GNU MPFR Library.
7
8The GNU MPFR Library is free software; you can redistribute it and/or modify
9it under the terms of the GNU Lesser General Public License as published by
10the Free Software Foundation; either version 3 of the License, or (at your
11option) any later version.
12
13The GNU MPFR Library is distributed in the hope that it will be useful, but
14WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
15or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
16License for more details.
17
18You should have received a copy of the GNU Lesser General Public License
19along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
20http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
2151 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"
77