contrib/mpfr/cot.c

   1 /* mpfr_cot - cotangent function.
   2
   3 Copyright 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
   4 Contributed by the Arenaire and Cacao 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 2.1 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.LIB.  If not, write to
  20 the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
  21 MA 02110-1301, USA. */
  22
  23 /* the cotangent is defined by cot(x) = 1/tan(x) = cos(x)/sin(x).
  24    cot (NaN) = NaN.
  25    cot (+Inf) = csc (-Inf) = NaN.
  26    cot (+0) = +Inf.
  27    cot (-0) = -Inf.
  28 */
  29
  30 #define FUNCTION mpfr_cot
  31 #define INVERSE  mpfr_tan
  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_RET(0); } while (1)
  36
  37 /* (This analysis is adapted from that for mpfr_coth.)
  38    Near x=0, cot(x) = 1/x - x/3 + ..., more precisely we have
  39    |cot(x) - 1/x| <= 0.36 for |x| <= 1. The error term has
  40    the opposite sign as 1/x, thus |cot(x)| <= |1/x|. Then:
  41    (i) either x is a power of two, then 1/x is exactly representable, and
  42        as long as 1/2*ulp(1/x) > 0.36, we can conclude;
  43    (ii) otherwise assume x has <= n bits, and y has <= n+1 bits, then
  44    |y - 1/x| >= 2^(-2n) ufp(y), where ufp means unit in first place.
  45    Since |cot(x) - 1/x| <= 0.36, if 2^(-2n) ufp(y) >= 0.72, then
  46    |y - cot(x)| >= 2^(-2n-1) ufp(y), and rounding 1/x gives the correct
  47    result. If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1).
  48    A sufficient condition is thus EXP(x) + 1 <= -2 MAX(PREC(x),PREC(Y)).
  49    The division can be inexact in case of underflow or overflow; but
  50    an underflow is not possible as emin = - emax. The overflow is a
  51    real overflow possibly except when |x| = 2^emin. */
  52 #define ACTION_TINY(y,x,r) \
  53   if (MPFR_EXP(x) + 1 <= -2 * (mp_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(y))) \
  54     {                                                                   \
  55       int two2emin;                                                     \
  56       int signx = MPFR_SIGN(x);                                         \
  57       MPFR_ASSERTN (MPFR_EMIN_MIN + MPFR_EMAX_MAX == 0);                \
  58       if ((two2emin = mpfr_get_exp (x) == __gmpfr_emin + 1 &&           \
  59            mpfr_powerof2_raw (x)))                                      \
  60         {                                                               \
  61           /* Case |x| = 2^emin. 1/x is not representable; so, compute   \
  62              1/(2x) instead (exact), and correct the result later. */   \
  63           mpfr_set_si_2exp (y, signx, __gmpfr_emax, GMP_RNDN);          \
  64           inexact = 0;                                                  \
  65         }                                                               \
  66       else                                                              \
  67         inexact = mpfr_ui_div (y, 1, x, r);                             \
  68       if (inexact == 0) /* x is a power of two */                       \
  69         { /* result always 1/x, except when rounding to zero */         \
  70           if (rnd_mode == GMP_RNDU || (rnd_mode == GMP_RNDZ && signx < 0)) \
  71             {                                                           \
  72               if (signx < 0)                                            \
  73                 mpfr_nextabove (y); /* -2^k + epsilon */                \
  74               inexact = 1;                                              \
  75             }                                                           \
  76           else if (rnd_mode == GMP_RNDD || rnd_mode == GMP_RNDZ)        \
  77             {                                                           \
  78               if (signx > 0)                                            \
  79                 mpfr_nextbelow (y); /* 2^k - epsilon */                 \
  80               inexact = -1;                                             \
  81             }                                                           \
  82           else /* round to nearest */                                   \
  83             inexact = signx;                                            \
  84           if (two2emin)                                                 \
  85             mpfr_mul_2ui (y, y, 1, r);  /* overflow in GMP_RNDN */      \
  86         }                                                               \
  87       /* Underflow is not possible with emin = - emax, but we cannot */ \
  88       /* add an assert as the underflow flag could have already been */ \
  89       /* set before the call to mpfr_cot.                            */ \
  90       MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);                \
  91       goto end;                                                         \
  92     }
  93
  94 #include "gen_inverse.h"