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Diffstat (limited to 'src/cot.c')
| -rw-r--r-- | src/cot.c | 96 |
1 files changed, 96 insertions, 0 deletions
diff --git a/src/cot.c b/src/cot.c new file mode 100644 index 0000000..21b786e --- /dev/null +++ b/src/cot.c @@ -0,0 +1,96 @@ +/* mpfr_cot - cotangent function. + +Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. +Contributed by the AriC and Caramel projects, INRIA. + +This file is part of the GNU MPFR Library. + +The GNU MPFR Library is free software; you can redistribute it and/or modify +it under the terms of the GNU Lesser General Public License as published by +the Free Software Foundation; either version 3 of the License, or (at your +option) any later version. + +The GNU MPFR Library is distributed in the hope that it will be useful, but +WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY +or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public +License for more details. + +You should have received a copy of the GNU Lesser General Public License +along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see +http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., +51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ + +/* the cotangent is defined by cot(x) = 1/tan(x) = cos(x)/sin(x). + cot (NaN) = NaN. + cot (+Inf) = csc (-Inf) = NaN. + cot (+0) = +Inf. + cot (-0) = -Inf. +*/ + +#define FUNCTION mpfr_cot +#define INVERSE mpfr_tan +#define ACTION_NAN(y) do { MPFR_SET_NAN(y); MPFR_RET_NAN; } while (1) +#define ACTION_INF(y) do { MPFR_SET_NAN(y); MPFR_RET_NAN; } while (1) +#define ACTION_ZERO(y,x) do { MPFR_SET_SAME_SIGN(y,x); MPFR_SET_INF(y); \ + mpfr_set_divby0 (); MPFR_RET(0); } while (1) + +/* (This analysis is adapted from that for mpfr_coth.) + Near x=0, cot(x) = 1/x - x/3 + ..., more precisely we have + |cot(x) - 1/x| <= 0.36 for |x| <= 1. The error term has + the opposite sign as 1/x, thus |cot(x)| <= |1/x|. Then: + (i) either x is a power of two, then 1/x is exactly representable, and + as long as 1/2*ulp(1/x) > 0.36, we can conclude; + (ii) otherwise assume x has <= n bits, and y has <= n+1 bits, then + |y - 1/x| >= 2^(-2n) ufp(y), where ufp means unit in first place. + Since |cot(x) - 1/x| <= 0.36, if 2^(-2n) ufp(y) >= 0.72, then + |y - cot(x)| >= 2^(-2n-1) ufp(y), and rounding 1/x gives the correct + result. If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1). + A sufficient condition is thus EXP(x) + 1 <= -2 MAX(PREC(x),PREC(Y)). + The division can be inexact in case of underflow or overflow; but + an underflow is not possible as emin = - emax. The overflow is a + real overflow possibly except when |x| = 2^emin. */ +#define ACTION_TINY(y,x,r) \ + if (MPFR_EXP(x) + 1 <= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(y))) \ + { \ + int two2emin; \ + int signx = MPFR_SIGN(x); \ + MPFR_ASSERTN (MPFR_EMIN_MIN + MPFR_EMAX_MAX == 0); \ + if ((two2emin = mpfr_get_exp (x) == __gmpfr_emin + 1 && \ + mpfr_powerof2_raw (x))) \ + { \ + /* Case |x| = 2^emin. 1/x is not representable; so, compute \ + 1/(2x) instead (exact), and correct the result later. */ \ + mpfr_set_si_2exp (y, signx, __gmpfr_emax, MPFR_RNDN); \ + inexact = 0; \ + } \ + else \ + inexact = mpfr_ui_div (y, 1, x, r); \ + if (inexact == 0) /* x is a power of two */ \ + { /* result always 1/x, except when rounding to zero */ \ + if (rnd_mode == MPFR_RNDA) \ + rnd_mode = (signx > 0) ? MPFR_RNDU : MPFR_RNDD; \ + if (rnd_mode == MPFR_RNDU || (rnd_mode == MPFR_RNDZ && signx < 0)) \ + { \ + if (signx < 0) \ + mpfr_nextabove (y); /* -2^k + epsilon */ \ + inexact = 1; \ + } \ + else if (rnd_mode == MPFR_RNDD || rnd_mode == MPFR_RNDZ) \ + { \ + if (signx > 0) \ + mpfr_nextbelow (y); /* 2^k - epsilon */ \ + inexact = -1; \ + } \ + else /* round to nearest */ \ + inexact = signx; \ + if (two2emin) \ + mpfr_mul_2ui (y, y, 1, r); /* overflow in MPFR_RNDN */ \ + } \ + /* Underflow is not possible with emin = - emax, but we cannot */ \ + /* add an assert as the underflow flag could have already been */ \ + /* set before the call to mpfr_cot. */ \ + MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); \ + goto end; \ + } + +#include "gen_inverse.h" |