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/* mpfr_atan -- arc-tangent of a floating-point number
Copyright 2001, 2002, 2003, 2004 Free Software Foundation.
This file is part of the MPFR Library, and was contributed by Mathieu Dutour.
The 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 2.1 of the License, or (at your
option) any later version.
The 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 MPFR Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
MA 02111-1307, USA. */
#include "mpfr-impl.h"
#define CST 2.27 /* CST=1+ln(2.4)/ln(2) */
#define CST2 1.45 /* CST2=1/ln(2) */
static int mpfr_atan_aux _MPFR_PROTO((mpfr_ptr, mpz_srcptr, long, int));
#undef B
#define A
#define A1 1
#define A2 2
#define C
#define C1 3
#define C2 2
#define NO_FACTORIAL
#define GENERIC mpfr_atan_aux
#include "generic.c"
#undef C
#undef C1
#undef C2
#undef A
#undef A1
#undef A2
#undef NO_FACTORIAL
#undef GENERIC
int
mpfr_atan (mpfr_ptr arctangent, mpfr_srcptr x, mp_rnd_t rnd_mode)
{
mpfr_t Pisur2;
mpfr_t xp;
mpfr_t arctgt;
int comparaison, sign, supplement, inexact;
mpfr_t t_arctan;
int i;
mpz_t ukz;
mpfr_t ukf;
mpfr_t sk,Ak;
mpz_t square;
mpfr_t tmp_arctan;
mpfr_t tmp, tmp2;
#ifdef DEBUG
mpfr_t tst;
#endif
int twopoweri;
int Prec;
int prec_x;
int prec_arctan;
int realprec;
int estimated_delta;
/* calculation of the floor */
mp_exp_t exptol;
int N0;
int logn;
/* Trivial cases */
if (MPFR_UNLIKELY( MPFR_IS_SINGULAR(x) ))
{
if (MPFR_IS_NAN(x))
{
MPFR_SET_NAN(arctangent);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF(x))
{
MPFR_CLEAR_FLAGS(arctangent);
if (MPFR_IS_POS(x))
/* arctan(+inf) = Pi/2 */
inexact = mpfr_const_pi (arctangent, rnd_mode);
else
/* arctan(-inf) = -Pi/2 */
{
if (rnd_mode == GMP_RNDU)
rnd_mode = GMP_RNDD;
else if (rnd_mode == GMP_RNDD)
rnd_mode = GMP_RNDU;
inexact = -mpfr_const_pi (arctangent, rnd_mode);
MPFR_CHANGE_SIGN (arctangent);
}
MPFR_SET_EXP (arctangent, MPFR_GET_EXP (arctangent) - 1);
return inexact;
}
else /* x is necessarily 0 */
{
MPFR_ASSERTD(MPFR_IS_ZERO(x));
mpfr_set_ui (arctangent, 0, GMP_RNDN);
return 0; /* exact result */
}
}
MPFR_CLEAR_FLAGS(arctangent);
sign = MPFR_SIGN(x);
prec_arctan = MPFR_PREC(arctangent);
/* Set x_p=|x| */
mpfr_init2 (xp, MPFR_PREC(x));
mpfr_abs (xp, x, rnd_mode);
/* Other simple case arctang(-+1)=-+pi/4 */
comparaison = mpfr_cmp_ui (xp, 1);
if (MPFR_UNLIKELY(comparaison == 0))
{
inexact = mpfr_const_pi (arctangent, MPFR_IS_POS_SIGN(sign) ? rnd_mode
: MPFR_INVERT_RND(rnd_mode));
MPFR_SET_EXP (arctangent, MPFR_GET_EXP (arctangent) - 2);
if (MPFR_IS_NEG_SIGN( sign ))
{
inexact = -inexact;
MPFR_CHANGE_SIGN(arctangent);
}
mpfr_clear (xp);
return inexact;
}
if (comparaison > 0)
supplement = 2;
else
supplement = 2 - MPFR_GET_EXP (xp);
mpfr_save_emin_emax ();
prec_x = __gmpfr_ceil_log2 ((double) MPFR_PREC(x) / BITS_PER_MP_LIMB);
logn = __gmpfr_ceil_log2 ((double) prec_x);
if (logn < 2)
logn = 2;
realprec = prec_arctan + __gmpfr_ceil_log2((double) prec_arctan) + 4;
mpz_init (ukz);
mpz_init (square);
/* Initialisation */
mpfr_init (sk);
mpfr_init (ukf);
mpfr_init (t_arctan);
mpfr_init (tmp_arctan);
mpfr_init (tmp);
mpfr_init (tmp2);
mpfr_init (Ak);
mpfr_init (arctgt);
mpfr_init (Pisur2);
while (1)
{
N0 = __gmpfr_ceil_log2((double) realprec + supplement + CST);
estimated_delta = 1 + supplement + __gmpfr_ceil_log2((double) (3*N0-2));
Prec = realprec + estimated_delta;
/* Initialisation */
mpfr_set_prec (sk,Prec);
mpfr_set_prec (ukf, Prec);
mpfr_set_prec (t_arctan, Prec);
mpfr_set_prec (tmp_arctan, Prec);
mpfr_set_prec (tmp, Prec);
mpfr_set_prec (tmp2, Prec);
mpfr_set_prec (Ak, Prec);
mpfr_set_prec (arctgt, Prec);
if (comparaison > 0)
{
mpfr_set_prec (Pisur2, Prec);
mpfr_const_pi (Pisur2, GMP_RNDN);
mpfr_div_2ui (Pisur2, Pisur2, 1, GMP_RNDN);
mpfr_ui_div (sk, 1, xp, GMP_RNDN);
}
else
mpfr_set (sk, xp, GMP_RNDN);
/* sk is 1/|x| if |x| > 1, and |x| otherwise, i.e. min(|x|, 1/|x|) */
/* Assignation */
mpfr_set_ui (tmp_arctan, 0, GMP_RNDN);
twopoweri = 1;
for (i = 0; i <= N0; i++)
{
mpfr_mul_2ui (tmp, sk, twopoweri, GMP_RNDN);
/* Calculation of trunc(tmp) --> mpz */
mpfr_trunc (ukf, tmp);
exptol = mpfr_get_z_exp (ukz, ukf);
/* since the s_k are decreasing (see algorithms.tex),
and s_0 = min(|x|, 1/|x|) < 1, we have sk < 1,
thus exptol < 0 */
MPFR_ASSERTD(exptol < 0);
mpz_tdiv_q_2exp (ukz, ukz, (unsigned long int) (-exptol));
/* Calculation of arctan(Ak) */
mpz_mul (square, ukz, ukz);
mpz_neg (square, square);
mpfr_atan_aux (t_arctan, square, 2*twopoweri, N0 - i);
mpfr_set_z (Ak, ukz, GMP_RNDN);
mpfr_div_2ui (Ak, Ak, twopoweri, GMP_RNDN);
mpfr_mul (t_arctan, t_arctan, Ak, GMP_RNDN);
/* Addition and iteration */
mpfr_add (tmp_arctan, tmp_arctan, t_arctan, GMP_RNDN);
if (i < N0)
{
mpfr_sub (tmp, sk, Ak, GMP_RNDN);
mpfr_mul (tmp2, sk, Ak, GMP_RNDN);
mpfr_add_ui (tmp2, tmp2, 1, GMP_RNDN);
mpfr_div (sk, tmp, tmp2, GMP_RNDN);
twopoweri <<= 1;
}
}
if (comparaison > 0)
mpfr_sub(arctgt, Pisur2, tmp_arctan, GMP_RNDN);
else
mpfr_set(arctgt, tmp_arctan, GMP_RNDN);
MPFR_SET_POS(arctgt);
if (!mpfr_can_round (arctgt, realprec, GMP_RNDN, GMP_RNDZ,
MPFR_PREC (arctangent) + (rnd_mode == GMP_RNDN)))
realprec += __gmpfr_ceil_log2 ((double) realprec);
else
break;
}
inexact = MPFR_IS_POS_SIGN(sign) ? mpfr_set (arctangent, arctgt, rnd_mode)
: mpfr_neg (arctangent, arctgt, rnd_mode);
mpfr_clear (sk);
mpfr_clear (ukf);
mpfr_clear (t_arctan);
mpfr_clear (tmp_arctan);
mpfr_clear (tmp);
mpfr_clear (tmp2);
mpfr_clear (Ak);
mpfr_clear (arctgt);
mpfr_clear (Pisur2);
mpfr_clear (xp);
mpz_clear (ukz);
mpz_clear (square);
mpfr_restore_emin_emax ();
return mpfr_check_range (arctgt, inexact, rnd_mode);
}
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