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/* mpfr_atan2 -- arc-tan 2 of a floating-point number
Copyright 2005-2015 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. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
static int
pi_div_2ui (mpfr_ptr dest, int i, int neg, mpfr_rnd_t rnd_mode)
{
int inexact;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_SAVE_EXPO_MARK (expo);
if (neg) /* -PI/2^i */
{
inexact = - mpfr_const_pi (dest, MPFR_INVERT_RND (rnd_mode));
MPFR_CHANGE_SIGN (dest);
}
else /* PI/2^i */
{
inexact = mpfr_const_pi (dest, rnd_mode);
}
mpfr_div_2ui (dest, dest, i, rnd_mode); /* exact */
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (dest, inexact, rnd_mode);
}
int
mpfr_atan2 (mpfr_ptr dest, mpfr_srcptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t tmp, pi;
int inexact;
mpfr_prec_t prec;
mpfr_exp_t e;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_LOG_FUNC
(("y[%Pu]=%.*Rg x[%Pu]=%.*Rg rnd=%d",
mpfr_get_prec (y), mpfr_log_prec, y,
mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
("atan[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (dest), mpfr_log_prec, dest, inexact));
/* Special cases */
if (MPFR_ARE_SINGULAR (x, y))
{
/* atan2(0, 0) does not raise the "invalid" floating-point
exception, nor does atan2(y, 0) raise the "divide-by-zero"
floating-point exception.
-- atan2(±0, -0) returns ±pi.313)
-- atan2(±0, +0) returns ±0.
-- atan2(±0, x) returns ±pi, for x < 0.
-- atan2(±0, x) returns ±0, for x > 0.
-- atan2(y, ±0) returns -pi/2 for y < 0.
-- atan2(y, ±0) returns pi/2 for y > 0.
-- atan2(±oo, -oo) returns ±3pi/4.
-- atan2(±oo, +oo) returns ±pi/4.
-- atan2(±oo, x) returns ±pi/2, for finite x.
-- atan2(±y, -oo) returns ±pi, for finite y > 0.
-- atan2(±y, +oo) returns ±0, for finite y > 0.
*/
if (MPFR_IS_NAN (x) || MPFR_IS_NAN (y))
{
MPFR_SET_NAN (dest);
MPFR_RET_NAN;
}
if (MPFR_IS_ZERO (y))
{
if (MPFR_IS_NEG (x)) /* +/- PI */
{
set_pi:
if (MPFR_IS_NEG (y))
{
inexact = mpfr_const_pi (dest, MPFR_INVERT_RND (rnd_mode));
MPFR_CHANGE_SIGN (dest);
return -inexact;
}
else
return mpfr_const_pi (dest, rnd_mode);
}
else /* +/- 0 */
{
set_zero:
MPFR_SET_ZERO (dest);
MPFR_SET_SAME_SIGN (dest, y);
return 0;
}
}
if (MPFR_IS_ZERO (x))
{
return pi_div_2ui (dest, 1, MPFR_IS_NEG (y), rnd_mode);
}
if (MPFR_IS_INF (y))
{
if (!MPFR_IS_INF (x)) /* +/- PI/2 */
return pi_div_2ui (dest, 1, MPFR_IS_NEG (y), rnd_mode);
else if (MPFR_IS_POS (x)) /* +/- PI/4 */
return pi_div_2ui (dest, 2, MPFR_IS_NEG (y), rnd_mode);
else /* +/- 3*PI/4: Ugly since we have to round properly */
{
mpfr_t tmp2;
MPFR_ZIV_DECL (loop2);
mpfr_prec_t prec2 = MPFR_PREC (dest) + 10;
MPFR_SAVE_EXPO_MARK (expo);
mpfr_init2 (tmp2, prec2);
MPFR_ZIV_INIT (loop2, prec2);
for (;;)
{
mpfr_const_pi (tmp2, MPFR_RNDN);
mpfr_mul_ui (tmp2, tmp2, 3, MPFR_RNDN); /* Error <= 2 */
mpfr_div_2ui (tmp2, tmp2, 2, MPFR_RNDN);
if (mpfr_round_p (MPFR_MANT (tmp2), MPFR_LIMB_SIZE (tmp2),
MPFR_PREC (tmp2) - 2,
MPFR_PREC (dest) + (rnd_mode == MPFR_RNDN)))
break;
MPFR_ZIV_NEXT (loop2, prec2);
mpfr_set_prec (tmp2, prec2);
}
MPFR_ZIV_FREE (loop2);
if (MPFR_IS_NEG (y))
MPFR_CHANGE_SIGN (tmp2);
inexact = mpfr_set (dest, tmp2, rnd_mode);
mpfr_clear (tmp2);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (dest, inexact, rnd_mode);
}
}
MPFR_ASSERTD (MPFR_IS_INF (x));
if (MPFR_IS_NEG (x))
goto set_pi;
else
goto set_zero;
}
/* When x is a power of two, we call directly atan(y/x) since y/x is
exact. */
if (MPFR_UNLIKELY (MPFR_IS_POWER_OF_2 (x)))
{
int r;
mpfr_t yoverx;
unsigned int saved_flags = __gmpfr_flags;
mpfr_init2 (yoverx, MPFR_PREC (y));
if (MPFR_LIKELY (mpfr_div_2si (yoverx, y, MPFR_GET_EXP (x) - 1,
MPFR_RNDN) == 0))
{
/* Here the flags have not changed due to mpfr_div_2si. */
r = mpfr_atan (dest, yoverx, rnd_mode);
mpfr_clear (yoverx);
return r;
}
else
{
/* Division is inexact because of a small exponent range */
mpfr_clear (yoverx);
__gmpfr_flags = saved_flags;
}
}
MPFR_SAVE_EXPO_MARK (expo);
/* Set up initial prec */
prec = MPFR_PREC (dest) + 3 + MPFR_INT_CEIL_LOG2 (MPFR_PREC (dest));
mpfr_init2 (tmp, prec);
MPFR_ZIV_INIT (loop, prec);
if (MPFR_IS_POS (x))
/* use atan2(y,x) = atan(y/x) */
for (;;)
{
int div_inex;
MPFR_BLOCK_DECL (flags);
MPFR_BLOCK (flags, div_inex = mpfr_div (tmp, y, x, MPFR_RNDN));
if (div_inex == 0)
{
/* Result is exact. */
inexact = mpfr_atan (dest, tmp, rnd_mode);
goto end;
}
/* Error <= ulp (tmp) except in case of underflow or overflow. */
/* If the division underflowed, since |atan(z)/z| < 1, we have
an underflow. */
if (MPFR_UNDERFLOW (flags))
{
int sign;
/* In the case MPFR_RNDN with 2^(emin-2) < |y/x| < 2^(emin-1):
The smallest significand value S > 1 of |y/x| is:
* 1 / (1 - 2^(-px)) if py <= px,
* (1 - 2^(-px) + 2^(-py)) / (1 - 2^(-px)) if py >= px.
Therefore S - 1 > 2^(-pz), where pz = max(px,py). We have:
atan(|y/x|) > atan(z), where z = 2^(emin-2) * (1 + 2^(-pz)).
> z - z^3 / 3.
> 2^(emin-2) * (1 + 2^(-pz) - 2^(2 emin - 5))
Assuming pz <= -2 emin + 5, we can round away from zero
(this is what mpfr_underflow always does on MPFR_RNDN).
In the case MPFR_RNDN with |y/x| <= 2^(emin-2), we round
toward zero, as |atan(z)/z| < 1. */
MPFR_ASSERTN (MPFR_PREC_MAX <=
2 * (mpfr_uexp_t) - MPFR_EMIN_MIN + 5);
if (rnd_mode == MPFR_RNDN && MPFR_IS_ZERO (tmp))
rnd_mode = MPFR_RNDZ;
sign = MPFR_SIGN (tmp);
mpfr_clear (tmp);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_underflow (dest, rnd_mode, sign);
}
mpfr_atan (tmp, tmp, MPFR_RNDN); /* Error <= 2*ulp (tmp) since
abs(D(arctan)) <= 1 */
/* TODO: check that the error bound is correct in case of overflow. */
/* FIXME: Error <= ulp(tmp) ? */
if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - 2, MPFR_PREC (dest),
rnd_mode)))
break;
MPFR_ZIV_NEXT (loop, prec);
mpfr_set_prec (tmp, prec);
}
else /* x < 0 */
/* Use sign(y)*(PI - atan (|y/x|)) */
{
mpfr_init2 (pi, prec);
for (;;)
{
mpfr_div (tmp, y, x, MPFR_RNDN); /* Error <= ulp (tmp) */
/* If tmp is 0, we have |y/x| <= 2^(-emin-2), thus
atan|y/x| < 2^(-emin-2). */
MPFR_SET_POS (tmp); /* no error */
mpfr_atan (tmp, tmp, MPFR_RNDN); /* Error <= 2*ulp (tmp) since
abs(D(arctan)) <= 1 */
mpfr_const_pi (pi, MPFR_RNDN); /* Error <= ulp(pi) /2 */
e = MPFR_NOTZERO(tmp) ? MPFR_GET_EXP (tmp) : __gmpfr_emin - 1;
mpfr_sub (tmp, pi, tmp, MPFR_RNDN); /* see above */
if (MPFR_IS_NEG (y))
MPFR_CHANGE_SIGN (tmp);
/* Error(tmp) <= (1/2+2^(EXP(pi)-EXP(tmp)-1)+2^(e-EXP(tmp)+1))*ulp
<= 2^(MAX (MAX (EXP(PI)-EXP(tmp)-1, e-EXP(tmp)+1),
-1)+2)*ulp(tmp) */
e = MAX (MAX (MPFR_GET_EXP (pi)-MPFR_GET_EXP (tmp) - 1,
e - MPFR_GET_EXP (tmp) + 1), -1) + 2;
if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - e, MPFR_PREC (dest),
rnd_mode)))
break;
MPFR_ZIV_NEXT (loop, prec);
mpfr_set_prec (tmp, prec);
mpfr_set_prec (pi, prec);
}
mpfr_clear (pi);
}
inexact = mpfr_set (dest, tmp, rnd_mode);
end:
MPFR_ZIV_FREE (loop);
mpfr_clear (tmp);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (dest, inexact, rnd_mode);
}
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