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+/* mpfr_atan2 -- arc-tan 2 of a floating-point number
+
+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. */
+
+#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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