/* mpfr_atan2 -- arc-tan 2 of a floating-point number Copyright 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc. Contributed by the Arenaire and Cacao projects, INRIA. This file is part of the GNU MPFR Library, and was contributed by Mathieu Dutour. 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 2.1 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.LIB. If not, 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" int mpfr_atan2 (mpfr_ptr dest, mpfr_srcptr y, mpfr_srcptr x, mp_rnd_t rnd_mode) { mpfr_t tmp, pi; int inexact; mp_prec_t prec; mp_exp_t e; MPFR_SAVE_EXPO_DECL (expo); MPFR_ZIV_DECL (loop); MPFR_LOG_FUNC (("y[%#R]=%R x[%#R]=%R rnd=%d", y, y, x, x, rnd_mode), ("atan[%#R]=%R inexact=%d", dest, 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)) { set_pi_2: if (MPFR_IS_NEG (y)) /* -PI/2 */ { inexact = mpfr_const_pi (dest, MPFR_INVERT_RND(rnd_mode)); MPFR_CHANGE_SIGN (dest); mpfr_div_2ui (dest, dest, 1, rnd_mode); return -inexact; } else /* PI/2 */ { inexact = mpfr_const_pi (dest, rnd_mode); mpfr_div_2ui (dest, dest, 1, rnd_mode); return inexact; } } if (MPFR_IS_INF (y)) { if (!MPFR_IS_INF (x)) /* +/- PI/2 */ goto set_pi_2; else if (MPFR_IS_POS (x)) /* +/- PI/4 */ { if (MPFR_IS_NEG (y)) { rnd_mode = MPFR_INVERT_RND (rnd_mode); inexact = mpfr_const_pi (dest, rnd_mode); MPFR_CHANGE_SIGN (dest); mpfr_div_2ui (dest, dest, 2, rnd_mode); return -inexact; } else { inexact = mpfr_const_pi (dest, rnd_mode); mpfr_div_2ui (dest, dest, 2, rnd_mode); return inexact; } } else /* +/- 3*PI/4: Ugly since we have to round properly */ { mpfr_t tmp2; MPFR_ZIV_DECL (loop2); mp_prec_t prec2 = MPFR_PREC (dest) + BITS_PER_MP_LIMB; mpfr_init2 (tmp2, prec2); MPFR_ZIV_INIT (loop2, prec2); for (;;) { mpfr_const_pi (tmp2, GMP_RNDN); mpfr_mul_ui (tmp2, tmp2, 3, GMP_RNDN); /* Error <= 2 */ mpfr_div_2ui (tmp2, tmp2, 2, GMP_RNDN); if (mpfr_round_p (MPFR_MANT (tmp2), MPFR_LIMB_SIZE (tmp2), MPFR_PREC (tmp2) - 2, MPFR_PREC (dest) + (rnd_mode == GMP_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); return inexact; } } MPFR_ASSERTD (MPFR_IS_INF (x)); if (MPFR_IS_NEG (x)) goto set_pi; else goto set_zero; } /* When x=1, atan2(y,x) = atan(y). FIXME: more generally, if x is a power of two, we could call directly atan(y/x) since y/x is exact. */ if (mpfr_cmp_ui (x, 1) == 0) return mpfr_atan (dest, y, rnd_mode); 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, GMP_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 GMP_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 GMP_RNDN). In the case GMP_RNDN with |y/x| <= 2^(emin-2), we round towards zero, as |atan(z)/z| < 1. */ MPFR_ASSERTN (MPFR_PREC_MAX <= 2 * (mpfr_uexp_t) - MPFR_EMIN_MIN + 5); if (rnd_mode == GMP_RNDN && MPFR_IS_ZERO (tmp)) rnd_mode = GMP_RNDZ; sign = MPFR_SIGN (tmp); mpfr_clear (tmp); MPFR_SAVE_EXPO_FREE (expo); return mpfr_underflow (dest, rnd_mode, sign); } mpfr_atan (tmp, tmp, GMP_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, GMP_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, GMP_RNDN); /* Error <= 2*ulp (tmp) since abs(D(arctan)) <= 1 */ mpfr_const_pi (pi, GMP_RNDN); /* Error <= ulp(pi) /2 */ e = MPFR_NOTZERO(tmp) ? MPFR_GET_EXP (tmp) : __gmpfr_emin - 1; mpfr_sub (tmp, pi, tmp, GMP_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); }