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280 lines
9.8 KiB
C
280 lines
9.8 KiB
C
/* mpfr_atan2 -- arc-tan 2 of a floating-point number
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Copyright 2005-2025 Free Software Foundation, Inc.
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Contributed by the Pascaline and Caramba projects, INRIA.
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This file is part of the GNU MPFR Library.
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The GNU MPFR Library is free software; you can redistribute it and/or modify
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it under the terms of the GNU Lesser General Public License as published by
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the Free Software Foundation; either version 3 of the License, or (at your
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option) any later version.
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The GNU MPFR Library is distributed in the hope that it will be useful, but
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WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
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License for more details.
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You should have received a copy of the GNU Lesser General Public License
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along with the GNU MPFR Library; see the file COPYING.LESSER.
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If not, see <https://www.gnu.org/licenses/>. */
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#define MPFR_NEED_LONGLONG_H
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#include "mpfr-impl.h"
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static int
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pi_div_2ui (mpfr_ptr dest, int i, int neg, mpfr_rnd_t rnd_mode)
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{
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int inexact;
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MPFR_SAVE_EXPO_DECL (expo);
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MPFR_SAVE_EXPO_MARK (expo);
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if (neg) /* -PI/2^i */
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{
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inexact = - mpfr_const_pi (dest, MPFR_INVERT_RND (rnd_mode));
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MPFR_CHANGE_SIGN (dest);
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}
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else /* PI/2^i */
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{
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inexact = mpfr_const_pi (dest, rnd_mode);
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}
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mpfr_div_2ui (dest, dest, i, rnd_mode); /* exact */
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MPFR_SAVE_EXPO_FREE (expo);
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return mpfr_check_range (dest, inexact, rnd_mode);
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}
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int
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mpfr_atan2 (mpfr_ptr dest, mpfr_srcptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
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{
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mpfr_t tmp, pi;
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int inexact;
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mpfr_prec_t prec;
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mpfr_exp_t e;
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MPFR_SAVE_EXPO_DECL (expo);
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MPFR_ZIV_DECL (loop);
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MPFR_LOG_FUNC
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(("y[%Pd]=%.*Rg x[%Pd]=%.*Rg rnd=%d",
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mpfr_get_prec (y), mpfr_log_prec, y,
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mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
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("atan[%Pd]=%.*Rg inexact=%d",
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mpfr_get_prec (dest), mpfr_log_prec, dest, inexact));
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/* Special cases */
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if (MPFR_ARE_SINGULAR (x, y))
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{
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/* atan2(0, 0) does not raise the "invalid" floating-point
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exception, nor does atan2(y, 0) raise the "divide-by-zero"
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floating-point exception.
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-- atan2(±0, -0) returns ±pi.313)
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-- atan2(±0, +0) returns ±0.
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-- atan2(±0, x) returns ±pi, for x < 0.
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-- atan2(±0, x) returns ±0, for x > 0.
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-- atan2(y, ±0) returns -pi/2 for y < 0.
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-- atan2(y, ±0) returns pi/2 for y > 0.
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-- atan2(±oo, -oo) returns ±3pi/4.
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-- atan2(±oo, +oo) returns ±pi/4.
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-- atan2(±oo, x) returns ±pi/2, for finite x.
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-- atan2(±y, -oo) returns ±pi, for finite y > 0.
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-- atan2(±y, +oo) returns ±0, for finite y > 0.
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*/
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if (MPFR_IS_NAN (x) || MPFR_IS_NAN (y))
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{
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MPFR_SET_NAN (dest);
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MPFR_RET_NAN;
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}
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if (MPFR_IS_ZERO (y))
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{
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if (MPFR_IS_NEG (x)) /* +/- PI */
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{
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set_pi:
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if (MPFR_IS_NEG (y))
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{
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inexact = mpfr_const_pi (dest, MPFR_INVERT_RND (rnd_mode));
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MPFR_CHANGE_SIGN (dest);
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return -inexact;
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}
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else
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return mpfr_const_pi (dest, rnd_mode);
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}
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else /* +/- 0 */
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{
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set_zero:
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MPFR_SET_ZERO (dest);
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MPFR_SET_SAME_SIGN (dest, y);
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return 0;
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}
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}
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if (MPFR_IS_ZERO (x))
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{
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return pi_div_2ui (dest, 1, MPFR_IS_NEG (y), rnd_mode);
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}
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if (MPFR_IS_INF (y))
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{
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if (!MPFR_IS_INF (x)) /* +/- PI/2 */
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return pi_div_2ui (dest, 1, MPFR_IS_NEG (y), rnd_mode);
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else if (MPFR_IS_POS (x)) /* +/- PI/4 */
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return pi_div_2ui (dest, 2, MPFR_IS_NEG (y), rnd_mode);
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else /* +/- 3*PI/4: Ugly since we have to round properly */
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{
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mpfr_t tmp2;
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MPFR_ZIV_DECL (loop2);
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mpfr_prec_t prec2 = MPFR_PREC (dest) + 10;
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MPFR_SAVE_EXPO_MARK (expo);
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mpfr_init2 (tmp2, prec2);
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MPFR_ZIV_INIT (loop2, prec2);
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for (;;)
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{
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mpfr_const_pi (tmp2, MPFR_RNDN);
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mpfr_mul_ui (tmp2, tmp2, 3, MPFR_RNDN); /* Error <= 2 */
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mpfr_div_2ui (tmp2, tmp2, 2, MPFR_RNDN);
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if (MPFR_CAN_ROUND (tmp2, MPFR_PREC (tmp2) - 2,
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MPFR_PREC (dest), rnd_mode))
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break;
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MPFR_ZIV_NEXT (loop2, prec2);
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mpfr_set_prec (tmp2, prec2);
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}
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MPFR_ZIV_FREE (loop2);
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if (MPFR_IS_NEG (y))
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MPFR_CHANGE_SIGN (tmp2);
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inexact = mpfr_set (dest, tmp2, rnd_mode);
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mpfr_clear (tmp2);
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MPFR_SAVE_EXPO_FREE (expo);
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return mpfr_check_range (dest, inexact, rnd_mode);
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}
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}
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MPFR_ASSERTD (MPFR_IS_INF (x));
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if (MPFR_IS_NEG (x))
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goto set_pi;
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else
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goto set_zero;
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}
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/* When x is a power of two, we call directly atan(y/x) since y/x is
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exact. */
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if (MPFR_UNLIKELY (MPFR_IS_POS (x) && mpfr_powerof2_raw (x)))
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{
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int r;
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mpfr_t yoverx;
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mpfr_flags_t saved_flags = __gmpfr_flags;
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mpfr_init2 (yoverx, MPFR_PREC (y));
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if (MPFR_LIKELY (mpfr_div_2si (yoverx, y, MPFR_GET_EXP (x) - 1,
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MPFR_RNDN) == 0))
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{
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/* Here the flags have not changed due to mpfr_div_2si. */
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r = mpfr_atan (dest, yoverx, rnd_mode);
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mpfr_clear (yoverx);
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return r;
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}
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else
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{
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/* Division is inexact because of a small exponent range */
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mpfr_clear (yoverx);
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__gmpfr_flags = saved_flags;
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}
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}
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MPFR_SAVE_EXPO_MARK (expo);
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/* Set up initial prec */
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prec = MPFR_PREC (dest) + 3 + MPFR_INT_CEIL_LOG2 (MPFR_PREC (dest));
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mpfr_init2 (tmp, prec);
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MPFR_ZIV_INIT (loop, prec);
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if (MPFR_IS_POS (x))
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/* use atan2(y,x) = atan(y/x) */
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for (;;)
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{
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int div_inex;
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MPFR_BLOCK_DECL (flags);
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MPFR_BLOCK (flags, div_inex = mpfr_div (tmp, y, x, MPFR_RNDN));
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if (div_inex == 0)
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{
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/* Result is exact. */
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inexact = mpfr_atan (dest, tmp, rnd_mode);
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goto end;
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}
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/* Error <= ulp (tmp) except in case of underflow or overflow. */
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/* If the division underflowed, since |atan(z)/z| < 1, we have
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an underflow. */
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if (MPFR_UNDERFLOW (flags))
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{
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int sign;
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/* In the case MPFR_RNDN with 2^(emin-2) < |y/x| < 2^(emin-1):
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The smallest significand value S > 1 of |y/x| is:
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* 1 / (1 - 2^(-px)) if py <= px,
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* (1 - 2^(-px) + 2^(-py)) / (1 - 2^(-px)) if py >= px.
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Therefore S - 1 > 2^(-pz), where pz = max(px,py). We have:
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atan(|y/x|) > atan(z), where z = 2^(emin-2) * (1 + 2^(-pz)).
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> z - z^3 / 3.
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> 2^(emin-2) * (1 + 2^(-pz) - 2^(2 emin - 5))
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Assuming pz <= -2 emin + 5, we can round away from zero
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(this is what mpfr_underflow always does on MPFR_RNDN).
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In the case MPFR_RNDN with |y/x| <= 2^(emin-2), we round
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toward zero, as |atan(z)/z| < 1. */
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MPFR_ASSERTN (MPFR_PREC_MAX <=
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2 * (mpfr_uexp_t) - MPFR_EMIN_MIN + 5);
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if (rnd_mode == MPFR_RNDN && MPFR_IS_ZERO (tmp))
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rnd_mode = MPFR_RNDZ;
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sign = MPFR_SIGN (tmp);
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mpfr_clear (tmp);
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MPFR_SAVE_EXPO_FREE (expo);
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return mpfr_underflow (dest, rnd_mode, sign);
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}
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mpfr_atan (tmp, tmp, MPFR_RNDN); /* Error <= 2*ulp (tmp) since
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abs(D(arctan)) <= 1 */
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/* TODO: check that the error bound is correct in case of overflow. */
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/* FIXME: Error <= ulp(tmp) ? */
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if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - 2, MPFR_PREC (dest),
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rnd_mode)))
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break;
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MPFR_ZIV_NEXT (loop, prec);
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mpfr_set_prec (tmp, prec);
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}
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else /* x < 0 */
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/* Use sign(y)*(PI - atan (|y/x|)) */
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{
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mpfr_init2 (pi, prec);
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for (;;)
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{
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mpfr_div (tmp, y, x, MPFR_RNDN); /* Error <= ulp (tmp) */
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/* If tmp is 0, we have |y/x| <= 2^(-emin-2), thus
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atan|y/x| < 2^(-emin-2). */
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MPFR_SET_POS (tmp); /* no error */
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mpfr_atan (tmp, tmp, MPFR_RNDN); /* Error <= 2*ulp (tmp) since
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abs(D(arctan)) <= 1 */
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mpfr_const_pi (pi, MPFR_RNDN); /* Error <= ulp(pi) /2 */
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e = MPFR_NOTZERO(tmp) ? MPFR_GET_EXP (tmp) : __gmpfr_emin - 1;
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mpfr_sub (tmp, pi, tmp, MPFR_RNDN); /* see above */
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if (MPFR_IS_NEG (y))
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MPFR_CHANGE_SIGN (tmp);
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/* Error(tmp) <= (1/2+2^(EXP(pi)-EXP(tmp)-1)+2^(e-EXP(tmp)+1))*ulp
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<= 2^(MAX (MAX (EXP(PI)-EXP(tmp)-1, e-EXP(tmp)+1),
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-1)+2)*ulp(tmp) */
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e = MAX (MAX (MPFR_GET_EXP (pi)-MPFR_GET_EXP (tmp) - 1,
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e - MPFR_GET_EXP (tmp) + 1), -1) + 2;
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if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - e, MPFR_PREC (dest),
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rnd_mode)))
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break;
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MPFR_ZIV_NEXT (loop, prec);
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mpfr_set_prec (tmp, prec);
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mpfr_set_prec (pi, prec);
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}
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mpfr_clear (pi);
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}
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inexact = mpfr_set (dest, tmp, rnd_mode);
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end:
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MPFR_ZIV_FREE (loop);
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mpfr_clear (tmp);
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MPFR_SAVE_EXPO_FREE (expo);
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return mpfr_check_range (dest, inexact, rnd_mode);
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}
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