1/* mpfr_atan2 -- arc-tan 2 of a floating-point number 2 3Copyright 2005-2023 Free Software Foundation, Inc. 4Contributed by the AriC and Caramba projects, INRIA. 5 6This file is part of the GNU MPFR Library. 7 8The GNU MPFR Library is free software; you can redistribute it and/or modify 9it under the terms of the GNU Lesser General Public License as published by 10the Free Software Foundation; either version 3 of the License, or (at your 11option) any later version. 12 13The GNU MPFR Library is distributed in the hope that it will be useful, but 14WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 15or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public 16License for more details. 17 18You should have received a copy of the GNU Lesser General Public License 19along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see 20https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., 2151 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ 22 23#define MPFR_NEED_LONGLONG_H 24#include "mpfr-impl.h" 25 26static int 27pi_div_2ui (mpfr_ptr dest, int i, int neg, mpfr_rnd_t rnd_mode) 28{ 29 int inexact; 30 MPFR_SAVE_EXPO_DECL (expo); 31 32 MPFR_SAVE_EXPO_MARK (expo); 33 if (neg) /* -PI/2^i */ 34 { 35 inexact = - mpfr_const_pi (dest, MPFR_INVERT_RND (rnd_mode)); 36 MPFR_CHANGE_SIGN (dest); 37 } 38 else /* PI/2^i */ 39 { 40 inexact = mpfr_const_pi (dest, rnd_mode); 41 } 42 mpfr_div_2ui (dest, dest, i, rnd_mode); /* exact */ 43 MPFR_SAVE_EXPO_FREE (expo); 44 return mpfr_check_range (dest, inexact, rnd_mode); 45} 46 47int 48mpfr_atan2 (mpfr_ptr dest, mpfr_srcptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) 49{ 50 mpfr_t tmp, pi; 51 int inexact; 52 mpfr_prec_t prec; 53 mpfr_exp_t e; 54 MPFR_SAVE_EXPO_DECL (expo); 55 MPFR_ZIV_DECL (loop); 56 57 MPFR_LOG_FUNC 58 (("y[%Pu]=%.*Rg x[%Pu]=%.*Rg rnd=%d", 59 mpfr_get_prec (y), mpfr_log_prec, y, 60 mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode), 61 ("atan[%Pu]=%.*Rg inexact=%d", 62 mpfr_get_prec (dest), mpfr_log_prec, dest, inexact)); 63 64 /* Special cases */ 65 if (MPFR_ARE_SINGULAR (x, y)) 66 { 67 /* atan2(0, 0) does not raise the "invalid" floating-point 68 exception, nor does atan2(y, 0) raise the "divide-by-zero" 69 floating-point exception. 70 -- atan2(��0, -0) returns ��pi.313) 71 -- atan2(��0, +0) returns ��0. 72 -- atan2(��0, x) returns ��pi, for x < 0. 73 -- atan2(��0, x) returns ��0, for x > 0. 74 -- atan2(y, ��0) returns -pi/2 for y < 0. 75 -- atan2(y, ��0) returns pi/2 for y > 0. 76 -- atan2(��oo, -oo) returns ��3pi/4. 77 -- atan2(��oo, +oo) returns ��pi/4. 78 -- atan2(��oo, x) returns ��pi/2, for finite x. 79 -- atan2(��y, -oo) returns ��pi, for finite y > 0. 80 -- atan2(��y, +oo) returns ��0, for finite y > 0. 81 */ 82 if (MPFR_IS_NAN (x) || MPFR_IS_NAN (y)) 83 { 84 MPFR_SET_NAN (dest); 85 MPFR_RET_NAN; 86 } 87 if (MPFR_IS_ZERO (y)) 88 { 89 if (MPFR_IS_NEG (x)) /* +/- PI */ 90 { 91 set_pi: 92 if (MPFR_IS_NEG (y)) 93 { 94 inexact = mpfr_const_pi (dest, MPFR_INVERT_RND (rnd_mode)); 95 MPFR_CHANGE_SIGN (dest); 96 return -inexact; 97 } 98 else 99 return mpfr_const_pi (dest, rnd_mode); 100 } 101 else /* +/- 0 */ 102 { 103 set_zero: 104 MPFR_SET_ZERO (dest); 105 MPFR_SET_SAME_SIGN (dest, y); 106 return 0; 107 } 108 } 109 if (MPFR_IS_ZERO (x)) 110 { 111 return pi_div_2ui (dest, 1, MPFR_IS_NEG (y), rnd_mode); 112 } 113 if (MPFR_IS_INF (y)) 114 { 115 if (!MPFR_IS_INF (x)) /* +/- PI/2 */ 116 return pi_div_2ui (dest, 1, MPFR_IS_NEG (y), rnd_mode); 117 else if (MPFR_IS_POS (x)) /* +/- PI/4 */ 118 return pi_div_2ui (dest, 2, MPFR_IS_NEG (y), rnd_mode); 119 else /* +/- 3*PI/4: Ugly since we have to round properly */ 120 { 121 mpfr_t tmp2; 122 MPFR_ZIV_DECL (loop2); 123 mpfr_prec_t prec2 = MPFR_PREC (dest) + 10; 124 125 MPFR_SAVE_EXPO_MARK (expo); 126 mpfr_init2 (tmp2, prec2); 127 MPFR_ZIV_INIT (loop2, prec2); 128 for (;;) 129 { 130 mpfr_const_pi (tmp2, MPFR_RNDN); 131 mpfr_mul_ui (tmp2, tmp2, 3, MPFR_RNDN); /* Error <= 2 */ 132 mpfr_div_2ui (tmp2, tmp2, 2, MPFR_RNDN); 133 if (MPFR_CAN_ROUND (tmp2, MPFR_PREC (tmp2) - 2, 134 MPFR_PREC (dest), rnd_mode)) 135 break; 136 MPFR_ZIV_NEXT (loop2, prec2); 137 mpfr_set_prec (tmp2, prec2); 138 } 139 MPFR_ZIV_FREE (loop2); 140 if (MPFR_IS_NEG (y)) 141 MPFR_CHANGE_SIGN (tmp2); 142 inexact = mpfr_set (dest, tmp2, rnd_mode); 143 mpfr_clear (tmp2); 144 MPFR_SAVE_EXPO_FREE (expo); 145 return mpfr_check_range (dest, inexact, rnd_mode); 146 } 147 } 148 MPFR_ASSERTD (MPFR_IS_INF (x)); 149 if (MPFR_IS_NEG (x)) 150 goto set_pi; 151 else 152 goto set_zero; 153 } 154 155 /* When x is a power of two, we call directly atan(y/x) since y/x is 156 exact. */ 157 if (MPFR_UNLIKELY (MPFR_IS_POS (x) && mpfr_powerof2_raw (x))) 158 { 159 int r; 160 mpfr_t yoverx; 161 mpfr_flags_t saved_flags = __gmpfr_flags; 162 163 mpfr_init2 (yoverx, MPFR_PREC (y)); 164 if (MPFR_LIKELY (mpfr_div_2si (yoverx, y, MPFR_GET_EXP (x) - 1, 165 MPFR_RNDN) == 0)) 166 { 167 /* Here the flags have not changed due to mpfr_div_2si. */ 168 r = mpfr_atan (dest, yoverx, rnd_mode); 169 mpfr_clear (yoverx); 170 return r; 171 } 172 else 173 { 174 /* Division is inexact because of a small exponent range */ 175 mpfr_clear (yoverx); 176 __gmpfr_flags = saved_flags; 177 } 178 } 179 180 MPFR_SAVE_EXPO_MARK (expo); 181 182 /* Set up initial prec */ 183 prec = MPFR_PREC (dest) + 3 + MPFR_INT_CEIL_LOG2 (MPFR_PREC (dest)); 184 mpfr_init2 (tmp, prec); 185 186 MPFR_ZIV_INIT (loop, prec); 187 if (MPFR_IS_POS (x)) 188 /* use atan2(y,x) = atan(y/x) */ 189 for (;;) 190 { 191 int div_inex; 192 MPFR_BLOCK_DECL (flags); 193 194 MPFR_BLOCK (flags, div_inex = mpfr_div (tmp, y, x, MPFR_RNDN)); 195 if (div_inex == 0) 196 { 197 /* Result is exact. */ 198 inexact = mpfr_atan (dest, tmp, rnd_mode); 199 goto end; 200 } 201 202 /* Error <= ulp (tmp) except in case of underflow or overflow. */ 203 204 /* If the division underflowed, since |atan(z)/z| < 1, we have 205 an underflow. */ 206 if (MPFR_UNDERFLOW (flags)) 207 { 208 int sign; 209 210 /* In the case MPFR_RNDN with 2^(emin-2) < |y/x| < 2^(emin-1): 211 The smallest significand value S > 1 of |y/x| is: 212 * 1 / (1 - 2^(-px)) if py <= px, 213 * (1 - 2^(-px) + 2^(-py)) / (1 - 2^(-px)) if py >= px. 214 Therefore S - 1 > 2^(-pz), where pz = max(px,py). We have: 215 atan(|y/x|) > atan(z), where z = 2^(emin-2) * (1 + 2^(-pz)). 216 > z - z^3 / 3. 217 > 2^(emin-2) * (1 + 2^(-pz) - 2^(2 emin - 5)) 218 Assuming pz <= -2 emin + 5, we can round away from zero 219 (this is what mpfr_underflow always does on MPFR_RNDN). 220 In the case MPFR_RNDN with |y/x| <= 2^(emin-2), we round 221 toward zero, as |atan(z)/z| < 1. */ 222 MPFR_ASSERTN (MPFR_PREC_MAX <= 223 2 * (mpfr_uexp_t) - MPFR_EMIN_MIN + 5); 224 if (rnd_mode == MPFR_RNDN && MPFR_IS_ZERO (tmp)) 225 rnd_mode = MPFR_RNDZ; 226 sign = MPFR_SIGN (tmp); 227 mpfr_clear (tmp); 228 MPFR_SAVE_EXPO_FREE (expo); 229 return mpfr_underflow (dest, rnd_mode, sign); 230 } 231 232 mpfr_atan (tmp, tmp, MPFR_RNDN); /* Error <= 2*ulp (tmp) since 233 abs(D(arctan)) <= 1 */ 234 /* TODO: check that the error bound is correct in case of overflow. */ 235 /* FIXME: Error <= ulp(tmp) ? */ 236 if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - 2, MPFR_PREC (dest), 237 rnd_mode))) 238 break; 239 MPFR_ZIV_NEXT (loop, prec); 240 mpfr_set_prec (tmp, prec); 241 } 242 else /* x < 0 */ 243 /* Use sign(y)*(PI - atan (|y/x|)) */ 244 { 245 mpfr_init2 (pi, prec); 246 for (;;) 247 { 248 mpfr_div (tmp, y, x, MPFR_RNDN); /* Error <= ulp (tmp) */ 249 /* If tmp is 0, we have |y/x| <= 2^(-emin-2), thus 250 atan|y/x| < 2^(-emin-2). */ 251 MPFR_SET_POS (tmp); /* no error */ 252 mpfr_atan (tmp, tmp, MPFR_RNDN); /* Error <= 2*ulp (tmp) since 253 abs(D(arctan)) <= 1 */ 254 mpfr_const_pi (pi, MPFR_RNDN); /* Error <= ulp(pi) /2 */ 255 e = MPFR_NOTZERO(tmp) ? MPFR_GET_EXP (tmp) : __gmpfr_emin - 1; 256 mpfr_sub (tmp, pi, tmp, MPFR_RNDN); /* see above */ 257 if (MPFR_IS_NEG (y)) 258 MPFR_CHANGE_SIGN (tmp); 259 /* Error(tmp) <= (1/2+2^(EXP(pi)-EXP(tmp)-1)+2^(e-EXP(tmp)+1))*ulp 260 <= 2^(MAX (MAX (EXP(PI)-EXP(tmp)-1, e-EXP(tmp)+1), 261 -1)+2)*ulp(tmp) */ 262 e = MAX (MAX (MPFR_GET_EXP (pi)-MPFR_GET_EXP (tmp) - 1, 263 e - MPFR_GET_EXP (tmp) + 1), -1) + 2; 264 if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - e, MPFR_PREC (dest), 265 rnd_mode))) 266 break; 267 MPFR_ZIV_NEXT (loop, prec); 268 mpfr_set_prec (tmp, prec); 269 mpfr_set_prec (pi, prec); 270 } 271 mpfr_clear (pi); 272 } 273 inexact = mpfr_set (dest, tmp, rnd_mode); 274 275 end: 276 MPFR_ZIV_FREE (loop); 277 mpfr_clear (tmp); 278 MPFR_SAVE_EXPO_FREE (expo); 279 return mpfr_check_range (dest, inexact, rnd_mode); 280} 281