1 /* mpc_atan -- arctangent of a complex number. 2 3 Copyright (C) 2009, 2010, 2011, 2012, 2013, 2017, 2020, 2022 INRIA 4 5 This file is part of GNU MPC. 6 7 GNU MPC is free software; you can redistribute it and/or modify it under 8 the terms of the GNU Lesser General Public License as published by the 9 Free Software Foundation; either version 3 of the License, or (at your 10 option) any later version. 11 12 GNU MPC is distributed in the hope that it will be useful, but WITHOUT ANY 13 WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS 14 FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for 15 more details. 16 17 You should have received a copy of the GNU Lesser General Public License 18 along with this program. If not, see http://www.gnu.org/licenses/ . 19 */ 20 21 #include <stdio.h> 22 #include "mpc-impl.h" 23 24 /* set rop to 25 -pi/2 if s < 0 26 +pi/2 else 27 rounded in the direction rnd 28 */ 29 int 30 set_pi_over_2 (mpfr_ptr rop, int s, mpfr_rnd_t rnd) 31 { 32 int inex; 33 34 inex = mpfr_const_pi (rop, s < 0 ? INV_RND (rnd) : rnd); 35 mpfr_div_2ui (rop, rop, 1, MPFR_RNDN); 36 if (s < 0) 37 { 38 inex = -inex; 39 mpfr_neg (rop, rop, MPFR_RNDN); 40 } 41 42 return inex; 43 } 44 45 int 46 mpc_atan (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd) 47 { 48 int s_re, s_im; 49 int inex_re, inex_im, inex; 50 mpfr_exp_t saved_emin, saved_emax; 51 52 inex_re = 0; 53 inex_im = 0; 54 s_re = mpfr_signbit (mpc_realref (op)); 55 s_im = mpfr_signbit (mpc_imagref (op)); 56 57 /* special values */ 58 if (mpfr_nan_p (mpc_realref (op)) || mpfr_nan_p (mpc_imagref (op))) 59 { 60 if (mpfr_nan_p (mpc_realref (op))) 61 { 62 mpfr_set_nan (mpc_realref (rop)); 63 if (mpfr_zero_p (mpc_imagref (op)) || mpfr_inf_p (mpc_imagref (op))) 64 { 65 mpfr_set_ui (mpc_imagref (rop), 0, MPFR_RNDN); 66 if (s_im) 67 mpc_conj (rop, rop, MPC_RNDNN); 68 } 69 else 70 mpfr_set_nan (mpc_imagref (rop)); 71 } 72 else 73 { 74 if (mpfr_inf_p (mpc_realref (op))) 75 { 76 inex_re = set_pi_over_2 (mpc_realref (rop), -s_re, MPC_RND_RE (rnd)); 77 mpfr_set_ui (mpc_imagref (rop), 0, MPFR_RNDN); 78 } 79 else 80 { 81 mpfr_set_nan (mpc_realref (rop)); 82 mpfr_set_nan (mpc_imagref (rop)); 83 } 84 } 85 return MPC_INEX (inex_re, 0); 86 } 87 88 if (mpfr_inf_p (mpc_realref (op)) || mpfr_inf_p (mpc_imagref (op))) 89 { 90 inex_re = set_pi_over_2 (mpc_realref (rop), -s_re, MPC_RND_RE (rnd)); 91 92 mpfr_set_ui (mpc_imagref (rop), 0, MPFR_RNDN); 93 if (s_im) 94 mpc_conj (rop, rop, MPFR_RNDN); 95 96 return MPC_INEX (inex_re, 0); 97 } 98 99 /* pure real argument */ 100 if (mpfr_zero_p (mpc_imagref (op))) 101 { 102 inex_re = mpfr_atan (mpc_realref (rop), mpc_realref (op), MPC_RND_RE (rnd)); 103 104 mpfr_set_ui (mpc_imagref (rop), 0, MPFR_RNDN); 105 if (s_im) 106 mpc_conj (rop, rop, MPFR_RNDN); 107 108 return MPC_INEX (inex_re, 0); 109 } 110 111 /* pure imaginary argument */ 112 if (mpfr_zero_p (mpc_realref (op))) 113 { 114 int cmp_1; 115 116 if (s_im) 117 cmp_1 = -mpfr_cmp_si (mpc_imagref (op), -1); 118 else 119 cmp_1 = mpfr_cmp_ui (mpc_imagref (op), +1); 120 121 if (cmp_1 < 0) 122 { 123 /* atan(+0+iy) = +0 +i*atanh(y), if |y| < 1 124 atan(-0+iy) = -0 +i*atanh(y), if |y| < 1 */ 125 126 mpfr_set_ui (mpc_realref (rop), 0, MPFR_RNDN); 127 if (s_re) 128 mpfr_neg (mpc_realref (rop), mpc_realref (rop), MPFR_RNDN); 129 130 inex_im = mpfr_atanh (mpc_imagref (rop), mpc_imagref (op), MPC_RND_IM (rnd)); 131 } 132 else if (cmp_1 == 0) 133 { 134 /* atan(+/-0 +i) = +/-0 +i*inf 135 atan(+/-0 -i) = +/-0 -i*inf */ 136 mpfr_set_zero (mpc_realref (rop), s_re ? -1 : +1); 137 mpfr_set_inf (mpc_imagref (rop), s_im ? -1 : +1); 138 } 139 else 140 { 141 /* atan(+0+iy) = +pi/2 +i*atanh(1/y), if |y| > 1 142 atan(-0+iy) = -pi/2 +i*atanh(1/y), if |y| > 1 */ 143 mpfr_rnd_t rnd_im; 144 mpfr_t y, z; 145 mpfr_prec_t p, p_im; 146 int ok = 0; 147 148 rnd_im = MPC_RND_IM (rnd); 149 mpfr_init (y); 150 mpfr_init (z); 151 p_im = mpfr_get_prec (mpc_imagref (rop)); 152 p = p_im; 153 154 /* a = o(1/y) with error(a) < ulp(a), rounded away 155 b = o(atanh(a)) with error(b) < ulp(b) + 1/|a^2-1|*ulp(a), 156 since if a = 1/y + eps, then atanh(a) = atanh(1/y) + eps * atanh'(t) 157 with t in (1/y, a). Since a is rounded away, we have 1/y <= a <= 1 158 if y > 1, and -1 <= a <= 1/y if y < -1, thus |atanh'(t)| = 159 1/|t^2-1| <= 1/|a^2-1|. 160 161 We round atanh(1/y) away from 0. 162 */ 163 do 164 { 165 mpfr_exp_t err, exp_a; 166 167 p += mpc_ceil_log2 (p) + 2; 168 mpfr_set_prec (y, p); 169 mpfr_set_prec (z, p); 170 inex_im = mpfr_ui_div (y, 1, mpc_imagref (op), MPFR_RNDA); 171 exp_a = mpfr_get_exp (y); 172 /* FIXME: should we consider the case with unreasonably huge 173 precision prec(y)>3*exp_min, where atanh(1/Im(op)) could be 174 representable while 1/Im(op) underflows ? 175 This corresponds to |y| = 0.5*2^emin, in which case the 176 result may be wrong. */ 177 178 /* We would like to compute a rounded-up error bound 1/|a^2-1|, 179 so we need to round down |a^2-1|, which means rounding up 180 a^2 since |a|<1. */ 181 mpfr_sqr (z, y, MPFR_RNDU); 182 /* since |y| > 1, we should have |a| <= 1, thus a^2 <= 1 */ 183 MPC_ASSERT(mpfr_cmp_ui (z, 1) <= 0); 184 /* in case z=1, we should try again with more precision */ 185 if (mpfr_cmp_ui (z, 1) == 0) 186 continue; 187 /* now z < 1 */ 188 mpfr_ui_sub (z, 1, z, MPFR_RNDZ); 189 190 /* atanh cannot underflow: |atanh(x)| > |x| for |x| < 1 */ 191 inex_im |= mpfr_atanh (y, y, MPFR_RNDA); 192 193 /* the error is now bounded by ulp(b) + 1/z*ulp(a), thus 194 ulp(b) + 2^(exp(a) - exp(b) + 1 - exp(z)) * ulp(b) */ 195 err = exp_a - mpfr_get_exp (y) + 1 - mpfr_get_exp (z); 196 if (err >= 0) /* 1 + 2^err <= 2^(err+1) */ 197 err = err + 1; 198 else 199 err = 1; /* 1 + 2^err <= 2^1 */ 200 201 /* the error is bounded by 2^err ulps */ 202 203 ok = inex_im == 0 204 || mpfr_can_round (y, p - err, MPFR_RNDA, MPFR_RNDZ, 205 p_im + (rnd_im == MPFR_RNDN)); 206 } while (ok == 0); 207 208 inex_re = set_pi_over_2 (mpc_realref (rop), -s_re, MPC_RND_RE (rnd)); 209 inex_im = mpfr_set (mpc_imagref (rop), y, rnd_im); 210 mpfr_clear (y); 211 mpfr_clear (z); 212 } 213 return MPC_INEX (inex_re, inex_im); 214 } 215 216 saved_emin = mpfr_get_emin (); 217 saved_emax = mpfr_get_emax (); 218 mpfr_set_emin (mpfr_get_emin_min ()); 219 mpfr_set_emax (mpfr_get_emax_max ()); 220 221 /* regular number argument */ 222 { 223 mpfr_t a, b, x, y; 224 mpfr_prec_t prec, p; 225 mpfr_exp_t err, expo; 226 int ok = 0; 227 mpfr_t minus_op_re; 228 mpfr_exp_t op_re_exp, op_im_exp; 229 mpfr_rnd_t rnd1, rnd2; 230 231 mpfr_inits2 (MPFR_PREC_MIN, a, b, x, y, (mpfr_ptr) 0); 232 233 /* real part: Re(arctan(x+i*y)) = [arctan2(x,1-y) - arctan2(-x,1+y)]/2 */ 234 minus_op_re[0] = mpc_realref (op)[0]; 235 MPFR_CHANGE_SIGN (minus_op_re); 236 op_re_exp = mpfr_get_exp (mpc_realref (op)); 237 op_im_exp = mpfr_get_exp (mpc_imagref (op)); 238 239 prec = mpfr_get_prec (mpc_realref (rop)); /* result precision */ 240 241 /* a = o(1-y) error(a) < 1 ulp(a) 242 b = o(atan2(x,a)) error(b) < [1+2^{3+Exp(x)-Exp(a)-Exp(b)}] ulp(b) 243 = kb ulp(b) 244 c = o(1+y) error(c) < 1 ulp(c) 245 d = o(atan2(-x,c)) error(d) < [1+2^{3+Exp(x)-Exp(c)-Exp(d)}] ulp(d) 246 = kd ulp(d) 247 e = o(b - d) error(e) < [1 + kb*2^{Exp(b}-Exp(e)} 248 + kd*2^{Exp(d)-Exp(e)}] ulp(e) 249 error(e) < [1 + 2^{4+Exp(x)-Exp(a)-Exp(e)} 250 + 2^{4+Exp(x)-Exp(c)-Exp(e)}] ulp(e) 251 because |atan(u)| < |u| 252 < [1 + 2^{5+Exp(x)-min(Exp(a),Exp(c)) 253 -Exp(e)}] ulp(e) 254 f = e/2 exact 255 */ 256 257 /* p: working precision */ 258 p = (op_im_exp > 0 || prec > SAFE_ABS (mpfr_prec_t, op_im_exp)) ? prec 259 : (prec - op_im_exp); 260 rnd1 = mpfr_sgn (mpc_realref (op)) > 0 ? MPFR_RNDD : MPFR_RNDU; 261 rnd2 = mpfr_sgn (mpc_realref (op)) < 0 ? MPFR_RNDU : MPFR_RNDD; 262 263 do 264 { 265 p += mpc_ceil_log2 (p) + 2; 266 mpfr_set_prec (a, p); 267 mpfr_set_prec (b, p); 268 mpfr_set_prec (x, p); 269 270 /* x = upper bound for atan (x/(1-y)). Since atan is increasing, we 271 need an upper bound on x/(1-y), i.e., a lower bound on 1-y for 272 x positive, and an upper bound on 1-y for x negative */ 273 mpfr_ui_sub (a, 1, mpc_imagref (op), rnd1); 274 if (mpfr_sgn (a) == 0) /* y is near 1, thus 1+y is near 2, and 275 expo will be 1 or 2 below */ 276 { 277 MPC_ASSERT (mpfr_cmp_ui (mpc_imagref(op), 1) == 0); 278 /* check for intermediate underflow */ 279 err = 2; /* ensures err will be expo below */ 280 } 281 else 282 err = mpfr_get_exp (a); /* err = Exp(a) with the notations above */ 283 mpfr_atan2 (x, mpc_realref (op), a, MPFR_RNDU); 284 285 /* b = lower bound for atan (-x/(1+y)): for x negative, we need a 286 lower bound on -x/(1+y), i.e., an upper bound on 1+y */ 287 mpfr_add_ui (a, mpc_imagref(op), 1, rnd2); 288 /* if a is exactly zero, i.e., Im(op) = -1, then the error on a is 0, 289 and we can simply ignore the terms involving Exp(a) in the error */ 290 if (mpfr_sgn (a) == 0) 291 { 292 MPC_ASSERT (mpfr_cmp_si (mpc_imagref(op), -1) == 0); 293 /* check for intermediate underflow */ 294 expo = err; /* will leave err unchanged below */ 295 } 296 else 297 expo = mpfr_get_exp (a); /* expo = Exp(c) with the notations above */ 298 mpfr_atan2 (b, minus_op_re, a, MPFR_RNDD); 299 300 err = err < expo ? err : expo; /* err = min(Exp(a),Exp(c)) */ 301 mpfr_sub (x, x, b, MPFR_RNDU); 302 303 err = 5 + op_re_exp - err - mpfr_get_exp (x); 304 /* error is bounded by [1 + 2^err] ulp(e) */ 305 err = err < 0 ? 1 : err + 1; 306 307 mpfr_div_2ui (x, x, 1, MPFR_RNDU); 308 309 /* Note: using RND2=RNDD guarantees that if x is exactly representable 310 on prec + ... bits, mpfr_can_round will return 0 */ 311 ok = mpfr_can_round (x, p - err, MPFR_RNDU, MPFR_RNDD, 312 prec + (MPC_RND_RE (rnd) == MPFR_RNDN)); 313 } while (ok == 0); 314 315 /* Imaginary part 316 Im(atan(x+I*y)) = 1/4 * [log(x^2+(1+y)^2) - log (x^2 +(1-y)^2)] */ 317 prec = mpfr_get_prec (mpc_imagref (rop)); /* result precision */ 318 319 /* a = o(1+y) error(a) < 1 ulp(a) 320 b = o(a^2) error(b) < 5 ulp(b) 321 c = o(x^2) error(c) < 1 ulp(c) 322 d = o(b+c) error(d) < 7 ulp(d) 323 e = o(log(d)) error(e) < [1 + 7*2^{2-Exp(e)}] ulp(e) = ke ulp(e) 324 f = o(1-y) error(f) < 1 ulp(f) 325 g = o(f^2) error(g) < 5 ulp(g) 326 h = o(c+f) error(h) < 7 ulp(h) 327 i = o(log(h)) error(i) < [1 + 7*2^{2-Exp(i)}] ulp(i) = ki ulp(i) 328 j = o(e-i) error(j) < [1 + ke*2^{Exp(e)-Exp(j)} 329 + ki*2^{Exp(i)-Exp(j)}] ulp(j) 330 error(j) < [1 + 2^{Exp(e)-Exp(j)} + 2^{Exp(i)-Exp(j)} 331 + 7*2^{3-Exp(j)}] ulp(j) 332 < [1 + 2^{max(Exp(e),Exp(i))-Exp(j)+1} 333 + 7*2^{3-Exp(j)}] ulp(j) 334 k = j/4 exact 335 */ 336 err = 2; 337 p = prec; /* working precision */ 338 339 do 340 { 341 p += mpc_ceil_log2 (p) + err; 342 mpfr_set_prec (a, p); 343 mpfr_set_prec (b, p); 344 mpfr_set_prec (y, p); 345 346 /* a = upper bound for log(x^2 + (1+y)^2) */ 347 mpfr_add_ui (a, mpc_imagref (op), 1, MPFR_RNDA); 348 mpfr_sqr (a, a, MPFR_RNDU); 349 mpfr_sqr (y, mpc_realref (op), MPFR_RNDU); 350 mpfr_add (a, a, y, MPFR_RNDU); 351 mpfr_log (a, a, MPFR_RNDU); 352 353 /* b = lower bound for log(x^2 + (1-y)^2) */ 354 mpfr_ui_sub (b, 1, mpc_imagref (op), MPFR_RNDZ); /* round to zero */ 355 mpfr_sqr (b, b, MPFR_RNDZ); 356 /* we could write mpfr_sqr (y, mpc_realref (op), MPFR_RNDZ) but it is 357 more efficient to reuse the value of y (x^2) above and subtract 358 one ulp */ 359 mpfr_nextbelow (y); 360 mpfr_add (b, b, y, MPFR_RNDZ); 361 mpfr_log (b, b, MPFR_RNDZ); 362 363 mpfr_sub (y, a, b, MPFR_RNDU); 364 365 if (mpfr_zero_p (y)) 366 /* FIXME: happens when x and y have very different magnitudes; 367 could be handled more efficiently */ 368 ok = 0; 369 else 370 { 371 expo = MPC_MAX (mpfr_get_exp (a), mpfr_get_exp (b)); 372 expo = expo - mpfr_get_exp (y) + 1; 373 err = 3 - mpfr_get_exp (y); 374 /* error(j) <= [1 + 2^expo + 7*2^err] ulp(j) */ 375 if (expo <= err) /* error(j) <= [1 + 2^{err+1}] ulp(j) */ 376 err = (err < 0) ? 1 : err + 2; 377 else 378 err = (expo < 0) ? 1 : expo + 2; 379 380 mpfr_div_2ui (y, y, 2, MPFR_RNDN); 381 MPC_ASSERT (!mpfr_zero_p (y)); 382 /* FIXME: underflow. Since the main term of the Taylor series 383 in y=0 is 1/(x^2+1) * y, this means that y is very small 384 and/or x very large; but then the mpfr_zero_p (y) above 385 should be true. This needs a proof, or better yet, 386 special code. */ 387 388 ok = mpfr_can_round (y, p - err, MPFR_RNDU, MPFR_RNDD, 389 prec + (MPC_RND_IM (rnd) == MPFR_RNDN)); 390 } 391 } while (ok == 0); 392 393 inex = mpc_set_fr_fr (rop, x, y, rnd); 394 395 mpfr_clears (a, b, x, y, (mpfr_ptr) 0); 396 397 /* restore the exponent range, and check the range of results */ 398 mpfr_set_emin (saved_emin); 399 mpfr_set_emax (saved_emax); 400 inex_re = mpfr_check_range (mpc_realref (rop), MPC_INEX_RE (inex), 401 MPC_RND_RE (rnd)); 402 inex_im = mpfr_check_range (mpc_imagref (rop), MPC_INEX_IM (inex), 403 MPC_RND_IM (rnd)); 404 405 return MPC_INEX (inex_re, inex_im); 406 } 407 } 408
Indexes created Thu Jul 09 00:25:44 UTC 2026