1 /* $NetBSD: catrigl.c,v 1.3 2022/04/19 20:32:16 rillig Exp $ */ 2 /*- 3 * Copyright (c) 2012 Stephen Montgomery-Smith <stephen (at) FreeBSD.ORG> 4 * All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND 16 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 17 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 18 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 19 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 20 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 21 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 22 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 23 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 24 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 25 * SUCH DAMAGE. 26 */ 27 28 /* 29 * The algorithm is very close to that in "Implementing the complex arcsine 30 * and arccosine functions using exception handling" by T. E. Hull, Thomas F. 31 * Fairgrieve, and Ping Tak Peter Tang, published in ACM Transactions on 32 * Mathematical Software, Volume 23 Issue 3, 1997, Pages 299-335, 33 * http://dl.acm.org/citation.cfm?id=275324. 34 * 35 * The code for catrig.c contains complete comments. 36 */ 37 #include <sys/cdefs.h> 38 __RCSID("$NetBSD: catrigl.c,v 1.3 2022/04/19 20:32:16 rillig Exp $"); 39 40 #include "namespace.h" 41 #ifdef __weak_alias 42 __weak_alias(casinl, _casinl) 43 #endif 44 #ifdef __weak_alias 45 __weak_alias(catanl, _catanl) 46 #endif 47 48 49 #include <sys/param.h> 50 #include <complex.h> 51 #include <float.h> 52 #include <math.h> 53 #ifdef notyet // missing log1pl __HAVE_LONG_DOUBLE 54 55 #include "math_private.h" 56 57 #undef isinf 58 #define isinf(x) (fabsl(x) == INFINITY) 59 #undef isnan 60 #define isnan(x) ((x) != (x)) 61 #define raise_inexact() do { volatile float junk __unused = /*LINTED*/1 + tiny; } while (0) 62 #undef signbit 63 #define signbit(x) (__builtin_signbitl(x)) 64 65 #if __HAVE_LONG_DOUBLE + 0 == 128 66 // Ok 67 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 68 // XXX: Byte order 69 #define EXT_EXPBITS 15 70 struct ieee_ext { 71 uint64_t ext_frac; 72 uint16_t ext_exp:EXT_EXPBITS; 73 uint16_t ext_sign:1; 74 uint16_t ext_pad; 75 }; 76 #define extu_exp extu_ext.ext_exp 77 #define extu_sign extu_ext.ext_sign 78 #define extu_frac extu_ext.ext_frac 79 union ieee_ext_u { 80 long double extu_ld; 81 struct ieee_ext extu_ext; 82 }; 83 #else 84 #error "unsupported long double format" 85 #endif 86 87 #define GET_LDBL_EXPSIGN(r, s) \ 88 do { \ 89 union ieee_ext_u u; \ 90 u.extu_ld = s; \ 91 r = u.extu_sign; \ 92 r >>= EXT_EXPBITS - 1; \ 93 } while (0) 94 #define SET_LDBL_EXPSIGN(s, r) \ 95 do { \ 96 union ieee_ext_u u; \ 97 u.extu_ld = s; \ 98 u.extu_exp &= __BITS(0, EXT_EXPBITS - 1); \ 99 u.extu_exp |= (r) << (EXT_EXPBITS - 1); \ 100 s = u.extu_ld; \ 101 } while (0) 102 103 static const long double 104 A_crossover = 10, 105 B_crossover = 0.6417, 106 FOUR_SQRT_MIN = 0x1p-8189L, 107 QUARTER_SQRT_MAX = 0x1p8189L, 108 RECIP_EPSILON = 1/LDBL_EPSILON, 109 SQRT_MIN = 0x1p-8191L; 110 111 static const long double 112 m_e = 2.71828182845904523536028747135266250e0L, /* 0x15bf0a8b1457695355fb8ac404e7a.0p-111 */ 113 m_ln2 = 6.93147180559945309417232121458176568e-1L, /* 0x162e42fefa39ef35793c7673007e6.0p-113 */ 114 pio2_hi = 1.5707963267948966192313216916397514L, /* pi/2 */ 115 SQRT_3_EPSILON = 2.40370335797945490975336727199878124e-17L, /* 0x1bb67ae8584caa73b25742d7078b8.0p-168 */ 116 SQRT_6_EPSILON = 3.39934988877629587239082586223300391e-17L; /* 0x13988e1409212e7d0321914321a55.0p-167 */ 117 118 static const volatile double 119 pio2_lo = 6.1232339957367659e-17; /* 0x11a62633145c07.0p-106 */ 120 static const volatile float 121 tiny = 0x1p-100; 122 123 static long double complex clog_for_large_values(long double complex z); 124 125 inline static long double 126 f(long double a, long double b, long double hypot_a_b) 127 { 128 if (b < 0) 129 return ((hypot_a_b - b) / 2); 130 if (b == 0) 131 return (a / 2); 132 return (a * a / (hypot_a_b + b) / 2); 133 } 134 135 inline static void 136 do_hard_work(long double x, long double y, long double *rx, int *B_is_usable, long double *B, long double *sqrt_A2my2, long double *new_y) 137 { 138 long double R, S, A; 139 long double Am1, Amy; 140 141 R = hypotl(x, y+1); 142 S = hypotl(x, y-1); 143 144 A = (R + S) / 2; 145 if (A < 1) 146 A = 1; 147 148 if (A < A_crossover) { 149 if (y == 1 && x < LDBL_EPSILON*LDBL_EPSILON/128) { 150 *rx = sqrtl(x); 151 } else if (x >= LDBL_EPSILON * fabsl(y-1)) { 152 Am1 = f(x, 1+y, R) + f(x, 1-y, S); 153 *rx = log1pl(Am1 + sqrtl(Am1*(A+1))); 154 } else if (y < 1) { 155 *rx = x/sqrtl((1-y)*(1+y)); 156 } else { 157 *rx = log1pl((y-1) + sqrtl((y-1)*(y+1))); 158 } 159 } else 160 *rx = logl(A + sqrtl(A*A-1)); 161 162 *new_y = y; 163 164 if (y < FOUR_SQRT_MIN) { 165 *B_is_usable = 0; 166 *sqrt_A2my2 = A * (2 / LDBL_EPSILON); 167 *new_y= y * (2 / LDBL_EPSILON); 168 return; 169 } 170 171 *B = y/A; 172 *B_is_usable = 1; 173 174 if (*B > B_crossover) { 175 *B_is_usable = 0; 176 if (y == 1 && x < LDBL_EPSILON/128) { 177 *sqrt_A2my2 = sqrtl(x)*sqrtl((A+y)/2); 178 } else if (x >= LDBL_EPSILON * fabsl(y-1)) { 179 Amy = f(x, y+1, R) + f(x, y-1, S); 180 *sqrt_A2my2 = sqrtl(Amy*(A+y)); 181 } else if (y > 1) { 182 *sqrt_A2my2 = x * (4/LDBL_EPSILON/LDBL_EPSILON) * y / 183 sqrtl((y+1)*(y-1)); 184 *new_y = y * (4/LDBL_EPSILON/LDBL_EPSILON); 185 } else { 186 *sqrt_A2my2 = sqrtl((1-y)*(1+y)); 187 } 188 } 189 } 190 191 long double complex 192 casinhl(long double complex z) 193 { 194 long double x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y; 195 int B_is_usable; 196 long double complex w; 197 198 x = creall(z); 199 y = cimagl(z); 200 ax = fabsl(x); 201 ay = fabsl(y); 202 203 if (isnan(x) || isnan(y)) { 204 if (isinf(x)) 205 return (CMPLXL(x, y+y)); 206 if (isinf(y)) 207 return (CMPLXL(y, x+x)); 208 if (y == 0) return (CMPLXL(x+x, y)); 209 return (CMPLXL(x+0.0L+(y+0), x+0.0L+(y+0))); 210 } 211 212 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) { 213 if (signbit(x) == 0) 214 w = clog_for_large_values(z) + m_ln2; 215 else 216 w = clog_for_large_values(-z) + m_ln2; 217 return (CMPLXL(copysignl(creall(w), x), copysignl(cimagl(w), y))); 218 } 219 220 if (x == 0 && y == 0) 221 return (z); 222 223 raise_inexact(); 224 225 if (ax < SQRT_6_EPSILON/4 && ay < SQRT_6_EPSILON/4) 226 return (z); 227 228 do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y); 229 if (B_is_usable) 230 ry = asinl(B); 231 else 232 ry = atan2l(new_y, sqrt_A2my2); 233 return (CMPLXL(copysignl(rx, x), copysignl(ry, y))); 234 } 235 236 long double complex 237 casinl(long double complex z) 238 { 239 long double complex w = casinhl(CMPLXL(cimagl(z), creall(z))); 240 return (CMPLXL(cimagl(w), creall(w))); 241 } 242 243 long double complex 244 cacosl(long double complex z) 245 { 246 long double x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x; 247 int sx, sy; 248 int B_is_usable; 249 long double complex w; 250 251 x = creall(z); 252 y = cimagl(z); 253 sx = signbit(x); 254 sy = signbit(y); 255 ax = fabsl(x); 256 ay = fabsl(y); 257 258 if (isnan(x) || isnan(y)) { 259 if (isinf(x)) 260 return (CMPLXL(y+y, -INFINITY)); 261 if (isinf(y)) 262 return (CMPLXL(x+x, -y)); 263 if (x == 0) return (CMPLXL(pio2_hi + pio2_lo, y+y)); 264 return (CMPLXL(x+0.0L+(y+0), x+0.0L+(y+0))); 265 } 266 267 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) { 268 w = clog_for_large_values(z); 269 rx = fabsl(cimagl(w)); 270 ry = creall(w) + m_ln2; 271 if (sy == 0) 272 ry = -ry; 273 return (CMPLXL(rx, ry)); 274 } 275 276 if (x == 1 && y == 0) 277 return (CMPLXL(0, -y)); 278 279 raise_inexact(); 280 281 if (ax < SQRT_6_EPSILON/4 && ay < SQRT_6_EPSILON/4) 282 return (CMPLXL(pio2_hi - (x - pio2_lo), -y)); 283 284 do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x); 285 if (B_is_usable) { 286 if (sx==0) 287 rx = acosl(B); 288 else 289 rx = acosl(-B); 290 } else { 291 if (sx==0) 292 rx = atan2l(sqrt_A2mx2, new_x); 293 else 294 rx = atan2l(sqrt_A2mx2, -new_x); 295 } 296 if (sy==0) 297 ry = -ry; 298 return (CMPLXL(rx, ry)); 299 } 300 301 long double complex 302 cacoshl(long double complex z) 303 { 304 long double complex w; 305 long double rx, ry; 306 307 w = cacosl(z); 308 rx = creall(w); 309 ry = cimagl(w); 310 if (isnan(rx) && isnan(ry)) 311 return (CMPLXL(ry, rx)); 312 if (isnan(rx)) 313 return (CMPLXL(fabsl(ry), rx)); 314 if (isnan(ry)) 315 return (CMPLXL(ry, ry)); 316 return (CMPLXL(fabsl(ry), copysignl(rx, cimagl(z)))); 317 } 318 319 static long double complex 320 clog_for_large_values(long double complex z) 321 { 322 long double x, y; 323 long double ax, ay, t; 324 325 x = creall(z); 326 y = cimagl(z); 327 ax = fabsl(x); 328 ay = fabsl(y); 329 if (ax < ay) { 330 t = ax; 331 ax = ay; 332 ay = t; 333 } 334 335 if (ax > LDBL_MAX / 2) 336 return (CMPLXL(logl(hypotl(x / m_e, y / m_e)) + 1, atan2l(y, x))); 337 338 if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN) 339 return (CMPLXL(logl(hypotl(x, y)), atan2l(y, x))); 340 341 return (CMPLXL(logl(ax*ax + ay*ay) / 2, atan2l(y, x))); 342 } 343 344 inline static long double 345 sum_squares(long double x, long double y) 346 { 347 if (y < SQRT_MIN) 348 return (x*x); 349 350 return (x*x + y*y); 351 } 352 353 inline static long double 354 real_part_reciprocal(long double x, long double y) 355 { 356 long double scale; 357 uint16_t hx, hy; 358 int16_t ix, iy; 359 360 GET_LDBL_EXPSIGN(hx, x); 361 ix = hx & 0x7fff; 362 GET_LDBL_EXPSIGN(hy, y); 363 iy = hy & 0x7fff; 364 #define BIAS (LDBL_MAX_EXP - 1) 365 #define CUTOFF (LDBL_MANT_DIG / 2 + 1) 366 if (ix - iy >= CUTOFF || isinf(x)) 367 return (1/x); 368 if (iy - ix >= CUTOFF) 369 return (x/y/y); 370 if (ix <= BIAS + LDBL_MAX_EXP / 2 - CUTOFF) 371 return (x/(x*x + y*y)); 372 scale = 1; 373 SET_LDBL_EXPSIGN(scale, 0x7fff - ix); 374 x *= scale; 375 y *= scale; 376 return (x/(x*x + y*y) * scale); 377 } 378 379 long double complex 380 catanhl(long double complex z) 381 { 382 long double x, y, ax, ay, rx, ry; 383 384 x = creall(z); 385 y = cimagl(z); 386 ax = fabsl(x); 387 ay = fabsl(y); 388 389 if (y == 0 && ax <= 1) 390 return (CMPLXL(atanhl(x), y)); /* XXX need atanhl() */ 391 392 if (x == 0) 393 return (CMPLXL(x, atanl(y))); 394 395 if (isnan(x) || isnan(y)) { 396 if (isinf(x)) 397 return (CMPLXL(copysignl(0, x), y+y)); 398 if (isinf(y)) 399 return (CMPLXL(copysignl(0, x), copysignl(pio2_hi + pio2_lo, y))); 400 return (CMPLXL(x+0.0L+(y+0), x+0.0L+(y+0))); 401 } 402 403 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) 404 return (CMPLXL(real_part_reciprocal(x, y), copysignl(pio2_hi + pio2_lo, y))); 405 406 if (ax < SQRT_3_EPSILON/2 && ay < SQRT_3_EPSILON/2) { 407 raise_inexact(); 408 return (z); 409 } 410 411 if (ax == 1 && ay < LDBL_EPSILON) { 412 #if 0 413 if (ay > 2*LDBL_MIN) 414 rx = - logl(ay/2) / 2; 415 else 416 #endif 417 rx = - (logl(ay) - m_ln2) / 2; 418 } else 419 rx = log1pl(4*ax / sum_squares(ax-1, ay)) / 4; 420 421 if (ax == 1) 422 ry = atan2l(2, -ay) / 2; 423 else if (ay < LDBL_EPSILON) 424 ry = atan2l(2*ay, (1-ax)*(1+ax)) / 2; 425 else 426 ry = atan2l(2*ay, (1-ax)*(1+ax) - ay*ay) / 2; 427 428 return (CMPLXL(copysignl(rx, x), copysignl(ry, y))); 429 } 430 431 long double complex 432 catanl(long double complex z) 433 { 434 long double complex w = catanhl(CMPLXL(cimagl(z), creall(z))); 435 return (CMPLXL(cimagl(w), creall(w))); 436 } 437 438 #else 439 __strong_alias(_casinl, casin) 440 __strong_alias(_catanl, catan) 441 __strong_alias(cacoshl, cacosh) 442 __strong_alias(cacosl, cacos) 443 __strong_alias(casinhl, casinh) 444 __strong_alias(catanhl, catanh) 445 #endif 446
Indexes created Sat Sep 20 22:09:52 GMT 2025