catrigl.c revision 1.1.2.2
1 /* $NetBSD: catrigl.c,v 1.1.2.2 2016/11/04 14:48:54 pgoyette 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 1.1.2.2 2016/11/04 14:48:54 pgoyette 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 <complex.h> 50 #include <float.h> 51 #ifdef __HAVE_LONG_DOUBLE 52 53 #include "math.h" 54 #include "math_private.h" 55 56 #undef isinf 57 #define isinf(x) (fabsl(x) == INFINITY) 58 #undef isnan 59 #define isnan(x) ((x) != (x)) 60 #define raise_inexact() do { volatile float junk __unused = /*LINTED*/1 + tiny; } while(/*CONSTCOND*/0) 61 #undef signbit 62 #define signbit(x) (__builtin_signbitl(x)) 63 64 #if __HAVE_LONG_DOUBLE + 0 == 128 65 // Ok 66 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 67 // XXX: Byte order 68 struct ieee_ext { 69 uint64_t ext_frac; 70 uint16_t ext_exp:15; 71 uint16_t ext_sign:1; 72 uint16_t ext_pad; 73 }; 74 #define extu_exp extu_ext.ext_exp 75 #define extu_sign extu_ext.ext_sign 76 #define extu_frac extu_ext.ext_frac 77 union ieee_ext_u { 78 long double extu_ld; 79 struct ieee_ext extu_ext; 80 }; 81 #else 82 #error "unsupported long double format" 83 #endif 84 85 #define GET_LDBL_EXPSIGN(r, s) \ 86 do { \ 87 union ieee_ext_u u; \ 88 u.extu_ld = s; \ 89 r = u.extu_sign; \ 90 r >>= EXT_EXPBITS - 1; 91 } while (/*CONSTCOND*/0) 92 #define SET_LDBL_EXPSIGN(r, s) \ 93 do { \ 94 union ieee_ext_u u; \ 95 u.extu_ld = s; \ 96 u.extu_exp &= __BITS(0, EXT_EXPBITS - 1); \ 97 u.extu_exp |= r << (EXT_EXPBITS - 1); \ 98 s = u.extu_ld; \ 99 } while (/*CONSTCOND*/0) 100 101 static const long double 102 A_crossover = 10, 103 B_crossover = 0.6417, 104 FOUR_SQRT_MIN = 0x1p-8189L, 105 QUARTER_SQRT_MAX = 0x1p8189L, 106 RECIP_EPSILON = 1/LDBL_EPSILON, 107 SQRT_MIN = 0x1p-8191L; 108 109 static const long double 110 m_e = 2.71828182845904523536028747135266250e0L, /* 0x15bf0a8b1457695355fb8ac404e7a.0p-111 */ 111 m_ln2 = 6.93147180559945309417232121458176568e-1L, /* 0x162e42fefa39ef35793c7673007e6.0p-113 */ 112 pio2_hi = 1.5707963267948966192313216916397514L, /* pi/2 */ 113 SQRT_3_EPSILON = 2.40370335797945490975336727199878124e-17L, /* 0x1bb67ae8584caa73b25742d7078b8.0p-168 */ 114 SQRT_6_EPSILON = 3.39934988877629587239082586223300391e-17L; /* 0x13988e1409212e7d0321914321a55.0p-167 */ 115 116 static const volatile double 117 pio2_lo = 6.1232339957367659e-17; /* 0x11a62633145c07.0p-106 */ 118 static const volatile float 119 tiny = 0x1p-100; 120 121 static long double complex clog_for_large_values(long double complex z); 122 123 inline static long double 124 f(long double a, long double b, long double hypot_a_b) 125 { 126 if (b < 0) 127 return ((hypot_a_b - b) / 2); 128 if (b == 0) 129 return (a / 2); 130 return (a * a / (hypot_a_b + b) / 2); 131 } 132 133 inline static void 134 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) 135 { 136 long double R, S, A; 137 long double Am1, Amy; 138 139 R = hypotl(x, y+1); 140 S = hypotl(x, y-1); 141 142 A = (R + S) / 2; 143 if (A < 1) 144 A = 1; 145 146 if (A < A_crossover) { 147 if (y == 1 && x < LDBL_EPSILON*LDBL_EPSILON/128) { 148 *rx = sqrtl(x); 149 } else if (x >= LDBL_EPSILON * fabsl(y-1)) { 150 Am1 = f(x, 1+y, R) + f(x, 1-y, S); 151 *rx = log1pl(Am1 + sqrtl(Am1*(A+1))); 152 } else if (y < 1) { 153 *rx = x/sqrtl((1-y)*(1+y)); 154 } else { 155 *rx = log1pl((y-1) + sqrtl((y-1)*(y+1))); 156 } 157 } else 158 *rx = logl(A + sqrtl(A*A-1)); 159 160 *new_y = y; 161 162 if (y < FOUR_SQRT_MIN) { 163 *B_is_usable = 0; 164 *sqrt_A2my2 = A * (2 / LDBL_EPSILON); 165 *new_y= y * (2 / LDBL_EPSILON); 166 return; 167 } 168 169 *B = y/A; 170 *B_is_usable = 1; 171 172 if (*B > B_crossover) { 173 *B_is_usable = 0; 174 if (y == 1 && x < LDBL_EPSILON/128) { 175 *sqrt_A2my2 = sqrtl(x)*sqrtl((A+y)/2); 176 } else if (x >= LDBL_EPSILON * fabsl(y-1)) { 177 Amy = f(x, y+1, R) + f(x, y-1, S); 178 *sqrt_A2my2 = sqrtl(Amy*(A+y)); 179 } else if (y > 1) { 180 *sqrt_A2my2 = x * (4/LDBL_EPSILON/LDBL_EPSILON) * y / 181 sqrtl((y+1)*(y-1)); 182 *new_y = y * (4/LDBL_EPSILON/LDBL_EPSILON); 183 } else { 184 *sqrt_A2my2 = sqrtl((1-y)*(1+y)); 185 } 186 } 187 } 188 189 long double complex 190 casinhl(long double complex z) 191 { 192 long double x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y; 193 int B_is_usable; 194 long double complex w; 195 196 x = creall(z); 197 y = cimagl(z); 198 ax = fabsl(x); 199 ay = fabsl(y); 200 201 if (isnan(x) || isnan(y)) { 202 if (isinf(x)) 203 return (CMPLXL(x, y+y)); 204 if (isinf(y)) 205 return (CMPLXL(y, x+x)); 206 if (y == 0) return (CMPLXL(x+x, y)); 207 return (CMPLXL(x+0.0L+(y+0), x+0.0L+(y+0))); 208 } 209 210 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) { 211 if (signbit(x) == 0) 212 w = clog_for_large_values(z) + m_ln2; 213 else 214 w = clog_for_large_values(-z) + m_ln2; 215 return (CMPLXL(copysignl(creall(w), x), copysignl(cimagl(w), y))); 216 } 217 218 if (x == 0 && y == 0) 219 return (z); 220 221 raise_inexact(); 222 223 if (ax < SQRT_6_EPSILON/4 && ay < SQRT_6_EPSILON/4) 224 return (z); 225 226 do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y); 227 if (B_is_usable) 228 ry = asinl(B); 229 else 230 ry = atan2l(new_y, sqrt_A2my2); 231 return (CMPLXL(copysignl(rx, x), copysignl(ry, y))); 232 } 233 234 long double complex 235 casinl(long double complex z) 236 { 237 long double complex w = casinhl(CMPLXL(cimagl(z), creall(z))); 238 return (CMPLXL(cimagl(w), creall(w))); 239 } 240 241 long double complex 242 cacosl(long double complex z) 243 { 244 long double x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x; 245 int sx, sy; 246 int B_is_usable; 247 long double complex w; 248 249 x = creall(z); 250 y = cimagl(z); 251 sx = signbit(x); 252 sy = signbit(y); 253 ax = fabsl(x); 254 ay = fabsl(y); 255 256 if (isnan(x) || isnan(y)) { 257 if (isinf(x)) 258 return (CMPLXL(y+y, -INFINITY)); 259 if (isinf(y)) 260 return (CMPLXL(x+x, -y)); 261 if (x == 0) return (CMPLXL(pio2_hi + pio2_lo, y+y)); 262 return (CMPLXL(x+0.0L+(y+0), x+0.0L+(y+0))); 263 } 264 265 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) { 266 w = clog_for_large_values(z); 267 rx = fabsl(cimagl(w)); 268 ry = creall(w) + m_ln2; 269 if (sy == 0) 270 ry = -ry; 271 return (CMPLXL(rx, ry)); 272 } 273 274 if (x == 1 && y == 0) 275 return (CMPLXL(0, -y)); 276 277 raise_inexact(); 278 279 if (ax < SQRT_6_EPSILON/4 && ay < SQRT_6_EPSILON/4) 280 return (CMPLXL(pio2_hi - (x - pio2_lo), -y)); 281 282 do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x); 283 if (B_is_usable) { 284 if (sx==0) 285 rx = acosl(B); 286 else 287 rx = acosl(-B); 288 } else { 289 if (sx==0) 290 rx = atan2l(sqrt_A2mx2, new_x); 291 else 292 rx = atan2l(sqrt_A2mx2, -new_x); 293 } 294 if (sy==0) 295 ry = -ry; 296 return (CMPLXL(rx, ry)); 297 } 298 299 long double complex 300 cacoshl(long double complex z) 301 { 302 long double complex w; 303 long double rx, ry; 304 305 w = cacosl(z); 306 rx = creall(w); 307 ry = cimagl(w); 308 if (isnan(rx) && isnan(ry)) 309 return (CMPLXL(ry, rx)); 310 if (isnan(rx)) 311 return (CMPLXL(fabsl(ry), rx)); 312 if (isnan(ry)) 313 return (CMPLXL(ry, ry)); 314 return (CMPLXL(fabsl(ry), copysignl(rx, cimagl(z)))); 315 } 316 317 static long double complex 318 clog_for_large_values(long double complex z) 319 { 320 long double x, y; 321 long double ax, ay, t; 322 323 x = creall(z); 324 y = cimagl(z); 325 ax = fabsl(x); 326 ay = fabsl(y); 327 if (ax < ay) { 328 t = ax; 329 ax = ay; 330 ay = t; 331 } 332 333 if (ax > LDBL_MAX / 2) 334 return (CMPLXL(logl(hypotl(x / m_e, y / m_e)) + 1, atan2l(y, x))); 335 336 if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN) 337 return (CMPLXL(logl(hypotl(x, y)), atan2l(y, x))); 338 339 return (CMPLXL(logl(ax*ax + ay*ay) / 2, atan2l(y, x))); 340 } 341 342 inline static long double 343 sum_squares(long double x, long double y) 344 { 345 if (y < SQRT_MIN) 346 return (x*x); 347 348 return (x*x + y*y); 349 } 350 351 inline static long double 352 real_part_reciprocal(long double x, long double y) 353 { 354 long double scale; 355 uint16_t hx, hy; 356 int16_t ix, iy; 357 358 GET_LDBL_EXPSIGN(hx, x); 359 ix = hx & 0x7fff; 360 GET_LDBL_EXPSIGN(hy, y); 361 iy = hy & 0x7fff; 362 #define BIAS (LDBL_MAX_EXP - 1) 363 #define CUTOFF (LDBL_MANT_DIG / 2 + 1) 364 if (ix - iy >= CUTOFF || isinf(x)) 365 return (1/x); 366 if (iy - ix >= CUTOFF) 367 return (x/y/y); 368 if (ix <= BIAS + LDBL_MAX_EXP / 2 - CUTOFF) 369 return (x/(x*x + y*y)); 370 scale = 1; 371 SET_LDBL_EXPSIGN(scale, 0x7fff - ix); 372 x *= scale; 373 y *= scale; 374 return (x/(x*x + y*y) * scale); 375 } 376 377 long double complex 378 catanhl(long double complex z) 379 { 380 long double x, y, ax, ay, rx, ry; 381 382 x = creall(z); 383 y = cimagl(z); 384 ax = fabsl(x); 385 ay = fabsl(y); 386 387 if (y == 0 && ax <= 1) 388 return (CMPLXL(atanhl(x), y)); /* XXX need atanhl() */ 389 390 if (x == 0) 391 return (CMPLXL(x, atanl(y))); 392 393 if (isnan(x) || isnan(y)) { 394 if (isinf(x)) 395 return (CMPLXL(copysignl(0, x), y+y)); 396 if (isinf(y)) 397 return (CMPLXL(copysignl(0, x), copysignl(pio2_hi + pio2_lo, y))); 398 return (CMPLXL(x+0.0L+(y+0), x+0.0L+(y+0))); 399 } 400 401 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) 402 return (CMPLXL(real_part_reciprocal(x, y), copysignl(pio2_hi + pio2_lo, y))); 403 404 if (ax < SQRT_3_EPSILON/2 && ay < SQRT_3_EPSILON/2) { 405 raise_inexact(); 406 return (z); 407 } 408 409 if (ax == 1 && ay < LDBL_EPSILON) { 410 #if 0 411 if (ay > 2*LDBL_MIN) 412 rx = - logl(ay/2) / 2; 413 else 414 #endif 415 rx = - (logl(ay) - m_ln2) / 2; 416 } else 417 rx = log1pl(4*ax / sum_squares(ax-1, ay)) / 4; 418 419 if (ax == 1) 420 ry = atan2l(2, -ay) / 2; 421 else if (ay < LDBL_EPSILON) 422 ry = atan2l(2*ay, (1-ax)*(1+ax)) / 2; 423 else 424 ry = atan2l(2*ay, (1-ax)*(1+ax) - ay*ay) / 2; 425 426 return (CMPLXL(copysignl(rx, x), copysignl(ry, y))); 427 } 428 429 long double complex 430 catanl(long double complex z) 431 { 432 long double complex w = catanhl(CMPLXL(cimagl(z), creall(z))); 433 return (CMPLXL(cimagl(w), creall(w))); 434 } 435 436 #else 437 __strong_alias(_casinl, casin) 438 __strong_alias(_catanl, catan) 439 __strong_alias(cacoshl, cacosh) 440 __strong_alias(cacosl, cacos) 441 __strong_alias(casinhl, casinh) 442 __strong_alias(catanhl, catanh) 443 #endif 444
Indexes created Tue Oct 14 00:09:49 GMT 2025