1 /*- 2 * Copyright (c) 2012 Stephen Montgomery-Smith <stephen@FreeBSD.ORG> 3 * Copyright (c) 2017 Mahdi Mokhtari <mmokhi@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 * See catrig.c for complete comments. 36 * 37 * XXX comments were removed automatically, and even short ones on the right 38 * of statements were removed (all of them), contrary to normal style. Only 39 * a few comments on the right of declarations remain. 40 */ 41 42 #include <sys/cdefs.h> 43 __FBSDID("$FreeBSD$"); 44 45 #include <complex.h> 46 #include <float.h> 47 48 #include "invtrig.h" 49 #include "math.h" 50 #include "math_private.h" 51 52 #undef isinf 53 #define isinf(x) (fabsl(x) == INFINITY) 54 #undef isnan 55 #define isnan(x) ((x) != (x)) 56 #define raise_inexact() do { volatile float junk __unused = 1 + tiny; } while(0) 57 #undef signbit 58 #define signbit(x) (__builtin_signbitl(x)) 59 60 #if LDBL_MAX_EXP != 0x4000 61 #error "Unsupported long double format" 62 #endif 63 64 static const long double 65 A_crossover = 10, 66 B_crossover = 0.6417, 67 FOUR_SQRT_MIN = 0x1p-8189L, 68 HALF_MAX = 0x1p16383L, 69 QUARTER_SQRT_MAX = 0x1p8189L, 70 RECIP_EPSILON = 1 / LDBL_EPSILON, 71 SQRT_MIN = 0x1p-8191L; 72 73 #if LDBL_MANT_DIG == 64 74 static const union IEEEl2bits 75 um_e = LD80C(0xadf85458a2bb4a9b, 1, 2.71828182845904523536e+0L), 76 um_ln2 = LD80C(0xb17217f7d1cf79ac, -1, 6.93147180559945309417e-1L); 77 #define m_e um_e.e 78 #define m_ln2 um_ln2.e 79 static const long double 80 /* The next 2 literals for non-i386. Misrounding them on i386 is harmless. */ 81 SQRT_3_EPSILON = 5.70316273435758915310e-10, /* 0x9cc470a0490973e8.0p-94 */ 82 SQRT_6_EPSILON = 8.06549008734932771664e-10; /* 0xddb3d742c265539e.0p-94 */ 83 #elif LDBL_MANT_DIG == 113 84 static const long double 85 m_e = 2.71828182845904523536028747135266250e0L, /* 0x15bf0a8b1457695355fb8ac404e7a.0p-111 */ 86 m_ln2 = 6.93147180559945309417232121458176568e-1L, /* 0x162e42fefa39ef35793c7673007e6.0p-113 */ 87 SQRT_3_EPSILON = 2.40370335797945490975336727199878124e-17, /* 0x1bb67ae8584caa73b25742d7078b8.0p-168 */ 88 SQRT_6_EPSILON = 3.39934988877629587239082586223300391e-17; /* 0x13988e1409212e7d0321914321a55.0p-167 */ 89 #else 90 #error "Unsupported long double format" 91 #endif 92 93 static const volatile float 94 tiny = 0x1p-100; 95 96 static long double complex clog_for_large_values(long double complex z); 97 98 static inline long double 99 f(long double a, long double b, long double hypot_a_b) 100 { 101 if (b < 0) 102 return ((hypot_a_b - b) / 2); 103 if (b == 0) 104 return (a / 2); 105 return (a * a / (hypot_a_b + b) / 2); 106 } 107 108 static inline void 109 do_hard_work(long double x, long double y, long double *rx, int *B_is_usable, 110 long double *B, long double *sqrt_A2my2, long double *new_y) 111 { 112 long double R, S, A; 113 long double Am1, Amy; 114 115 R = hypotl(x, y + 1); 116 S = hypotl(x, y - 1); 117 118 A = (R + S) / 2; 119 if (A < 1) 120 A = 1; 121 122 if (A < A_crossover) { 123 if (y == 1 && x < LDBL_EPSILON * LDBL_EPSILON / 128) { 124 *rx = sqrtl(x); 125 } else if (x >= LDBL_EPSILON * fabsl(y - 1)) { 126 Am1 = f(x, 1 + y, R) + f(x, 1 - y, S); 127 *rx = log1pl(Am1 + sqrtl(Am1 * (A + 1))); 128 } else if (y < 1) { 129 *rx = x / sqrtl((1 - y) * (1 + y)); 130 } else { 131 *rx = log1pl((y - 1) + sqrtl((y - 1) * (y + 1))); 132 } 133 } else { 134 *rx = logl(A + sqrtl(A * A - 1)); 135 } 136 137 *new_y = y; 138 139 if (y < FOUR_SQRT_MIN) { 140 *B_is_usable = 0; 141 *sqrt_A2my2 = A * (2 / LDBL_EPSILON); 142 *new_y = y * (2 / LDBL_EPSILON); 143 return; 144 } 145 146 *B = y / A; 147 *B_is_usable = 1; 148 149 if (*B > B_crossover) { 150 *B_is_usable = 0; 151 if (y == 1 && x < LDBL_EPSILON / 128) { 152 *sqrt_A2my2 = sqrtl(x) * sqrtl((A + y) / 2); 153 } else if (x >= LDBL_EPSILON * fabsl(y - 1)) { 154 Amy = f(x, y + 1, R) + f(x, y - 1, S); 155 *sqrt_A2my2 = sqrtl(Amy * (A + y)); 156 } else if (y > 1) { 157 *sqrt_A2my2 = x * (4 / LDBL_EPSILON / LDBL_EPSILON) * y / 158 sqrtl((y + 1) * (y - 1)); 159 *new_y = y * (4 / LDBL_EPSILON / LDBL_EPSILON); 160 } else { 161 *sqrt_A2my2 = sqrtl((1 - y) * (1 + y)); 162 } 163 } 164 } 165 166 long double complex 167 casinhl(long double complex z) 168 { 169 long double x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y; 170 int B_is_usable; 171 long double complex w; 172 173 x = creall(z); 174 y = cimagl(z); 175 ax = fabsl(x); 176 ay = fabsl(y); 177 178 if (isnan(x) || isnan(y)) { 179 if (isinf(x)) 180 return (CMPLXL(x, y + y)); 181 if (isinf(y)) 182 return (CMPLXL(y, x + x)); 183 if (y == 0) 184 return (CMPLXL(x + x, y)); 185 return (CMPLXL(nan_mix(x, y), nan_mix(x, y))); 186 } 187 188 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) { 189 if (signbit(x) == 0) 190 w = clog_for_large_values(z) + m_ln2; 191 else 192 w = clog_for_large_values(-z) + m_ln2; 193 return (CMPLXL(copysignl(creall(w), x), 194 copysignl(cimagl(w), y))); 195 } 196 197 if (x == 0 && y == 0) 198 return (z); 199 200 raise_inexact(); 201 202 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4) 203 return (z); 204 205 do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y); 206 if (B_is_usable) 207 ry = asinl(B); 208 else 209 ry = atan2l(new_y, sqrt_A2my2); 210 return (CMPLXL(copysignl(rx, x), copysignl(ry, y))); 211 } 212 213 long double complex 214 casinl(long double complex z) 215 { 216 long double complex w; 217 218 w = casinhl(CMPLXL(cimagl(z), creall(z))); 219 return (CMPLXL(cimagl(w), creall(w))); 220 } 221 222 long double complex 223 cacosl(long double complex z) 224 { 225 long double x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x; 226 int sx, sy; 227 int B_is_usable; 228 long double complex w; 229 230 x = creall(z); 231 y = cimagl(z); 232 sx = signbit(x); 233 sy = signbit(y); 234 ax = fabsl(x); 235 ay = fabsl(y); 236 237 if (isnan(x) || isnan(y)) { 238 if (isinf(x)) 239 return (CMPLXL(y + y, -INFINITY)); 240 if (isinf(y)) 241 return (CMPLXL(x + x, -y)); 242 if (x == 0) 243 return (CMPLXL(pio2_hi + pio2_lo, y + y)); 244 return (CMPLXL(nan_mix(x, y), nan_mix(x, y))); 245 } 246 247 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) { 248 w = clog_for_large_values(z); 249 rx = fabsl(cimagl(w)); 250 ry = creall(w) + m_ln2; 251 if (sy == 0) 252 ry = -ry; 253 return (CMPLXL(rx, ry)); 254 } 255 256 if (x == 1 && y == 0) 257 return (CMPLXL(0, -y)); 258 259 raise_inexact(); 260 261 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4) 262 return (CMPLXL(pio2_hi - (x - pio2_lo), -y)); 263 264 do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x); 265 if (B_is_usable) { 266 if (sx == 0) 267 rx = acosl(B); 268 else 269 rx = acosl(-B); 270 } else { 271 if (sx == 0) 272 rx = atan2l(sqrt_A2mx2, new_x); 273 else 274 rx = atan2l(sqrt_A2mx2, -new_x); 275 } 276 if (sy == 0) 277 ry = -ry; 278 return (CMPLXL(rx, ry)); 279 } 280 281 long double complex 282 cacoshl(long double complex z) 283 { 284 long double complex w; 285 long double rx, ry; 286 287 w = cacosl(z); 288 rx = creall(w); 289 ry = cimagl(w); 290 if (isnan(rx) && isnan(ry)) 291 return (CMPLXL(ry, rx)); 292 if (isnan(rx)) 293 return (CMPLXL(fabsl(ry), rx)); 294 if (isnan(ry)) 295 return (CMPLXL(ry, ry)); 296 return (CMPLXL(fabsl(ry), copysignl(rx, cimagl(z)))); 297 } 298 299 static long double complex 300 clog_for_large_values(long double complex z) 301 { 302 long double x, y; 303 long double ax, ay, t; 304 305 x = creall(z); 306 y = cimagl(z); 307 ax = fabsl(x); 308 ay = fabsl(y); 309 if (ax < ay) { 310 t = ax; 311 ax = ay; 312 ay = t; 313 } 314 315 if (ax > HALF_MAX) 316 return (CMPLXL(logl(hypotl(x / m_e, y / m_e)) + 1, 317 atan2l(y, x))); 318 319 if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN) 320 return (CMPLXL(logl(hypotl(x, y)), atan2l(y, x))); 321 322 return (CMPLXL(logl(ax * ax + ay * ay) / 2, atan2l(y, x))); 323 } 324 325 static inline long double 326 sum_squares(long double x, long double y) 327 { 328 329 if (y < SQRT_MIN) 330 return (x * x); 331 332 return (x * x + y * y); 333 } 334 335 static inline long double 336 real_part_reciprocal(long double x, long double y) 337 { 338 long double scale; 339 uint16_t hx, hy; 340 int16_t ix, iy; 341 342 GET_LDBL_EXPSIGN(hx, x); 343 ix = hx & 0x7fff; 344 GET_LDBL_EXPSIGN(hy, y); 345 iy = hy & 0x7fff; 346 #define BIAS (LDBL_MAX_EXP - 1) 347 #define CUTOFF (LDBL_MANT_DIG / 2 + 1) 348 if (ix - iy >= CUTOFF || isinf(x)) 349 return (1 / x); 350 if (iy - ix >= CUTOFF) 351 return (x / y / y); 352 if (ix <= BIAS + LDBL_MAX_EXP / 2 - CUTOFF) 353 return (x / (x * x + y * y)); 354 scale = 1; 355 SET_LDBL_EXPSIGN(scale, 0x7fff - ix); 356 x *= scale; 357 y *= scale; 358 return (x / (x * x + y * y) * scale); 359 } 360 361 long double complex 362 catanhl(long double complex z) 363 { 364 long double x, y, ax, ay, rx, ry; 365 366 x = creall(z); 367 y = cimagl(z); 368 ax = fabsl(x); 369 ay = fabsl(y); 370 371 if (y == 0 && ax <= 1) 372 return (CMPLXL(atanhl(x), y)); 373 374 if (x == 0) 375 return (CMPLXL(x, atanl(y))); 376 377 if (isnan(x) || isnan(y)) { 378 if (isinf(x)) 379 return (CMPLXL(copysignl(0, x), y + y)); 380 if (isinf(y)) 381 return (CMPLXL(copysignl(0, x), 382 copysignl(pio2_hi + pio2_lo, y))); 383 return (CMPLXL(nan_mix(x, y), nan_mix(x, y))); 384 } 385 386 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) 387 return (CMPLXL(real_part_reciprocal(x, y), 388 copysignl(pio2_hi + pio2_lo, y))); 389 390 if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) { 391 raise_inexact(); 392 return (z); 393 } 394 395 if (ax == 1 && ay < LDBL_EPSILON) 396 rx = (m_ln2 - logl(ay)) / 2; 397 else 398 rx = log1pl(4 * ax / sum_squares(ax - 1, ay)) / 4; 399 400 if (ax == 1) 401 ry = atan2l(2, -ay) / 2; 402 else if (ay < LDBL_EPSILON) 403 ry = atan2l(2 * ay, (1 - ax) * (1 + ax)) / 2; 404 else 405 ry = atan2l(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2; 406 407 return (CMPLXL(copysignl(rx, x), copysignl(ry, y))); 408 } 409 410 long double complex 411 catanl(long double complex z) 412 { 413 long double complex w; 414 415 w = catanhl(CMPLXL(cimagl(z), creall(z))); 416 return (CMPLXL(cimagl(w), creall(w))); 417 } 418