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