1 /*- 2 * SPDX-License-Identifier: BSD-2-Clause-FreeBSD 3 * 4 * Copyright (c) 2012 Stephen Montgomery-Smith <stephen@FreeBSD.ORG> 5 * All rights reserved. 6 * 7 * Redistribution and use in source and binary forms, with or without 8 * modification, are permitted provided that the following conditions 9 * are met: 10 * 1. Redistributions of source code must retain the above copyright 11 * notice, this list of conditions and the following disclaimer. 12 * 2. Redistributions in binary form must reproduce the above copyright 13 * notice, this list of conditions and the following disclaimer in the 14 * documentation and/or other materials provided with the distribution. 15 * 16 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND 17 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 18 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 19 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 20 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 21 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 22 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 23 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 24 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 25 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 26 * SUCH DAMAGE. 27 */ 28 29 /* 30 * The algorithm is very close to that in "Implementing the complex arcsine 31 * and arccosine functions using exception handling" by T. E. Hull, Thomas F. 32 * Fairgrieve, and Ping Tak Peter Tang, published in ACM Transactions on 33 * Mathematical Software, Volume 23 Issue 3, 1997, Pages 299-335, 34 * http://dl.acm.org/citation.cfm?id=275324. 35 * 36 * See catrig.c for complete comments. 37 * 38 * XXX comments were removed automatically, and even short ones on the right 39 * of statements were removed (all of them), contrary to normal style. Only 40 * a few comments on the right of declarations remain. 41 */ 42 43 #include <sys/cdefs.h> 44 __FBSDID("$FreeBSD$"); 45 46 #include <complex.h> 47 #include <float.h> 48 49 #include "math.h" 50 #include "math_private.h" 51 52 #undef isinf 53 #define isinf(x) (fabsf(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_signbitf(x)) 59 60 static const float 61 A_crossover = 10, 62 B_crossover = 0.6417, 63 FOUR_SQRT_MIN = 0x1p-61, 64 QUARTER_SQRT_MAX = 0x1p61, 65 m_e = 2.7182818285e0, /* 0xadf854.0p-22 */ 66 m_ln2 = 6.9314718056e-1, /* 0xb17218.0p-24 */ 67 pio2_hi = 1.5707962513e0, /* 0xc90fda.0p-23 */ 68 RECIP_EPSILON = 1 / FLT_EPSILON, 69 SQRT_3_EPSILON = 5.9801995673e-4, /* 0x9cc471.0p-34 */ 70 SQRT_6_EPSILON = 8.4572793338e-4, /* 0xddb3d7.0p-34 */ 71 SQRT_MIN = 0x1p-63; 72 73 static const volatile float 74 pio2_lo = 7.5497899549e-8, /* 0xa22169.0p-47 */ 75 tiny = 0x1p-100; 76 77 static float complex clog_for_large_values(float complex z); 78 79 static inline float 80 f(float a, float b, float hypot_a_b) 81 { 82 if (b < 0) 83 return ((hypot_a_b - b) / 2); 84 if (b == 0) 85 return (a / 2); 86 return (a * a / (hypot_a_b + b) / 2); 87 } 88 89 static inline void 90 do_hard_work(float x, float y, float *rx, int *B_is_usable, float *B, 91 float *sqrt_A2my2, float *new_y) 92 { 93 float R, S, A; 94 float Am1, Amy; 95 96 R = hypotf(x, y + 1); 97 S = hypotf(x, y - 1); 98 99 A = (R + S) / 2; 100 if (A < 1) 101 A = 1; 102 103 if (A < A_crossover) { 104 if (y == 1 && x < FLT_EPSILON * FLT_EPSILON / 128) { 105 *rx = sqrtf(x); 106 } else if (x >= FLT_EPSILON * fabsf(y - 1)) { 107 Am1 = f(x, 1 + y, R) + f(x, 1 - y, S); 108 *rx = log1pf(Am1 + sqrtf(Am1 * (A + 1))); 109 } else if (y < 1) { 110 *rx = x / sqrtf((1 - y) * (1 + y)); 111 } else { 112 *rx = log1pf((y - 1) + sqrtf((y - 1) * (y + 1))); 113 } 114 } else { 115 *rx = logf(A + sqrtf(A * A - 1)); 116 } 117 118 *new_y = y; 119 120 if (y < FOUR_SQRT_MIN) { 121 *B_is_usable = 0; 122 *sqrt_A2my2 = A * (2 / FLT_EPSILON); 123 *new_y = y * (2 / FLT_EPSILON); 124 return; 125 } 126 127 *B = y / A; 128 *B_is_usable = 1; 129 130 if (*B > B_crossover) { 131 *B_is_usable = 0; 132 if (y == 1 && x < FLT_EPSILON / 128) { 133 *sqrt_A2my2 = sqrtf(x) * sqrtf((A + y) / 2); 134 } else if (x >= FLT_EPSILON * fabsf(y - 1)) { 135 Amy = f(x, y + 1, R) + f(x, y - 1, S); 136 *sqrt_A2my2 = sqrtf(Amy * (A + y)); 137 } else if (y > 1) { 138 *sqrt_A2my2 = x * (4 / FLT_EPSILON / FLT_EPSILON) * y / 139 sqrtf((y + 1) * (y - 1)); 140 *new_y = y * (4 / FLT_EPSILON / FLT_EPSILON); 141 } else { 142 *sqrt_A2my2 = sqrtf((1 - y) * (1 + y)); 143 } 144 } 145 } 146 147 float complex 148 casinhf(float complex z) 149 { 150 float x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y; 151 int B_is_usable; 152 float complex w; 153 154 x = crealf(z); 155 y = cimagf(z); 156 ax = fabsf(x); 157 ay = fabsf(y); 158 159 if (isnan(x) || isnan(y)) { 160 if (isinf(x)) 161 return (CMPLXF(x, y + y)); 162 if (isinf(y)) 163 return (CMPLXF(y, x + x)); 164 if (y == 0) 165 return (CMPLXF(x + x, y)); 166 return (CMPLXF(nan_mix(x, y), nan_mix(x, y))); 167 } 168 169 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) { 170 if (signbit(x) == 0) 171 w = clog_for_large_values(z) + m_ln2; 172 else 173 w = clog_for_large_values(-z) + m_ln2; 174 return (CMPLXF(copysignf(crealf(w), x), 175 copysignf(cimagf(w), y))); 176 } 177 178 if (x == 0 && y == 0) 179 return (z); 180 181 raise_inexact(); 182 183 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4) 184 return (z); 185 186 do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y); 187 if (B_is_usable) 188 ry = asinf(B); 189 else 190 ry = atan2f(new_y, sqrt_A2my2); 191 return (CMPLXF(copysignf(rx, x), copysignf(ry, y))); 192 } 193 194 float complex 195 casinf(float complex z) 196 { 197 float complex w = casinhf(CMPLXF(cimagf(z), crealf(z))); 198 199 return (CMPLXF(cimagf(w), crealf(w))); 200 } 201 202 float complex 203 cacosf(float complex z) 204 { 205 float x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x; 206 int sx, sy; 207 int B_is_usable; 208 float complex w; 209 210 x = crealf(z); 211 y = cimagf(z); 212 sx = signbit(x); 213 sy = signbit(y); 214 ax = fabsf(x); 215 ay = fabsf(y); 216 217 if (isnan(x) || isnan(y)) { 218 if (isinf(x)) 219 return (CMPLXF(y + y, -INFINITY)); 220 if (isinf(y)) 221 return (CMPLXF(x + x, -y)); 222 if (x == 0) 223 return (CMPLXF(pio2_hi + pio2_lo, y + y)); 224 return (CMPLXF(nan_mix(x, y), nan_mix(x, y))); 225 } 226 227 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) { 228 w = clog_for_large_values(z); 229 rx = fabsf(cimagf(w)); 230 ry = crealf(w) + m_ln2; 231 if (sy == 0) 232 ry = -ry; 233 return (CMPLXF(rx, ry)); 234 } 235 236 if (x == 1 && y == 0) 237 return (CMPLXF(0, -y)); 238 239 raise_inexact(); 240 241 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4) 242 return (CMPLXF(pio2_hi - (x - pio2_lo), -y)); 243 244 do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x); 245 if (B_is_usable) { 246 if (sx == 0) 247 rx = acosf(B); 248 else 249 rx = acosf(-B); 250 } else { 251 if (sx == 0) 252 rx = atan2f(sqrt_A2mx2, new_x); 253 else 254 rx = atan2f(sqrt_A2mx2, -new_x); 255 } 256 if (sy == 0) 257 ry = -ry; 258 return (CMPLXF(rx, ry)); 259 } 260 261 float complex 262 cacoshf(float complex z) 263 { 264 float complex w; 265 float rx, ry; 266 267 w = cacosf(z); 268 rx = crealf(w); 269 ry = cimagf(w); 270 if (isnan(rx) && isnan(ry)) 271 return (CMPLXF(ry, rx)); 272 if (isnan(rx)) 273 return (CMPLXF(fabsf(ry), rx)); 274 if (isnan(ry)) 275 return (CMPLXF(ry, ry)); 276 return (CMPLXF(fabsf(ry), copysignf(rx, cimagf(z)))); 277 } 278 279 static float complex 280 clog_for_large_values(float complex z) 281 { 282 float x, y; 283 float ax, ay, t; 284 285 x = crealf(z); 286 y = cimagf(z); 287 ax = fabsf(x); 288 ay = fabsf(y); 289 if (ax < ay) { 290 t = ax; 291 ax = ay; 292 ay = t; 293 } 294 295 if (ax > FLT_MAX / 2) 296 return (CMPLXF(logf(hypotf(x / m_e, y / m_e)) + 1, 297 atan2f(y, x))); 298 299 if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN) 300 return (CMPLXF(logf(hypotf(x, y)), atan2f(y, x))); 301 302 return (CMPLXF(logf(ax * ax + ay * ay) / 2, atan2f(y, x))); 303 } 304 305 static inline float 306 sum_squares(float x, float y) 307 { 308 309 if (y < SQRT_MIN) 310 return (x * x); 311 312 return (x * x + y * y); 313 } 314 315 static inline float 316 real_part_reciprocal(float x, float y) 317 { 318 float scale; 319 uint32_t hx, hy; 320 int32_t ix, iy; 321 322 GET_FLOAT_WORD(hx, x); 323 ix = hx & 0x7f800000; 324 GET_FLOAT_WORD(hy, y); 325 iy = hy & 0x7f800000; 326 #define BIAS (FLT_MAX_EXP - 1) 327 #define CUTOFF (FLT_MANT_DIG / 2 + 1) 328 if (ix - iy >= CUTOFF << 23 || isinf(x)) 329 return (1 / x); 330 if (iy - ix >= CUTOFF << 23) 331 return (x / y / y); 332 if (ix <= (BIAS + FLT_MAX_EXP / 2 - CUTOFF) << 23) 333 return (x / (x * x + y * y)); 334 SET_FLOAT_WORD(scale, 0x7f800000 - ix); 335 x *= scale; 336 y *= scale; 337 return (x / (x * x + y * y) * scale); 338 } 339 340 float complex 341 catanhf(float complex z) 342 { 343 float x, y, ax, ay, rx, ry; 344 345 x = crealf(z); 346 y = cimagf(z); 347 ax = fabsf(x); 348 ay = fabsf(y); 349 350 if (y == 0 && ax <= 1) 351 return (CMPLXF(atanhf(x), y)); 352 353 if (x == 0) 354 return (CMPLXF(x, atanf(y))); 355 356 if (isnan(x) || isnan(y)) { 357 if (isinf(x)) 358 return (CMPLXF(copysignf(0, x), y + y)); 359 if (isinf(y)) 360 return (CMPLXF(copysignf(0, x), 361 copysignf(pio2_hi + pio2_lo, y))); 362 return (CMPLXF(nan_mix(x, y), nan_mix(x, y))); 363 } 364 365 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) 366 return (CMPLXF(real_part_reciprocal(x, y), 367 copysignf(pio2_hi + pio2_lo, y))); 368 369 if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) { 370 raise_inexact(); 371 return (z); 372 } 373 374 if (ax == 1 && ay < FLT_EPSILON) 375 rx = (m_ln2 - logf(ay)) / 2; 376 else 377 rx = log1pf(4 * ax / sum_squares(ax - 1, ay)) / 4; 378 379 if (ax == 1) 380 ry = atan2f(2, -ay) / 2; 381 else if (ay < FLT_EPSILON) 382 ry = atan2f(2 * ay, (1 - ax) * (1 + ax)) / 2; 383 else 384 ry = atan2f(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2; 385 386 return (CMPLXF(copysignf(rx, x), copysignf(ry, y))); 387 } 388 389 float complex 390 catanf(float complex z) 391 { 392 float complex w = catanhf(CMPLXF(cimagf(z), crealf(z))); 393 394 return (CMPLXF(cimagf(w), crealf(w))); 395 } 396