1 /*- 2 * SPDX-License-Identifier: BSD-2-Clause 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 <complex.h> 44 #include <float.h> 45 46 #include "math.h" 47 #include "math_private.h" 48 49 #undef isinf 50 #define isinf(x) (fabsf(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_signbitf(x)) 56 57 static const float 58 A_crossover = 10, 59 B_crossover = 0.6417, 60 FOUR_SQRT_MIN = 0x1p-61, 61 QUARTER_SQRT_MAX = 0x1p61, 62 m_e = 2.7182818285e0, /* 0xadf854.0p-22 */ 63 m_ln2 = 6.9314718056e-1, /* 0xb17218.0p-24 */ 64 pio2_hi = 1.5707962513e0, /* 0xc90fda.0p-23 */ 65 RECIP_EPSILON = 1 / FLT_EPSILON, 66 SQRT_3_EPSILON = 5.9801995673e-4, /* 0x9cc471.0p-34 */ 67 SQRT_6_EPSILON = 8.4572793338e-4, /* 0xddb3d7.0p-34 */ 68 SQRT_MIN = 0x1p-63; 69 70 static const volatile float 71 pio2_lo = 7.5497899549e-8, /* 0xa22169.0p-47 */ 72 tiny = 0x1p-100; 73 74 static float complex clog_for_large_values(float complex z); 75 76 static inline float 77 f(float a, float b, float hypot_a_b) 78 { 79 if (b < 0) 80 return ((hypot_a_b - b) / 2); 81 if (b == 0) 82 return (a / 2); 83 return (a * a / (hypot_a_b + b) / 2); 84 } 85 86 static inline void 87 do_hard_work(float x, float y, float *rx, int *B_is_usable, float *B, 88 float *sqrt_A2my2, float *new_y) 89 { 90 float R, S, A; 91 float Am1, Amy; 92 93 R = hypotf(x, y + 1); 94 S = hypotf(x, y - 1); 95 96 A = (R + S) / 2; 97 if (A < 1) 98 A = 1; 99 100 if (A < A_crossover) { 101 if (y == 1 && x < FLT_EPSILON * FLT_EPSILON / 128) { 102 *rx = sqrtf(x); 103 } else if (x >= FLT_EPSILON * fabsf(y - 1)) { 104 Am1 = f(x, 1 + y, R) + f(x, 1 - y, S); 105 *rx = log1pf(Am1 + sqrtf(Am1 * (A + 1))); 106 } else if (y < 1) { 107 *rx = x / sqrtf((1 - y) * (1 + y)); 108 } else { 109 *rx = log1pf((y - 1) + sqrtf((y - 1) * (y + 1))); 110 } 111 } else { 112 *rx = logf(A + sqrtf(A * A - 1)); 113 } 114 115 *new_y = y; 116 117 if (y < FOUR_SQRT_MIN) { 118 *B_is_usable = 0; 119 *sqrt_A2my2 = A * (2 / FLT_EPSILON); 120 *new_y = y * (2 / FLT_EPSILON); 121 return; 122 } 123 124 *B = y / A; 125 *B_is_usable = 1; 126 127 if (*B > B_crossover) { 128 *B_is_usable = 0; 129 if (y == 1 && x < FLT_EPSILON / 128) { 130 *sqrt_A2my2 = sqrtf(x) * sqrtf((A + y) / 2); 131 } else if (x >= FLT_EPSILON * fabsf(y - 1)) { 132 Amy = f(x, y + 1, R) + f(x, y - 1, S); 133 *sqrt_A2my2 = sqrtf(Amy * (A + y)); 134 } else if (y > 1) { 135 *sqrt_A2my2 = x * (4 / FLT_EPSILON / FLT_EPSILON) * y / 136 sqrtf((y + 1) * (y - 1)); 137 *new_y = y * (4 / FLT_EPSILON / FLT_EPSILON); 138 } else { 139 *sqrt_A2my2 = sqrtf((1 - y) * (1 + y)); 140 } 141 } 142 } 143 144 float complex 145 casinhf(float complex z) 146 { 147 float x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y; 148 int B_is_usable; 149 float complex w; 150 151 x = crealf(z); 152 y = cimagf(z); 153 ax = fabsf(x); 154 ay = fabsf(y); 155 156 if (isnan(x) || isnan(y)) { 157 if (isinf(x)) 158 return (CMPLXF(x, y + y)); 159 if (isinf(y)) 160 return (CMPLXF(y, x + x)); 161 if (y == 0) 162 return (CMPLXF(x + x, y)); 163 return (CMPLXF(nan_mix(x, y), nan_mix(x, y))); 164 } 165 166 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) { 167 if (signbit(x) == 0) 168 w = clog_for_large_values(z) + m_ln2; 169 else 170 w = clog_for_large_values(-z) + m_ln2; 171 return (CMPLXF(copysignf(crealf(w), x), 172 copysignf(cimagf(w), y))); 173 } 174 175 if (x == 0 && y == 0) 176 return (z); 177 178 raise_inexact(); 179 180 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4) 181 return (z); 182 183 do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y); 184 if (B_is_usable) 185 ry = asinf(B); 186 else 187 ry = atan2f(new_y, sqrt_A2my2); 188 return (CMPLXF(copysignf(rx, x), copysignf(ry, y))); 189 } 190 191 float complex 192 casinf(float complex z) 193 { 194 float complex w = casinhf(CMPLXF(cimagf(z), crealf(z))); 195 196 return (CMPLXF(cimagf(w), crealf(w))); 197 } 198 199 float complex 200 cacosf(float complex z) 201 { 202 float x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x; 203 int sx, sy; 204 int B_is_usable; 205 float complex w; 206 207 x = crealf(z); 208 y = cimagf(z); 209 sx = signbit(x); 210 sy = signbit(y); 211 ax = fabsf(x); 212 ay = fabsf(y); 213 214 if (isnan(x) || isnan(y)) { 215 if (isinf(x)) 216 return (CMPLXF(y + y, -INFINITY)); 217 if (isinf(y)) 218 return (CMPLXF(x + x, -y)); 219 if (x == 0) 220 return (CMPLXF(pio2_hi + pio2_lo, y + y)); 221 return (CMPLXF(nan_mix(x, y), nan_mix(x, y))); 222 } 223 224 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) { 225 w = clog_for_large_values(z); 226 rx = fabsf(cimagf(w)); 227 ry = crealf(w) + m_ln2; 228 if (sy == 0) 229 ry = -ry; 230 return (CMPLXF(rx, ry)); 231 } 232 233 if (x == 1 && y == 0) 234 return (CMPLXF(0, -y)); 235 236 raise_inexact(); 237 238 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4) 239 return (CMPLXF(pio2_hi - (x - pio2_lo), -y)); 240 241 do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x); 242 if (B_is_usable) { 243 if (sx == 0) 244 rx = acosf(B); 245 else 246 rx = acosf(-B); 247 } else { 248 if (sx == 0) 249 rx = atan2f(sqrt_A2mx2, new_x); 250 else 251 rx = atan2f(sqrt_A2mx2, -new_x); 252 } 253 if (sy == 0) 254 ry = -ry; 255 return (CMPLXF(rx, ry)); 256 } 257 258 float complex 259 cacoshf(float complex z) 260 { 261 float complex w; 262 float rx, ry; 263 264 w = cacosf(z); 265 rx = crealf(w); 266 ry = cimagf(w); 267 if (isnan(rx) && isnan(ry)) 268 return (CMPLXF(ry, rx)); 269 if (isnan(rx)) 270 return (CMPLXF(fabsf(ry), rx)); 271 if (isnan(ry)) 272 return (CMPLXF(ry, ry)); 273 return (CMPLXF(fabsf(ry), copysignf(rx, cimagf(z)))); 274 } 275 276 static float complex 277 clog_for_large_values(float complex z) 278 { 279 float x, y; 280 float ax, ay, t; 281 282 x = crealf(z); 283 y = cimagf(z); 284 ax = fabsf(x); 285 ay = fabsf(y); 286 if (ax < ay) { 287 t = ax; 288 ax = ay; 289 ay = t; 290 } 291 292 if (ax > FLT_MAX / 2) 293 return (CMPLXF(logf(hypotf(x / m_e, y / m_e)) + 1, 294 atan2f(y, x))); 295 296 if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN) 297 return (CMPLXF(logf(hypotf(x, y)), atan2f(y, x))); 298 299 return (CMPLXF(logf(ax * ax + ay * ay) / 2, atan2f(y, x))); 300 } 301 302 static inline float 303 sum_squares(float x, float y) 304 { 305 306 if (y < SQRT_MIN) 307 return (x * x); 308 309 return (x * x + y * y); 310 } 311 312 static inline float 313 real_part_reciprocal(float x, float y) 314 { 315 float scale; 316 uint32_t hx, hy; 317 int32_t ix, iy; 318 319 GET_FLOAT_WORD(hx, x); 320 ix = hx & 0x7f800000; 321 GET_FLOAT_WORD(hy, y); 322 iy = hy & 0x7f800000; 323 #define BIAS (FLT_MAX_EXP - 1) 324 #define CUTOFF (FLT_MANT_DIG / 2 + 1) 325 if (ix - iy >= CUTOFF << 23 || isinf(x)) 326 return (1 / x); 327 if (iy - ix >= CUTOFF << 23) 328 return (x / y / y); 329 if (ix <= (BIAS + FLT_MAX_EXP / 2 - CUTOFF) << 23) 330 return (x / (x * x + y * y)); 331 SET_FLOAT_WORD(scale, 0x7f800000 - ix); 332 x *= scale; 333 y *= scale; 334 return (x / (x * x + y * y) * scale); 335 } 336 337 float complex 338 catanhf(float complex z) 339 { 340 float x, y, ax, ay, rx, ry; 341 342 x = crealf(z); 343 y = cimagf(z); 344 ax = fabsf(x); 345 ay = fabsf(y); 346 347 if (y == 0 && ax <= 1) 348 return (CMPLXF(atanhf(x), y)); 349 350 if (x == 0) 351 return (CMPLXF(x, atanf(y))); 352 353 if (isnan(x) || isnan(y)) { 354 if (isinf(x)) 355 return (CMPLXF(copysignf(0, x), y + y)); 356 if (isinf(y)) 357 return (CMPLXF(copysignf(0, x), 358 copysignf(pio2_hi + pio2_lo, y))); 359 return (CMPLXF(nan_mix(x, y), nan_mix(x, y))); 360 } 361 362 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) 363 return (CMPLXF(real_part_reciprocal(x, y), 364 copysignf(pio2_hi + pio2_lo, y))); 365 366 if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) { 367 raise_inexact(); 368 return (z); 369 } 370 371 if (ax == 1 && ay < FLT_EPSILON) 372 rx = (m_ln2 - logf(ay)) / 2; 373 else 374 rx = log1pf(4 * ax / sum_squares(ax - 1, ay)) / 4; 375 376 if (ax == 1) 377 ry = atan2f(2, -ay) / 2; 378 else if (ay < FLT_EPSILON) 379 ry = atan2f(2 * ay, (1 - ax) * (1 + ax)) / 2; 380 else 381 ry = atan2f(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2; 382 383 return (CMPLXF(copysignf(rx, x), copysignf(ry, y))); 384 } 385 386 float complex 387 catanf(float complex z) 388 { 389 float complex w = catanhf(CMPLXF(cimagf(z), crealf(z))); 390 391 return (CMPLXF(cimagf(w), crealf(w))); 392 } 393