1 /* 2 * CDDL HEADER START 3 * 4 * The contents of this file are subject to the terms of the 5 * Common Development and Distribution License (the "License"). 6 * You may not use this file except in compliance with the License. 7 * 8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE 9 * or http://www.opensolaris.org/os/licensing. 10 * See the License for the specific language governing permissions 11 * and limitations under the License. 12 * 13 * When distributing Covered Code, include this CDDL HEADER in each 14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE. 15 * If applicable, add the following below this CDDL HEADER, with the 16 * fields enclosed by brackets "[]" replaced with your own identifying 17 * information: Portions Copyright [yyyy] [name of copyright owner] 18 * 19 * CDDL HEADER END 20 */ 21 22 /* 23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved. 24 */ 25 /* 26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved. 27 * Use is subject to license terms. 28 */ 29 30 #pragma weak __cacos = cacos 31 32 /* INDENT OFF */ 33 /* 34 * dcomplex cacos(dcomplex z); 35 * 36 * Alogrithm 37 * (based on T.E.Hull, Thomas F. Fairgrieve and Ping Tak Peter Tang's 38 * paper "Implementing the Complex Arcsine and Arccosine Functins Using 39 * Exception Handling", ACM TOMS, Vol 23, pp 299-335) 40 * 41 * The principal value of complex inverse cosine function cacos(z), 42 * where z = x+iy, can be defined by 43 * 44 * cacos(z) = acos(B) - i sign(y) log (A + sqrt(A*A-1)), 45 * 46 * where the log function is the natural log, and 47 * ____________ ____________ 48 * 1 / 2 2 1 / 2 2 49 * A = --- / (x+1) + y + --- / (x-1) + y 50 * 2 \/ 2 \/ 51 * ____________ ____________ 52 * 1 / 2 2 1 / 2 2 53 * B = --- / (x+1) + y - --- / (x-1) + y . 54 * 2 \/ 2 \/ 55 * 56 * The Branch cuts are on the real line from -inf to -1 and from 1 to inf. 57 * The real and imaginary parts are based on Abramowitz and Stegun 58 * [Handbook of Mathematic Functions, 1972]. The sign of the imaginary 59 * part is chosen to be the generally considered the principal value of 60 * this function. 61 * 62 * Notes:1. A is the average of the distances from z to the points (1,0) 63 * and (-1,0) in the complex z-plane, and in particular A>=1. 64 * 2. B is in [-1,1], and A*B = x 65 * 66 * Basic relations 67 * cacos(conj(z)) = conj(cacos(z)) 68 * cacos(-z) = pi - cacos(z) 69 * cacos( z) = pi/2 - casin(z) 70 * 71 * Special cases (conform to ISO/IEC 9899:1999(E)): 72 * cacos(+-0 + i y ) = pi/2 - i y for y is +-0, +-inf, NaN 73 * cacos( x + i inf) = pi/2 - i inf for all x 74 * cacos( x + i NaN) = NaN + i NaN with invalid for non-zero finite x 75 * cacos(-inf + i y ) = pi - i inf for finite +y 76 * cacos( inf + i y ) = 0 - i inf for finite +y 77 * cacos(-inf + i inf) = 3pi/4- i inf 78 * cacos( inf + i inf) = pi/4 - i inf 79 * cacos(+-inf+ i NaN) = NaN - i inf (sign of imaginary is unspecified) 80 * cacos(NaN + i y ) = NaN + i NaN with invalid for finite y 81 * cacos(NaN + i inf) = NaN - i inf 82 * cacos(NaN + i NaN) = NaN + i NaN 83 * 84 * Special Regions (better formula for accuracy and for avoiding spurious 85 * overflow or underflow) (all x and y are assumed nonnegative): 86 * case 1: y = 0 87 * case 2: tiny y relative to x-1: y <= ulp(0.5)*|x-1| 88 * case 3: tiny y: y < 4 sqrt(u), where u = minimum normal number 89 * case 4: huge y relative to x+1: y >= (1+x)/ulp(0.5) 90 * case 5: huge x and y: x and y >= sqrt(M)/8, where M = maximum normal number 91 * case 6: tiny x: x < 4 sqrt(u) 92 * -------- 93 * case 1 & 2. y=0 or y/|x-1| is tiny. We have 94 * ____________ _____________ 95 * / 2 2 / y 2 96 * / (x+-1) + y = |x+-1| / 1 + (------) 97 * \/ \/ |x+-1| 98 * 99 * 1 y 2 100 * ~ |x+-1| ( 1 + --- (------) ) 101 * 2 |x+-1| 102 * 103 * 2 104 * y 105 * = |x+-1| + --------. 106 * 2|x+-1| 107 * 108 * Consequently, it is not difficult to see that 109 * 2 110 * y 111 * [ 1 + ------------ , if x < 1, 112 * [ 2(1+x)(1-x) 113 * [ 114 * [ 115 * [ x, if x = 1 (y = 0), 116 * [ 117 * A ~= [ 2 118 * [ x * y 119 * [ x + ------------ ~ x, if x > 1 120 * [ 2(x+1)(x-1) 121 * 122 * and hence 123 * ______ 2 124 * / 2 y y 125 * A + \/ A - 1 ~ 1 + ---------------- + -----------, if x < 1, 126 * sqrt((x+1)(1-x)) 2(x+1)(1-x) 127 * 128 * 129 * ~ x + sqrt((x-1)*(x+1)), if x >= 1. 130 * 131 * 2 132 * y 133 * [ x(1 - -----------) ~ x, if x < 1, 134 * [ 2(1+x)(1-x) 135 * B = x/A ~ [ 136 * [ 1, if x = 1, 137 * [ 138 * [ 2 139 * [ y 140 * [ 1 - ------------ , if x > 1, 141 * [ 2(x+1)(x-1) 142 * Thus 143 * [ acos(x) - i y/sqrt((x-1)*(x+1)), if x < 1, 144 * [ 145 * cacos(x+i*y)~ [ 0 - i 0, if x = 1, 146 * [ 147 * [ y/sqrt(x*x-1) - i log(x+sqrt(x*x-1)), if x > 1. 148 * 149 * Note: y/sqrt(x*x-1) ~ y/x when x >= 2**26. 150 * case 3. y < 4 sqrt(u), where u = minimum normal x. 151 * After case 1 and 2, this will only occurs when x=1. When x=1, we have 152 * A = (sqrt(4+y*y)+y)/2 ~ 1 + y/2 + y^2/8 + ... 153 * and 154 * B = 1/A = 1 - y/2 + y^2/8 + ... 155 * Since 156 * cos(sqrt(y)) ~ 1 - y/2 + ... 157 * we have, for the real part, 158 * acos(B) ~ acos(1 - y/2) ~ sqrt(y) 159 * For the imaginary part, 160 * log(A+sqrt(A*A-1)) ~ log(1+y/2+sqrt(2*y/2)) 161 * = log(1+y/2+sqrt(y)) 162 * = (y/2+sqrt(y)) - (y/2+sqrt(y))^2/2 + ... 163 * ~ sqrt(y) - y*(sqrt(y)+y/2)/2 164 * ~ sqrt(y) 165 * 166 * case 4. y >= (x+1)/ulp(0.5). In this case, A ~ y and B ~ x/y. Thus 167 * real part = acos(B) ~ pi/2 168 * and 169 * imag part = log(y+sqrt(y*y-one)) 170 * 171 * case 5. Both x and y are large: x and y > sqrt(M)/8, where M = maximum x 172 * In this case, 173 * A ~ sqrt(x*x+y*y) 174 * B ~ x/sqrt(x*x+y*y). 175 * Thus 176 * real part = acos(B) = atan(y/x), 177 * imag part = log(A+sqrt(A*A-1)) ~ log(2A) 178 * = log(2) + 0.5*log(x*x+y*y) 179 * = log(2) + log(y) + 0.5*log(1+(x/y)^2) 180 * 181 * case 6. x < 4 sqrt(u). In this case, we have 182 * A ~ sqrt(1+y*y), B = x/sqrt(1+y*y). 183 * Since B is tiny, we have 184 * real part = acos(B) ~ pi/2 185 * imag part = log(A+sqrt(A*A-1)) = log (A+sqrt(y*y)) 186 * = log(y+sqrt(1+y*y)) 187 * = 0.5*log(y^2+2ysqrt(1+y^2)+1+y^2) 188 * = 0.5*log(1+2y(y+sqrt(1+y^2))); 189 * = 0.5*log1p(2y(y+A)); 190 * 191 * cacos(z) = acos(B) - i sign(y) log (A + sqrt(A*A-1)), 192 */ 193 /* INDENT ON */ 194 195 #include "libm.h" 196 #include "complex_wrapper.h" 197 198 /* INDENT OFF */ 199 static const double 200 zero = 0.0, 201 one = 1.0, 202 E = 1.11022302462515654042e-16, /* 2**-53 */ 203 ln2 = 6.93147180559945286227e-01, 204 pi = 3.1415926535897931159979634685, 205 pi_l = 1.224646799147353177e-16, 206 pi_2 = 1.570796326794896558e+00, 207 pi_2_l = 6.123233995736765886e-17, 208 pi_4 = 0.78539816339744827899949, 209 pi_4_l = 3.061616997868382943e-17, 210 pi3_4 = 2.356194490192344836998, 211 pi3_4_l = 9.184850993605148829195e-17, 212 Foursqrtu = 5.96667258496016539463e-154, /* 2**(-509) */ 213 Acrossover = 1.5, 214 Bcrossover = 0.6417, 215 half = 0.5; 216 /* INDENT ON */ 217 218 dcomplex 219 cacos(dcomplex z) { 220 double x, y, t, R, S, A, Am1, B, y2, xm1, xp1, Apx; 221 int ix, iy, hx, hy; 222 unsigned lx, ly; 223 dcomplex ans; 224 225 x = D_RE(z); 226 y = D_IM(z); 227 hx = HI_WORD(x); 228 lx = LO_WORD(x); 229 hy = HI_WORD(y); 230 ly = LO_WORD(y); 231 ix = hx & 0x7fffffff; 232 iy = hy & 0x7fffffff; 233 234 /* x is 0 */ 235 if ((ix | lx) == 0) { 236 if (((iy | ly) == 0) || (iy >= 0x7ff00000)) { 237 D_RE(ans) = pi_2; 238 D_IM(ans) = -y; 239 return (ans); 240 } 241 } 242 243 /* |y| is inf or NaN */ 244 if (iy >= 0x7ff00000) { 245 if (ISINF(iy, ly)) { /* cacos(x + i inf) = pi/2 - i inf */ 246 D_IM(ans) = -y; 247 if (ix < 0x7ff00000) { 248 D_RE(ans) = pi_2 + pi_2_l; 249 } else if (ISINF(ix, lx)) { 250 if (hx >= 0) 251 D_RE(ans) = pi_4 + pi_4_l; 252 else 253 D_RE(ans) = pi3_4 + pi3_4_l; 254 } else { 255 D_RE(ans) = x; 256 } 257 } else { /* cacos(x + i NaN) = NaN + i NaN */ 258 D_RE(ans) = y + x; 259 if (ISINF(ix, lx)) 260 D_IM(ans) = -fabs(x); 261 else 262 D_IM(ans) = y; 263 } 264 return (ans); 265 } 266 267 x = fabs(x); 268 y = fabs(y); 269 270 /* x is inf or NaN */ 271 if (ix >= 0x7ff00000) { /* x is inf or NaN */ 272 if (ISINF(ix, lx)) { /* x is INF */ 273 D_IM(ans) = -x; 274 if (iy >= 0x7ff00000) { 275 if (ISINF(iy, ly)) { 276 /* INDENT OFF */ 277 /* cacos(inf + i inf) = pi/4 - i inf */ 278 /* cacos(-inf+ i inf) =3pi/4 - i inf */ 279 /* INDENT ON */ 280 if (hx >= 0) 281 D_RE(ans) = pi_4 + pi_4_l; 282 else 283 D_RE(ans) = pi3_4 + pi3_4_l; 284 } else 285 /* INDENT OFF */ 286 /* cacos(inf + i NaN) = NaN - i inf */ 287 /* INDENT ON */ 288 D_RE(ans) = y + y; 289 } else 290 /* INDENT OFF */ 291 /* cacos(inf + iy ) = 0 - i inf */ 292 /* cacos(-inf+ iy ) = pi - i inf */ 293 /* INDENT ON */ 294 if (hx >= 0) 295 D_RE(ans) = zero; 296 else 297 D_RE(ans) = pi + pi_l; 298 } else { /* x is NaN */ 299 /* INDENT OFF */ 300 /* 301 * cacos(NaN + i inf) = NaN - i inf 302 * cacos(NaN + i y ) = NaN + i NaN 303 * cacos(NaN + i NaN) = NaN + i NaN 304 */ 305 /* INDENT ON */ 306 D_RE(ans) = x + y; 307 if (iy >= 0x7ff00000) { 308 D_IM(ans) = -y; 309 } else { 310 D_IM(ans) = x; 311 } 312 } 313 if (hy < 0) 314 D_IM(ans) = -D_IM(ans); 315 return (ans); 316 } 317 318 if ((iy | ly) == 0) { /* region 1: y=0 */ 319 if (ix < 0x3ff00000) { /* |x| < 1 */ 320 D_RE(ans) = acos(x); 321 D_IM(ans) = zero; 322 } else { 323 D_RE(ans) = zero; 324 if (ix >= 0x43500000) /* |x| >= 2**54 */ 325 D_IM(ans) = ln2 + log(x); 326 else if (ix >= 0x3ff80000) /* x > Acrossover */ 327 D_IM(ans) = log(x + sqrt((x - one) * (x + 328 one))); 329 else { 330 xm1 = x - one; 331 D_IM(ans) = log1p(xm1 + sqrt(xm1 * (x + one))); 332 } 333 } 334 } else if (y <= E * fabs(x - one)) { /* region 2: y < tiny*|x-1| */ 335 if (ix < 0x3ff00000) { /* x < 1 */ 336 D_RE(ans) = acos(x); 337 D_IM(ans) = y / sqrt((one + x) * (one - x)); 338 } else if (ix >= 0x43500000) { /* |x| >= 2**54 */ 339 D_RE(ans) = y / x; 340 D_IM(ans) = ln2 + log(x); 341 } else { 342 t = sqrt((x - one) * (x + one)); 343 D_RE(ans) = y / t; 344 if (ix >= 0x3ff80000) /* x > Acrossover */ 345 D_IM(ans) = log(x + t); 346 else 347 D_IM(ans) = log1p((x - one) + t); 348 } 349 } else if (y < Foursqrtu) { /* region 3 */ 350 t = sqrt(y); 351 D_RE(ans) = t; 352 D_IM(ans) = t; 353 } else if (E * y - one >= x) { /* region 4 */ 354 D_RE(ans) = pi_2; 355 D_IM(ans) = ln2 + log(y); 356 } else if (ix >= 0x5fc00000 || iy >= 0x5fc00000) { /* x,y>2**509 */ 357 /* region 5: x+1 or y is very large (>= sqrt(max)/8) */ 358 t = x / y; 359 D_RE(ans) = atan(y / x); 360 D_IM(ans) = ln2 + log(y) + half * log1p(t * t); 361 } else if (x < Foursqrtu) { 362 /* region 6: x is very small, < 4sqrt(min) */ 363 D_RE(ans) = pi_2; 364 A = sqrt(one + y * y); 365 if (iy >= 0x3ff80000) /* if y > Acrossover */ 366 D_IM(ans) = log(y + A); 367 else 368 D_IM(ans) = half * log1p((y + y) * (y + A)); 369 } else { /* safe region */ 370 y2 = y * y; 371 xp1 = x + one; 372 xm1 = x - one; 373 R = sqrt(xp1 * xp1 + y2); 374 S = sqrt(xm1 * xm1 + y2); 375 A = half * (R + S); 376 B = x / A; 377 if (B <= Bcrossover) 378 D_RE(ans) = acos(B); 379 else { /* use atan and an accurate approx to a-x */ 380 Apx = A + x; 381 if (x <= one) 382 D_RE(ans) = atan(sqrt(half * Apx * (y2 / (R + 383 xp1) + (S - xm1))) / x); 384 else 385 D_RE(ans) = atan((y * sqrt(half * (Apx / (R + 386 xp1) + Apx / (S + xm1)))) / x); 387 } 388 if (A <= Acrossover) { 389 /* use log1p and an accurate approx to A-1 */ 390 if (x < one) 391 Am1 = half * (y2 / (R + xp1) + y2 / (S - xm1)); 392 else 393 Am1 = half * (y2 / (R + xp1) + (S + xm1)); 394 D_IM(ans) = log1p(Am1 + sqrt(Am1 * (A + one))); 395 } else { 396 D_IM(ans) = log(A + sqrt(A * A - one)); 397 } 398 } 399 if (hx < 0) 400 D_RE(ans) = pi - D_RE(ans); 401 if (hy >= 0) 402 D_IM(ans) = -D_IM(ans); 403 return (ans); 404 } 405