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 __cpow = cpow
  31 
  32 /* INDENT OFF */
  33 /*
  34  * dcomplex cpow(dcomplex z);
  35  *
  36  * z**w analytically equivalent to
  37  *
  38  * cpow(z,w) = cexp(w clog(z))
  39  *
  40  * Let z = x+iy, w = u+iv.
  41  * Since
  42  *                        _________
  43  *                       / 2    2            -1   y
  44  *     log(x+iy) = log(\/ x  + y    ) + i tan   (---)
  45  *                                                x
  46  *
  47  *                  1       2    2         -1   y
  48  *               = --- log(x  + y ) + i tan   (---)
  49  *                  2                           x
  50  *                       u       2    2         -1  y
  51  * (u+iv)* log(x+iy) =  --- log(x  + y ) - v tan  (---)  +          (1)
  52  *                       2                          x
  53  *
  54  *                            v       2    2         -1  y
  55  *                     i * [ --- log(x  + y ) + u tan  (---) ]      (2)
  56  *                            2                          x
  57  *
  58  *                   = r + i q
  59  *
  60  * Therefore,
  61  *      w     r+iq    r
  62  *     z  =  e     = e  (cos(q)+i*sin(q))
  63  *                                   _______
  64  *                                  / 2   2
  65  *       r                        \/ x + y     -v*atan2(y,x)
  66  * Here e  can be expressed as:  u          * e
  67  *
  68  * Special cases (in the order of appearance):
  69  *      1.  (anything) ** 0  is 1
  70  *      2.  (anything) ** 1  is itself
  71  *      3.  When v = 0, y = 0:
  72  *            If x is finite and negative, and u is finite, then
  73  *               x ** u = exp(u*pi i) * pow(|x|, u);
  74  *            otherwise,
  75  *               x ** u = pow(x, u);
  76  *      4.  When v = 0, x = 0 or |x| = |y| or x is inf or y is inf:
  77  *               (x + y i) ** u = r * exp(q i)
  78  *          where
  79  *               r = hypot(x,y) ** u
  80  *               q = u * atan2pi(y, x)
  81  *
  82  *      5.  otherwise, z**w is NAN if any x, y, u, v is a Nan or inf
  83  *
  84  *      Note: many results of special cases are obtained in terms of
  85  *      polar coordinate. In the conversion from polar to rectangle:
  86  *                  r exp(q i) = r * cos(q) + r * sin(q) i,
  87  *      we regard r * 0 is 0 except when r is a NaN.
  88  */
  89 /* INDENT ON */
  90 
  91 #include "libm.h"       /* atan2/exp/fabs/hypot/log/pow/scalbn */
  92                         /* atan2pi/exp2/sincos/sincospi/__k_clog_r/__k_atan2 */
  93 #include "complex_wrapper.h"
  94 
  95 extern void sincospi(double, double *, double *);
  96 
  97 static const double
  98         huge = 1e300,
  99         tiny = 1e-300,
 100         invln2 = 1.44269504088896338700e+00,
 101         ln2hi = 6.93147180369123816490e-01,   /* 0x3fe62e42, 0xfee00000 */
 102         ln2lo = 1.90821492927058770002e-10,   /* 0x3dea39ef, 0x35793c76 */
 103         one = 1.0,
 104         zero = 0.0;
 105 
 106 static const int hiinf = 0x7ff00000;
 107 extern double atan2pi(double, double);
 108 
 109 /*
 110  * Assuming |t[0]| > |t[1]| and |t[2]| > |t[3]|, sum4fp subroutine
 111  * compute t[0] + t[1] + t[2] + t[3] into two double fp numbers.
 112  */
 113 static double
 114 sum4fp(double ta[], double *w) {
 115         double t1, t2, t3, t4, w1, w2, t;
 116         t1 = ta[0]; t2 = ta[1]; t3 = ta[2]; t4 = ta[3];
 117         /*
 118          * Rearrange ti so that |t1| >= |t2| >= |t3| >= |t4|
 119          */
 120         if (fabs(t4) > fabs(t1)) {
 121                 t = t1; t1 = t3; t3 = t;
 122                 t = t2; t2 = t4; t4 = t;
 123         } else if (fabs(t3) > fabs(t1)) {
 124                 t = t1; t1 = t3;
 125                 if (fabs(t4) > fabs(t2)) {
 126                         t3 = t4; t4 = t2; t2 = t;
 127                 } else {
 128                         t3 = t2; t2 = t;
 129                 }
 130         } else if (fabs(t3) > fabs(t2)) {
 131                 t = t2; t2 = t3;
 132                 if (fabs(t4) > fabs(t2)) {
 133                         t3 = t4; t4 = t;
 134                 } else
 135                         t3 = t;
 136         }
 137         /* summing r = t1 + t2 + t3 + t4 to w1 + w2 */
 138         w1 = t3 + t4;
 139         w2 = t4 - (w1 - t3);
 140         t  = t2 + w1;
 141         w2 += w1 - (t - t2);
 142         w1 = t + w2;
 143         w2 += t - w1;
 144         t  = t1 + w1;
 145         w2 += w1 - (t - t1);
 146         w1 = t + w2;
 147         *w = w2 - (w1 - t);
 148         return (w1);
 149 }
 150 
 151 dcomplex
 152 cpow(dcomplex z, dcomplex w) {
 153         dcomplex ans;
 154         double x, y, u, v, t, c, s, r, x2, y2;
 155         double b[4], t1, t2, t3, t4, w1, w2, u1, v1, x1, y1;
 156         int ix, iy, hx, lx, hy, ly, hv, hu, iu, iv, lu, lv;
 157         int i, j, k;
 158 
 159         x = D_RE(z);
 160         y = D_IM(z);
 161         u = D_RE(w);
 162         v = D_IM(w);
 163         hx = ((int *) &x)[HIWORD];
 164         lx = ((int *) &x)[LOWORD];
 165         hy = ((int *) &y)[HIWORD];
 166         ly = ((int *) &y)[LOWORD];
 167         hu = ((int *) &u)[HIWORD];
 168         lu = ((int *) &u)[LOWORD];
 169         hv = ((int *) &v)[HIWORD];
 170         lv = ((int *) &v)[LOWORD];
 171         ix = hx & 0x7fffffff;
 172         iy = hy & 0x7fffffff;
 173         iu = hu & 0x7fffffff;
 174         iv = hv & 0x7fffffff;
 175 
 176         j = 0;
 177         if ((iv | lv) == 0) {   /* z**(real) */
 178                 if (((hu - 0x3ff00000) | lu) == 0) {    /* z ** 1 = z */
 179                         D_RE(ans) = x;
 180                         D_IM(ans) = y;
 181                 } else if ((iu | lu) == 0) {    /* z ** 0 = 1 */
 182                         D_RE(ans) = one;
 183                         D_IM(ans) = zero;
 184                 } else if ((iy | ly) == 0) {    /* (real)**(real) */
 185                         D_IM(ans) = zero;
 186                         if (hx < 0 && ix < hiinf && iu < hiinf) {
 187                                 /* -x ** u  is exp(i*pi*u)*pow(x,u) */
 188                                 r = pow(-x, u);
 189                                 sincospi(u, &s, &c);
 190                                 D_RE(ans) = (c == zero)? c: c * r;
 191                                 D_IM(ans) = (s == zero)? s: s * r;
 192                         } else
 193                                 D_RE(ans) = pow(x, u);
 194                 } else if (((ix | lx) == 0) || ix >= hiinf || iy >= hiinf) {
 195                         if (isnan(x) || isnan(y) || isnan(u))
 196                                 D_RE(ans) = D_IM(ans) = x + y + u;
 197                         else {
 198                                 if ((ix | lx) == 0)
 199                                         r = fabs(y);
 200                                 else
 201                                         r = fabs(x) + fabs(y);
 202                                 t = atan2pi(y, x);
 203                                 sincospi(t * u, &s, &c);
 204                                 D_RE(ans) = (c == zero)? c: c * r;
 205                                 D_IM(ans) = (s == zero)? s: s * r;
 206                         }
 207                 } else if (((ix - iy) | (lx - ly)) == 0) {   /* |x| = |y| */
 208                         if (hx >= 0) {
 209                                 t = (hy >= 0)? 0.25 : -0.25;
 210                                 sincospi(t * u, &s, &c);
 211                         } else if ((lu & 3) == 0) {
 212                                 t = (hy >= 0)? 0.75 : -0.75;
 213                                 sincospi(t * u, &s, &c);
 214                         } else {
 215                                 r = (hy >= 0)? u : -u;
 216                                 t = -0.25 * r;
 217                                 w1 = r + t;
 218                                 w2 = t - (w1 - r);
 219                                 sincospi(w1, &t1, &t2);
 220                                 sincospi(w2, &t3, &t4);
 221                                 s = t1 * t4 + t3 * t2;
 222                                 c = t2 * t4 - t1 * t3;
 223                         }
 224                         if (ix < 0x3fe00000) /* |x| < 1/2 */
 225                                 r = pow(fabs(x + x), u) * exp2(-0.5 * u);
 226                         else if (ix >= 0x3ff00000 || iu < 0x408ff800)
 227                                 /* |x| >= 1 or |u| < 1023 */
 228                                 r = pow(fabs(x), u) * exp2(0.5 * u);
 229                         else   /* special treatment */
 230                                 j = 2;
 231                         if (j == 0) {
 232                                 D_RE(ans) = (c == zero)? c: c * r;
 233                                 D_IM(ans) = (s == zero)? s: s * r;
 234                         }
 235                 } else
 236                         j = 1;
 237                 if (j == 0)
 238                         return (ans);
 239         }
 240         if (iu >= hiinf || iv >= hiinf || ix >= hiinf || iy >= hiinf) {
 241                 /*
 242                  * non-zero imag part(s) with inf component(s) yields NaN
 243                  */
 244                 t = fabs(x) + fabs(y) + fabs(u) + fabs(v);
 245                 D_RE(ans) = D_IM(ans) = t - t;
 246         } else {
 247                 k = 0;  /* no scaling */
 248                 if (iu > 0x7f000000 || iv > 0x7f000000) {
 249                         u *= .0009765625; /* scale 2**-10 to avoid overflow */
 250                         v *= .0009765625;
 251                         k = 1;  /* scale by 2**-10 */
 252                 }
 253                 /*
 254                  * Use similated higher precision arithmetic to compute:
 255                  * r = u * log(hypot(x, y)) - v * atan2(y, x)
 256                  * q = u * atan2(y, x) + v * log(hypot(x, y))
 257                  */
 258                 t1 = __k_clog_r(x, y, &t2);
 259                 t3 = __k_atan2(y, x, &t4);
 260                 x1 = t1;
 261                 y1 = t3;
 262                 u1 = u;
 263                 v1 = v;
 264                 ((int *) &u1)[LOWORD] &= 0xf8000000;
 265                 ((int *) &v1)[LOWORD] &= 0xf8000000;
 266                 ((int *) &x1)[LOWORD] &= 0xf8000000;
 267                 ((int *) &y1)[LOWORD] &= 0xf8000000;
 268                 x2 = t2 - (x1 - t1);    /* log(hypot(x,y)) = x1 + x2 */
 269                 y2 = t4 - (y1 - t3);    /* atan2(y,x) = y1 + y2 */
 270                 /* compute q = u * atan2(y, x) + v * log(hypot(x, y)) */
 271                 if (j != 2) {
 272                         b[0] = u1 * y1;
 273                         b[1] = (u - u1) * y1 + u * y2;
 274                         if (j == 1) {   /* v = 0 */
 275                                 w1 = b[0] + b[1];
 276                                 w2 = b[1] - (w1 - b[0]);
 277                         } else {
 278                                 b[2] = v1 * x1;
 279                                 b[3] = (v - v1) * x1 + v * x2;
 280                                 w1 = sum4fp(b, &w2);
 281                         }
 282                         sincos(w1, &t1, &t2);
 283                         sincos(w2, &t3, &t4);
 284                         s = t1 * t4 + t3 * t2;
 285                         c = t2 * t4 - t1 * t3;
 286                         if (k == 1)
 287                         /*
 288                          * square (cos(q) + i sin(q)) k times to get
 289                          * (cos(2^k * q + i sin(2^k * q)
 290                          */
 291                                 for (i = 0; i < 10; i++) {
 292                                         t1 = s * c;
 293                                         c = (c + s) * (c - s);
 294                                         s = t1 + t1;
 295                                 }
 296                 }
 297                 /* compute r = u * (t1, t2) - v * (t3, t4) */
 298                 b[0] = u1 * x1;
 299                 b[1] = (u - u1) * x1 + u * x2;
 300                 if (j == 1) {   /* v = 0 */
 301                         w1 = b[0] + b[1];
 302                         w2 = b[1] - (w1 - b[0]);
 303                 } else {
 304                         b[2] = -v1 * y1;
 305                         b[3] = (v1 - v) * y1 - v * y2;
 306                         w1 = sum4fp(b, &w2);
 307                 }
 308                 /* check over/underflow for exp(w1 + w2) */
 309                 if (k && fabs(w1) < 1000.0) {
 310                         w1 *= 1024; w2 *= 1024; k = 0;
 311                 }
 312                 hx = ((int *) &w1)[HIWORD];
 313                 lx = ((int *) &w1)[LOWORD];
 314                 ix = hx & 0x7fffffff;
 315                 /* compute exp(w1 + w2) */
 316                 if (ix < 0x3c900000) /* exp(tiny < 2**-54) = 1 */
 317                         r = one;
 318                 else if (ix >= 0x40880000) /* overflow/underflow */
 319                         r = (hx < 0)? tiny * tiny : huge * huge;
 320                 else {  /* compute exp(w1 + w2) */
 321                         k = (int) (invln2 * w1 + ((hx >= 0)? 0.5 : -0.5));
 322                         t1 = (double) k;
 323                         t2 = w1 - t1 * ln2hi;
 324                         t3 = w2 - t1 * ln2lo;
 325                         r = exp(t2 + t3);
 326                 }
 327                 if (c != zero) c *= r;
 328                 if (s != zero) s *= r;
 329                 if (k != 0) {
 330                         c = scalbn(c, k);
 331                         s = scalbn(s, k);
 332                 }
 333                 D_RE(ans) = c;
 334                 D_IM(ans) = s;
 335         }
 336         return (ans);
 337 }