usr/src/lib/libm/common/complex/cpowl.c

   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 __cpowl = cpowl
  31
  32 #include "libm.h"                               /* __k_clog_rl/__k_atan2l */
  33 /* atan2l/atan2pil/exp2l/expl/fabsl/hypotl/isinfl/logl/powl/sincosl/sincospil */
  34 #include "complex_wrapper.h"
  35 #include "longdouble.h"
  36
  37 #if defined(__sparc)
  38 #define HALF(x)  ((int *) &x)[3] = 0; ((int *) &x)[2] &= 0xfe000000
  39 #define LAST(x)  ((int *) &x)[3]
  40 #elif defined(__x86)
  41 #define HALF(x)  ((int *) &x)[0] = 0
  42 #define LAST(x)  ((int *) &x)[0]
  43 #endif
  44
  45 /* INDENT OFF */
  46 static const int hiinf = 0x7fff0000;
  47 static const long double
  48         tiny = 1.0e-4000L,
  49         huge = 1.0e4000L,
  50 #if defined(__x86)
  51                 /* 43 significant bits, 21 trailing zeros */
  52         ln2hil = 0.693147180559890330187045037746429443359375L,
  53         ln2lol = 5.497923018708371174712471612513436025525412068e-14L,
  54 #else   /* sparc */
  55                 /* 0x3FF962E4 2FEFA39E F35793C7 00000000 */
  56         ln2hil = 0.693147180559945309417231592858066493070671489074L,
  57         ln2lol = 5.28600110075004828645286235820646730106802446566153e-25L,
  58 #endif
  59         invln2  = 1.442695040888963407359924681001892137427e+0000L,
  60         one = 1.0L,
  61         zero = 0.0L;
  62 /* INDENT ON */
  63
  64 /*
  65  * Assuming |t[0]| > |t[1]| and |t[2]| > |t[3]|, sum4fpl subroutine
  66  * compute t[0] + t[1] + t[2] + t[3] into two long double fp numbers.
  67  */
  68 static long double sum4fpl(long double ta[], long double *w)
  69 {
  70         long double t1, t2, t3, t4, w1, w2, t;
  71         t1 = ta[0]; t2 = ta[1]; t3 = ta[2]; t4 = ta[3];
  72         /*
  73          * Rearrange ti so that |t1| >= |t2| >= |t3| >= |t4|
  74          */
  75         if (fabsl(t4) > fabsl(t1)) {
  76                 t = t1; t1 = t3; t3 = t;
  77                 t = t2; t2 = t4; t4 = t;
  78         } else if (fabsl(t3) > fabsl(t1)) {
  79                 t = t1; t1 = t3;
  80                 if (fabsl(t4) > fabsl(t2)) {
  81                         t3 = t4; t4 = t2; t2 = t;
  82                 } else {
  83                         t3 = t2; t2 = t;
  84                 }
  85         } else if (fabsl(t3) > fabsl(t2)) {
  86                 t = t2; t2 = t3;
  87                 if (fabsl(t4) > fabsl(t2)) {
  88                         t3 = t4; t4 = t;
  89                 } else
  90                         t3 = t;
  91         }
  92         /* summing r = t1 + t2 + t3 + t4 to w1 + w2 */
  93         w1 = t3 + t4;
  94         w2 = t4 - (w1 - t3);
  95         t  = t2 + w1;
  96         w2 += w1 - (t - t2);
  97         w1 = t + w2;
  98         w2 += t - w1;
  99         t  = t1 + w1;
 100         w2 += w1 - (t - t1);
 101         w1 = t + w2;
 102         *w = w2 - (w1 - t);
 103         return (w1);
 104 }
 105
 106 ldcomplex
 107 cpowl(ldcomplex z, ldcomplex w) {
 108         ldcomplex ans;
 109         long double x, y, u, v, t, c, s, r;
 110         long double t1, t2, t3, t4, x1, x2, y1, y2, u1, v1, b[4], w1, w2;
 111         int ix, iy, hx, hy, hv, hu, iu, iv, i, j, k;
 112
 113         x = LD_RE(z);
 114         y = LD_IM(z);
 115         u = LD_RE(w);
 116         v = LD_IM(w);
 117         hx = HI_XWORD(x);
 118         hy = HI_XWORD(y);
 119         hu = HI_XWORD(u);
 120         hv = HI_XWORD(v);
 121         ix = hx & 0x7fffffff;
 122         iy = hy & 0x7fffffff;
 123         iu = hu & 0x7fffffff;
 124         iv = hv & 0x7fffffff;
 125
 126         j = 0;
 127         if (v == zero) {        /* z**(real) */
 128                 if (u == one) { /* (anything) ** 1  is itself */
 129                         LD_RE(ans) = x;
 130                         LD_IM(ans) = y;
 131                 } else if (u == zero) { /* (anything) ** 0  is 1 */
 132                         LD_RE(ans) = one;
 133                         LD_IM(ans) = zero;
 134                 } else if (y == zero) { /* real ** real */
 135                         LD_IM(ans) = zero;
 136                         if (hx < 0 && ix < hiinf && iu < hiinf) {
 137                         /* -x ** u  is exp(i*pi*u)*pow(x,u) */
 138                                 r = powl(-x, u);
 139                                 sincospil(u, &s, &c);
 140                                 LD_RE(ans) = (c == zero)? c: c * r;
 141                                 LD_IM(ans) = (s == zero)? s: s * r;
 142                         } else
 143                                 LD_RE(ans) = powl(x, u);
 144                 } else if (x == zero || ix >= hiinf || iy >= hiinf) {
 145                         if (isnanl(x) || isnanl(y) || isnanl(u))
 146                                 LD_RE(ans) = LD_IM(ans) = x + y + u;
 147                         else {
 148                                 if (x == zero)
 149                                         r = fabsl(y);
 150                                 else
 151                                         r = fabsl(x) + fabsl(y);
 152                                 t = atan2pil(y, x);
 153                                 sincospil(t * u, &s, &c);
 154                                 LD_RE(ans) = (c == zero)? c: c * r;
 155                                 LD_IM(ans) = (s == zero)? s: s * r;
 156                         }
 157                 } else if (fabsl(x) == fabsl(y)) {    /* |x| = |y| */
 158                         if (hx >= 0) {
 159                                 t = (hy >= 0)? 0.25L : -0.25L;
 160                                 sincospil(t * u, &s, &c);
 161                         } else if ((LAST(u) & 3) == 0) {
 162                                 t = (hy >= 0)? 0.75L : -0.75L;
 163                                 sincospil(t * u, &s, &c);
 164                         } else {
 165                                 r = (hy >= 0)? u : -u;
 166                                 t = -0.25L * r;
 167                                 w1 = r + t;
 168                                 w2 = t - (w1 - r);
 169                                 sincospil(w1, &t1, &t2);
 170                                 sincospil(w2, &t3, &t4);
 171                                 s = t1 * t4 + t3 * t2;
 172                                 c = t2 * t4 - t1 * t3;
 173                         }
 174                         if (ix < 0x3ffe0000)    /* |x| < 1/2 */
 175                                 r = powl(fabsl(x + x), u) * exp2l(-0.5L * u);
 176                         else if (ix >= 0x3fff0000 || iu < 0x400cfff8)
 177                                 /* |x| >= 1 or |u| < 16383 */
 178                                 r = powl(fabsl(x), u) * exp2l(0.5L * u);
 179                         else   /* special treatment */
 180                                 j = 2;
 181                         if (j == 0) {
 182                                 LD_RE(ans) = (c == zero)? c: c * r;
 183                                 LD_IM(ans) = (s == zero)? s: s * r;
 184                         }
 185                 } else
 186                         j = 1;
 187                 if (j == 0)
 188                         return (ans);
 189         }
 190         if (iu >= hiinf || iv >= hiinf || ix >= hiinf || iy >= hiinf) {
 191                 /*
 192                  * non-zero imag part(s) with inf component(s) yields NaN
 193                  */
 194                 t = fabsl(x) + fabsl(y) + fabsl(u) + fabsl(v);
 195                 LD_RE(ans) = LD_IM(ans) = t - t;
 196         } else {
 197                 k = 0;  /* no scaling */
 198                 if (iu > 0x7ffe0000 || iv > 0x7ffe0000) {
 199                         u *= 1.52587890625000000000e-05L;
 200                         v *= 1.52587890625000000000e-05L;
 201                         k = 1;  /* scale u and v by 2**-16 */
 202                 }
 203                 /*
 204                  * Use similated higher precision arithmetic to compute:
 205                  * r = u * log(hypot(x, y)) - v * atan2(y, x)
 206                  * q = u * atan2(y, x) + v * log(hypot(x, y))
 207                  */
 208
 209                 t1 = __k_clog_rl(x, y, &t2);
 210                 t3 = __k_atan2l(y, x, &t4);
 211                 x1 = t1; HALF(x1);
 212                 y1 = t3; HALF(y1);
 213                 u1 = u; HALF(u1);
 214                 v1 = v; HALF(v1);
 215                 x2 = t2 - (x1 - t1);    /* log(hypot(x,y)) = x1 + x2 */
 216                 y2 = t4 - (y1 - t3);    /* atan2(y,x) = y1 + y2 */
 217                 /* compute q = u * atan2(y, x) + v * log(hypot(x, y)) */
 218                 if (j != 2) {
 219                         b[0] = u1 * y1;
 220                         b[1] = (u - u1) * y1 + u * y2;
 221                         if (j == 1) {   /* v = 0 */
 222                                 w1 = b[0] + b[1];
 223                                 w2 = b[1] - (w1 - b[0]);
 224                         } else {
 225                                 b[2] = v1 * x1;
 226                                 b[3] = (v - v1) * x1 + v * x2;
 227                                 w1 = sum4fpl(b, &w2);
 228                         }
 229                         sincosl(w1, &t1, &t2);
 230                         sincosl(w2, &t3, &t4);
 231                         s = t1 * t4 + t3 * t2;
 232                         c = t2 * t4 - t1 * t3;
 233                         if (k == 1)     /* square j times */
 234                                 for (i = 0; i < 10; i++) {
 235                                         t1 = s * c;
 236                                         c = (c + s) * (c - s);
 237                                         s = t1 + t1;
 238                                 }
 239                 }
 240                 /* compute r = u * (t1, t2) - v * (t3, t4) */
 241                 b[0] = u1 * x1;
 242                 b[1] = (u - u1) * x1 + u * x2;
 243                 if (j == 1) {   /* v = 0 */
 244                         w1 = b[0] + b[1];
 245                         w2 = b[1] - (w1 - b[0]);
 246                 } else {
 247                         b[2] = -v1 * y1;
 248                         b[3] = (v1 - v) * y1 - v * y2;
 249                         w1 = sum4fpl(b, &w2);
 250                 }
 251                 /* scale back unless w1 is large enough to cause exception */
 252                 if (k != 0 && fabsl(w1) < 20000.0L) {
 253                         w1 *= 65536.0L; w2 *= 65536.0L;
 254                 }
 255                 hx = HI_XWORD(w1);
 256                 ix = hx & 0x7fffffff;
 257                 /* compute exp(w1 + w2) */
 258                 k = 0;
 259                 if (ix < 0x3f8c0000) /* exp(tiny < 2**-115) = 1 */
 260                         r = one;
 261                 else if (ix >= 0x400c6760) /* overflow/underflow */
 262                         r = (hx < 0)? tiny * tiny : huge * huge;
 263                 else {  /* compute exp(w1 + w2) */
 264                         k = (int) (invln2 * w1 + ((hx >= 0)? 0.5L : -0.5L));
 265                         t1 = (long double) k;
 266                         t2 = w1 - t1 * ln2hil;
 267                         t3 = w2 - t1 * ln2lol;
 268                         r = expl(t2 + t3);
 269                 }
 270                 if (c != zero) c *= r;
 271                 if (s != zero) s *= r;
 272                 if (k != 0) {
 273                         c = scalbnl(c, k);
 274                         s = scalbnl(s, k);
 275                 }
 276                 LD_RE(ans) = c;
 277                 LD_IM(ans) = s;
 278         }
 279         return (ans);
 280 }