1 /*
   2  * Copyright (c) 1998, 2015, Oracle and/or its affiliates. All rights reserved.
   3  * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
   4  *
   5  * This code is free software; you can redistribute it and/or modify it
   6  * under the terms of the GNU General Public License version 2 only, as
   7  * published by the Free Software Foundation.  Oracle designates this
   8  * particular file as subject to the "Classpath" exception as provided
   9  * by Oracle in the LICENSE file that accompanied this code.
  10  *
  11  * This code is distributed in the hope that it will be useful, but WITHOUT
  12  * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
  13  * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
  14  * version 2 for more details (a copy is included in the LICENSE file that
  15  * accompanied this code).
  16  *
  17  * You should have received a copy of the GNU General Public License version
  18  * 2 along with this work; if not, write to the Free Software Foundation,
  19  * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
  20  *
  21  * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
  22  * or visit www.oracle.com if you need additional information or have any
  23  * questions.
  24  */
  25 
  26 package java.lang;
  27 
  28 /**
  29  * Port of the "Freely Distributable Math Library", version 5.3, from
  30  * C to Java.
  31  *
  32  * <p>The C version of fdlibm relied on the idiom of pointer aliasing
  33  * a 64-bit double floating-point value as a two-element array of
  34  * 32-bit integers and reading and writing the two halves of the
  35  * double independently. This coding pattern was problematic to C
  36  * optimizers and not directly expressible in Java. Therefore, rather
  37  * than a memory level overlay, if portions of a double need to be
  38  * operated on as integer values, the standard library methods for
  39  * bitwise floating-point to integer conversion,
  40  * Double.longBitsToDouble and Double.doubleToRawLongBits, are directly
  41  * or indirectly used.
  42  *
  43  * <p>The C version of fdlibm also took some pains to signal the
  44  * correct IEEE 754 exceptional conditions divide by zero, invalid,
  45  * overflow and underflow. For example, overflow would be signaled by
  46  * {@code huge * huge} where {@code huge} was a large constant that
  47  * would overflow when squared. Since IEEE floating-point exceptional
  48  * handling is not supported natively in the JVM, such coding patterns
  49  * have been omitted from this port. For example, rather than {@code
  50  * return huge * huge}, this port will use {@code return INFINITY}.
  51  *
  52  * <p>Various comparison and arithmetic operations in fdlibm could be
  53  * done either based on the integer view of a value or directly on the
  54  * floating-point representation. Which idiom is faster may depend on
  55  * platform specific factors. However, for code clarity if no other
  56  * reason, this port will favor expressing the semantics of those
  57  * operations in terms of floating-point operations when convenient to
  58  * do so.
  59  */
  60 class FdLibm {
  61     // Constants used by multiple algorithms
  62     private static final double INFINITY = Double.POSITIVE_INFINITY;
  63 
  64     private FdLibm() {
  65         throw new UnsupportedOperationException("No FdLibm instances for you.");
  66     }
  67 
  68     /**
  69      * Return the low-order 32 bits of the double argument as an int.
  70      */
  71     private static int __LO(double x) {
  72         long transducer = Double.doubleToRawLongBits(x);
  73         return (int)transducer;
  74     }
  75 
  76     /**
  77      * Return a double with its low-order bits of the second argument
  78      * and the high-order bits of the first argument..
  79      */
  80     private static double __LO(double x, int low) {
  81         long transX = Double.doubleToRawLongBits(x);
  82         return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L)|low );
  83     }
  84 
  85     /**
  86      * Return the high-order 32 bits of the double argument as an int.
  87      */
  88     private static int __HI(double x) {
  89         long transducer = Double.doubleToRawLongBits(x);
  90         return (int)(transducer >> 32);
  91     }
  92 
  93     /**
  94      * Return a double with its high-order bits of the second argument
  95      * and the low-order bits of the first argument..
  96      */
  97     private static double __HI(double x, int high) {
  98         long transX = Double.doubleToRawLongBits(x);
  99         return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 );
 100     }
 101 
 102     /**
 103      * hypot(x,y)
 104      *
 105      * Method :
 106      *      If (assume round-to-nearest) z = x*x + y*y
 107      *      has error less than sqrt(2)/2 ulp, than
 108      *      sqrt(z) has error less than 1 ulp (exercise).
 109      *
 110      *      So, compute sqrt(x*x + y*y) with some care as
 111      *      follows to get the error below 1 ulp:
 112      *
 113      *      Assume x > y > 0;
 114      *      (if possible, set rounding to round-to-nearest)
 115      *      1. if x > 2y  use
 116      *              x1*x1 + (y*y + (x2*(x + x1))) for x*x + y*y
 117      *      where x1 = x with lower 32 bits cleared, x2 = x - x1; else
 118      *      2. if x <= 2y use
 119      *              t1*y1 + ((x-y) * (x-y) + (t1*y2 + t2*y))
 120      *      where t1 = 2x with lower 32 bits cleared, t2 = 2x - t1,
 121      *      y1= y with lower 32 bits chopped, y2 = y - y1.
 122      *
 123      *      NOTE: scaling may be necessary if some argument is too
 124      *            large or too tiny
 125      *
 126      * Special cases:
 127      *      hypot(x,y) is INF if x or y is +INF or -INF; else
 128      *      hypot(x,y) is NAN if x or y is NAN.
 129      *
 130      * Accuracy:
 131      *      hypot(x,y) returns sqrt(x^2 + y^2) with error less
 132      *      than 1 ulp (unit in the last place)
 133      */
 134     public static class Hypot {
 135         public static final double TWO_MINUS_600 = 0x1.0p-600;
 136         public static final double TWO_PLUS_600  = 0x1.0p+600;
 137 
 138         public static strictfp double compute(double x, double y) {
 139             double a = Math.abs(x);
 140             double b = Math.abs(y);
 141 
 142             if (!Double.isFinite(a) || !Double.isFinite(b)) {
 143                 if (a == INFINITY || b == INFINITY)
 144                     return INFINITY;
 145                 else
 146                     return a + b; // Propagate NaN significand bits
 147             }
 148 
 149             if (b > a) {
 150                 double tmp = a;
 151                 a = b;
 152                 b = tmp;
 153             }
 154             assert a >= b;
 155 
 156             // Doing bitwise conversion after screening for NaN allows
 157             // the code to not worry about the possibility of
 158             // "negative" NaN values.
 159 
 160             // Note: the ha and hb variables are the high-order
 161             // 32-bits of a and b stored as integer values. The ha and
 162             // hb values are used first for a rough magnitude
 163             // comparison of a and b and second for simulating higher
 164             // precision by allowing a and b, respectively, to be
 165             // decomposed into non-overlapping portions. Both of these
 166             // uses could be eliminated. The magnitude comparison
 167             // could be eliminated by extracting and comparing the
 168             // exponents of a and b or just be performing a
 169             // floating-point divide.  Splitting a floating-point
 170             // number into non-overlapping portions can be
 171             // accomplished by judicious use of multiplies and
 172             // additions. For details see T. J. Dekker, A Floating
 173             // Point Technique for Extending the Available Precision ,
 174             // Numerische Mathematik, vol. 18, 1971, pp.224-242 and
 175             // subsequent work.
 176 
 177             int ha = __HI(a);        // high word of a
 178             int hb = __HI(b);        // high word of b
 179 
 180             if ((ha - hb) > 0x3c00000) {
 181                 return a + b;  // x / y > 2**60
 182             }
 183 
 184             int k = 0;
 185             if (a > 0x1.00000_ffff_ffffp500) {   // a > ~2**500
 186                 // scale a and b by 2**-600
 187                 ha -= 0x25800000;
 188                 hb -= 0x25800000;
 189                 a = a * TWO_MINUS_600;
 190                 b = b * TWO_MINUS_600;
 191                 k += 600;
 192             }
 193             double t1, t2;
 194             if (b < 0x1.0p-500) {   // b < 2**-500
 195                 if (b < Double.MIN_NORMAL) {      // subnormal b or 0 */
 196                     if (b == 0.0)
 197                         return a;
 198                     t1 = 0x1.0p1022;   // t1 = 2^1022
 199                     b *= t1;
 200                     a *= t1;
 201                     k -= 1022;
 202                 } else {            // scale a and b by 2^600
 203                     ha += 0x25800000;       // a *= 2^600
 204                     hb += 0x25800000;       // b *= 2^600
 205                     a = a * TWO_PLUS_600;
 206                     b = b * TWO_PLUS_600;
 207                     k -= 600;
 208                 }
 209             }
 210             // medium size a and b
 211             double w = a - b;
 212             if (w > b) {
 213                 t1 = 0;
 214                 t1 = __HI(t1, ha);
 215                 t2 = a - t1;
 216                 w  = Math.sqrt(t1*t1 - (b*(-b) - t2 * (a + t1)));
 217             } else {
 218                 double y1, y2;
 219                 a  = a + a;
 220                 y1 = 0;
 221                 y1 = __HI(y1, hb);
 222                 y2 = b - y1;
 223                 t1 = 0;
 224                 t1 = __HI(t1, ha + 0x00100000);
 225                 t2 = a - t1;
 226                 w  = Math.sqrt(t1*y1 - (w*(-w) - (t1*y2 + t2*b)));
 227             }
 228             if (k != 0) {
 229                 return Math.powerOfTwoD(k) * w;
 230             } else
 231                 return w;
 232         }
 233     }
 234 
 235     /**
 236      * Compute x**y
 237      *                    n
 238      * Method:  Let x =  2   * (1+f)
 239      *      1. Compute and return log2(x) in two pieces:
 240      *              log2(x) = w1 + w2,
 241      *         where w1 has 53 - 24 = 29 bit trailing zeros.
 242      *      2. Perform y*log2(x) = n+y' by simulating multi-precision
 243      *         arithmetic, where |y'| <= 0.5.
 244      *      3. Return x**y = 2**n*exp(y'*log2)
 245      *
 246      * Special cases:
 247      *      1.  (anything) ** 0  is 1
 248      *      2.  (anything) ** 1  is itself
 249      *      3.  (anything) ** NAN is NAN
 250      *      4.  NAN ** (anything except 0) is NAN
 251      *      5.  +-(|x| > 1) **  +INF is +INF
 252      *      6.  +-(|x| > 1) **  -INF is +0
 253      *      7.  +-(|x| < 1) **  +INF is +0
 254      *      8.  +-(|x| < 1) **  -INF is +INF
 255      *      9.  +-1         ** +-INF is NAN
 256      *      10. +0 ** (+anything except 0, NAN)               is +0
 257      *      11. -0 ** (+anything except 0, NAN, odd integer)  is +0
 258      *      12. +0 ** (-anything except 0, NAN)               is +INF
 259      *      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
 260      *      14. -0 ** (odd integer) = -( +0 ** (odd integer) )
 261      *      15. +INF ** (+anything except 0,NAN) is +INF
 262      *      16. +INF ** (-anything except 0,NAN) is +0
 263      *      17. -INF ** (anything)  = -0 ** (-anything)
 264      *      18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
 265      *      19. (-anything except 0 and inf) ** (non-integer) is NAN
 266      *
 267      * Accuracy:
 268      *      pow(x,y) returns x**y nearly rounded. In particular
 269      *                      pow(integer,integer)
 270      *      always returns the correct integer provided it is
 271      *      representable.
 272      */
 273     public static class Pow {
 274         public static strictfp double compute(final double x, final double y) {
 275             double z;
 276             double r, s, t, u, v, w;
 277             int i, j, k, n;
 278 
 279             // y == zero: x**0 = 1
 280             if (y == 0.0)
 281                 return 1.0;
 282 
 283             // +/-NaN return x + y to propagate NaN significands
 284             if (Double.isNaN(x) || Double.isNaN(y))
 285                 return x + y;
 286 
 287             final double y_abs = Math.abs(y);
 288             double x_abs   = Math.abs(x);
 289             // Special values of y
 290             if (y == 2.0) {
 291                 return x * x;
 292             } else if (y == 0.5) {
 293                 if (x >= -Double.MAX_VALUE) // Handle x == -infinity later
 294                     return Math.sqrt(x + 0.0); // Add 0.0 to properly handle x == -0.0
 295             } else if (y_abs == 1.0) {        // y is  +/-1
 296                 return (y == 1.0) ? x : 1.0 / x;
 297             } else if (y_abs == INFINITY) {       // y is +/-infinity
 298                 if (x_abs == 1.0)
 299                     return  y - y;         // inf**+/-1 is NaN
 300                 else if (x_abs > 1.0) // (|x| > 1)**+/-inf = inf, 0
 301                     return (y >= 0) ? y : 0.0;
 302                 else                       // (|x| < 1)**-/+inf = inf, 0
 303                     return (y < 0) ? -y : 0.0;
 304             }
 305 
 306             final int hx = __HI(x);
 307             int ix = hx & 0x7fffffff;
 308 
 309             /*
 310              * When x < 0, determine if y is an odd integer:
 311              * y_is_int = 0       ... y is not an integer
 312              * y_is_int = 1       ... y is an odd int
 313              * y_is_int = 2       ... y is an even int
 314              */
 315             int y_is_int  = 0;
 316             if (hx < 0) {
 317                 if (y_abs >= 0x1.0p53)   // |y| >= 2^53 = 9.007199254740992E15
 318                     y_is_int = 2; // y is an even integer since ulp(2^53) = 2.0
 319                 else if (y_abs >= 1.0) { // |y| >= 1.0
 320                     long y_abs_as_long = (long) y_abs;
 321                     if ( ((double) y_abs_as_long) == y_abs) {
 322                         y_is_int = 2 -  (int)(y_abs_as_long & 0x1L);
 323                     }
 324                 }
 325             }
 326 
 327             // Special value of x
 328             if (x_abs == 0.0 ||
 329                 x_abs == INFINITY ||
 330                 x_abs == 1.0) {
 331                 z = x_abs;                 // x is +/-0, +/-inf, +/-1
 332                 if (y < 0.0)
 333                     z = 1.0/z;     // z = (1/|x|)
 334                 if (hx < 0) {
 335                     if (((ix - 0x3ff00000) | y_is_int) == 0) {
 336                         z = (z-z)/(z-z); // (-1)**non-int is NaN
 337                     } else if (y_is_int == 1)
 338                         z = -1.0 * z;             // (x < 0)**odd = -(|x|**odd)
 339                 }
 340                 return z;
 341             }
 342 
 343             n = (hx >> 31) + 1;
 344 
 345             // (x < 0)**(non-int) is NaN
 346             if ((n | y_is_int) == 0)
 347                 return (x-x)/(x-x);
 348 
 349             s = 1.0; // s (sign of result -ve**odd) = -1 else = 1
 350             if ( (n | (y_is_int - 1)) == 0)
 351                 s = -1.0; // (-ve)**(odd int)
 352 
 353             double p_h, p_l, t1, t2;
 354             // |y| is huge
 355             if (y_abs > 0x1.00000_ffff_ffffp31) { // if |y| > ~2**31
 356                 final double INV_LN2   =  0x1.7154_7652_b82fep0;   //  1.44269504088896338700e+00 = 1/ln2
 357                 final double INV_LN2_H =  0x1.715476p0;            //  1.44269502162933349609e+00 = 24 bits of 1/ln2
 358                 final double INV_LN2_L =  0x1.4ae0_bf85_ddf44p-26; //  1.92596299112661746887e-08 = 1/ln2 tail
 359 
 360                 // Over/underflow if x is not close to one
 361                 if (x_abs < 0x1.fffff_0000_0000p-1) // |x| < ~0.9999995231628418
 362                     return (y < 0.0) ? s * INFINITY : s * 0.0;
 363                 if (x_abs > 0x1.00000_ffff_ffffp0)         // |x| > ~1.0
 364                     return (y > 0.0) ? s * INFINITY : s * 0.0;
 365                 /*
 366                  * now |1-x| is tiny <= 2**-20, sufficient to compute
 367                  * log(x) by x - x^2/2 + x^3/3 - x^4/4
 368                  */
 369                 t = x_abs - 1.0;        // t has 20 trailing zeros
 370                 w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
 371                 u = INV_LN2_H * t;      // INV_LN2_H has 21 sig. bits
 372                 v =  t * INV_LN2_L - w * INV_LN2;
 373                 t1 = u + v;
 374                 t1 =__LO(t1, 0);
 375                 t2 = v - (t1 - u);
 376             } else {
 377                 final double CP      =  0x1.ec70_9dc3_a03fdp-1;  //  9.61796693925975554329e-01 = 2/(3ln2)
 378                 final double CP_H    =  0x1.ec709ep-1;           //  9.61796700954437255859e-01 = (float)cp
 379                 final double CP_L    = -0x1.e2fe_0145_b01f5p-28; // -7.02846165095275826516e-09 = tail of CP_H
 380 
 381                 double z_h, z_l, ss, s2, s_h, s_l, t_h, t_l;
 382                 n = 0;
 383                 // Take care of subnormal numbers
 384                 if (ix < 0x00100000) {
 385                     x_abs *= 0x1.0p53; // 2^53 = 9007199254740992.0
 386                     n -= 53;
 387                     ix = __HI(x_abs);
 388                 }
 389                 n  += ((ix) >> 20) - 0x3ff;
 390                 j  = ix & 0x000fffff;
 391                 // Determine interval
 392                 ix = j | 0x3ff00000;          // Normalize ix
 393                 if (j <= 0x3988E)
 394                     k = 0;         // |x| <sqrt(3/2)
 395                 else if (j < 0xBB67A)
 396                     k = 1;         // |x| <sqrt(3)
 397                 else {
 398                     k = 0;
 399                     n += 1;
 400                     ix -= 0x00100000;
 401                 }
 402                 x_abs = __HI(x_abs, ix);
 403 
 404                 // Compute ss = s_h + s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5)
 405 
 406                 final double BP[]    = {1.0,
 407                                        1.5};
 408                 final double DP_H[]  = {0.0,
 409                                         0x1.2b80_34p-1};        // 5.84962487220764160156e-01
 410                 final double DP_L[]  = {0.0,
 411                                         0x1.cfde_b43c_fd006p-27};// 1.35003920212974897128e-08
 412 
 413                 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
 414                 final double L1      =  0x1.3333_3333_33303p-1;  //  5.99999999999994648725e-01
 415                 final double L2      =  0x1.b6db_6db6_fabffp-2;  //  4.28571428578550184252e-01
 416                 final double L3      =  0x1.5555_5518_f264dp-2;  //  3.33333329818377432918e-01
 417                 final double L4      =  0x1.1746_0a91_d4101p-2;  //  2.72728123808534006489e-01
 418                 final double L5      =  0x1.d864_a93c_9db65p-3;  //  2.30660745775561754067e-01
 419                 final double L6      =  0x1.a7e2_84a4_54eefp-3;  //  2.06975017800338417784e-01
 420                 u = x_abs - BP[k];               // BP[0]=1.0, BP[1]=1.5
 421                 v = 1.0 / (x_abs + BP[k]);
 422                 ss = u * v;
 423                 s_h = ss;
 424                 s_h = __LO(s_h, 0);
 425                 // t_h=x_abs + BP[k] High
 426                 t_h = 0.0;
 427                 t_h = __HI(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18) );
 428                 t_l = x_abs - (t_h - BP[k]);
 429                 s_l = v * ((u - s_h * t_h) - s_h * t_l);
 430                 // Compute log(x_abs)
 431                 s2 = ss * ss;
 432                 r = s2 * s2* (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
 433                 r += s_l * (s_h + ss);
 434                 s2  = s_h * s_h;
 435                 t_h = 3.0 + s2 + r;
 436                 t_h = __LO(t_h, 0);
 437                 t_l = r - ((t_h - 3.0) - s2);
 438                 // u+v = ss*(1+...)
 439                 u = s_h * t_h;
 440                 v = s_l * t_h + t_l * ss;
 441                 // 2/(3log2)*(ss + ...)
 442                 p_h = u + v;
 443                 p_h = __LO(p_h, 0);
 444                 p_l = v - (p_h - u);
 445                 z_h = CP_H * p_h;             // CP_H + CP_L = 2/(3*log2)
 446                 z_l = CP_L * p_h + p_l * CP + DP_L[k];
 447                 // log2(x_abs) = (ss + ..)*2/(3*log2) = n + DP_H + z_h + z_l
 448                 t = (double)n;
 449                 t1 = (((z_h + z_l) + DP_H[k]) + t);
 450                 t1 = __LO(t1, 0);
 451                 t2 = z_l - (((t1 - t) - DP_H[k]) - z_h);
 452             }
 453 
 454             // Split up y into (y1 + y2) and compute (y1 + y2) * (t1 + t2)
 455             double y1  = y;
 456             y1 = __LO(y1, 0);
 457             p_l = (y - y1) * t1 + y * t2;
 458             p_h = y1 * t1;
 459             z = p_l + p_h;
 460             j = __HI(z);
 461             i = __LO(z);
 462             if (j >= 0x40900000) {                           // z >= 1024
 463                 if (((j - 0x40900000) | i)!=0)               // if z > 1024
 464                     return s * INFINITY;                     // Overflow
 465                 else {
 466                     final double OVT     =  8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp))
 467                     if (p_l + OVT > z - p_h)
 468                         return s * INFINITY;   // Overflow
 469                 }
 470             } else if ((j & 0x7fffffff) >= 0x4090cc00 ) {        // z <= -1075
 471                 if (((j - 0xc090cc00) | i)!=0)           // z < -1075
 472                     return s * 0.0;           // Underflow
 473                 else {
 474                     if (p_l <= z - p_h)
 475                         return s * 0.0;      // Underflow
 476                 }
 477             }
 478             /*
 479              * Compute 2**(p_h+p_l)
 480              */
 481             // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
 482             final double P1      =  0x1.5555_5555_5553ep-3;  //  1.66666666666666019037e-01
 483             final double P2      = -0x1.6c16_c16b_ebd93p-9;  // -2.77777777770155933842e-03
 484             final double P3      =  0x1.1566_aaf2_5de2cp-14; //  6.61375632143793436117e-05
 485             final double P4      = -0x1.bbd4_1c5d_26bf1p-20; // -1.65339022054652515390e-06
 486             final double P5      =  0x1.6376_972b_ea4d0p-25; //  4.13813679705723846039e-08
 487             final double LG2     =  0x1.62e4_2fef_a39efp-1;  //  6.93147180559945286227e-01
 488             final double LG2_H   =  0x1.62e43p-1;            //  6.93147182464599609375e-01
 489             final double LG2_L   = -0x1.05c6_10ca_86c39p-29; // -1.90465429995776804525e-09
 490             i = j & 0x7fffffff;
 491             k = (i >> 20) - 0x3ff;
 492             n = 0;
 493             if (i > 0x3fe00000) {              // if |z| > 0.5, set n = [z + 0.5]
 494                 n = j + (0x00100000 >> (k + 1));
 495                 k = ((n & 0x7fffffff) >> 20) - 0x3ff;     // new k for n
 496                 t = 0.0;
 497                 t = __HI(t, (n & ~(0x000fffff >> k)) );
 498                 n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
 499                 if (j < 0)
 500                     n = -n;
 501                 p_h -= t;
 502             }
 503             t = p_l + p_h;
 504             t = __LO(t, 0);
 505             u = t * LG2_H;
 506             v = (p_l - (t - p_h)) * LG2 + t * LG2_L;
 507             z = u + v;
 508             w = v - (z - u);
 509             t  = z * z;
 510             t1  = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
 511             r  = (z * t1)/(t1 - 2.0) - (w + z * w);
 512             z  = 1.0 - (r - z);
 513             j  = __HI(z);
 514             j += (n << 20);
 515             if ((j >> 20) <= 0)
 516                 z = Math.scalb(z, n); // subnormal output
 517             else {
 518                 int z_hi = __HI(z);
 519                 z_hi += (n << 20);
 520                 z = __HI(z, z_hi);
 521             }
 522             return s * z;
 523         }
 524     }
 525 }