1 /*
   2  * Copyright (c) 2005, 2014, 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.
   8  *
   9  * This code is distributed in the hope that it will be useful, but WITHOUT
  10  * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
  11  * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
  12  * version 2 for more details (a copy is included in the LICENSE file that
  13  * accompanied this code).
  14  *
  15  * You should have received a copy of the GNU General Public License version
  16  * 2 along with this work; if not, write to the Free Software Foundation,
  17  * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
  18  *
  19  * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
  20  * or visit www.oracle.com if you need additional information or have any
  21  * questions.
  22  *
  23  */
  24 
  25 #include "precompiled.hpp"
  26 #include "prims/jni.h"
  27 #include "runtime/interfaceSupport.hpp"
  28 #include "runtime/sharedRuntime.hpp"
  29 
  30 // This file contains copies of the fdlibm routines used by
  31 // StrictMath. It turns out that it is almost always required to use
  32 // these runtime routines; the Intel CPU doesn't meet the Java
  33 // specification for sin/cos outside a certain limited argument range,
  34 // and the SPARC CPU doesn't appear to have sin/cos instructions. It
  35 // also turns out that avoiding the indirect call through function
  36 // pointer out to libjava.so in SharedRuntime speeds these routines up
  37 // by roughly 15% on both Win32/x86 and Solaris/SPARC.
  38 
  39 // Enabling optimizations in this file causes incorrect code to be
  40 // generated; can not figure out how to turn down optimization for one
  41 // file in the IDE on Windows
  42 #ifdef WIN32
  43 # pragma optimize ( "", off )
  44 #endif
  45 
  46 #include <math.h>
  47 
  48 // VM_LITTLE_ENDIAN is #defined appropriately in the Makefiles
  49 // [jk] this is not 100% correct because the float word order may different
  50 // from the byte order (e.g. on ARM)
  51 #ifdef VM_LITTLE_ENDIAN
  52 # define __HI(x) *(1+(int*)&x)
  53 # define __LO(x) *(int*)&x
  54 #else
  55 # define __HI(x) *(int*)&x
  56 # define __LO(x) *(1+(int*)&x)
  57 #endif
  58 
  59 static double copysignA(double x, double y) {
  60   __HI(x) = (__HI(x)&0x7fffffff)|(__HI(y)&0x80000000);
  61   return x;
  62 }
  63 
  64 /*
  65  * ====================================================
  66  * Copyright (c) 1998 Oracle and/or its affiliates. All rights reserved.
  67  *
  68  * Developed at SunSoft, a Sun Microsystems, Inc. business.
  69  * Permission to use, copy, modify, and distribute this
  70  * software is freely granted, provided that this notice
  71  * is preserved.
  72  * ====================================================
  73  */
  74 
  75 /*
  76  * scalbn (double x, int n)
  77  * scalbn(x,n) returns x* 2**n  computed by  exponent
  78  * manipulation rather than by actually performing an
  79  * exponentiation or a multiplication.
  80  */
  81 
  82 static const double
  83 two54   =  1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
  84   twom54  =  5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */
  85   hugeX   = 1.0e+300,
  86   tiny   = 1.0e-300;
  87 
  88 static double scalbnA (double x, int n) {
  89   int  k,hx,lx;
  90   hx = __HI(x);
  91   lx = __LO(x);
  92   k = (hx&0x7ff00000)>>20;              /* extract exponent */
  93   if (k==0) {                           /* 0 or subnormal x */
  94     if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */
  95     x *= two54;
  96     hx = __HI(x);
  97     k = ((hx&0x7ff00000)>>20) - 54;
  98     if (n< -50000) return tiny*x;       /*underflow*/
  99   }
 100   if (k==0x7ff) return x+x;             /* NaN or Inf */
 101   k = k+n;
 102   if (k >  0x7fe) return hugeX*copysignA(hugeX,x); /* overflow  */
 103   if (k > 0)                            /* normal result */
 104     {__HI(x) = (hx&0x800fffff)|(k<<20); return x;}
 105   if (k <= -54) {
 106     if (n > 50000)      /* in case integer overflow in n+k */
 107       return hugeX*copysignA(hugeX,x);   /*overflow*/
 108     else return tiny*copysignA(tiny,x);  /*underflow*/
 109   }
 110   k += 54;                              /* subnormal result */
 111   __HI(x) = (hx&0x800fffff)|(k<<20);
 112   return x*twom54;
 113 }
 114 
 115 /* __ieee754_log(x)
 116  * Return the logarithm of x
 117  *
 118  * Method :
 119  *   1. Argument Reduction: find k and f such that
 120  *                    x = 2^k * (1+f),
 121  *       where  sqrt(2)/2 < 1+f < sqrt(2) .
 122  *
 123  *   2. Approximation of log(1+f).
 124  *    Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
 125  *             = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 126  *             = 2s + s*R
 127  *      We use a special Reme algorithm on [0,0.1716] to generate
 128  *    a polynomial of degree 14 to approximate R The maximum error
 129  *    of this polynomial approximation is bounded by 2**-58.45. In
 130  *    other words,
 131  *                    2      4      6      8      10      12      14
 132  *        R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
 133  *    (the values of Lg1 to Lg7 are listed in the program)
 134  *    and
 135  *        |      2          14          |     -58.45
 136  *        | Lg1*s +...+Lg7*s    -  R(z) | <= 2
 137  *        |                             |
 138  *    Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
 139  *    In order to guarantee error in log below 1ulp, we compute log
 140  *    by
 141  *            log(1+f) = f - s*(f - R)        (if f is not too large)
 142  *            log(1+f) = f - (hfsq - s*(hfsq+R)).     (better accuracy)
 143  *
 144  *    3. Finally,  log(x) = k*ln2 + log(1+f).
 145  *                        = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
 146  *       Here ln2 is split into two floating point number:
 147  *                    ln2_hi + ln2_lo,
 148  *       where n*ln2_hi is always exact for |n| < 2000.
 149  *
 150  * Special cases:
 151  *    log(x) is NaN with signal if x < 0 (including -INF) ;
 152  *    log(+INF) is +INF; log(0) is -INF with signal;
 153  *    log(NaN) is that NaN with no signal.
 154  *
 155  * Accuracy:
 156  *    according to an error analysis, the error is always less than
 157  *    1 ulp (unit in the last place).
 158  *
 159  * Constants:
 160  * The hexadecimal values are the intended ones for the following
 161  * constants. The decimal values may be used, provided that the
 162  * compiler will convert from decimal to binary accurately enough
 163  * to produce the hexadecimal values shown.
 164  */
 165 
 166 static const double
 167 ln2_hi  =  6.93147180369123816490e-01,        /* 3fe62e42 fee00000 */
 168   ln2_lo  =  1.90821492927058770002e-10,        /* 3dea39ef 35793c76 */
 169   Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
 170   Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
 171   Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
 172   Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
 173   Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
 174   Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
 175   Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
 176 
 177 static double zero = 0.0;
 178 
 179 static double __ieee754_log(double x) {
 180   double hfsq,f,s,z,R,w,t1,t2,dk;
 181   int k,hx,i,j;
 182   unsigned lx;
 183 
 184   hx = __HI(x);               /* high word of x */
 185   lx = __LO(x);               /* low  word of x */
 186 
 187   k=0;
 188   if (hx < 0x00100000) {                   /* x < 2**-1022  */
 189     if (((hx&0x7fffffff)|lx)==0)
 190       return -two54/zero;             /* log(+-0)=-inf */
 191     if (hx<0) return (x-x)/zero;   /* log(-#) = NaN */
 192     k -= 54; x *= two54; /* subnormal number, scale up x */
 193     hx = __HI(x);             /* high word of x */
 194   }
 195   if (hx >= 0x7ff00000) return x+x;
 196   k += (hx>>20)-1023;
 197   hx &= 0x000fffff;
 198   i = (hx+0x95f64)&0x100000;
 199   __HI(x) = hx|(i^0x3ff00000);        /* normalize x or x/2 */
 200   k += (i>>20);
 201   f = x-1.0;
 202   if((0x000fffff&(2+hx))<3) {  /* |f| < 2**-20 */
 203     if(f==zero) {
 204       if (k==0) return zero;
 205       else {dk=(double)k; return dk*ln2_hi+dk*ln2_lo;}
 206     }
 207     R = f*f*(0.5-0.33333333333333333*f);
 208     if(k==0) return f-R; else {dk=(double)k;
 209     return dk*ln2_hi-((R-dk*ln2_lo)-f);}
 210   }
 211   s = f/(2.0+f);
 212   dk = (double)k;
 213   z = s*s;
 214   i = hx-0x6147a;
 215   w = z*z;
 216   j = 0x6b851-hx;
 217   t1= w*(Lg2+w*(Lg4+w*Lg6));
 218   t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
 219   i |= j;
 220   R = t2+t1;
 221   if(i>0) {
 222     hfsq=0.5*f*f;
 223     if(k==0) return f-(hfsq-s*(hfsq+R)); else
 224       return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
 225   } else {
 226     if(k==0) return f-s*(f-R); else
 227       return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
 228   }
 229 }
 230 
 231 JRT_LEAF(jdouble, SharedRuntime::dlog(jdouble x))
 232   return __ieee754_log(x);
 233 JRT_END
 234 
 235 /* __ieee754_log10(x)
 236  * Return the base 10 logarithm of x
 237  *
 238  * Method :
 239  *    Let log10_2hi = leading 40 bits of log10(2) and
 240  *        log10_2lo = log10(2) - log10_2hi,
 241  *        ivln10   = 1/log(10) rounded.
 242  *    Then
 243  *            n = ilogb(x),
 244  *            if(n<0)  n = n+1;
 245  *            x = scalbn(x,-n);
 246  *            log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))
 247  *
 248  * Note 1:
 249  *    To guarantee log10(10**n)=n, where 10**n is normal, the rounding
 250  *    mode must set to Round-to-Nearest.
 251  * Note 2:
 252  *    [1/log(10)] rounded to 53 bits has error  .198   ulps;
 253  *    log10 is monotonic at all binary break points.
 254  *
 255  * Special cases:
 256  *    log10(x) is NaN with signal if x < 0;
 257  *    log10(+INF) is +INF with no signal; log10(0) is -INF with signal;
 258  *    log10(NaN) is that NaN with no signal;
 259  *    log10(10**N) = N  for N=0,1,...,22.
 260  *
 261  * Constants:
 262  * The hexadecimal values are the intended ones for the following constants.
 263  * The decimal values may be used, provided that the compiler will convert
 264  * from decimal to binary accurately enough to produce the hexadecimal values
 265  * shown.
 266  */
 267 
 268 static const double
 269 ivln10     =  4.34294481903251816668e-01, /* 0x3FDBCB7B, 0x1526E50E */
 270   log10_2hi  =  3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
 271   log10_2lo  =  3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */
 272 
 273 static double __ieee754_log10(double x) {
 274   double y,z;
 275   int i,k,hx;
 276   unsigned lx;
 277 
 278   hx = __HI(x);       /* high word of x */
 279   lx = __LO(x);       /* low word of x */
 280 
 281   k=0;
 282   if (hx < 0x00100000) {                  /* x < 2**-1022  */
 283     if (((hx&0x7fffffff)|lx)==0)
 284       return -two54/zero;             /* log(+-0)=-inf */
 285     if (hx<0) return (x-x)/zero;        /* log(-#) = NaN */
 286     k -= 54; x *= two54; /* subnormal number, scale up x */
 287     hx = __HI(x);                /* high word of x */
 288   }
 289   if (hx >= 0x7ff00000) return x+x;
 290   k += (hx>>20)-1023;
 291   i  = ((unsigned)k&0x80000000)>>31;
 292   hx = (hx&0x000fffff)|((0x3ff-i)<<20);
 293   y  = (double)(k+i);
 294   __HI(x) = hx;
 295   z  = y*log10_2lo + ivln10*__ieee754_log(x);
 296   return  z+y*log10_2hi;
 297 }
 298 
 299 JRT_LEAF(jdouble, SharedRuntime::dlog10(jdouble x))
 300   return __ieee754_log10(x);
 301 JRT_END
 302 
 303 
 304 /* __ieee754_exp(x)
 305  * Returns the exponential of x.
 306  *
 307  * Method
 308  *   1. Argument reduction:
 309  *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
 310  *      Given x, find r and integer k such that
 311  *
 312  *               x = k*ln2 + r,  |r| <= 0.5*ln2.
 313  *
 314  *      Here r will be represented as r = hi-lo for better
 315  *      accuracy.
 316  *
 317  *   2. Approximation of exp(r) by a special rational function on
 318  *      the interval [0,0.34658]:
 319  *      Write
 320  *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
 321  *      We use a special Reme algorithm on [0,0.34658] to generate
 322  *      a polynomial of degree 5 to approximate R. The maximum error
 323  *      of this polynomial approximation is bounded by 2**-59. In
 324  *      other words,
 325  *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
 326  *      (where z=r*r, and the values of P1 to P5 are listed below)
 327  *      and
 328  *          |                  5          |     -59
 329  *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
 330  *          |                             |
 331  *      The computation of exp(r) thus becomes
 332  *                             2*r
 333  *              exp(r) = 1 + -------
 334  *                            R - r
 335  *                                 r*R1(r)
 336  *                     = 1 + r + ----------- (for better accuracy)
 337  *                                2 - R1(r)
 338  *      where
 339  *                               2       4             10
 340  *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
 341  *
 342  *   3. Scale back to obtain exp(x):
 343  *      From step 1, we have
 344  *         exp(x) = 2^k * exp(r)
 345  *
 346  * Special cases:
 347  *      exp(INF) is INF, exp(NaN) is NaN;
 348  *      exp(-INF) is 0, and
 349  *      for finite argument, only exp(0)=1 is exact.
 350  *
 351  * Accuracy:
 352  *      according to an error analysis, the error is always less than
 353  *      1 ulp (unit in the last place).
 354  *
 355  * Misc. info.
 356  *      For IEEE double
 357  *          if x >  7.09782712893383973096e+02 then exp(x) overflow
 358  *          if x < -7.45133219101941108420e+02 then exp(x) underflow
 359  *
 360  * Constants:
 361  * The hexadecimal values are the intended ones for the following
 362  * constants. The decimal values may be used, provided that the
 363  * compiler will convert from decimal to binary accurately enough
 364  * to produce the hexadecimal values shown.
 365  */
 366 
 367 static const double
 368 one     = 1.0,
 369   halF[2]       = {0.5,-0.5,},
 370   twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/
 371     o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
 372     u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
 373     ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
 374                   -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
 375     ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
 376                   -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
 377       invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
 378         P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
 379         P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
 380         P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
 381         P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
 382         P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
 383 
 384 static double __ieee754_exp(double x) {
 385   double y,hi=0,lo=0,c,t;
 386   int k=0,xsb;
 387   unsigned hx;
 388 
 389   hx  = __HI(x);        /* high word of x */
 390   xsb = (hx>>31)&1;             /* sign bit of x */
 391   hx &= 0x7fffffff;             /* high word of |x| */
 392 
 393   /* filter out non-finite argument */
 394   if(hx >= 0x40862E42) {                        /* if |x|>=709.78... */
 395     if(hx>=0x7ff00000) {
 396       if(((hx&0xfffff)|__LO(x))!=0)
 397         return x+x;             /* NaN */
 398       else return (xsb==0)? x:0.0;      /* exp(+-inf)={inf,0} */
 399     }
 400     if(x > o_threshold) return hugeX*hugeX; /* overflow */
 401     if(x < u_threshold) return twom1000*twom1000; /* underflow */
 402   }
 403 
 404   /* argument reduction */
 405   if(hx > 0x3fd62e42) {         /* if  |x| > 0.5 ln2 */
 406     if(hx < 0x3FF0A2B2) {       /* and |x| < 1.5 ln2 */
 407       hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
 408     } else {
 409       k  = (int)(invln2*x+halF[xsb]);
 410       t  = k;
 411       hi = x - t*ln2HI[0];      /* t*ln2HI is exact here */
 412       lo = t*ln2LO[0];
 413     }
 414     x  = hi - lo;
 415   }
 416   else if(hx < 0x3e300000)  {   /* when |x|<2**-28 */
 417     if(hugeX+x>one) return one+x;/* trigger inexact */
 418   }
 419   else k = 0;
 420 
 421   /* x is now in primary range */
 422   t  = x*x;
 423   c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
 424   if(k==0)      return one-((x*c)/(c-2.0)-x);
 425   else          y = one-((lo-(x*c)/(2.0-c))-hi);
 426   if(k >= -1021) {
 427     __HI(y) += (k<<20); /* add k to y's exponent */
 428     return y;
 429   } else {
 430     __HI(y) += ((k+1000)<<20);/* add k to y's exponent */
 431     return y*twom1000;
 432   }
 433 }
 434 
 435 JRT_LEAF(jdouble, SharedRuntime::dexp(jdouble x))
 436   return __ieee754_exp(x);
 437 JRT_END
 438 
 439 /* __ieee754_pow(x,y) return x**y
 440  *
 441  *                    n
 442  * Method:  Let x =  2   * (1+f)
 443  *      1. Compute and return log2(x) in two pieces:
 444  *              log2(x) = w1 + w2,
 445  *         where w1 has 53-24 = 29 bit trailing zeros.
 446  *      2. Perform y*log2(x) = n+y' by simulating muti-precision
 447  *         arithmetic, where |y'|<=0.5.
 448  *      3. Return x**y = 2**n*exp(y'*log2)
 449  *
 450  * Special cases:
 451  *      1.  (anything) ** 0  is 1
 452  *      2.  (anything) ** 1  is itself
 453  *      3.  (anything) ** NAN is NAN
 454  *      4.  NAN ** (anything except 0) is NAN
 455  *      5.  +-(|x| > 1) **  +INF is +INF
 456  *      6.  +-(|x| > 1) **  -INF is +0
 457  *      7.  +-(|x| < 1) **  +INF is +0
 458  *      8.  +-(|x| < 1) **  -INF is +INF
 459  *      9.  +-1         ** +-INF is NAN
 460  *      10. +0 ** (+anything except 0, NAN)               is +0
 461  *      11. -0 ** (+anything except 0, NAN, odd integer)  is +0
 462  *      12. +0 ** (-anything except 0, NAN)               is +INF
 463  *      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
 464  *      14. -0 ** (odd integer) = -( +0 ** (odd integer) )
 465  *      15. +INF ** (+anything except 0,NAN) is +INF
 466  *      16. +INF ** (-anything except 0,NAN) is +0
 467  *      17. -INF ** (anything)  = -0 ** (-anything)
 468  *      18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
 469  *      19. (-anything except 0 and inf) ** (non-integer) is NAN
 470  *
 471  * Accuracy:
 472  *      pow(x,y) returns x**y nearly rounded. In particular
 473  *                      pow(integer,integer)
 474  *      always returns the correct integer provided it is
 475  *      representable.
 476  *
 477  * Constants :
 478  * The hexadecimal values are the intended ones for the following
 479  * constants. The decimal values may be used, provided that the
 480  * compiler will convert from decimal to binary accurately enough
 481  * to produce the hexadecimal values shown.
 482  */
 483 
 484 static const double
 485 bp[] = {1.0, 1.5,},
 486   dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
 487     dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
 488       zeroX    =  0.0,
 489         two     =  2.0,
 490         two53   =  9007199254740992.0,  /* 0x43400000, 0x00000000 */
 491         /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
 492         L1X  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
 493         L2X  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
 494         L3X  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
 495         L4X  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
 496         L5X  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
 497         L6X  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
 498         lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
 499         lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
 500         lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
 501         ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
 502         cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
 503         cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
 504         cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
 505         ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
 506         ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
 507         ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
 508 
 509 double __ieee754_pow(double x, double y) {
 510   double z,ax,z_h,z_l,p_h,p_l;
 511   double y1,t1,t2,r,s,t,u,v,w;
 512   int i0,i1,i,j,k,yisint,n;
 513   int hx,hy,ix,iy;
 514   unsigned lx,ly;
 515 
 516   i0 = ((*(int*)&one)>>29)^1; i1=1-i0;
 517   hx = __HI(x); lx = __LO(x);
 518   hy = __HI(y); ly = __LO(y);
 519   ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
 520 
 521   /* y==zero: x**0 = 1 */
 522   if((iy|ly)==0) return one;
 523 
 524   /* +-NaN return x+y */
 525   if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
 526      iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
 527     return x+y;
 528 
 529   /* determine if y is an odd int when x < 0
 530    * yisint = 0 ... y is not an integer
 531    * yisint = 1 ... y is an odd int
 532    * yisint = 2 ... y is an even int
 533    */
 534   yisint  = 0;
 535   if(hx<0) {
 536     if(iy>=0x43400000) yisint = 2; /* even integer y */
 537     else if(iy>=0x3ff00000) {
 538       k = (iy>>20)-0x3ff;          /* exponent */
 539       if(k>20) {
 540         j = ly>>(52-k);
 541         if((unsigned)(j<<(52-k))==ly) yisint = 2-(j&1);
 542       } else if(ly==0) {
 543         j = iy>>(20-k);
 544         if((j<<(20-k))==iy) yisint = 2-(j&1);
 545       }
 546     }
 547   }
 548 
 549   /* special value of y */
 550   if(ly==0) {
 551     if (iy==0x7ff00000) {       /* y is +-inf */
 552       if(((ix-0x3ff00000)|lx)==0)
 553         return  y - y;  /* inf**+-1 is NaN */
 554       else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
 555         return (hy>=0)? y: zeroX;
 556       else                      /* (|x|<1)**-,+inf = inf,0 */
 557         return (hy<0)?-y: zeroX;
 558     }
 559     if(iy==0x3ff00000) {        /* y is  +-1 */
 560       if(hy<0) return one/x; else return x;
 561     }
 562     if(hy==0x40000000) return x*x; /* y is  2 */
 563     if(hy==0x3fe00000) {        /* y is  0.5 */
 564       if(hx>=0) /* x >= +0 */
 565         return sqrt(x);
 566     }
 567   }
 568 
 569   ax   = fabsd(x);
 570   /* special value of x */
 571   if(lx==0) {
 572     if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
 573       z = ax;                   /*x is +-0,+-inf,+-1*/
 574       if(hy<0) z = one/z;       /* z = (1/|x|) */
 575       if(hx<0) {
 576         if(((ix-0x3ff00000)|yisint)==0) {
 577 #ifdef CAN_USE_NAN_DEFINE
 578           z = NAN;
 579 #else
 580           z = (z-z)/(z-z); /* (-1)**non-int is NaN */
 581 #endif
 582         } else if(yisint==1)
 583           z = -1.0*z;           /* (x<0)**odd = -(|x|**odd) */
 584       }
 585       return z;
 586     }
 587   }
 588 
 589   n = (hx>>31)+1;
 590 
 591   /* (x<0)**(non-int) is NaN */
 592   if((n|yisint)==0)
 593 #ifdef CAN_USE_NAN_DEFINE
 594     return NAN;
 595 #else
 596     return (x-x)/(x-x);
 597 #endif
 598 
 599   s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
 600   if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
 601 
 602   /* |y| is huge */
 603   if(iy>0x41e00000) { /* if |y| > 2**31 */
 604     if(iy>0x43f00000){  /* if |y| > 2**64, must o/uflow */
 605       if(ix<=0x3fefffff) return (hy<0)? hugeX*hugeX:tiny*tiny;
 606       if(ix>=0x3ff00000) return (hy>0)? hugeX*hugeX:tiny*tiny;
 607     }
 608     /* over/underflow if x is not close to one */
 609     if(ix<0x3fefffff) return (hy<0)? s*hugeX*hugeX:s*tiny*tiny;
 610     if(ix>0x3ff00000) return (hy>0)? s*hugeX*hugeX:s*tiny*tiny;
 611     /* now |1-x| is tiny <= 2**-20, suffice to compute
 612        log(x) by x-x^2/2+x^3/3-x^4/4 */
 613     t = ax-one;         /* t has 20 trailing zeros */
 614     w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
 615     u = ivln2_h*t;      /* ivln2_h has 21 sig. bits */
 616     v = t*ivln2_l-w*ivln2;
 617     t1 = u+v;
 618     __LO(t1) = 0;
 619     t2 = v-(t1-u);
 620   } else {
 621     double ss,s2,s_h,s_l,t_h,t_l;
 622     n = 0;
 623     /* take care subnormal number */
 624     if(ix<0x00100000)
 625       {ax *= two53; n -= 53; ix = __HI(ax); }
 626     n  += ((ix)>>20)-0x3ff;
 627     j  = ix&0x000fffff;
 628     /* determine interval */
 629     ix = j|0x3ff00000;          /* normalize ix */
 630     if(j<=0x3988E) k=0;         /* |x|<sqrt(3/2) */
 631     else if(j<0xBB67A) k=1;     /* |x|<sqrt(3)   */
 632     else {k=0;n+=1;ix -= 0x00100000;}
 633     __HI(ax) = ix;
 634 
 635     /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
 636     u = ax-bp[k];               /* bp[0]=1.0, bp[1]=1.5 */
 637     v = one/(ax+bp[k]);
 638     ss = u*v;
 639     s_h = ss;
 640     __LO(s_h) = 0;
 641     /* t_h=ax+bp[k] High */
 642     t_h = zeroX;
 643     __HI(t_h)=((ix>>1)|0x20000000)+0x00080000+(k<<18);
 644     t_l = ax - (t_h-bp[k]);
 645     s_l = v*((u-s_h*t_h)-s_h*t_l);
 646     /* compute log(ax) */
 647     s2 = ss*ss;
 648     r = s2*s2*(L1X+s2*(L2X+s2*(L3X+s2*(L4X+s2*(L5X+s2*L6X)))));
 649     r += s_l*(s_h+ss);
 650     s2  = s_h*s_h;
 651     t_h = 3.0+s2+r;
 652     __LO(t_h) = 0;
 653     t_l = r-((t_h-3.0)-s2);
 654     /* u+v = ss*(1+...) */
 655     u = s_h*t_h;
 656     v = s_l*t_h+t_l*ss;
 657     /* 2/(3log2)*(ss+...) */
 658     p_h = u+v;
 659     __LO(p_h) = 0;
 660     p_l = v-(p_h-u);
 661     z_h = cp_h*p_h;             /* cp_h+cp_l = 2/(3*log2) */
 662     z_l = cp_l*p_h+p_l*cp+dp_l[k];
 663     /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
 664     t = (double)n;
 665     t1 = (((z_h+z_l)+dp_h[k])+t);
 666     __LO(t1) = 0;
 667     t2 = z_l-(((t1-t)-dp_h[k])-z_h);
 668   }
 669 
 670   /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
 671   y1  = y;
 672   __LO(y1) = 0;
 673   p_l = (y-y1)*t1+y*t2;
 674   p_h = y1*t1;
 675   z = p_l+p_h;
 676   j = __HI(z);
 677   i = __LO(z);
 678   if (j>=0x40900000) {                          /* z >= 1024 */
 679     if(((j-0x40900000)|i)!=0)                   /* if z > 1024 */
 680       return s*hugeX*hugeX;                     /* overflow */
 681     else {
 682       if(p_l+ovt>z-p_h) return s*hugeX*hugeX;   /* overflow */
 683     }
 684   } else if((j&0x7fffffff)>=0x4090cc00 ) {      /* z <= -1075 */
 685     if(((j-0xc090cc00)|i)!=0)           /* z < -1075 */
 686       return s*tiny*tiny;               /* underflow */
 687     else {
 688       if(p_l<=z-p_h) return s*tiny*tiny;        /* underflow */
 689     }
 690   }
 691   /*
 692    * compute 2**(p_h+p_l)
 693    */
 694   i = j&0x7fffffff;
 695   k = (i>>20)-0x3ff;
 696   n = 0;
 697   if(i>0x3fe00000) {            /* if |z| > 0.5, set n = [z+0.5] */
 698     n = j+(0x00100000>>(k+1));
 699     k = ((n&0x7fffffff)>>20)-0x3ff;     /* new k for n */
 700     t = zeroX;
 701     __HI(t) = (n&~(0x000fffff>>k));
 702     n = ((n&0x000fffff)|0x00100000)>>(20-k);
 703     if(j<0) n = -n;
 704     p_h -= t;
 705   }
 706   t = p_l+p_h;
 707   __LO(t) = 0;
 708   u = t*lg2_h;
 709   v = (p_l-(t-p_h))*lg2+t*lg2_l;
 710   z = u+v;
 711   w = v-(z-u);
 712   t  = z*z;
 713   t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
 714   r  = (z*t1)/(t1-two)-(w+z*w);
 715   z  = one-(r-z);
 716   j  = __HI(z);
 717   j += (n<<20);
 718   if((j>>20)<=0) z = scalbnA(z,n);       /* subnormal output */
 719   else __HI(z) += (n<<20);
 720   return s*z;
 721 }
 722 
 723 
 724 JRT_LEAF(jdouble, SharedRuntime::dpow(jdouble x, jdouble y))
 725   return __ieee754_pow(x, y);
 726 JRT_END
 727 
 728 #ifdef WIN32
 729 # pragma optimize ( "", on )
 730 #endif