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 warning( disable: 4748 ) // /GS can not protect parameters and local variables from local buffer overrun because optimizations are disabled in function
  44 # pragma optimize ( "", off )
  45 #endif
  46 
  47 #include "runtime/sharedRuntimeMath.hpp"
  48 
  49 /* __ieee754_log(x)
  50  * Return the logarithm of x
  51  *
  52  * Method :
  53  *   1. Argument Reduction: find k and f such that
  54  *                    x = 2^k * (1+f),
  55  *       where  sqrt(2)/2 < 1+f < sqrt(2) .
  56  *
  57  *   2. Approximation of log(1+f).
  58  *    Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  59  *             = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  60  *             = 2s + s*R
  61  *      We use a special Reme algorithm on [0,0.1716] to generate
  62  *    a polynomial of degree 14 to approximate R The maximum error
  63  *    of this polynomial approximation is bounded by 2**-58.45. In
  64  *    other words,
  65  *                    2      4      6      8      10      12      14
  66  *        R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
  67  *    (the values of Lg1 to Lg7 are listed in the program)
  68  *    and
  69  *        |      2          14          |     -58.45
  70  *        | Lg1*s +...+Lg7*s    -  R(z) | <= 2
  71  *        |                             |
  72  *    Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  73  *    In order to guarantee error in log below 1ulp, we compute log
  74  *    by
  75  *            log(1+f) = f - s*(f - R)        (if f is not too large)
  76  *            log(1+f) = f - (hfsq - s*(hfsq+R)).     (better accuracy)
  77  *
  78  *    3. Finally,  log(x) = k*ln2 + log(1+f).
  79  *                        = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  80  *       Here ln2 is split into two floating point number:
  81  *                    ln2_hi + ln2_lo,
  82  *       where n*ln2_hi is always exact for |n| < 2000.
  83  *
  84  * Special cases:
  85  *    log(x) is NaN with signal if x < 0 (including -INF) ;
  86  *    log(+INF) is +INF; log(0) is -INF with signal;
  87  *    log(NaN) is that NaN with no signal.
  88  *
  89  * Accuracy:
  90  *    according to an error analysis, the error is always less than
  91  *    1 ulp (unit in the last place).
  92  *
  93  * Constants:
  94  * The hexadecimal values are the intended ones for the following
  95  * constants. The decimal values may be used, provided that the
  96  * compiler will convert from decimal to binary accurately enough
  97  * to produce the hexadecimal values shown.
  98  */
  99 
 100 static const double
 101 ln2_hi  =  6.93147180369123816490e-01,        /* 3fe62e42 fee00000 */
 102   ln2_lo  =  1.90821492927058770002e-10,        /* 3dea39ef 35793c76 */
 103   Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
 104   Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
 105   Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
 106   Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
 107   Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
 108   Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
 109   Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
 110 
 111 static double zero = 0.0;
 112 
 113 static double __ieee754_log(double x) {
 114   double hfsq,f,s,z,R,w,t1,t2,dk;
 115   int k,hx,i,j;
 116   unsigned lx;
 117 
 118   hx = high(x);               /* high word of x */
 119   lx = low(x);                /* low  word of x */
 120 
 121   k=0;
 122   if (hx < 0x00100000) {                   /* x < 2**-1022  */
 123     if (((hx&0x7fffffff)|lx)==0)
 124       return -two54/zero;             /* log(+-0)=-inf */
 125     if (hx<0) return (x-x)/zero;   /* log(-#) = NaN */
 126     k -= 54; x *= two54; /* subnormal number, scale up x */
 127     hx = high(x);             /* high word of x */
 128   }
 129   if (hx >= 0x7ff00000) return x+x;
 130   k += (hx>>20)-1023;
 131   hx &= 0x000fffff;
 132   i = (hx+0x95f64)&0x100000;
 133   set_high(&x, hx|(i^0x3ff00000)); /* normalize x or x/2 */
 134   k += (i>>20);
 135   f = x-1.0;
 136   if((0x000fffff&(2+hx))<3) {  /* |f| < 2**-20 */
 137     if(f==zero) {
 138       if (k==0) return zero;
 139       else {dk=(double)k; return dk*ln2_hi+dk*ln2_lo;}
 140     }
 141     R = f*f*(0.5-0.33333333333333333*f);
 142     if(k==0) return f-R; else {dk=(double)k;
 143     return dk*ln2_hi-((R-dk*ln2_lo)-f);}
 144   }
 145   s = f/(2.0+f);
 146   dk = (double)k;
 147   z = s*s;
 148   i = hx-0x6147a;
 149   w = z*z;
 150   j = 0x6b851-hx;
 151   t1= w*(Lg2+w*(Lg4+w*Lg6));
 152   t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
 153   i |= j;
 154   R = t2+t1;
 155   if(i>0) {
 156     hfsq=0.5*f*f;
 157     if(k==0) return f-(hfsq-s*(hfsq+R)); else
 158       return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
 159   } else {
 160     if(k==0) return f-s*(f-R); else
 161       return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
 162   }
 163 }
 164 
 165 JRT_LEAF(jdouble, SharedRuntime::dlog(jdouble x))
 166   return __ieee754_log(x);
 167 JRT_END
 168 
 169 /* __ieee754_log10(x)
 170  * Return the base 10 logarithm of x
 171  *
 172  * Method :
 173  *    Let log10_2hi = leading 40 bits of log10(2) and
 174  *        log10_2lo = log10(2) - log10_2hi,
 175  *        ivln10   = 1/log(10) rounded.
 176  *    Then
 177  *            n = ilogb(x),
 178  *            if(n<0)  n = n+1;
 179  *            x = scalbn(x,-n);
 180  *            log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))
 181  *
 182  * Note 1:
 183  *    To guarantee log10(10**n)=n, where 10**n is normal, the rounding
 184  *    mode must set to Round-to-Nearest.
 185  * Note 2:
 186  *    [1/log(10)] rounded to 53 bits has error  .198   ulps;
 187  *    log10 is monotonic at all binary break points.
 188  *
 189  * Special cases:
 190  *    log10(x) is NaN with signal if x < 0;
 191  *    log10(+INF) is +INF with no signal; log10(0) is -INF with signal;
 192  *    log10(NaN) is that NaN with no signal;
 193  *    log10(10**N) = N  for N=0,1,...,22.
 194  *
 195  * Constants:
 196  * The hexadecimal values are the intended ones for the following constants.
 197  * The decimal values may be used, provided that the compiler will convert
 198  * from decimal to binary accurately enough to produce the hexadecimal values
 199  * shown.
 200  */
 201 
 202 static const double
 203 ivln10     =  4.34294481903251816668e-01, /* 0x3FDBCB7B, 0x1526E50E */
 204   log10_2hi  =  3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
 205   log10_2lo  =  3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */
 206 
 207 static double __ieee754_log10(double x) {
 208   double y,z;
 209   int i,k,hx;
 210   unsigned lx;
 211 
 212   hx = high(x);       /* high word of x */
 213   lx = low(x);        /* low word of x */
 214 
 215   k=0;
 216   if (hx < 0x00100000) {                  /* x < 2**-1022  */
 217     if (((hx&0x7fffffff)|lx)==0)
 218       return -two54/zero;             /* log(+-0)=-inf */
 219     if (hx<0) return (x-x)/zero;        /* log(-#) = NaN */
 220     k -= 54; x *= two54; /* subnormal number, scale up x */
 221     hx = high(x);                /* high word of x */
 222   }
 223   if (hx >= 0x7ff00000) return x+x;
 224   k += (hx>>20)-1023;
 225   i  = ((unsigned)k&0x80000000)>>31;
 226   hx = (hx&0x000fffff)|((0x3ff-i)<<20);
 227   y  = (double)(k+i);
 228   set_high(&x, hx);
 229   z  = y*log10_2lo + ivln10*__ieee754_log(x);
 230   return  z+y*log10_2hi;
 231 }
 232 
 233 JRT_LEAF(jdouble, SharedRuntime::dlog10(jdouble x))
 234   return __ieee754_log10(x);
 235 JRT_END
 236 
 237 
 238 /* __ieee754_exp(x)
 239  * Returns the exponential of x.
 240  *
 241  * Method
 242  *   1. Argument reduction:
 243  *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
 244  *      Given x, find r and integer k such that
 245  *
 246  *               x = k*ln2 + r,  |r| <= 0.5*ln2.
 247  *
 248  *      Here r will be represented as r = hi-lo for better
 249  *      accuracy.
 250  *
 251  *   2. Approximation of exp(r) by a special rational function on
 252  *      the interval [0,0.34658]:
 253  *      Write
 254  *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
 255  *      We use a special Reme algorithm on [0,0.34658] to generate
 256  *      a polynomial of degree 5 to approximate R. The maximum error
 257  *      of this polynomial approximation is bounded by 2**-59. In
 258  *      other words,
 259  *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
 260  *      (where z=r*r, and the values of P1 to P5 are listed below)
 261  *      and
 262  *          |                  5          |     -59
 263  *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
 264  *          |                             |
 265  *      The computation of exp(r) thus becomes
 266  *                             2*r
 267  *              exp(r) = 1 + -------
 268  *                            R - r
 269  *                                 r*R1(r)
 270  *                     = 1 + r + ----------- (for better accuracy)
 271  *                                2 - R1(r)
 272  *      where
 273  *                               2       4             10
 274  *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
 275  *
 276  *   3. Scale back to obtain exp(x):
 277  *      From step 1, we have
 278  *         exp(x) = 2^k * exp(r)
 279  *
 280  * Special cases:
 281  *      exp(INF) is INF, exp(NaN) is NaN;
 282  *      exp(-INF) is 0, and
 283  *      for finite argument, only exp(0)=1 is exact.
 284  *
 285  * Accuracy:
 286  *      according to an error analysis, the error is always less than
 287  *      1 ulp (unit in the last place).
 288  *
 289  * Misc. info.
 290  *      For IEEE double
 291  *          if x >  7.09782712893383973096e+02 then exp(x) overflow
 292  *          if x < -7.45133219101941108420e+02 then exp(x) underflow
 293  *
 294  * Constants:
 295  * The hexadecimal values are the intended ones for the following
 296  * constants. The decimal values may be used, provided that the
 297  * compiler will convert from decimal to binary accurately enough
 298  * to produce the hexadecimal values shown.
 299  */
 300 
 301 static const double
 302 one     = 1.0,
 303   halF[2]       = {0.5,-0.5,},
 304   twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/
 305     o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
 306     u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
 307     ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
 308                   -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
 309     ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
 310                   -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
 311       invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
 312         P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
 313         P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
 314         P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
 315         P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
 316         P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
 317 
 318 static double __ieee754_exp(double x) {
 319   double y,hi=0,lo=0,c,t;
 320   int k=0,xsb;
 321   unsigned hx;
 322 
 323   hx  = high(x);                /* high word of x */
 324   xsb = (hx>>31)&1;             /* sign bit of x */
 325   hx &= 0x7fffffff;             /* high word of |x| */
 326 
 327   /* filter out non-finite argument */
 328   if(hx >= 0x40862E42) {                        /* if |x|>=709.78... */
 329     if(hx>=0x7ff00000) {
 330       if(((hx&0xfffff)|low(x))!=0)
 331         return x+x;             /* NaN */
 332       else return (xsb==0)? x:0.0;      /* exp(+-inf)={inf,0} */
 333     }
 334     if(x > o_threshold) return hugeX*hugeX; /* overflow */
 335     if(x < u_threshold) return twom1000*twom1000; /* underflow */
 336   }
 337 
 338   /* argument reduction */
 339   if(hx > 0x3fd62e42) {         /* if  |x| > 0.5 ln2 */
 340     if(hx < 0x3FF0A2B2) {       /* and |x| < 1.5 ln2 */
 341       hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
 342     } else {
 343       k  = (int)(invln2*x+halF[xsb]);
 344       t  = k;
 345       hi = x - t*ln2HI[0];      /* t*ln2HI is exact here */
 346       lo = t*ln2LO[0];
 347     }
 348     x  = hi - lo;
 349   }
 350   else if(hx < 0x3e300000)  {   /* when |x|<2**-28 */
 351     if(hugeX+x>one) return one+x;/* trigger inexact */
 352   }
 353   else k = 0;
 354 
 355   /* x is now in primary range */
 356   t  = x*x;
 357   c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
 358   if(k==0)      return one-((x*c)/(c-2.0)-x);
 359   else          y = one-((lo-(x*c)/(2.0-c))-hi);
 360   if(k >= -1021) {
 361     set_high(&y, high(y) + (k<<20)); /* add k to y's exponent */
 362     return y;
 363   } else {
 364     set_high(&y, high(y) + ((k+1000)<<20)); /* add k to y's exponent */
 365     return y*twom1000;
 366   }
 367 }
 368 
 369 JRT_LEAF(jdouble, SharedRuntime::dexp(jdouble x))
 370   return __ieee754_exp(x);
 371 JRT_END
 372 
 373 /* __ieee754_pow(x,y) return x**y
 374  *
 375  *                    n
 376  * Method:  Let x =  2   * (1+f)
 377  *      1. Compute and return log2(x) in two pieces:
 378  *              log2(x) = w1 + w2,
 379  *         where w1 has 53-24 = 29 bit trailing zeros.
 380  *      2. Perform y*log2(x) = n+y' by simulating muti-precision
 381  *         arithmetic, where |y'|<=0.5.
 382  *      3. Return x**y = 2**n*exp(y'*log2)
 383  *
 384  * Special cases:
 385  *      1.  (anything) ** 0  is 1
 386  *      2.  (anything) ** 1  is itself
 387  *      3.  (anything) ** NAN is NAN
 388  *      4.  NAN ** (anything except 0) is NAN
 389  *      5.  +-(|x| > 1) **  +INF is +INF
 390  *      6.  +-(|x| > 1) **  -INF is +0
 391  *      7.  +-(|x| < 1) **  +INF is +0
 392  *      8.  +-(|x| < 1) **  -INF is +INF
 393  *      9.  +-1         ** +-INF is NAN
 394  *      10. +0 ** (+anything except 0, NAN)               is +0
 395  *      11. -0 ** (+anything except 0, NAN, odd integer)  is +0
 396  *      12. +0 ** (-anything except 0, NAN)               is +INF
 397  *      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
 398  *      14. -0 ** (odd integer) = -( +0 ** (odd integer) )
 399  *      15. +INF ** (+anything except 0,NAN) is +INF
 400  *      16. +INF ** (-anything except 0,NAN) is +0
 401  *      17. -INF ** (anything)  = -0 ** (-anything)
 402  *      18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
 403  *      19. (-anything except 0 and inf) ** (non-integer) is NAN
 404  *
 405  * Accuracy:
 406  *      pow(x,y) returns x**y nearly rounded. In particular
 407  *                      pow(integer,integer)
 408  *      always returns the correct integer provided it is
 409  *      representable.
 410  *
 411  * Constants :
 412  * The hexadecimal values are the intended ones for the following
 413  * constants. The decimal values may be used, provided that the
 414  * compiler will convert from decimal to binary accurately enough
 415  * to produce the hexadecimal values shown.
 416  */
 417 
 418 static const double
 419 bp[] = {1.0, 1.5,},
 420   dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
 421     dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
 422       zeroX    =  0.0,
 423         two     =  2.0,
 424         two53   =  9007199254740992.0,  /* 0x43400000, 0x00000000 */
 425         /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
 426         L1X  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
 427         L2X  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
 428         L3X  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
 429         L4X  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
 430         L5X  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
 431         L6X  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
 432         lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
 433         lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
 434         lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
 435         ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
 436         cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
 437         cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
 438         cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
 439         ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
 440         ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
 441         ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
 442 
 443 double __ieee754_pow(double x, double y) {
 444   double z,ax,z_h,z_l,p_h,p_l;
 445   double y1,t1,t2,r,s,t,u,v,w;
 446   int i0,i1,i,j,k,yisint,n;
 447   int hx,hy,ix,iy;
 448   unsigned lx,ly;
 449 
 450   i0 = ((*(int*)&one)>>29)^1; i1=1-i0;
 451   hx = high(x); lx = low(x);
 452   hy = high(y); ly = low(y);
 453   ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
 454 
 455   /* y==zero: x**0 = 1 */
 456   if((iy|ly)==0) return one;
 457 
 458   /* +-NaN return x+y */
 459   if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
 460      iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
 461     return x+y;
 462 
 463   /* determine if y is an odd int when x < 0
 464    * yisint = 0 ... y is not an integer
 465    * yisint = 1 ... y is an odd int
 466    * yisint = 2 ... y is an even int
 467    */
 468   yisint  = 0;
 469   if(hx<0) {
 470     if(iy>=0x43400000) yisint = 2; /* even integer y */
 471     else if(iy>=0x3ff00000) {
 472       k = (iy>>20)-0x3ff;          /* exponent */
 473       if(k>20) {
 474         j = ly>>(52-k);
 475         if((unsigned)(j<<(52-k))==ly) yisint = 2-(j&1);
 476       } else if(ly==0) {
 477         j = iy>>(20-k);
 478         if((j<<(20-k))==iy) yisint = 2-(j&1);
 479       }
 480     }
 481   }
 482 
 483   /* special value of y */
 484   if(ly==0) {
 485     if (iy==0x7ff00000) {       /* y is +-inf */
 486       if(((ix-0x3ff00000)|lx)==0)
 487         return  y - y;  /* inf**+-1 is NaN */
 488       else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
 489         return (hy>=0)? y: zeroX;
 490       else                      /* (|x|<1)**-,+inf = inf,0 */
 491         return (hy<0)?-y: zeroX;
 492     }
 493     if(iy==0x3ff00000) {        /* y is  +-1 */
 494       if(hy<0) return one/x; else return x;
 495     }
 496     if(hy==0x40000000) return x*x; /* y is  2 */
 497     if(hy==0x3fe00000) {        /* y is  0.5 */
 498       if(hx>=0) /* x >= +0 */
 499         return sqrt(x);
 500     }
 501   }
 502 
 503   ax   = fabsd(x);
 504   /* special value of x */
 505   if(lx==0) {
 506     if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
 507       z = ax;                   /*x is +-0,+-inf,+-1*/
 508       if(hy<0) z = one/z;       /* z = (1/|x|) */
 509       if(hx<0) {
 510         if(((ix-0x3ff00000)|yisint)==0) {
 511 #ifdef CAN_USE_NAN_DEFINE
 512           z = NAN;
 513 #else
 514           z = (z-z)/(z-z); /* (-1)**non-int is NaN */
 515 #endif
 516         } else if(yisint==1)
 517           z = -1.0*z;           /* (x<0)**odd = -(|x|**odd) */
 518       }
 519       return z;
 520     }
 521   }
 522 
 523   n = (hx>>31)+1;
 524 
 525   /* (x<0)**(non-int) is NaN */
 526   if((n|yisint)==0)
 527 #ifdef CAN_USE_NAN_DEFINE
 528     return NAN;
 529 #else
 530     return (x-x)/(x-x);
 531 #endif
 532 
 533   s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
 534   if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
 535 
 536   /* |y| is huge */
 537   if(iy>0x41e00000) { /* if |y| > 2**31 */
 538     if(iy>0x43f00000){  /* if |y| > 2**64, must o/uflow */
 539       if(ix<=0x3fefffff) return (hy<0)? hugeX*hugeX:tiny*tiny;
 540       if(ix>=0x3ff00000) return (hy>0)? hugeX*hugeX:tiny*tiny;
 541     }
 542     /* over/underflow if x is not close to one */
 543     if(ix<0x3fefffff) return (hy<0)? s*hugeX*hugeX:s*tiny*tiny;
 544     if(ix>0x3ff00000) return (hy>0)? s*hugeX*hugeX:s*tiny*tiny;
 545     /* now |1-x| is tiny <= 2**-20, suffice to compute
 546        log(x) by x-x^2/2+x^3/3-x^4/4 */
 547     t = ax-one;         /* t has 20 trailing zeros */
 548     w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
 549     u = ivln2_h*t;      /* ivln2_h has 21 sig. bits */
 550     v = t*ivln2_l-w*ivln2;
 551     t1 = u+v;
 552     set_low(&t1, 0);
 553     t2 = v-(t1-u);
 554   } else {
 555     double ss,s2,s_h,s_l,t_h,t_l;
 556     n = 0;
 557     /* take care subnormal number */
 558     if(ix<0x00100000)
 559       {ax *= two53; n -= 53; ix = high(ax); }
 560     n  += ((ix)>>20)-0x3ff;
 561     j  = ix&0x000fffff;
 562     /* determine interval */
 563     ix = j|0x3ff00000;          /* normalize ix */
 564     if(j<=0x3988E) k=0;         /* |x|<sqrt(3/2) */
 565     else if(j<0xBB67A) k=1;     /* |x|<sqrt(3)   */
 566     else {k=0;n+=1;ix -= 0x00100000;}
 567     set_high(&ax, ix);
 568 
 569     /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
 570     u = ax-bp[k];               /* bp[0]=1.0, bp[1]=1.5 */
 571     v = one/(ax+bp[k]);
 572     ss = u*v;
 573     s_h = ss;
 574     set_low(&s_h, 0);
 575     /* t_h=ax+bp[k] High */
 576     t_h = zeroX;
 577     set_high(&t_h, ((ix>>1)|0x20000000)+0x00080000+(k<<18));
 578     t_l = ax - (t_h-bp[k]);
 579     s_l = v*((u-s_h*t_h)-s_h*t_l);
 580     /* compute log(ax) */
 581     s2 = ss*ss;
 582     r = s2*s2*(L1X+s2*(L2X+s2*(L3X+s2*(L4X+s2*(L5X+s2*L6X)))));
 583     r += s_l*(s_h+ss);
 584     s2  = s_h*s_h;
 585     t_h = 3.0+s2+r;
 586     set_low(&t_h, 0);
 587     t_l = r-((t_h-3.0)-s2);
 588     /* u+v = ss*(1+...) */
 589     u = s_h*t_h;
 590     v = s_l*t_h+t_l*ss;
 591     /* 2/(3log2)*(ss+...) */
 592     p_h = u+v;
 593     set_low(&p_h, 0);
 594     p_l = v-(p_h-u);
 595     z_h = cp_h*p_h;             /* cp_h+cp_l = 2/(3*log2) */
 596     z_l = cp_l*p_h+p_l*cp+dp_l[k];
 597     /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
 598     t = (double)n;
 599     t1 = (((z_h+z_l)+dp_h[k])+t);
 600     set_low(&t1, 0);
 601     t2 = z_l-(((t1-t)-dp_h[k])-z_h);
 602   }
 603 
 604   /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
 605   y1  = y;
 606   set_low(&y1, 0);
 607   p_l = (y-y1)*t1+y*t2;
 608   p_h = y1*t1;
 609   z = p_l+p_h;
 610   j = high(z);
 611   i = low(z);
 612   if (j>=0x40900000) {                          /* z >= 1024 */
 613     if(((j-0x40900000)|i)!=0)                   /* if z > 1024 */
 614       return s*hugeX*hugeX;                     /* overflow */
 615     else {
 616       if(p_l+ovt>z-p_h) return s*hugeX*hugeX;   /* overflow */
 617     }
 618   } else if((j&0x7fffffff)>=0x4090cc00 ) {      /* z <= -1075 */
 619     if(((j-0xc090cc00)|i)!=0)           /* z < -1075 */
 620       return s*tiny*tiny;               /* underflow */
 621     else {
 622       if(p_l<=z-p_h) return s*tiny*tiny;        /* underflow */
 623     }
 624   }
 625   /*
 626    * compute 2**(p_h+p_l)
 627    */
 628   i = j&0x7fffffff;
 629   k = (i>>20)-0x3ff;
 630   n = 0;
 631   if(i>0x3fe00000) {            /* if |z| > 0.5, set n = [z+0.5] */
 632     n = j+(0x00100000>>(k+1));
 633     k = ((n&0x7fffffff)>>20)-0x3ff;     /* new k for n */
 634     t = zeroX;
 635     set_high(&t, (n&~(0x000fffff>>k)));
 636     n = ((n&0x000fffff)|0x00100000)>>(20-k);
 637     if(j<0) n = -n;
 638     p_h -= t;
 639   }
 640   t = p_l+p_h;
 641   set_low(&t, 0);
 642   u = t*lg2_h;
 643   v = (p_l-(t-p_h))*lg2+t*lg2_l;
 644   z = u+v;
 645   w = v-(z-u);
 646   t  = z*z;
 647   t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
 648   r  = (z*t1)/(t1-two)-(w+z*w);
 649   z  = one-(r-z);
 650   j  = high(z);
 651   j += (n<<20);
 652   if((j>>20)<=0) z = scalbnA(z,n);       /* subnormal output */
 653   else set_high(&z, high(z) + (n<<20));
 654   return s*z;
 655 }
 656 
 657 
 658 JRT_LEAF(jdouble, SharedRuntime::dpow(jdouble x, jdouble y))
 659   return __ieee754_pow(x, y);
 660 JRT_END
 661 
 662 #ifdef WIN32
 663 # pragma optimize ( "", on )
 664 #endif