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 /*

  * Copyright (c) 2005, 2014, Oracle and/or its affiliates. All rights reserved.

  * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.

  *

  * This code is free software; you can redistribute it and/or modify it

  * under the terms of the GNU General Public License version 2 only, as

  * published by the Free Software Foundation.

  *

  * This code is distributed in the hope that it will be useful, but WITHOUT

  * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or

  * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License

  * version 2 for more details (a copy is included in the LICENSE file that

  * accompanied this code).

  *

  * You should have received a copy of the GNU General Public License version

  * 2 along with this work; if not, write to the Free Software Foundation,

  * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.

  *

  * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA

  * or visit www.oracle.com if you need additional information or have any

  * questions.

  *

  */

 #include "precompiled.hpp"

 #include "prims/jni.h"

 #include "runtime/interfaceSupport.hpp"

 #include "runtime/sharedRuntime.hpp"

 // This file contains copies of the fdlibm routines used by

 // StrictMath. It turns out that it is almost always required to use

 // these runtime routines; the Intel CPU doesn't meet the Java

 // specification for sin/cos outside a certain limited argument range,

 // and the SPARC CPU doesn't appear to have sin/cos instructions. It

 // also turns out that avoiding the indirect call through function

 // pointer out to libjava.so in SharedRuntime speeds these routines up

 // by roughly 15% on both Win32/x86 and Solaris/SPARC.

 // Enabling optimizations in this file causes incorrect code to be

 // generated; can not figure out how to turn down optimization for one

 // file in the IDE on Windows

 #ifdef WIN32

 # pragma warning( disable: 4748 ) // /GS can not protect parameters and local variables from local buffer overrun because optimizations are disabled in function

 # pragma optimize ( "", off )

 #endif

 #include "runtime/sharedRuntimeMath.hpp"

 /* __ieee754_log(x)

  * Return the logrithm of x

  *

  * Method :

  *   1. Argument Reduction: find k and f such that

  *                    x = 2^k * (1+f),

  *       where  sqrt(2)/2 < 1+f < sqrt(2) .

  *

  *   2. Approximation of log(1+f).

  *    Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)

  *             = 2s + 2/3 s**3 + 2/5 s**5 + .....,

  *             = 2s + s*R

  *      We use a special Reme algorithm on [0,0.1716] to generate

  *    a polynomial of degree 14 to approximate R The maximum error

  *    of this polynomial approximation is bounded by 2**-58.45. In

  *    other words,

  *                    2      4      6      8      10      12      14

  *        R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s

  *    (the values of Lg1 to Lg7 are listed in the program)

  *    and

  *        |      2          14          |     -58.45

  *        | Lg1*s +...+Lg7*s    -  R(z) | <= 2

  *        |                             |

  *    Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.

  *    In order to guarantee error in log below 1ulp, we compute log

  *    by

  *            log(1+f) = f - s*(f - R)        (if f is not too large)

  *            log(1+f) = f - (hfsq - s*(hfsq+R)).     (better accuracy)

  *

  *    3. Finally,  log(x) = k*ln2 + log(1+f).

  *                        = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))

  *       Here ln2 is split into two floating point number:

  *                    ln2_hi + ln2_lo,

  *       where n*ln2_hi is always exact for |n| < 2000.

  *

  * Special cases:

  *    log(x) is NaN with signal if x < 0 (including -INF) ;

  *    log(+INF) is +INF; log(0) is -INF with signal;

  *    log(NaN) is that NaN with no signal.

  *

  * Accuracy:

  *    according to an error analysis, the error is always less than

  *    1 ulp (unit in the last place).

  *

  * Constants:

  * The hexadecimal values are the intended ones for the following

  * constants. The decimal values may be used, provided that the

  * compiler will convert from decimal to binary accurately enough

  * to produce the hexadecimal values shown.

  */

 static const double

 ln2_hi  =  6.93147180369123816490e-01,        /* 3fe62e42 fee00000 */

   ln2_lo  =  1.90821492927058770002e-10,        /* 3dea39ef 35793c76 */

   Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */

   Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */

   Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */

   Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */

   Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */

   Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */

   Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */

 static double zero = 0.0;

 static double __ieee754_log(double x) {

   double hfsq,f,s,z,R,w,t1,t2,dk;

   int k,hx,i,j;

   unsigned lx;

   hx = high(x);               /* high word of x */

   lx = low(x);                /* low  word of x */

   k=0;

   if (hx < 0x00100000) {                   /* x < 2**-1022  */

     if (((hx&0x7fffffff)|lx)==0)

       return -two54/zero;             /* log(+-0)=-inf */

     if (hx<0) return (x-x)/zero;   /* log(-#) = NaN */

     k -= 54; x *= two54; /* subnormal number, scale up x */

     hx = high(x);             /* high word of x */

   }

   if (hx >= 0x7ff00000) return x+x;

   k += (hx>>20)-1023;

   hx &= 0x000fffff;

   i = (hx+0x95f64)&0x100000;

   set_high(&x, hx|(i^0x3ff00000)); /* normalize x or x/2 */

   k += (i>>20);

   f = x-1.0;

   if((0x000fffff&(2+hx))<3) {  /* |f| < 2**-20 */

     if(f==zero) {

       if (k==0) return zero;

       else {dk=(double)k; return dk*ln2_hi+dk*ln2_lo;}

     }

     R = f*f*(0.5-0.33333333333333333*f);

     if(k==0) return f-R; else {dk=(double)k;

     return dk*ln2_hi-((R-dk*ln2_lo)-f);}

   }

   s = f/(2.0+f);

   dk = (double)k;

   z = s*s;

   i = hx-0x6147a;

   w = z*z;

   j = 0x6b851-hx;

   t1= w*(Lg2+w*(Lg4+w*Lg6));

   t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));

   i |= j;

   R = t2+t1;

   if(i>0) {

     hfsq=0.5*f*f;

     if(k==0) return f-(hfsq-s*(hfsq+R)); else

       return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);

   } else {

     if(k==0) return f-s*(f-R); else

       return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);

   }

 }

 JRT_LEAF(jdouble, SharedRuntime::dlog(jdouble x))

   return __ieee754_log(x);

 JRT_END

 /* __ieee754_log10(x)

  * Return the base 10 logarithm of x

  *

  * Method :

  *    Let log10_2hi = leading 40 bits of log10(2) and

  *        log10_2lo = log10(2) - log10_2hi,

  *        ivln10   = 1/log(10) rounded.

  *    Then

  *            n = ilogb(x),

  *            if(n<0)  n = n+1;

  *            x = scalbn(x,-n);

  *            log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))

  *

  * Note 1:

  *    To guarantee log10(10**n)=n, where 10**n is normal, the rounding

  *    mode must set to Round-to-Nearest.

  * Note 2:

  *    [1/log(10)] rounded to 53 bits has error  .198   ulps;

  *    log10 is monotonic at all binary break points.

  *

  * Special cases:

  *    log10(x) is NaN with signal if x < 0;

  *    log10(+INF) is +INF with no signal; log10(0) is -INF with signal;

  *    log10(NaN) is that NaN with no signal;

  *    log10(10**N) = N  for N=0,1,...,22.

  *

  * Constants:

  * The hexadecimal values are the intended ones for the following constants.

  * The decimal values may be used, provided that the compiler will convert

  * from decimal to binary accurately enough to produce the hexadecimal values

  * shown.

  */

 static const double

 ivln10     =  4.34294481903251816668e-01, /* 0x3FDBCB7B, 0x1526E50E */

   log10_2hi  =  3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */

   log10_2lo  =  3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */

 static double __ieee754_log10(double x) {

   double y,z;

   int i,k,hx;

   unsigned lx;

   hx = high(x);       /* high word of x */

   lx = low(x);        /* low word of x */

   k=0;

   if (hx < 0x00100000) {                  /* x < 2**-1022  */

     if (((hx&0x7fffffff)|lx)==0)

       return -two54/zero;             /* log(+-0)=-inf */

     if (hx<0) return (x-x)/zero;        /* log(-#) = NaN */

     k -= 54; x *= two54; /* subnormal number, scale up x */

     hx = high(x);                /* high word of x */

   }

   if (hx >= 0x7ff00000) return x+x;

   k += (hx>>20)-1023;

   i  = ((unsigned)k&0x80000000)>>31;

   hx = (hx&0x000fffff)|((0x3ff-i)<<20);

   y  = (double)(k+i);

   set_high(&x, hx);

   z  = y*log10_2lo + ivln10*__ieee754_log(x);

   return  z+y*log10_2hi;

 }

 JRT_LEAF(jdouble, SharedRuntime::dlog10(jdouble x))

   return __ieee754_log10(x);

 JRT_END

 /* __ieee754_exp(x)

  * Returns the exponential of x.

  *

  * Method

  *   1. Argument reduction:

  *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.

  *      Given x, find r and integer k such that

  *

  *               x = k*ln2 + r,  |r| <= 0.5*ln2.

  *

  *      Here r will be represented as r = hi-lo for better

  *      accuracy.

  *

  *   2. Approximation of exp(r) by a special rational function on

  *      the interval [0,0.34658]:

  *      Write

  *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...

  *      We use a special Reme algorithm on [0,0.34658] to generate

  *      a polynomial of degree 5 to approximate R. The maximum error

  *      of this polynomial approximation is bounded by 2**-59. In

  *      other words,

  *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5

  *      (where z=r*r, and the values of P1 to P5 are listed below)

  *      and

  *          |                  5          |     -59

  *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2

  *          |                             |

  *      The computation of exp(r) thus becomes

  *                             2*r

  *              exp(r) = 1 + -------

  *                            R - r

  *                                 r*R1(r)

  *                     = 1 + r + ----------- (for better accuracy)

  *                                2 - R1(r)

  *      where

  *                               2       4             10

  *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).

  *

  *   3. Scale back to obtain exp(x):

  *      From step 1, we have

  *         exp(x) = 2^k * exp(r)

  *

  * Special cases:

  *      exp(INF) is INF, exp(NaN) is NaN;

  *      exp(-INF) is 0, and

  *      for finite argument, only exp(0)=1 is exact.

  *

  * Accuracy:

  *      according to an error analysis, the error is always less than

  *      1 ulp (unit in the last place).

  *

  * Misc. info.

  *      For IEEE double

  *          if x >  7.09782712893383973096e+02 then exp(x) overflow

  *          if x < -7.45133219101941108420e+02 then exp(x) underflow

  *

  * Constants:

  * The hexadecimal values are the intended ones for the following

  * constants. The decimal values may be used, provided that the

  * compiler will convert from decimal to binary accurately enough

  * to produce the hexadecimal values shown.

  */

 static const double

 one     = 1.0,

   halF[2]       = {0.5,-0.5,},

   twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/

     o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */

     u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */

     ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */

                   -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */

     ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */

                   -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */

       invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */

         P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */

         P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */

         P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */

         P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */

         P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */

 static double __ieee754_exp(double x) {

   double y,hi=0,lo=0,c,t;

   int k=0,xsb;

   unsigned hx;

   hx  = high(x);                /* high word of x */

   xsb = (hx>>31)&1;             /* sign bit of x */

   hx &= 0x7fffffff;             /* high word of |x| */

   /* filter out non-finite argument */

   if(hx >= 0x40862E42) {                        /* if |x|>=709.78... */

     if(hx>=0x7ff00000) {

       if(((hx&0xfffff)|low(x))!=0)

         return x+x;             /* NaN */

       else return (xsb==0)? x:0.0;      /* exp(+-inf)={inf,0} */

     }

     if(x > o_threshold) return hugeX*hugeX; /* overflow */

     if(x < u_threshold) return twom1000*twom1000; /* underflow */

   }

   /* argument reduction */

   if(hx > 0x3fd62e42) {         /* if  |x| > 0.5 ln2 */

     if(hx < 0x3FF0A2B2) {       /* and |x| < 1.5 ln2 */

       hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;

     } else {

       k  = (int)(invln2*x+halF[xsb]);

       t  = k;

       hi = x - t*ln2HI[0];      /* t*ln2HI is exact here */

       lo = t*ln2LO[0];

     }

     x  = hi - lo;

   }

   else if(hx < 0x3e300000)  {   /* when |x|<2**-28 */

     if(hugeX+x>one) return one+x;/* trigger inexact */

   }

   else k = 0;

   /* x is now in primary range */

   t  = x*x;

   c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));

   if(k==0)      return one-((x*c)/(c-2.0)-x);

   else          y = one-((lo-(x*c)/(2.0-c))-hi);

   if(k >= -1021) {

     set_high(&y, high(y) + (k<<20)); /* add k to y's exponent */

     return y;

   } else {

     set_high(&y, high(y) + ((k+1000)<<20)); /* add k to y's exponent */

     return y*twom1000;

   }

 }

 JRT_LEAF(jdouble, SharedRuntime::dexp(jdouble x))

   return __ieee754_exp(x);

 JRT_END

 /* __ieee754_pow(x,y) return x**y

  *

  *                    n

  * Method:  Let x =  2   * (1+f)

  *      1. Compute and return log2(x) in two pieces:

  *              log2(x) = w1 + w2,

  *         where w1 has 53-24 = 29 bit trailing zeros.

  *      2. Perform y*log2(x) = n+y' by simulating muti-precision

  *         arithmetic, where |y'|<=0.5.

  *      3. Return x**y = 2**n*exp(y'*log2)

  *

  * Special cases:

  *      1.  (anything) ** 0  is 1

  *      2.  (anything) ** 1  is itself

  *      3.  (anything) ** NAN is NAN

  *      4.  NAN ** (anything except 0) is NAN

  *      5.  +-(|x| > 1) **  +INF is +INF

  *      6.  +-(|x| > 1) **  -INF is +0

  *      7.  +-(|x| < 1) **  +INF is +0

  *      8.  +-(|x| < 1) **  -INF is +INF

  *      9.  +-1         ** +-INF is NAN

  *      10. +0 ** (+anything except 0, NAN)               is +0

  *      11. -0 ** (+anything except 0, NAN, odd integer)  is +0

  *      12. +0 ** (-anything except 0, NAN)               is +INF

  *      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF

  *      14. -0 ** (odd integer) = -( +0 ** (odd integer) )

  *      15. +INF ** (+anything except 0,NAN) is +INF

  *      16. +INF ** (-anything except 0,NAN) is +0

  *      17. -INF ** (anything)  = -0 ** (-anything)

  *      18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)

  *      19. (-anything except 0 and inf) ** (non-integer) is NAN

  *

  * Accuracy:

  *      pow(x,y) returns x**y nearly rounded. In particular

  *                      pow(integer,integer)

  *      always returns the correct integer provided it is

  *      representable.

  *

  * Constants :

  * The hexadecimal values are the intended ones for the following

  * constants. The decimal values may be used, provided that the

  * compiler will convert from decimal to binary accurately enough

  * to produce the hexadecimal values shown.

  */

 static const double

 bp[] = {1.0, 1.5,},

   dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */

     dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */

       zeroX    =  0.0,

         two     =  2.0,

         two53   =  9007199254740992.0,  /* 0x43400000, 0x00000000 */

         /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */

         L1X  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */

         L2X  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */

         L3X  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */

         L4X  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */

         L5X  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */

         L6X  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */

         lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */

         lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */

         lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */

         ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */

         cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */

         cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */

         cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/

         ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */

         ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/

         ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/

 double __ieee754_pow(double x, double y) {

   double z,ax,z_h,z_l,p_h,p_l;

   double y1,t1,t2,r,s,t,u,v,w;

   int i0,i1,i,j,k,yisint,n;

   int hx,hy,ix,iy;

   unsigned lx,ly;

   i0 = ((*(int*)&one)>>29)^1; i1=1-i0;

   hx = high(x); lx = low(x);

   hy = high(y); ly = low(y);

   ix = hx&0x7fffffff;  iy = hy&0x7fffffff;

   /* y==zero: x**0 = 1 */

   if((iy|ly)==0) return one;

   /* +-NaN return x+y */

   if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||

      iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))

     return x+y;

   /* determine if y is an odd int when x < 0

    * yisint = 0 ... y is not an integer

    * yisint = 1 ... y is an odd int

    * yisint = 2 ... y is an even int

    */

   yisint  = 0;

   if(hx<0) {

     if(iy>=0x43400000) yisint = 2; /* even integer y */

     else if(iy>=0x3ff00000) {

       k = (iy>>20)-0x3ff;          /* exponent */

       if(k>20) {

         j = ly>>(52-k);

         if((unsigned)(j<<(52-k))==ly) yisint = 2-(j&1);

       } else if(ly==0) {

         j = iy>>(20-k);

         if((j<<(20-k))==iy) yisint = 2-(j&1);

       }

     }

   }

   /* special value of y */

   if(ly==0) {

     if (iy==0x7ff00000) {       /* y is +-inf */

       if(((ix-0x3ff00000)|lx)==0)

         return  y - y;  /* inf**+-1 is NaN */

       else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */

         return (hy>=0)? y: zeroX;

       else                      /* (|x|<1)**-,+inf = inf,0 */

         return (hy<0)?-y: zeroX;

     }

     if(iy==0x3ff00000) {        /* y is  +-1 */

       if(hy<0) return one/x; else return x;

     }

     if(hy==0x40000000) return x*x; /* y is  2 */

     if(hy==0x3fe00000) {        /* y is  0.5 */

       if(hx>=0) /* x >= +0 */

         return sqrt(x);

     }

   }

   ax   = fabsd(x);

   /* special value of x */

   if(lx==0) {

     if(ix==0x7ff00000||ix==0||ix==0x3ff00000){

       z = ax;                   /*x is +-0,+-inf,+-1*/

       if(hy<0) z = one/z;       /* z = (1/|x|) */

       if(hx<0) {

         if(((ix-0x3ff00000)|yisint)==0) {

 #ifdef CAN_USE_NAN_DEFINE

           z = NAN;

 #else

           z = (z-z)/(z-z); /* (-1)**non-int is NaN */

 #endif

         } else if(yisint==1)

           z = -1.0*z;           /* (x<0)**odd = -(|x|**odd) */

       }

       return z;

     }

   }

   n = (hx>>31)+1;

   /* (x<0)**(non-int) is NaN */

   if((n|yisint)==0)

 #ifdef CAN_USE_NAN_DEFINE

     return NAN;

 #else

     return (x-x)/(x-x);

 #endif

   s = one; /* s (sign of result -ve**odd) = -1 else = 1 */

   if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */

   /* |y| is huge */

   if(iy>0x41e00000) { /* if |y| > 2**31 */

     if(iy>0x43f00000){  /* if |y| > 2**64, must o/uflow */

       if(ix<=0x3fefffff) return (hy<0)? hugeX*hugeX:tiny*tiny;

       if(ix>=0x3ff00000) return (hy>0)? hugeX*hugeX:tiny*tiny;

     }

     /* over/underflow if x is not close to one */

     if(ix<0x3fefffff) return (hy<0)? s*hugeX*hugeX:s*tiny*tiny;

     if(ix>0x3ff00000) return (hy>0)? s*hugeX*hugeX:s*tiny*tiny;

     /* now |1-x| is tiny <= 2**-20, suffice to compute

        log(x) by x-x^2/2+x^3/3-x^4/4 */

     t = ax-one;         /* t has 20 trailing zeros */

     w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));

     u = ivln2_h*t;      /* ivln2_h has 21 sig. bits */

     v = t*ivln2_l-w*ivln2;

     t1 = u+v;

     set_low(&t1, 0);

     t2 = v-(t1-u);

   } else {

     double ss,s2,s_h,s_l,t_h,t_l;

     n = 0;

     /* take care subnormal number */

     if(ix<0x00100000)

       {ax *= two53; n -= 53; ix = high(ax); }

     n  += ((ix)>>20)-0x3ff;

     j  = ix&0x000fffff;

     /* determine interval */

     ix = j|0x3ff00000;          /* normalize ix */

     if(j<=0x3988E) k=0;         /* |x|<sqrt(3/2) */

     else if(j<0xBB67A) k=1;     /* |x|<sqrt(3)   */

     else {k=0;n+=1;ix -= 0x00100000;}

     set_high(&ax, ix);

     /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */

     u = ax-bp[k];               /* bp[0]=1.0, bp[1]=1.5 */

     v = one/(ax+bp[k]);

     ss = u*v;

     s_h = ss;

     set_low(&s_h, 0);

     /* t_h=ax+bp[k] High */

     t_h = zeroX;

     set_high(&t_h, ((ix>>1)|0x20000000)+0x00080000+(k<<18));

     t_l = ax - (t_h-bp[k]);

     s_l = v*((u-s_h*t_h)-s_h*t_l);

     /* compute log(ax) */

     s2 = ss*ss;

     r = s2*s2*(L1X+s2*(L2X+s2*(L3X+s2*(L4X+s2*(L5X+s2*L6X)))));

     r += s_l*(s_h+ss);

     s2  = s_h*s_h;

     t_h = 3.0+s2+r;

     set_low(&t_h, 0);

     t_l = r-((t_h-3.0)-s2);

     /* u+v = ss*(1+...) */

     u = s_h*t_h;

     v = s_l*t_h+t_l*ss;

     /* 2/(3log2)*(ss+...) */

     p_h = u+v;

     set_low(&p_h, 0);

     p_l = v-(p_h-u);

     z_h = cp_h*p_h;             /* cp_h+cp_l = 2/(3*log2) */

     z_l = cp_l*p_h+p_l*cp+dp_l[k];

     /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */

     t = (double)n;

     t1 = (((z_h+z_l)+dp_h[k])+t);

     set_low(&t1, 0);

     t2 = z_l-(((t1-t)-dp_h[k])-z_h);

   }

   /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */

   y1  = y;

   set_low(&y1, 0);

   p_l = (y-y1)*t1+y*t2;

   p_h = y1*t1;

   z = p_l+p_h;

   j = high(z);

   i = low(z);

   if (j>=0x40900000) {                          /* z >= 1024 */

     if(((j-0x40900000)|i)!=0)                   /* if z > 1024 */

       return s*hugeX*hugeX;                     /* overflow */

     else {

       if(p_l+ovt>z-p_h) return s*hugeX*hugeX;   /* overflow */

     }

   } else if((j&0x7fffffff)>=0x4090cc00 ) {      /* z <= -1075 */

     if(((j-0xc090cc00)|i)!=0)           /* z < -1075 */

       return s*tiny*tiny;               /* underflow */

     else {

       if(p_l<=z-p_h) return s*tiny*tiny;        /* underflow */

     }

   }

   /*

    * compute 2**(p_h+p_l)

    */

   i = j&0x7fffffff;

   k = (i>>20)-0x3ff;

   n = 0;

   if(i>0x3fe00000) {            /* if |z| > 0.5, set n = [z+0.5] */

     n = j+(0x00100000>>(k+1));

     k = ((n&0x7fffffff)>>20)-0x3ff;     /* new k for n */

     t = zeroX;

     set_high(&t, (n&~(0x000fffff>>k)));

     n = ((n&0x000fffff)|0x00100000)>>(20-k);

     if(j<0) n = -n;

     p_h -= t;

   }

   t = p_l+p_h;

   set_low(&t, 0);

   u = t*lg2_h;

   v = (p_l-(t-p_h))*lg2+t*lg2_l;

   z = u+v;

   w = v-(z-u);

   t  = z*z;

   t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));

   r  = (z*t1)/(t1-two)-(w+z*w);

   z  = one-(r-z);

   j  = high(z);

   j += (n<<20);

   if((j>>20)<=0) z = scalbnA(z,n);       /* subnormal output */

   else set_high(&z, high(z) + (n<<20));

   return s*z;

 }

 JRT_LEAF(jdouble, SharedRuntime::dpow(jdouble x, jdouble y))

   return __ieee754_pow(x, y);

 JRT_END

 #ifdef WIN32

 # pragma optimize ( "", on )

 #endif