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 /*

  * Copyright (c) 2001, 2005, 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 "incls/_precompiled.incl"

 #include "incls/_sharedRuntimeTrig.cpp.incl"

 // 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 optimize ( "", off )

 #endif

 /* The above workaround now causes more problems with the latest MS compiler.

  * Visual Studio 2010's /GS option tries to guard against buffer overruns.

  * /GS is on by default if you specify optimizations, which we do globally

  * via /W3 /O2. However the above selective turning off of optimizations means

  * that /GS issues a warning "4748". And since we treat warnings as errors (/WX)

  * then the compilation fails. There are several possible solutions

  * (1) Remove that pragma above as obsolete with VS2010 - requires testing.

  * (2) Stop treating warnings as errors - would be a backward step

  * (3) Disable /GS - may help performance but you lose the security checks

  * (4) Disable the warning with "#pragma warning( disable : 4748 )"

  * (5) Disable planting the code with  __declspec(safebuffers)

  * I've opted for (5) although we should investigate the local performance

  * benefits of (1) and global performance benefit of (3).

  */

 #if defined(WIN32) && (defined(_MSC_VER) && (_MSC_VER >= 1600))

 #define SAFEBUF __declspec(safebuffers)

 #else

 #define SAFEBUF

 #endif

 #include <math.h>

 // VM_LITTLE_ENDIAN is #defined appropriately in the Makefiles

 // [jk] this is not 100% correct because the float word order may different

 // from the byte order (e.g. on ARM)

 #ifdef VM_LITTLE_ENDIAN

 # define __HI(x) *(1+(int*)&x)

 # define __LO(x) *(int*)&x

 #else

 # define __HI(x) *(int*)&x

 # define __LO(x) *(1+(int*)&x)

 #endif

 static double copysignA(double x, double y) {

   __HI(x) = (__HI(x)&0x7fffffff)|(__HI(y)&0x80000000);

   return x;

 }

 /*

  * scalbn (double x, int n)

  * scalbn(x,n) returns x* 2**n  computed by  exponent

  * manipulation rather than by actually performing an

  * exponentiation or a multiplication.

  */

 static const double

 two54   =  1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */

 twom54  =  5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */

 hugeX  = 1.0e+300,

 tiny   = 1.0e-300;

 static double scalbnA (double x, int n) {

   int  k,hx,lx;

   hx = __HI(x);

   lx = __LO(x);

   k = (hx&0x7ff00000)>>20;              /* extract exponent */

   if (k==0) {                           /* 0 or subnormal x */

     if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */

     x *= two54;

     hx = __HI(x);

     k = ((hx&0x7ff00000)>>20) - 54;

     if (n< -50000) return tiny*x;       /*underflow*/

   }

   if (k==0x7ff) return x+x;             /* NaN or Inf */

   k = k+n;

   if (k >  0x7fe) return hugeX*copysignA(hugeX,x); /* overflow  */

   if (k > 0)                            /* normal result */

     {__HI(x) = (hx&0x800fffff)|(k<<20); return x;}

   if (k <= -54) {

     if (n > 50000)      /* in case integer overflow in n+k */

       return hugeX*copysignA(hugeX,x);  /*overflow*/

     else return tiny*copysignA(tiny,x); /*underflow*/

   }

   k += 54;                              /* subnormal result */

   __HI(x) = (hx&0x800fffff)|(k<<20);

   return x*twom54;

 }

 /*

  * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)

  * double x[],y[]; int e0,nx,prec; int ipio2[];

  *

  * __kernel_rem_pio2 return the last three digits of N with

  *              y = x - N*pi/2

  * so that |y| < pi/2.

  *

  * The method is to compute the integer (mod 8) and fraction parts of

  * (2/pi)*x without doing the full multiplication. In general we

  * skip the part of the product that are known to be a huge integer (

  * more accurately, = 0 mod 8 ). Thus the number of operations are

  * independent of the exponent of the input.

  *

  * (2/pi) is represented by an array of 24-bit integers in ipio2[].

  *

  * Input parameters:

  *      x[]     The input value (must be positive) is broken into nx

  *              pieces of 24-bit integers in double precision format.

  *              x[i] will be the i-th 24 bit of x. The scaled exponent

  *              of x[0] is given in input parameter e0 (i.e., x[0]*2^e0

  *              match x's up to 24 bits.

  *

  *              Example of breaking a double positive z into x[0]+x[1]+x[2]:

  *                      e0 = ilogb(z)-23

  *                      z  = scalbn(z,-e0)

  *              for i = 0,1,2

  *                      x[i] = floor(z)

  *                      z    = (z-x[i])*2**24

  *

  *

  *      y[]     ouput result in an array of double precision numbers.

  *              The dimension of y[] is:

  *                      24-bit  precision       1

  *                      53-bit  precision       2

  *                      64-bit  precision       2

  *                      113-bit precision       3

  *              The actual value is the sum of them. Thus for 113-bit

  *              precsion, one may have to do something like:

  *

  *              long double t,w,r_head, r_tail;

  *              t = (long double)y[2] + (long double)y[1];

  *              w = (long double)y[0];

  *              r_head = t+w;

  *              r_tail = w - (r_head - t);

  *

  *      e0      The exponent of x[0]

  *

  *      nx      dimension of x[]

  *

  *      prec    an interger indicating the precision:

  *                      0       24  bits (single)

  *                      1       53  bits (double)

  *                      2       64  bits (extended)

  *                      3       113 bits (quad)

  *

  *      ipio2[]

  *              integer array, contains the (24*i)-th to (24*i+23)-th

  *              bit of 2/pi after binary point. The corresponding

  *              floating value is

  *

  *                      ipio2[i] * 2^(-24(i+1)).

  *

  * External function:

  *      double scalbn(), floor();

  *

  *

  * Here is the description of some local variables:

  *

  *      jk      jk+1 is the initial number of terms of ipio2[] needed

  *              in the computation. The recommended value is 2,3,4,

  *              6 for single, double, extended,and quad.

  *

  *      jz      local integer variable indicating the number of

  *              terms of ipio2[] used.

  *

  *      jx      nx - 1

  *

  *      jv      index for pointing to the suitable ipio2[] for the

  *              computation. In general, we want

  *                      ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8

  *              is an integer. Thus

  *                      e0-3-24*jv >= 0 or (e0-3)/24 >= jv

  *              Hence jv = max(0,(e0-3)/24).

  *

  *      jp      jp+1 is the number of terms in PIo2[] needed, jp = jk.

  *

  *      q[]     double array with integral value, representing the

  *              24-bits chunk of the product of x and 2/pi.

  *

  *      q0      the corresponding exponent of q[0]. Note that the

  *              exponent for q[i] would be q0-24*i.

  *

  *      PIo2[]  double precision array, obtained by cutting pi/2

  *              into 24 bits chunks.

  *

  *      f[]     ipio2[] in floating point

  *

  *      iq[]    integer array by breaking up q[] in 24-bits chunk.

  *

  *      fq[]    final product of x*(2/pi) in fq[0],..,fq[jk]

  *

  *      ih      integer. If >0 it indicats q[] is >= 0.5, hence

  *              it also indicates the *sign* of the result.

  *

  */

 /*

  * 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 int init_jk[] = {2,3,4,6}; /* initial value for jk */

 static const double PIo2[] = {

   1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */

   7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */

   5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */

   3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */

   1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */

   1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */

   2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */

   2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */

 };

 static const double

 zeroB   = 0.0,

 one     = 1.0,

 two24B  = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */

 twon24  = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */

 static SAFEBUF int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2) {

   int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;

   double z,fw,f[20],fq[20],q[20];

   /* initialize jk*/

   jk = init_jk[prec];

   jp = jk;

   /* determine jx,jv,q0, note that 3>q0 */

   jx =  nx-1;

   jv = (e0-3)/24; if(jv<0) jv=0;

   q0 =  e0-24*(jv+1);

   /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */

   j = jv-jx; m = jx+jk;

   for(i=0;i<=m;i++,j++) f[i] = (j<0)? zeroB : (double) ipio2[j];

   /* compute q[0],q[1],...q[jk] */

   for (i=0;i<=jk;i++) {

     for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;

   }

   jz = jk;

 recompute:

   /* distill q[] into iq[] reversingly */

   for(i=0,j=jz,z=q[jz];j>0;i++,j--) {

     fw    =  (double)((int)(twon24* z));

     iq[i] =  (int)(z-two24B*fw);

     z     =  q[j-1]+fw;

   }

   /* compute n */

   z  = scalbnA(z,q0);           /* actual value of z */

   z -= 8.0*floor(z*0.125);              /* trim off integer >= 8 */

   n  = (int) z;

   z -= (double)n;

   ih = 0;

   if(q0>0) {    /* need iq[jz-1] to determine n */

     i  = (iq[jz-1]>>(24-q0)); n += i;

     iq[jz-1] -= i<<(24-q0);

     ih = iq[jz-1]>>(23-q0);

   }

   else if(q0==0) ih = iq[jz-1]>>23;

   else if(z>=0.5) ih=2;

   if(ih>0) {    /* q > 0.5 */

     n += 1; carry = 0;

     for(i=0;i<jz ;i++) {        /* compute 1-q */

       j = iq[i];

       if(carry==0) {

         if(j!=0) {

           carry = 1; iq[i] = 0x1000000- j;

         }

       } else  iq[i] = 0xffffff - j;

     }

     if(q0>0) {          /* rare case: chance is 1 in 12 */

       switch(q0) {

       case 1:

         iq[jz-1] &= 0x7fffff; break;

       case 2:

         iq[jz-1] &= 0x3fffff; break;

       }

     }

     if(ih==2) {

       z = one - z;

       if(carry!=0) z -= scalbnA(one,q0);

     }

   }

   /* check if recomputation is needed */

   if(z==zeroB) {

     j = 0;

     for (i=jz-1;i>=jk;i--) j |= iq[i];

     if(j==0) { /* need recomputation */

       for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */

       for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */

         f[jx+i] = (double) ipio2[jv+i];

         for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];

         q[i] = fw;

       }

       jz += k;

       goto recompute;

     }

   }

   /* chop off zero terms */

   if(z==0.0) {

     jz -= 1; q0 -= 24;

     while(iq[jz]==0) { jz--; q0-=24;}

   } else { /* break z into 24-bit if neccessary */

     z = scalbnA(z,-q0);

     if(z>=two24B) {

       fw = (double)((int)(twon24*z));

       iq[jz] = (int)(z-two24B*fw);

       jz += 1; q0 += 24;

       iq[jz] = (int) fw;

     } else iq[jz] = (int) z ;

   }

   /* convert integer "bit" chunk to floating-point value */

   fw = scalbnA(one,q0);

   for(i=jz;i>=0;i--) {

     q[i] = fw*(double)iq[i]; fw*=twon24;

   }

   /* compute PIo2[0,...,jp]*q[jz,...,0] */

   for(i=jz;i>=0;i--) {

     for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];

     fq[jz-i] = fw;

   }

   /* compress fq[] into y[] */

   switch(prec) {

   case 0:

     fw = 0.0;

     for (i=jz;i>=0;i--) fw += fq[i];

     y[0] = (ih==0)? fw: -fw;

     break;

   case 1:

   case 2:

     fw = 0.0;

     for (i=jz;i>=0;i--) fw += fq[i];

     y[0] = (ih==0)? fw: -fw;

     fw = fq[0]-fw;

     for (i=1;i<=jz;i++) fw += fq[i];

     y[1] = (ih==0)? fw: -fw;

     break;

   case 3:       /* painful */

     for (i=jz;i>0;i--) {

       fw      = fq[i-1]+fq[i];

       fq[i]  += fq[i-1]-fw;

       fq[i-1] = fw;

     }

     for (i=jz;i>1;i--) {

       fw      = fq[i-1]+fq[i];

       fq[i]  += fq[i-1]-fw;

       fq[i-1] = fw;

     }

     for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];

     if(ih==0) {

       y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;

     } else {

       y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;

     }

   }

   return n&7;

 }

 /*

  * ====================================================

  * Copyright (c) 1993 Oracle and/or its affilates. All rights reserved.

  *

  * Developed at SunPro, a Sun Microsystems, Inc. business.

  * Permission to use, copy, modify, and distribute this

  * software is freely granted, provided that this notice

  * is preserved.

  * ====================================================

  *

  */

 /* __ieee754_rem_pio2(x,y)

  *

  * return the remainder of x rem pi/2 in y[0]+y[1]

  * use __kernel_rem_pio2()

  */

 /*

  * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi

  */

 static const int two_over_pi[] = {

   0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,

   0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,

   0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,

   0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,

   0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,

   0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,

   0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,

   0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,

   0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,

   0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,

   0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,

 };

 static const int npio2_hw[] = {

   0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,

   0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,

   0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,

   0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,

   0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,

   0x404858EB, 0x404921FB,

 };

 /*

  * invpio2:  53 bits of 2/pi

  * pio2_1:   first  33 bit of pi/2

  * pio2_1t:  pi/2 - pio2_1

  * pio2_2:   second 33 bit of pi/2

  * pio2_2t:  pi/2 - (pio2_1+pio2_2)

  * pio2_3:   third  33 bit of pi/2

  * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)

  */

 static const double

 zeroA =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */

 half =  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */

 two24A =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */

 invpio2 =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */

 pio2_1  =  1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */

 pio2_1t =  6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */

 pio2_2  =  6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */

 pio2_2t =  2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */

 pio2_3  =  2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */

 pio2_3t =  8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */

 static SAFEBUF int __ieee754_rem_pio2(double x, double *y) {

   double z,w,t,r,fn;

   double tx[3];

   int e0,i,j,nx,n,ix,hx,i0;

   i0 = ((*(int*)&two24A)>>30)^1;        /* high word index */

   hx = *(i0+(int*)&x);          /* high word of x */

   ix = hx&0x7fffffff;

   if(ix<=0x3fe921fb)   /* |x| ~<= pi/4 , no need for reduction */

     {y[0] = x; y[1] = 0; return 0;}

   if(ix<0x4002d97c) {  /* |x| < 3pi/4, special case with n=+-1 */

     if(hx>0) {

       z = x - pio2_1;

       if(ix!=0x3ff921fb) {    /* 33+53 bit pi is good enough */

         y[0] = z - pio2_1t;

         y[1] = (z-y[0])-pio2_1t;

       } else {                /* near pi/2, use 33+33+53 bit pi */

         z -= pio2_2;

         y[0] = z - pio2_2t;

         y[1] = (z-y[0])-pio2_2t;

       }

       return 1;

     } else {    /* negative x */

       z = x + pio2_1;

       if(ix!=0x3ff921fb) {    /* 33+53 bit pi is good enough */

         y[0] = z + pio2_1t;

         y[1] = (z-y[0])+pio2_1t;

       } else {                /* near pi/2, use 33+33+53 bit pi */

         z += pio2_2;

         y[0] = z + pio2_2t;

         y[1] = (z-y[0])+pio2_2t;

       }

       return -1;

     }

   }

   if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */

     t  = fabsd(x);

     n  = (int) (t*invpio2+half);

     fn = (double)n;

     r  = t-fn*pio2_1;

     w  = fn*pio2_1t;    /* 1st round good to 85 bit */

     if(n<32&&ix!=npio2_hw[n-1]) {

       y[0] = r-w;       /* quick check no cancellation */

     } else {

       j  = ix>>20;

       y[0] = r-w;

       i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff);

       if(i>16) {  /* 2nd iteration needed, good to 118 */

         t  = r;

         w  = fn*pio2_2;

         r  = t-w;

         w  = fn*pio2_2t-((t-r)-w);

         y[0] = r-w;

         i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff);

         if(i>49)  {     /* 3rd iteration need, 151 bits acc */

           t  = r;       /* will cover all possible cases */

           w  = fn*pio2_3;

           r  = t-w;

           w  = fn*pio2_3t-((t-r)-w);

           y[0] = r-w;

         }

       }

     }

     y[1] = (r-y[0])-w;

     if(hx<0)    {y[0] = -y[0]; y[1] = -y[1]; return -n;}

     else         return n;

   }

   /*

    * all other (large) arguments

    */

   if(ix>=0x7ff00000) {          /* x is inf or NaN */

     y[0]=y[1]=x-x; return 0;

   }

   /* set z = scalbn(|x|,ilogb(x)-23) */

   *(1-i0+(int*)&z) = *(1-i0+(int*)&x);

   e0    = (ix>>20)-1046;        /* e0 = ilogb(z)-23; */

   *(i0+(int*)&z) = ix - (e0<<20);

   for(i=0;i<2;i++) {

     tx[i] = (double)((int)(z));

     z     = (z-tx[i])*two24A;

   }

   tx[2] = z;

   nx = 3;

   while(tx[nx-1]==zeroA) nx--;  /* skip zero term */

   n  =  __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi);

   if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}

   return n;

 }

 /* __kernel_sin( x, y, iy)

  * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854

  * Input x is assumed to be bounded by ~pi/4 in magnitude.

  * Input y is the tail of x.

  * Input iy indicates whether y is 0. (if iy=0, y assume to be 0).

  *

  * Algorithm

  *      1. Since sin(-x) = -sin(x), we need only to consider positive x.

  *      2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0.

  *      3. sin(x) is approximated by a polynomial of degree 13 on

  *         [0,pi/4]

  *                               3            13

  *              sin(x) ~ x + S1*x + ... + S6*x

  *         where

  *

  *      |sin(x)         2     4     6     8     10     12  |     -58

  *      |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x  +S6*x   )| <= 2

  *      |  x                                               |

  *

  *      4. sin(x+y) = sin(x) + sin'(x')*y

  *                  ~ sin(x) + (1-x*x/2)*y

  *         For better accuracy, let

  *                   3      2      2      2      2

  *              r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6))))

  *         then                   3    2

  *              sin(x) = x + (S1*x + (x *(r-y/2)+y))

  */

 static const double

 S1  = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */

 S2  =  8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */

 S3  = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */

 S4  =  2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */

 S5  = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */

 S6  =  1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */

 static double __kernel_sin(double x, double y, int iy)

 {

         double z,r,v;

         int ix;

         ix = __HI(x)&0x7fffffff;        /* high word of x */

         if(ix<0x3e400000)                       /* |x| < 2**-27 */

            {if((int)x==0) return x;}            /* generate inexact */

         z       =  x*x;

         v       =  z*x;

         r       =  S2+z*(S3+z*(S4+z*(S5+z*S6)));

         if(iy==0) return x+v*(S1+z*r);

         else      return x-((z*(half*y-v*r)-y)-v*S1);

 }

 /*

  * __kernel_cos( x,  y )

  * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164

  * Input x is assumed to be bounded by ~pi/4 in magnitude.

  * Input y is the tail of x.

  *

  * Algorithm

  *      1. Since cos(-x) = cos(x), we need only to consider positive x.

  *      2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.

  *      3. cos(x) is approximated by a polynomial of degree 14 on

  *         [0,pi/4]

  *                                       4            14

  *              cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x

  *         where the remez error is

  *

  *      |              2     4     6     8     10    12     14 |     -58

  *      |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2

  *      |                                                      |

  *

  *                     4     6     8     10    12     14

  *      4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then

  *             cos(x) = 1 - x*x/2 + r

  *         since cos(x+y) ~ cos(x) - sin(x)*y

  *                        ~ cos(x) - x*y,

  *         a correction term is necessary in cos(x) and hence

  *              cos(x+y) = 1 - (x*x/2 - (r - x*y))

  *         For better accuracy when x > 0.3, let qx = |x|/4 with

  *         the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.

  *         Then

  *              cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).

  *         Note that 1-qx and (x*x/2-qx) is EXACT here, and the

  *         magnitude of the latter is at least a quarter of x*x/2,

  *         thus, reducing the rounding error in the subtraction.

  */

 static const double

 C1  =  4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */

 C2  = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */

 C3  =  2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */

 C4  = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */

 C5  =  2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */

 C6  = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */

 static double __kernel_cos(double x, double y)

 {

   double a,hz,z,r,qx;

   int ix;

   ix = __HI(x)&0x7fffffff;      /* ix = |x|'s high word*/

   if(ix<0x3e400000) {                   /* if x < 2**27 */

     if(((int)x)==0) return one;         /* generate inexact */

   }

   z  = x*x;

   r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));

   if(ix < 0x3FD33333)                   /* if |x| < 0.3 */

     return one - (0.5*z - (z*r - x*y));

   else {

     if(ix > 0x3fe90000) {               /* x > 0.78125 */

       qx = 0.28125;

     } else {

       __HI(qx) = ix-0x00200000; /* x/4 */

       __LO(qx) = 0;

     }

     hz = 0.5*z-qx;

     a  = one-qx;

     return a - (hz - (z*r-x*y));

   }

 }

 /* __kernel_tan( x, y, k )

  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854

  * Input x is assumed to be bounded by ~pi/4 in magnitude.

  * Input y is the tail of x.

  * Input k indicates whether tan (if k=1) or

  * -1/tan (if k= -1) is returned.

  *

  * Algorithm

  *      1. Since tan(-x) = -tan(x), we need only to consider positive x.

  *      2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.

  *      3. tan(x) is approximated by a odd polynomial of degree 27 on

  *         [0,0.67434]

  *                               3             27

  *              tan(x) ~ x + T1*x + ... + T13*x

  *         where

  *

  *              |tan(x)         2     4            26   |     -59.2

  *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2

  *              |  x                                    |

  *

  *         Note: tan(x+y) = tan(x) + tan'(x)*y

  *                        ~ tan(x) + (1+x*x)*y

  *         Therefore, for better accuracy in computing tan(x+y), let

  *                   3      2      2       2       2

  *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))

  *         then

  *                                  3    2

  *              tan(x+y) = x + (T1*x + (x *(r+y)+y))

  *

  *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then

  *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))

  *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))

  */

 static const double

 pio4  =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */

 pio4lo=  3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */

 T[] =  {

   3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */

   1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */

   5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */

   2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */

   8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */

   3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */

   1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */

   5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */

   2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */

   7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */

   7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */

  -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */

   2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */

 };

 static double __kernel_tan(double x, double y, int iy)

 {

   double z,r,v,w,s;

   int ix,hx;

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

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

   if(ix<0x3e300000) {                     /* x < 2**-28 */

     if((int)x==0) {                       /* generate inexact */

       if (((ix | __LO(x)) | (iy + 1)) == 0)

         return one / fabsd(x);

       else {

         if (iy == 1)

           return x;

         else {    /* compute -1 / (x+y) carefully */

           double a, t;

           z = w = x + y;

           __LO(z) = 0;

           v = y - (z - x);

           t = a = -one / w;

           __LO(t) = 0;

           s = one + t * z;

           return t + a * (s + t * v);

         }

       }

     }

   }

   if(ix>=0x3FE59428) {                    /* |x|>=0.6744 */

     if(hx<0) {x = -x; y = -y;}

     z = pio4-x;

     w = pio4lo-y;

     x = z+w; y = 0.0;

   }

   z       =  x*x;

   w       =  z*z;

   /* Break x^5*(T[1]+x^2*T[2]+...) into

    *    x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +

    *    x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))

    */

   r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));

   v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));

   s = z*x;

   r = y + z*(s*(r+v)+y);

   r += T[0]*s;

   w = x+r;

   if(ix>=0x3FE59428) {

     v = (double)iy;

     return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));

   }

   if(iy==1) return w;

   else {          /* if allow error up to 2 ulp,

                      simply return -1.0/(x+r) here */

     /*  compute -1.0/(x+r) accurately */

     double a,t;

     z  = w;

     __LO(z) = 0;

     v  = r-(z - x);     /* z+v = r+x */

     t = a  = -1.0/w;    /* a = -1.0/w */

     __LO(t) = 0;

     s  = 1.0+t*z;

     return t+a*(s+t*v);

   }

 }

 //----------------------------------------------------------------------

 //

 // Routines for new sin/cos implementation

 //

 //----------------------------------------------------------------------

 /* sin(x)

  * Return sine function of x.

  *

  * kernel function:

  *      __kernel_sin            ... sine function on [-pi/4,pi/4]

  *      __kernel_cos            ... cose function on [-pi/4,pi/4]

  *      __ieee754_rem_pio2      ... argument reduction routine

  *

  * Method.

  *      Let S,C and T denote the sin, cos and tan respectively on

  *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2

  *      in [-pi/4 , +pi/4], and let n = k mod 4.

  *      We have

  *

  *          n        sin(x)      cos(x)        tan(x)

  *     ----------------------------------------------------------

  *          0          S           C             T

  *          1          C          -S            -1/T

  *          2         -S          -C             T

  *          3         -C           S            -1/T

  *     ----------------------------------------------------------

  *

  * Special cases:

  *      Let trig be any of sin, cos, or tan.

  *      trig(+-INF)  is NaN, with signals;

  *      trig(NaN)    is that NaN;

  *

  * Accuracy:

  *      TRIG(x) returns trig(x) nearly rounded

  */

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

   double y[2],z=0.0;

   int n, ix;

   /* High word of x. */

   ix = __HI(x);

   /* |x| ~< pi/4 */

   ix &= 0x7fffffff;

   if(ix <= 0x3fe921fb) return __kernel_sin(x,z,0);

   /* sin(Inf or NaN) is NaN */

   else if (ix>=0x7ff00000) return x-x;

   /* argument reduction needed */

   else {

     n = __ieee754_rem_pio2(x,y);

     switch(n&3) {

     case 0: return  __kernel_sin(y[0],y[1],1);

     case 1: return  __kernel_cos(y[0],y[1]);

     case 2: return -__kernel_sin(y[0],y[1],1);

     default:

       return -__kernel_cos(y[0],y[1]);

     }

   }

 JRT_END

 /* cos(x)

  * Return cosine function of x.

  *

  * kernel function:

  *      __kernel_sin            ... sine function on [-pi/4,pi/4]

  *      __kernel_cos            ... cosine function on [-pi/4,pi/4]

  *      __ieee754_rem_pio2      ... argument reduction routine

  *

  * Method.

  *      Let S,C and T denote the sin, cos and tan respectively on

  *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2

  *      in [-pi/4 , +pi/4], and let n = k mod 4.

  *      We have

  *

  *          n        sin(x)      cos(x)        tan(x)

  *     ----------------------------------------------------------

  *          0          S           C             T

  *          1          C          -S            -1/T

  *          2         -S          -C             T

  *          3         -C           S            -1/T

  *     ----------------------------------------------------------

  *

  * Special cases:

  *      Let trig be any of sin, cos, or tan.

  *      trig(+-INF)  is NaN, with signals;

  *      trig(NaN)    is that NaN;

  *

  * Accuracy:

  *      TRIG(x) returns trig(x) nearly rounded

  */

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

   double y[2],z=0.0;

   int n, ix;

   /* High word of x. */

   ix = __HI(x);

   /* |x| ~< pi/4 */

   ix &= 0x7fffffff;

   if(ix <= 0x3fe921fb) return __kernel_cos(x,z);

   /* cos(Inf or NaN) is NaN */

   else if (ix>=0x7ff00000) return x-x;

   /* argument reduction needed */

   else {

     n = __ieee754_rem_pio2(x,y);

     switch(n&3) {

     case 0: return  __kernel_cos(y[0],y[1]);

     case 1: return -__kernel_sin(y[0],y[1],1);

     case 2: return -__kernel_cos(y[0],y[1]);

     default:

       return  __kernel_sin(y[0],y[1],1);

     }

   }

 JRT_END

 /* tan(x)

  * Return tangent function of x.

  *

  * kernel function:

  *      __kernel_tan            ... tangent function on [-pi/4,pi/4]

  *      __ieee754_rem_pio2      ... argument reduction routine

  *

  * Method.

  *      Let S,C and T denote the sin, cos and tan respectively on

  *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2

  *      in [-pi/4 , +pi/4], and let n = k mod 4.

  *      We have

  *

  *          n        sin(x)      cos(x)        tan(x)

  *     ----------------------------------------------------------

  *          0          S           C             T

  *          1          C          -S            -1/T

  *          2         -S          -C             T

  *          3         -C           S            -1/T

  *     ----------------------------------------------------------

  *

  * Special cases:

  *      Let trig be any of sin, cos, or tan.

  *      trig(+-INF)  is NaN, with signals;

  *      trig(NaN)    is that NaN;

  *

  * Accuracy:

  *      TRIG(x) returns trig(x) nearly rounded

  */

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

   double y[2],z=0.0;

   int n, ix;

   /* High word of x. */

   ix = __HI(x);

   /* |x| ~< pi/4 */

   ix &= 0x7fffffff;

   if(ix <= 0x3fe921fb) return __kernel_tan(x,z,1);

   /* tan(Inf or NaN) is NaN */

   else if (ix>=0x7ff00000) return x-x;            /* NaN */

   /* argument reduction needed */

   else {

     n = __ieee754_rem_pio2(x,y);

     return __kernel_tan(y[0],y[1],1-((n&1)<<1)); /*   1 -- n even

                                                      -1 -- n odd */

   }

 JRT_END

 #ifdef WIN32

 # pragma optimize ( "", on )

 #endif