jdk8-mips64-public/hotspot: src/share/vm/runtime/sharedRuntimeTrig.cpp@7ae4e26cb1e0 (annotated)

src/share/vm/runtime/sharedRuntimeTrig.cpp@7ae4e26cb1e0 (annotated)

src/share/vm/runtime/sharedRuntimeTrig.cpp

Thu, 12 Oct 2017 21:27:07 +0800

author: aoqi
date: Thu, 12 Oct 2017 21:27:07 +0800
changeset 7535: 7ae4e26cb1e0
parent 7002: a073be2ce5c2
parent 6876: 710a3c8b516e
child 9138: b56ab8e56604
permissions: -rw-r--r--

merge

 /*
  * Copyright (c) 2001, 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.
  *
  */
 /*
  * This file has been modified by Loongson Technology in 2015. These
  * modifications are Copyright (c) 2015 Loongson Technology, and are made
  * available on the same license terms set forth above.
  */
 #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 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 "runtime/sharedRuntimeMath.hpp"
 /*
  * __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[] = {
 .57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
 .54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
 .39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
 .28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
 .27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
 .22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
 .73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
 .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[] = {
 xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
 x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
 x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
 xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
 x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
 x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
 x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
 xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
 x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
 x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
 x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
 };
 static const int npio2_hw[] = {
 x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
 x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
 x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
 x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
 x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
 x404858EB, 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))
  */
 #ifdef MIPS64
 #undef S1
 #undef S2
 #undef S3
 #undef S4
 #undef S5
 #undef S6
 #endif
 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 = high(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,h,z,r,qx=0;
   int ix;
   ix = high(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 {
       set_high(&qx, ix-0x00200000); /* x/4 */
       set_low(&qx, 0);
     }
     h = 0.5*z-qx;
     a = one-qx;
     return a - (h - (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[] =  {
 .33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
 .33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
 .39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
 .18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
 .86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
 .59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
 .45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
 .88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
 .46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
 .81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
 .14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
  -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
 .59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
 };
 static double __kernel_tan(double x, double y, int iy)
 {
   double z,r,v,w,s;
   int ix,hx;
   hx = high(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 | low(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;
           set_low(&z, 0);
           v = y - (z - x);
           t = a = -one / w;
           set_low(&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;
     set_low(&z, 0);
     v  = r-(z - x);     /* z+v = r+x */
     t = a  = -1.0/w;    /* a = -1.0/w */
     set_low(&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 = high(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 = high(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 = high(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

Mercurial > jdk8-mips64-public > hotspot / annotate

src/share/vm/runtime/sharedRuntimeTrig.cpp@7ae4e26cb1e0 (annotated)

src/share/vm/runtime/sharedRuntimeTrig.cpp