Mercurial > hg > graal-compiler
view src/share/vm/runtime/sharedRuntimeTrig.cpp @ 2149:7e37af9d69ef
7011379: G1: overly long concurrent marking cycles
Summary: This changeset introduces filtering of SATB buffers at the point when they are about to be enqueued. If this filtering clears enough entries on each buffer, the buffer can then be re-used and not enqueued. This cuts down the number of SATB buffers that need to be processed by the concurrent marking threads.
Reviewed-by: johnc, ysr
author | tonyp |
---|---|
date | Wed, 19 Jan 2011 09:35:17 -0500 |
parents | f95d63e2154a |
children | 63a4eb8bcd23 bdd155477289 |
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/* * Copyright (c) 2001, 2010, 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 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