Mercurial > hg > openjdk > hsx17
view src/share/vm/runtime/sharedRuntimeTrig.cpp @ 1334:193a468093fa hs17-b15
6955813: Fix incorrect Oracle rebranding headers from 6941466
Summary: Redo the header changes to fix new copyright notice style
Reviewed-by: ohair
| author | trims |
|---|---|
| date | Wed, 26 May 2010 00:30:39 -0700 |
| parents | 885e7f460925 |
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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 #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 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 affiliates. 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 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