| /* |
| * 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. |
| * |
| */ |
| |
| #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[] = { |
| 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 = 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[] = { |
| 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 = 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 |
