duke@435: /* lfoltan@7000: * Copyright (c) 2001, 2014, Oracle and/or its affiliates. All rights reserved. duke@435: * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. duke@435: * duke@435: * This code is free software; you can redistribute it and/or modify it duke@435: * under the terms of the GNU General Public License version 2 only, as duke@435: * published by the Free Software Foundation. duke@435: * duke@435: * This code is distributed in the hope that it will be useful, but WITHOUT duke@435: * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or duke@435: * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License duke@435: * version 2 for more details (a copy is included in the LICENSE file that duke@435: * accompanied this code). duke@435: * duke@435: * You should have received a copy of the GNU General Public License version duke@435: * 2 along with this work; if not, write to the Free Software Foundation, duke@435: * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. duke@435: * trims@1907: * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA trims@1907: * or visit www.oracle.com if you need additional information or have any trims@1907: * questions. duke@435: * duke@435: */ duke@435: stefank@2314: #include "precompiled.hpp" stefank@2314: #include "prims/jni.h" stefank@2314: #include "runtime/interfaceSupport.hpp" stefank@2314: #include "runtime/sharedRuntime.hpp" duke@435: duke@435: // This file contains copies of the fdlibm routines used by duke@435: // StrictMath. It turns out that it is almost always required to use duke@435: // these runtime routines; the Intel CPU doesn't meet the Java duke@435: // specification for sin/cos outside a certain limited argument range, duke@435: // and the SPARC CPU doesn't appear to have sin/cos instructions. It duke@435: // also turns out that avoiding the indirect call through function duke@435: // pointer out to libjava.so in SharedRuntime speeds these routines up duke@435: // by roughly 15% on both Win32/x86 and Solaris/SPARC. duke@435: duke@435: // Enabling optimizations in this file causes incorrect code to be duke@435: // generated; can not figure out how to turn down optimization for one duke@435: // file in the IDE on Windows duke@435: #ifdef WIN32 duke@435: # pragma optimize ( "", off ) duke@435: #endif duke@435: prr@1840: /* The above workaround now causes more problems with the latest MS compiler. prr@1840: * Visual Studio 2010's /GS option tries to guard against buffer overruns. prr@1840: * /GS is on by default if you specify optimizations, which we do globally prr@1840: * via /W3 /O2. However the above selective turning off of optimizations means prr@1840: * that /GS issues a warning "4748". And since we treat warnings as errors (/WX) prr@1840: * then the compilation fails. There are several possible solutions prr@1840: * (1) Remove that pragma above as obsolete with VS2010 - requires testing. prr@1840: * (2) Stop treating warnings as errors - would be a backward step prr@1840: * (3) Disable /GS - may help performance but you lose the security checks prr@1840: * (4) Disable the warning with "#pragma warning( disable : 4748 )" prr@1840: * (5) Disable planting the code with __declspec(safebuffers) prr@1840: * I've opted for (5) although we should investigate the local performance prr@1840: * benefits of (1) and global performance benefit of (3). prr@1840: */ prr@1840: #if defined(WIN32) && (defined(_MSC_VER) && (_MSC_VER >= 1600)) prr@1840: #define SAFEBUF __declspec(safebuffers) prr@1840: #else prr@1840: #define SAFEBUF prr@1840: #endif prr@1840: lfoltan@7000: #include "runtime/sharedRuntimeMath.hpp" duke@435: duke@435: /* duke@435: * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) duke@435: * double x[],y[]; int e0,nx,prec; int ipio2[]; duke@435: * duke@435: * __kernel_rem_pio2 return the last three digits of N with duke@435: * y = x - N*pi/2 duke@435: * so that |y| < pi/2. duke@435: * duke@435: * The method is to compute the integer (mod 8) and fraction parts of duke@435: * (2/pi)*x without doing the full multiplication. In general we duke@435: * skip the part of the product that are known to be a huge integer ( duke@435: * more accurately, = 0 mod 8 ). Thus the number of operations are duke@435: * independent of the exponent of the input. duke@435: * duke@435: * (2/pi) is represented by an array of 24-bit integers in ipio2[]. duke@435: * duke@435: * Input parameters: duke@435: * x[] The input value (must be positive) is broken into nx duke@435: * pieces of 24-bit integers in double precision format. duke@435: * x[i] will be the i-th 24 bit of x. The scaled exponent duke@435: * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 duke@435: * match x's up to 24 bits. duke@435: * duke@435: * Example of breaking a double positive z into x[0]+x[1]+x[2]: duke@435: * e0 = ilogb(z)-23 duke@435: * z = scalbn(z,-e0) duke@435: * for i = 0,1,2 duke@435: * x[i] = floor(z) duke@435: * z = (z-x[i])*2**24 duke@435: * duke@435: * duke@435: * y[] ouput result in an array of double precision numbers. duke@435: * The dimension of y[] is: duke@435: * 24-bit precision 1 duke@435: * 53-bit precision 2 duke@435: * 64-bit precision 2 duke@435: * 113-bit precision 3 duke@435: * The actual value is the sum of them. Thus for 113-bit duke@435: * precsion, one may have to do something like: duke@435: * duke@435: * long double t,w,r_head, r_tail; duke@435: * t = (long double)y[2] + (long double)y[1]; duke@435: * w = (long double)y[0]; duke@435: * r_head = t+w; duke@435: * r_tail = w - (r_head - t); duke@435: * duke@435: * e0 The exponent of x[0] duke@435: * duke@435: * nx dimension of x[] duke@435: * duke@435: * prec an interger indicating the precision: duke@435: * 0 24 bits (single) duke@435: * 1 53 bits (double) duke@435: * 2 64 bits (extended) duke@435: * 3 113 bits (quad) duke@435: * duke@435: * ipio2[] duke@435: * integer array, contains the (24*i)-th to (24*i+23)-th duke@435: * bit of 2/pi after binary point. The corresponding duke@435: * floating value is duke@435: * duke@435: * ipio2[i] * 2^(-24(i+1)). duke@435: * duke@435: * External function: duke@435: * double scalbn(), floor(); duke@435: * duke@435: * duke@435: * Here is the description of some local variables: duke@435: * duke@435: * jk jk+1 is the initial number of terms of ipio2[] needed duke@435: * in the computation. The recommended value is 2,3,4, duke@435: * 6 for single, double, extended,and quad. duke@435: * duke@435: * jz local integer variable indicating the number of duke@435: * terms of ipio2[] used. duke@435: * duke@435: * jx nx - 1 duke@435: * duke@435: * jv index for pointing to the suitable ipio2[] for the duke@435: * computation. In general, we want duke@435: * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 duke@435: * is an integer. Thus duke@435: * e0-3-24*jv >= 0 or (e0-3)/24 >= jv duke@435: * Hence jv = max(0,(e0-3)/24). duke@435: * duke@435: * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. duke@435: * duke@435: * q[] double array with integral value, representing the duke@435: * 24-bits chunk of the product of x and 2/pi. duke@435: * duke@435: * q0 the corresponding exponent of q[0]. Note that the duke@435: * exponent for q[i] would be q0-24*i. duke@435: * duke@435: * PIo2[] double precision array, obtained by cutting pi/2 duke@435: * into 24 bits chunks. duke@435: * duke@435: * f[] ipio2[] in floating point duke@435: * duke@435: * iq[] integer array by breaking up q[] in 24-bits chunk. duke@435: * duke@435: * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] duke@435: * duke@435: * ih integer. If >0 it indicats q[] is >= 0.5, hence duke@435: * it also indicates the *sign* of the result. duke@435: * duke@435: */ duke@435: duke@435: duke@435: /* duke@435: * Constants: duke@435: * The hexadecimal values are the intended ones for the following duke@435: * constants. The decimal values may be used, provided that the duke@435: * compiler will convert from decimal to binary accurately enough duke@435: * to produce the hexadecimal values shown. duke@435: */ duke@435: duke@435: duke@435: static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ duke@435: duke@435: static const double PIo2[] = { duke@435: 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ duke@435: 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ duke@435: 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ duke@435: 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ duke@435: 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ duke@435: 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ duke@435: 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ duke@435: 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ duke@435: }; duke@435: duke@435: static const double duke@435: zeroB = 0.0, duke@435: one = 1.0, duke@435: two24B = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ duke@435: twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ duke@435: prr@1840: static SAFEBUF int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2) { duke@435: int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; duke@435: double z,fw,f[20],fq[20],q[20]; duke@435: duke@435: /* initialize jk*/ duke@435: jk = init_jk[prec]; duke@435: jp = jk; duke@435: duke@435: /* determine jx,jv,q0, note that 3>q0 */ duke@435: jx = nx-1; duke@435: jv = (e0-3)/24; if(jv<0) jv=0; duke@435: q0 = e0-24*(jv+1); duke@435: duke@435: /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ duke@435: j = jv-jx; m = jx+jk; duke@435: for(i=0;i<=m;i++,j++) f[i] = (j<0)? zeroB : (double) ipio2[j]; duke@435: duke@435: /* compute q[0],q[1],...q[jk] */ duke@435: for (i=0;i<=jk;i++) { duke@435: for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; duke@435: } duke@435: duke@435: jz = jk; duke@435: recompute: duke@435: /* distill q[] into iq[] reversingly */ duke@435: for(i=0,j=jz,z=q[jz];j>0;i++,j--) { duke@435: fw = (double)((int)(twon24* z)); duke@435: iq[i] = (int)(z-two24B*fw); duke@435: z = q[j-1]+fw; duke@435: } duke@435: duke@435: /* compute n */ duke@435: z = scalbnA(z,q0); /* actual value of z */ duke@435: z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ duke@435: n = (int) z; duke@435: z -= (double)n; duke@435: ih = 0; duke@435: if(q0>0) { /* need iq[jz-1] to determine n */ duke@435: i = (iq[jz-1]>>(24-q0)); n += i; duke@435: iq[jz-1] -= i<<(24-q0); duke@435: ih = iq[jz-1]>>(23-q0); duke@435: } duke@435: else if(q0==0) ih = iq[jz-1]>>23; duke@435: else if(z>=0.5) ih=2; duke@435: duke@435: if(ih>0) { /* q > 0.5 */ duke@435: n += 1; carry = 0; duke@435: for(i=0;i0) { /* rare case: chance is 1 in 12 */ duke@435: switch(q0) { duke@435: case 1: duke@435: iq[jz-1] &= 0x7fffff; break; duke@435: case 2: duke@435: iq[jz-1] &= 0x3fffff; break; duke@435: } duke@435: } duke@435: if(ih==2) { duke@435: z = one - z; duke@435: if(carry!=0) z -= scalbnA(one,q0); duke@435: } duke@435: } duke@435: duke@435: /* check if recomputation is needed */ duke@435: if(z==zeroB) { duke@435: j = 0; duke@435: for (i=jz-1;i>=jk;i--) j |= iq[i]; duke@435: if(j==0) { /* need recomputation */ duke@435: for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ duke@435: duke@435: for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ duke@435: f[jx+i] = (double) ipio2[jv+i]; duke@435: for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; duke@435: q[i] = fw; duke@435: } duke@435: jz += k; duke@435: goto recompute; duke@435: } duke@435: } duke@435: duke@435: /* chop off zero terms */ duke@435: if(z==0.0) { duke@435: jz -= 1; q0 -= 24; duke@435: while(iq[jz]==0) { jz--; q0-=24;} duke@435: } else { /* break z into 24-bit if neccessary */ duke@435: z = scalbnA(z,-q0); duke@435: if(z>=two24B) { duke@435: fw = (double)((int)(twon24*z)); duke@435: iq[jz] = (int)(z-two24B*fw); duke@435: jz += 1; q0 += 24; duke@435: iq[jz] = (int) fw; duke@435: } else iq[jz] = (int) z ; duke@435: } duke@435: duke@435: /* convert integer "bit" chunk to floating-point value */ duke@435: fw = scalbnA(one,q0); duke@435: for(i=jz;i>=0;i--) { duke@435: q[i] = fw*(double)iq[i]; fw*=twon24; duke@435: } duke@435: duke@435: /* compute PIo2[0,...,jp]*q[jz,...,0] */ duke@435: for(i=jz;i>=0;i--) { duke@435: for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; duke@435: fq[jz-i] = fw; duke@435: } duke@435: duke@435: /* compress fq[] into y[] */ duke@435: switch(prec) { duke@435: case 0: duke@435: fw = 0.0; duke@435: for (i=jz;i>=0;i--) fw += fq[i]; duke@435: y[0] = (ih==0)? fw: -fw; duke@435: break; duke@435: case 1: duke@435: case 2: duke@435: fw = 0.0; duke@435: for (i=jz;i>=0;i--) fw += fq[i]; duke@435: y[0] = (ih==0)? fw: -fw; duke@435: fw = fq[0]-fw; duke@435: for (i=1;i<=jz;i++) fw += fq[i]; duke@435: y[1] = (ih==0)? fw: -fw; duke@435: break; duke@435: case 3: /* painful */ duke@435: for (i=jz;i>0;i--) { duke@435: fw = fq[i-1]+fq[i]; duke@435: fq[i] += fq[i-1]-fw; duke@435: fq[i-1] = fw; duke@435: } duke@435: for (i=jz;i>1;i--) { duke@435: fw = fq[i-1]+fq[i]; duke@435: fq[i] += fq[i-1]-fw; duke@435: fq[i-1] = fw; duke@435: } duke@435: for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; duke@435: if(ih==0) { duke@435: y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; duke@435: } else { duke@435: y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; duke@435: } duke@435: } duke@435: return n&7; duke@435: } duke@435: duke@435: duke@435: /* duke@435: * ==================================================== trims@1907: * Copyright (c) 1993 Oracle and/or its affilates. All rights reserved. duke@435: * duke@435: * Developed at SunPro, a Sun Microsystems, Inc. business. duke@435: * Permission to use, copy, modify, and distribute this duke@435: * software is freely granted, provided that this notice duke@435: * is preserved. duke@435: * ==================================================== duke@435: * duke@435: */ duke@435: duke@435: /* __ieee754_rem_pio2(x,y) duke@435: * duke@435: * return the remainder of x rem pi/2 in y[0]+y[1] duke@435: * use __kernel_rem_pio2() duke@435: */ duke@435: duke@435: /* duke@435: * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi duke@435: */ duke@435: static const int two_over_pi[] = { duke@435: 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, duke@435: 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, duke@435: 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, duke@435: 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, duke@435: 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, duke@435: 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF, duke@435: 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, duke@435: 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, duke@435: 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, duke@435: 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, duke@435: 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B, duke@435: }; duke@435: duke@435: static const int npio2_hw[] = { duke@435: 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, duke@435: 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C, duke@435: 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A, duke@435: 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C, duke@435: 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB, duke@435: 0x404858EB, 0x404921FB, duke@435: }; duke@435: duke@435: /* duke@435: * invpio2: 53 bits of 2/pi duke@435: * pio2_1: first 33 bit of pi/2 duke@435: * pio2_1t: pi/2 - pio2_1 duke@435: * pio2_2: second 33 bit of pi/2 duke@435: * pio2_2t: pi/2 - (pio2_1+pio2_2) duke@435: * pio2_3: third 33 bit of pi/2 duke@435: * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) duke@435: */ duke@435: duke@435: static const double duke@435: zeroA = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ duke@435: half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ duke@435: two24A = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ duke@435: invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ duke@435: pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */ duke@435: pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */ duke@435: pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */ duke@435: pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */ duke@435: pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */ duke@435: pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */ duke@435: prr@1840: static SAFEBUF int __ieee754_rem_pio2(double x, double *y) { duke@435: double z,w,t,r,fn; duke@435: double tx[3]; duke@435: int e0,i,j,nx,n,ix,hx,i0; duke@435: duke@435: i0 = ((*(int*)&two24A)>>30)^1; /* high word index */ duke@435: hx = *(i0+(int*)&x); /* high word of x */ duke@435: ix = hx&0x7fffffff; duke@435: if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */ duke@435: {y[0] = x; y[1] = 0; return 0;} duke@435: if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */ duke@435: if(hx>0) { duke@435: z = x - pio2_1; duke@435: if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ duke@435: y[0] = z - pio2_1t; duke@435: y[1] = (z-y[0])-pio2_1t; duke@435: } else { /* near pi/2, use 33+33+53 bit pi */ duke@435: z -= pio2_2; duke@435: y[0] = z - pio2_2t; duke@435: y[1] = (z-y[0])-pio2_2t; duke@435: } duke@435: return 1; duke@435: } else { /* negative x */ duke@435: z = x + pio2_1; duke@435: if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ duke@435: y[0] = z + pio2_1t; duke@435: y[1] = (z-y[0])+pio2_1t; duke@435: } else { /* near pi/2, use 33+33+53 bit pi */ duke@435: z += pio2_2; duke@435: y[0] = z + pio2_2t; duke@435: y[1] = (z-y[0])+pio2_2t; duke@435: } duke@435: return -1; duke@435: } duke@435: } duke@435: if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */ duke@435: t = fabsd(x); duke@435: n = (int) (t*invpio2+half); duke@435: fn = (double)n; duke@435: r = t-fn*pio2_1; duke@435: w = fn*pio2_1t; /* 1st round good to 85 bit */ duke@435: if(n<32&&ix!=npio2_hw[n-1]) { duke@435: y[0] = r-w; /* quick check no cancellation */ duke@435: } else { duke@435: j = ix>>20; duke@435: y[0] = r-w; duke@435: i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff); duke@435: if(i>16) { /* 2nd iteration needed, good to 118 */ duke@435: t = r; duke@435: w = fn*pio2_2; duke@435: r = t-w; duke@435: w = fn*pio2_2t-((t-r)-w); duke@435: y[0] = r-w; duke@435: i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff); duke@435: if(i>49) { /* 3rd iteration need, 151 bits acc */ duke@435: t = r; /* will cover all possible cases */ duke@435: w = fn*pio2_3; duke@435: r = t-w; duke@435: w = fn*pio2_3t-((t-r)-w); duke@435: y[0] = r-w; duke@435: } duke@435: } duke@435: } duke@435: y[1] = (r-y[0])-w; duke@435: if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} duke@435: else return n; duke@435: } duke@435: /* duke@435: * all other (large) arguments duke@435: */ duke@435: if(ix>=0x7ff00000) { /* x is inf or NaN */ duke@435: y[0]=y[1]=x-x; return 0; duke@435: } duke@435: /* set z = scalbn(|x|,ilogb(x)-23) */ duke@435: *(1-i0+(int*)&z) = *(1-i0+(int*)&x); duke@435: e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */ duke@435: *(i0+(int*)&z) = ix - (e0<<20); duke@435: for(i=0;i<2;i++) { duke@435: tx[i] = (double)((int)(z)); duke@435: z = (z-tx[i])*two24A; duke@435: } duke@435: tx[2] = z; duke@435: nx = 3; duke@435: while(tx[nx-1]==zeroA) nx--; /* skip zero term */ duke@435: n = __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi); duke@435: if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} duke@435: return n; duke@435: } duke@435: duke@435: duke@435: /* __kernel_sin( x, y, iy) duke@435: * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 duke@435: * Input x is assumed to be bounded by ~pi/4 in magnitude. duke@435: * Input y is the tail of x. duke@435: * Input iy indicates whether y is 0. (if iy=0, y assume to be 0). duke@435: * duke@435: * Algorithm duke@435: * 1. Since sin(-x) = -sin(x), we need only to consider positive x. duke@435: * 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0. duke@435: * 3. sin(x) is approximated by a polynomial of degree 13 on duke@435: * [0,pi/4] duke@435: * 3 13 duke@435: * sin(x) ~ x + S1*x + ... + S6*x duke@435: * where duke@435: * duke@435: * |sin(x) 2 4 6 8 10 12 | -58 duke@435: * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 duke@435: * | x | duke@435: * duke@435: * 4. sin(x+y) = sin(x) + sin'(x')*y duke@435: * ~ sin(x) + (1-x*x/2)*y duke@435: * For better accuracy, let duke@435: * 3 2 2 2 2 duke@435: * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) duke@435: * then 3 2 duke@435: * sin(x) = x + (S1*x + (x *(r-y/2)+y)) duke@435: */ duke@435: duke@435: static const double duke@435: S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */ duke@435: S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */ duke@435: S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */ duke@435: S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */ duke@435: S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */ duke@435: S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */ duke@435: duke@435: static double __kernel_sin(double x, double y, int iy) duke@435: { duke@435: double z,r,v; duke@435: int ix; thartmann@7002: ix = high(x)&0x7fffffff; /* high word of x */ duke@435: if(ix<0x3e400000) /* |x| < 2**-27 */ duke@435: {if((int)x==0) return x;} /* generate inexact */ duke@435: z = x*x; duke@435: v = z*x; duke@435: r = S2+z*(S3+z*(S4+z*(S5+z*S6))); duke@435: if(iy==0) return x+v*(S1+z*r); duke@435: else return x-((z*(half*y-v*r)-y)-v*S1); duke@435: } duke@435: duke@435: /* duke@435: * __kernel_cos( x, y ) duke@435: * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 duke@435: * Input x is assumed to be bounded by ~pi/4 in magnitude. duke@435: * Input y is the tail of x. duke@435: * duke@435: * Algorithm duke@435: * 1. Since cos(-x) = cos(x), we need only to consider positive x. duke@435: * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. duke@435: * 3. cos(x) is approximated by a polynomial of degree 14 on duke@435: * [0,pi/4] duke@435: * 4 14 duke@435: * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x duke@435: * where the remez error is duke@435: * duke@435: * | 2 4 6 8 10 12 14 | -58 duke@435: * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 duke@435: * | | duke@435: * duke@435: * 4 6 8 10 12 14 duke@435: * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then duke@435: * cos(x) = 1 - x*x/2 + r duke@435: * since cos(x+y) ~ cos(x) - sin(x)*y duke@435: * ~ cos(x) - x*y, duke@435: * a correction term is necessary in cos(x) and hence duke@435: * cos(x+y) = 1 - (x*x/2 - (r - x*y)) duke@435: * For better accuracy when x > 0.3, let qx = |x|/4 with duke@435: * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. duke@435: * Then duke@435: * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). duke@435: * Note that 1-qx and (x*x/2-qx) is EXACT here, and the duke@435: * magnitude of the latter is at least a quarter of x*x/2, duke@435: * thus, reducing the rounding error in the subtraction. duke@435: */ duke@435: duke@435: static const double duke@435: C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ duke@435: C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ duke@435: C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ duke@435: C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ duke@435: C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ duke@435: C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ duke@435: duke@435: static double __kernel_cos(double x, double y) duke@435: { thartmann@7002: double a,h,z,r,qx=0; duke@435: int ix; thartmann@7002: ix = high(x)&0x7fffffff; /* ix = |x|'s high word*/ duke@435: if(ix<0x3e400000) { /* if x < 2**27 */ duke@435: if(((int)x)==0) return one; /* generate inexact */ duke@435: } duke@435: z = x*x; duke@435: r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6))))); duke@435: if(ix < 0x3FD33333) /* if |x| < 0.3 */ duke@435: return one - (0.5*z - (z*r - x*y)); duke@435: else { duke@435: if(ix > 0x3fe90000) { /* x > 0.78125 */ duke@435: qx = 0.28125; duke@435: } else { thartmann@7002: set_high(&qx, ix-0x00200000); /* x/4 */ thartmann@7002: set_low(&qx, 0); duke@435: } goetz@6461: h = 0.5*z-qx; goetz@6461: a = one-qx; goetz@6461: return a - (h - (z*r-x*y)); duke@435: } duke@435: } duke@435: duke@435: /* __kernel_tan( x, y, k ) duke@435: * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 duke@435: * Input x is assumed to be bounded by ~pi/4 in magnitude. duke@435: * Input y is the tail of x. duke@435: * Input k indicates whether tan (if k=1) or duke@435: * -1/tan (if k= -1) is returned. duke@435: * duke@435: * Algorithm duke@435: * 1. Since tan(-x) = -tan(x), we need only to consider positive x. duke@435: * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. duke@435: * 3. tan(x) is approximated by a odd polynomial of degree 27 on duke@435: * [0,0.67434] duke@435: * 3 27 duke@435: * tan(x) ~ x + T1*x + ... + T13*x duke@435: * where duke@435: * duke@435: * |tan(x) 2 4 26 | -59.2 duke@435: * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 duke@435: * | x | duke@435: * duke@435: * Note: tan(x+y) = tan(x) + tan'(x)*y duke@435: * ~ tan(x) + (1+x*x)*y duke@435: * Therefore, for better accuracy in computing tan(x+y), let duke@435: * 3 2 2 2 2 duke@435: * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) duke@435: * then duke@435: * 3 2 duke@435: * tan(x+y) = x + (T1*x + (x *(r+y)+y)) duke@435: * duke@435: * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then duke@435: * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) duke@435: * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) duke@435: */ duke@435: duke@435: static const double duke@435: pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ duke@435: pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */ duke@435: T[] = { duke@435: 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */ duke@435: 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */ duke@435: 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */ duke@435: 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */ duke@435: 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */ duke@435: 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */ duke@435: 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */ duke@435: 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */ duke@435: 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */ duke@435: 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */ duke@435: 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */ duke@435: -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */ duke@435: 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */ duke@435: }; duke@435: duke@435: static double __kernel_tan(double x, double y, int iy) duke@435: { duke@435: double z,r,v,w,s; duke@435: int ix,hx; thartmann@7002: hx = high(x); /* high word of x */ duke@435: ix = hx&0x7fffffff; /* high word of |x| */ duke@435: if(ix<0x3e300000) { /* x < 2**-28 */ duke@435: if((int)x==0) { /* generate inexact */ thartmann@7002: if (((ix | low(x)) | (iy + 1)) == 0) duke@435: return one / fabsd(x); duke@435: else { duke@435: if (iy == 1) duke@435: return x; duke@435: else { /* compute -1 / (x+y) carefully */ duke@435: double a, t; duke@435: duke@435: z = w = x + y; thartmann@7002: set_low(&z, 0); duke@435: v = y - (z - x); duke@435: t = a = -one / w; thartmann@7002: set_low(&t, 0); duke@435: s = one + t * z; duke@435: return t + a * (s + t * v); duke@435: } duke@435: } duke@435: } duke@435: } duke@435: if(ix>=0x3FE59428) { /* |x|>=0.6744 */ duke@435: if(hx<0) {x = -x; y = -y;} duke@435: z = pio4-x; duke@435: w = pio4lo-y; duke@435: x = z+w; y = 0.0; duke@435: } duke@435: z = x*x; duke@435: w = z*z; duke@435: /* Break x^5*(T[1]+x^2*T[2]+...) into duke@435: * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + duke@435: * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) duke@435: */ duke@435: r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11])))); duke@435: v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12]))))); duke@435: s = z*x; duke@435: r = y + z*(s*(r+v)+y); duke@435: r += T[0]*s; duke@435: w = x+r; duke@435: if(ix>=0x3FE59428) { duke@435: v = (double)iy; duke@435: return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r))); duke@435: } duke@435: if(iy==1) return w; duke@435: else { /* if allow error up to 2 ulp, duke@435: simply return -1.0/(x+r) here */ duke@435: /* compute -1.0/(x+r) accurately */ duke@435: double a,t; duke@435: z = w; thartmann@7002: set_low(&z, 0); duke@435: v = r-(z - x); /* z+v = r+x */ duke@435: t = a = -1.0/w; /* a = -1.0/w */ thartmann@7002: set_low(&t, 0); duke@435: s = 1.0+t*z; duke@435: return t+a*(s+t*v); duke@435: } duke@435: } duke@435: duke@435: duke@435: //---------------------------------------------------------------------- duke@435: // duke@435: // Routines for new sin/cos implementation duke@435: // duke@435: //---------------------------------------------------------------------- duke@435: duke@435: /* sin(x) duke@435: * Return sine function of x. duke@435: * duke@435: * kernel function: duke@435: * __kernel_sin ... sine function on [-pi/4,pi/4] duke@435: * __kernel_cos ... cose function on [-pi/4,pi/4] duke@435: * __ieee754_rem_pio2 ... argument reduction routine duke@435: * duke@435: * Method. duke@435: * Let S,C and T denote the sin, cos and tan respectively on duke@435: * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 duke@435: * in [-pi/4 , +pi/4], and let n = k mod 4. duke@435: * We have duke@435: * duke@435: * n sin(x) cos(x) tan(x) duke@435: * ---------------------------------------------------------- duke@435: * 0 S C T duke@435: * 1 C -S -1/T duke@435: * 2 -S -C T duke@435: * 3 -C S -1/T duke@435: * ---------------------------------------------------------- duke@435: * duke@435: * Special cases: duke@435: * Let trig be any of sin, cos, or tan. duke@435: * trig(+-INF) is NaN, with signals; duke@435: * trig(NaN) is that NaN; duke@435: * duke@435: * Accuracy: duke@435: * TRIG(x) returns trig(x) nearly rounded duke@435: */ duke@435: duke@435: JRT_LEAF(jdouble, SharedRuntime::dsin(jdouble x)) duke@435: double y[2],z=0.0; duke@435: int n, ix; duke@435: duke@435: /* High word of x. */ thartmann@7002: ix = high(x); duke@435: duke@435: /* |x| ~< pi/4 */ duke@435: ix &= 0x7fffffff; duke@435: if(ix <= 0x3fe921fb) return __kernel_sin(x,z,0); duke@435: duke@435: /* sin(Inf or NaN) is NaN */ duke@435: else if (ix>=0x7ff00000) return x-x; duke@435: duke@435: /* argument reduction needed */ duke@435: else { duke@435: n = __ieee754_rem_pio2(x,y); duke@435: switch(n&3) { duke@435: case 0: return __kernel_sin(y[0],y[1],1); duke@435: case 1: return __kernel_cos(y[0],y[1]); duke@435: case 2: return -__kernel_sin(y[0],y[1],1); duke@435: default: duke@435: return -__kernel_cos(y[0],y[1]); duke@435: } duke@435: } duke@435: JRT_END duke@435: duke@435: /* cos(x) duke@435: * Return cosine function of x. duke@435: * duke@435: * kernel function: duke@435: * __kernel_sin ... sine function on [-pi/4,pi/4] duke@435: * __kernel_cos ... cosine function on [-pi/4,pi/4] duke@435: * __ieee754_rem_pio2 ... argument reduction routine duke@435: * duke@435: * Method. duke@435: * Let S,C and T denote the sin, cos and tan respectively on duke@435: * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 duke@435: * in [-pi/4 , +pi/4], and let n = k mod 4. duke@435: * We have duke@435: * duke@435: * n sin(x) cos(x) tan(x) duke@435: * ---------------------------------------------------------- duke@435: * 0 S C T duke@435: * 1 C -S -1/T duke@435: * 2 -S -C T duke@435: * 3 -C S -1/T duke@435: * ---------------------------------------------------------- duke@435: * duke@435: * Special cases: duke@435: * Let trig be any of sin, cos, or tan. duke@435: * trig(+-INF) is NaN, with signals; duke@435: * trig(NaN) is that NaN; duke@435: * duke@435: * Accuracy: duke@435: * TRIG(x) returns trig(x) nearly rounded duke@435: */ duke@435: duke@435: JRT_LEAF(jdouble, SharedRuntime::dcos(jdouble x)) duke@435: double y[2],z=0.0; duke@435: int n, ix; duke@435: duke@435: /* High word of x. */ thartmann@7002: ix = high(x); duke@435: duke@435: /* |x| ~< pi/4 */ duke@435: ix &= 0x7fffffff; duke@435: if(ix <= 0x3fe921fb) return __kernel_cos(x,z); duke@435: duke@435: /* cos(Inf or NaN) is NaN */ duke@435: else if (ix>=0x7ff00000) return x-x; duke@435: duke@435: /* argument reduction needed */ duke@435: else { duke@435: n = __ieee754_rem_pio2(x,y); duke@435: switch(n&3) { duke@435: case 0: return __kernel_cos(y[0],y[1]); duke@435: case 1: return -__kernel_sin(y[0],y[1],1); duke@435: case 2: return -__kernel_cos(y[0],y[1]); duke@435: default: duke@435: return __kernel_sin(y[0],y[1],1); duke@435: } duke@435: } duke@435: JRT_END duke@435: duke@435: /* tan(x) duke@435: * Return tangent function of x. duke@435: * duke@435: * kernel function: duke@435: * __kernel_tan ... tangent function on [-pi/4,pi/4] duke@435: * __ieee754_rem_pio2 ... argument reduction routine duke@435: * duke@435: * Method. duke@435: * Let S,C and T denote the sin, cos and tan respectively on duke@435: * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 duke@435: * in [-pi/4 , +pi/4], and let n = k mod 4. duke@435: * We have duke@435: * duke@435: * n sin(x) cos(x) tan(x) duke@435: * ---------------------------------------------------------- duke@435: * 0 S C T duke@435: * 1 C -S -1/T duke@435: * 2 -S -C T duke@435: * 3 -C S -1/T duke@435: * ---------------------------------------------------------- duke@435: * duke@435: * Special cases: duke@435: * Let trig be any of sin, cos, or tan. duke@435: * trig(+-INF) is NaN, with signals; duke@435: * trig(NaN) is that NaN; duke@435: * duke@435: * Accuracy: duke@435: * TRIG(x) returns trig(x) nearly rounded duke@435: */ duke@435: duke@435: JRT_LEAF(jdouble, SharedRuntime::dtan(jdouble x)) duke@435: double y[2],z=0.0; duke@435: int n, ix; duke@435: duke@435: /* High word of x. */ thartmann@7002: ix = high(x); duke@435: duke@435: /* |x| ~< pi/4 */ duke@435: ix &= 0x7fffffff; duke@435: if(ix <= 0x3fe921fb) return __kernel_tan(x,z,1); duke@435: duke@435: /* tan(Inf or NaN) is NaN */ duke@435: else if (ix>=0x7ff00000) return x-x; /* NaN */ duke@435: duke@435: /* argument reduction needed */ duke@435: else { duke@435: n = __ieee754_rem_pio2(x,y); duke@435: return __kernel_tan(y[0],y[1],1-((n&1)<<1)); /* 1 -- n even duke@435: -1 -- n odd */ duke@435: } duke@435: JRT_END duke@435: duke@435: duke@435: #ifdef WIN32 duke@435: # pragma optimize ( "", on ) duke@435: #endif