1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/share/vm/runtime/sharedRuntimeTrig.cpp Sat Dec 01 00:00:00 2007 +0000
1.3 @@ -0,0 +1,957 @@
1.4 +/*
1.5 + * Copyright 2001-2005 Sun Microsystems, Inc. All Rights Reserved.
1.6 + * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
1.7 + *
1.8 + * This code is free software; you can redistribute it and/or modify it
1.9 + * under the terms of the GNU General Public License version 2 only, as
1.10 + * published by the Free Software Foundation.
1.11 + *
1.12 + * This code is distributed in the hope that it will be useful, but WITHOUT
1.13 + * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
1.14 + * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
1.15 + * version 2 for more details (a copy is included in the LICENSE file that
1.16 + * accompanied this code).
1.17 + *
1.18 + * You should have received a copy of the GNU General Public License version
1.19 + * 2 along with this work; if not, write to the Free Software Foundation,
1.20 + * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
1.21 + *
1.22 + * Please contact Sun Microsystems, Inc., 4150 Network Circle, Santa Clara,
1.23 + * CA 95054 USA or visit www.sun.com if you need additional information or
1.24 + * have any questions.
1.25 + *
1.26 + */
1.27 +
1.28 +#include "incls/_precompiled.incl"
1.29 +#include "incls/_sharedRuntimeTrig.cpp.incl"
1.30 +
1.31 +// This file contains copies of the fdlibm routines used by
1.32 +// StrictMath. It turns out that it is almost always required to use
1.33 +// these runtime routines; the Intel CPU doesn't meet the Java
1.34 +// specification for sin/cos outside a certain limited argument range,
1.35 +// and the SPARC CPU doesn't appear to have sin/cos instructions. It
1.36 +// also turns out that avoiding the indirect call through function
1.37 +// pointer out to libjava.so in SharedRuntime speeds these routines up
1.38 +// by roughly 15% on both Win32/x86 and Solaris/SPARC.
1.39 +
1.40 +// Enabling optimizations in this file causes incorrect code to be
1.41 +// generated; can not figure out how to turn down optimization for one
1.42 +// file in the IDE on Windows
1.43 +#ifdef WIN32
1.44 +# pragma optimize ( "", off )
1.45 +#endif
1.46 +
1.47 +#include <math.h>
1.48 +
1.49 +// VM_LITTLE_ENDIAN is #defined appropriately in the Makefiles
1.50 +// [jk] this is not 100% correct because the float word order may different
1.51 +// from the byte order (e.g. on ARM)
1.52 +#ifdef VM_LITTLE_ENDIAN
1.53 +# define __HI(x) *(1+(int*)&x)
1.54 +# define __LO(x) *(int*)&x
1.55 +#else
1.56 +# define __HI(x) *(int*)&x
1.57 +# define __LO(x) *(1+(int*)&x)
1.58 +#endif
1.59 +
1.60 +static double copysignA(double x, double y) {
1.61 + __HI(x) = (__HI(x)&0x7fffffff)|(__HI(y)&0x80000000);
1.62 + return x;
1.63 +}
1.64 +
1.65 +/*
1.66 + * scalbn (double x, int n)
1.67 + * scalbn(x,n) returns x* 2**n computed by exponent
1.68 + * manipulation rather than by actually performing an
1.69 + * exponentiation or a multiplication.
1.70 + */
1.71 +
1.72 +static const double
1.73 +two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
1.74 +twom54 = 5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */
1.75 +hugeX = 1.0e+300,
1.76 +tiny = 1.0e-300;
1.77 +
1.78 +static double scalbnA (double x, int n) {
1.79 + int k,hx,lx;
1.80 + hx = __HI(x);
1.81 + lx = __LO(x);
1.82 + k = (hx&0x7ff00000)>>20; /* extract exponent */
1.83 + if (k==0) { /* 0 or subnormal x */
1.84 + if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */
1.85 + x *= two54;
1.86 + hx = __HI(x);
1.87 + k = ((hx&0x7ff00000)>>20) - 54;
1.88 + if (n< -50000) return tiny*x; /*underflow*/
1.89 + }
1.90 + if (k==0x7ff) return x+x; /* NaN or Inf */
1.91 + k = k+n;
1.92 + if (k > 0x7fe) return hugeX*copysignA(hugeX,x); /* overflow */
1.93 + if (k > 0) /* normal result */
1.94 + {__HI(x) = (hx&0x800fffff)|(k<<20); return x;}
1.95 + if (k <= -54) {
1.96 + if (n > 50000) /* in case integer overflow in n+k */
1.97 + return hugeX*copysignA(hugeX,x); /*overflow*/
1.98 + else return tiny*copysignA(tiny,x); /*underflow*/
1.99 + }
1.100 + k += 54; /* subnormal result */
1.101 + __HI(x) = (hx&0x800fffff)|(k<<20);
1.102 + return x*twom54;
1.103 +}
1.104 +
1.105 +/*
1.106 + * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
1.107 + * double x[],y[]; int e0,nx,prec; int ipio2[];
1.108 + *
1.109 + * __kernel_rem_pio2 return the last three digits of N with
1.110 + * y = x - N*pi/2
1.111 + * so that |y| < pi/2.
1.112 + *
1.113 + * The method is to compute the integer (mod 8) and fraction parts of
1.114 + * (2/pi)*x without doing the full multiplication. In general we
1.115 + * skip the part of the product that are known to be a huge integer (
1.116 + * more accurately, = 0 mod 8 ). Thus the number of operations are
1.117 + * independent of the exponent of the input.
1.118 + *
1.119 + * (2/pi) is represented by an array of 24-bit integers in ipio2[].
1.120 + *
1.121 + * Input parameters:
1.122 + * x[] The input value (must be positive) is broken into nx
1.123 + * pieces of 24-bit integers in double precision format.
1.124 + * x[i] will be the i-th 24 bit of x. The scaled exponent
1.125 + * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
1.126 + * match x's up to 24 bits.
1.127 + *
1.128 + * Example of breaking a double positive z into x[0]+x[1]+x[2]:
1.129 + * e0 = ilogb(z)-23
1.130 + * z = scalbn(z,-e0)
1.131 + * for i = 0,1,2
1.132 + * x[i] = floor(z)
1.133 + * z = (z-x[i])*2**24
1.134 + *
1.135 + *
1.136 + * y[] ouput result in an array of double precision numbers.
1.137 + * The dimension of y[] is:
1.138 + * 24-bit precision 1
1.139 + * 53-bit precision 2
1.140 + * 64-bit precision 2
1.141 + * 113-bit precision 3
1.142 + * The actual value is the sum of them. Thus for 113-bit
1.143 + * precsion, one may have to do something like:
1.144 + *
1.145 + * long double t,w,r_head, r_tail;
1.146 + * t = (long double)y[2] + (long double)y[1];
1.147 + * w = (long double)y[0];
1.148 + * r_head = t+w;
1.149 + * r_tail = w - (r_head - t);
1.150 + *
1.151 + * e0 The exponent of x[0]
1.152 + *
1.153 + * nx dimension of x[]
1.154 + *
1.155 + * prec an interger indicating the precision:
1.156 + * 0 24 bits (single)
1.157 + * 1 53 bits (double)
1.158 + * 2 64 bits (extended)
1.159 + * 3 113 bits (quad)
1.160 + *
1.161 + * ipio2[]
1.162 + * integer array, contains the (24*i)-th to (24*i+23)-th
1.163 + * bit of 2/pi after binary point. The corresponding
1.164 + * floating value is
1.165 + *
1.166 + * ipio2[i] * 2^(-24(i+1)).
1.167 + *
1.168 + * External function:
1.169 + * double scalbn(), floor();
1.170 + *
1.171 + *
1.172 + * Here is the description of some local variables:
1.173 + *
1.174 + * jk jk+1 is the initial number of terms of ipio2[] needed
1.175 + * in the computation. The recommended value is 2,3,4,
1.176 + * 6 for single, double, extended,and quad.
1.177 + *
1.178 + * jz local integer variable indicating the number of
1.179 + * terms of ipio2[] used.
1.180 + *
1.181 + * jx nx - 1
1.182 + *
1.183 + * jv index for pointing to the suitable ipio2[] for the
1.184 + * computation. In general, we want
1.185 + * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
1.186 + * is an integer. Thus
1.187 + * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
1.188 + * Hence jv = max(0,(e0-3)/24).
1.189 + *
1.190 + * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
1.191 + *
1.192 + * q[] double array with integral value, representing the
1.193 + * 24-bits chunk of the product of x and 2/pi.
1.194 + *
1.195 + * q0 the corresponding exponent of q[0]. Note that the
1.196 + * exponent for q[i] would be q0-24*i.
1.197 + *
1.198 + * PIo2[] double precision array, obtained by cutting pi/2
1.199 + * into 24 bits chunks.
1.200 + *
1.201 + * f[] ipio2[] in floating point
1.202 + *
1.203 + * iq[] integer array by breaking up q[] in 24-bits chunk.
1.204 + *
1.205 + * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
1.206 + *
1.207 + * ih integer. If >0 it indicats q[] is >= 0.5, hence
1.208 + * it also indicates the *sign* of the result.
1.209 + *
1.210 + */
1.211 +
1.212 +
1.213 +/*
1.214 + * Constants:
1.215 + * The hexadecimal values are the intended ones for the following
1.216 + * constants. The decimal values may be used, provided that the
1.217 + * compiler will convert from decimal to binary accurately enough
1.218 + * to produce the hexadecimal values shown.
1.219 + */
1.220 +
1.221 +
1.222 +static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
1.223 +
1.224 +static const double PIo2[] = {
1.225 + 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
1.226 + 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
1.227 + 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
1.228 + 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
1.229 + 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
1.230 + 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
1.231 + 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
1.232 + 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
1.233 +};
1.234 +
1.235 +static const double
1.236 +zeroB = 0.0,
1.237 +one = 1.0,
1.238 +two24B = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
1.239 +twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
1.240 +
1.241 +static int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2) {
1.242 + int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
1.243 + double z,fw,f[20],fq[20],q[20];
1.244 +
1.245 + /* initialize jk*/
1.246 + jk = init_jk[prec];
1.247 + jp = jk;
1.248 +
1.249 + /* determine jx,jv,q0, note that 3>q0 */
1.250 + jx = nx-1;
1.251 + jv = (e0-3)/24; if(jv<0) jv=0;
1.252 + q0 = e0-24*(jv+1);
1.253 +
1.254 + /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
1.255 + j = jv-jx; m = jx+jk;
1.256 + for(i=0;i<=m;i++,j++) f[i] = (j<0)? zeroB : (double) ipio2[j];
1.257 +
1.258 + /* compute q[0],q[1],...q[jk] */
1.259 + for (i=0;i<=jk;i++) {
1.260 + for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
1.261 + }
1.262 +
1.263 + jz = jk;
1.264 +recompute:
1.265 + /* distill q[] into iq[] reversingly */
1.266 + for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
1.267 + fw = (double)((int)(twon24* z));
1.268 + iq[i] = (int)(z-two24B*fw);
1.269 + z = q[j-1]+fw;
1.270 + }
1.271 +
1.272 + /* compute n */
1.273 + z = scalbnA(z,q0); /* actual value of z */
1.274 + z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
1.275 + n = (int) z;
1.276 + z -= (double)n;
1.277 + ih = 0;
1.278 + if(q0>0) { /* need iq[jz-1] to determine n */
1.279 + i = (iq[jz-1]>>(24-q0)); n += i;
1.280 + iq[jz-1] -= i<<(24-q0);
1.281 + ih = iq[jz-1]>>(23-q0);
1.282 + }
1.283 + else if(q0==0) ih = iq[jz-1]>>23;
1.284 + else if(z>=0.5) ih=2;
1.285 +
1.286 + if(ih>0) { /* q > 0.5 */
1.287 + n += 1; carry = 0;
1.288 + for(i=0;i<jz ;i++) { /* compute 1-q */
1.289 + j = iq[i];
1.290 + if(carry==0) {
1.291 + if(j!=0) {
1.292 + carry = 1; iq[i] = 0x1000000- j;
1.293 + }
1.294 + } else iq[i] = 0xffffff - j;
1.295 + }
1.296 + if(q0>0) { /* rare case: chance is 1 in 12 */
1.297 + switch(q0) {
1.298 + case 1:
1.299 + iq[jz-1] &= 0x7fffff; break;
1.300 + case 2:
1.301 + iq[jz-1] &= 0x3fffff; break;
1.302 + }
1.303 + }
1.304 + if(ih==2) {
1.305 + z = one - z;
1.306 + if(carry!=0) z -= scalbnA(one,q0);
1.307 + }
1.308 + }
1.309 +
1.310 + /* check if recomputation is needed */
1.311 + if(z==zeroB) {
1.312 + j = 0;
1.313 + for (i=jz-1;i>=jk;i--) j |= iq[i];
1.314 + if(j==0) { /* need recomputation */
1.315 + for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
1.316 +
1.317 + for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
1.318 + f[jx+i] = (double) ipio2[jv+i];
1.319 + for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
1.320 + q[i] = fw;
1.321 + }
1.322 + jz += k;
1.323 + goto recompute;
1.324 + }
1.325 + }
1.326 +
1.327 + /* chop off zero terms */
1.328 + if(z==0.0) {
1.329 + jz -= 1; q0 -= 24;
1.330 + while(iq[jz]==0) { jz--; q0-=24;}
1.331 + } else { /* break z into 24-bit if neccessary */
1.332 + z = scalbnA(z,-q0);
1.333 + if(z>=two24B) {
1.334 + fw = (double)((int)(twon24*z));
1.335 + iq[jz] = (int)(z-two24B*fw);
1.336 + jz += 1; q0 += 24;
1.337 + iq[jz] = (int) fw;
1.338 + } else iq[jz] = (int) z ;
1.339 + }
1.340 +
1.341 + /* convert integer "bit" chunk to floating-point value */
1.342 + fw = scalbnA(one,q0);
1.343 + for(i=jz;i>=0;i--) {
1.344 + q[i] = fw*(double)iq[i]; fw*=twon24;
1.345 + }
1.346 +
1.347 + /* compute PIo2[0,...,jp]*q[jz,...,0] */
1.348 + for(i=jz;i>=0;i--) {
1.349 + for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
1.350 + fq[jz-i] = fw;
1.351 + }
1.352 +
1.353 + /* compress fq[] into y[] */
1.354 + switch(prec) {
1.355 + case 0:
1.356 + fw = 0.0;
1.357 + for (i=jz;i>=0;i--) fw += fq[i];
1.358 + y[0] = (ih==0)? fw: -fw;
1.359 + break;
1.360 + case 1:
1.361 + case 2:
1.362 + fw = 0.0;
1.363 + for (i=jz;i>=0;i--) fw += fq[i];
1.364 + y[0] = (ih==0)? fw: -fw;
1.365 + fw = fq[0]-fw;
1.366 + for (i=1;i<=jz;i++) fw += fq[i];
1.367 + y[1] = (ih==0)? fw: -fw;
1.368 + break;
1.369 + case 3: /* painful */
1.370 + for (i=jz;i>0;i--) {
1.371 + fw = fq[i-1]+fq[i];
1.372 + fq[i] += fq[i-1]-fw;
1.373 + fq[i-1] = fw;
1.374 + }
1.375 + for (i=jz;i>1;i--) {
1.376 + fw = fq[i-1]+fq[i];
1.377 + fq[i] += fq[i-1]-fw;
1.378 + fq[i-1] = fw;
1.379 + }
1.380 + for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
1.381 + if(ih==0) {
1.382 + y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
1.383 + } else {
1.384 + y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
1.385 + }
1.386 + }
1.387 + return n&7;
1.388 +}
1.389 +
1.390 +
1.391 +/*
1.392 + * ====================================================
1.393 + * Copyright 13 Dec 1993 Sun Microsystems, Inc. All Rights Reserved.
1.394 + *
1.395 + * Developed at SunPro, a Sun Microsystems, Inc. business.
1.396 + * Permission to use, copy, modify, and distribute this
1.397 + * software is freely granted, provided that this notice
1.398 + * is preserved.
1.399 + * ====================================================
1.400 + *
1.401 + */
1.402 +
1.403 +/* __ieee754_rem_pio2(x,y)
1.404 + *
1.405 + * return the remainder of x rem pi/2 in y[0]+y[1]
1.406 + * use __kernel_rem_pio2()
1.407 + */
1.408 +
1.409 +/*
1.410 + * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
1.411 + */
1.412 +static const int two_over_pi[] = {
1.413 + 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
1.414 + 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
1.415 + 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
1.416 + 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
1.417 + 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
1.418 + 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
1.419 + 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
1.420 + 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
1.421 + 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
1.422 + 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
1.423 + 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
1.424 +};
1.425 +
1.426 +static const int npio2_hw[] = {
1.427 + 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
1.428 + 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
1.429 + 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
1.430 + 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
1.431 + 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
1.432 + 0x404858EB, 0x404921FB,
1.433 +};
1.434 +
1.435 +/*
1.436 + * invpio2: 53 bits of 2/pi
1.437 + * pio2_1: first 33 bit of pi/2
1.438 + * pio2_1t: pi/2 - pio2_1
1.439 + * pio2_2: second 33 bit of pi/2
1.440 + * pio2_2t: pi/2 - (pio2_1+pio2_2)
1.441 + * pio2_3: third 33 bit of pi/2
1.442 + * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
1.443 + */
1.444 +
1.445 +static const double
1.446 +zeroA = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
1.447 +half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
1.448 +two24A = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
1.449 +invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
1.450 +pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
1.451 +pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
1.452 +pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
1.453 +pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
1.454 +pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
1.455 +pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
1.456 +
1.457 +static int __ieee754_rem_pio2(double x, double *y) {
1.458 + double z,w,t,r,fn;
1.459 + double tx[3];
1.460 + int e0,i,j,nx,n,ix,hx,i0;
1.461 +
1.462 + i0 = ((*(int*)&two24A)>>30)^1; /* high word index */
1.463 + hx = *(i0+(int*)&x); /* high word of x */
1.464 + ix = hx&0x7fffffff;
1.465 + if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
1.466 + {y[0] = x; y[1] = 0; return 0;}
1.467 + if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */
1.468 + if(hx>0) {
1.469 + z = x - pio2_1;
1.470 + if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
1.471 + y[0] = z - pio2_1t;
1.472 + y[1] = (z-y[0])-pio2_1t;
1.473 + } else { /* near pi/2, use 33+33+53 bit pi */
1.474 + z -= pio2_2;
1.475 + y[0] = z - pio2_2t;
1.476 + y[1] = (z-y[0])-pio2_2t;
1.477 + }
1.478 + return 1;
1.479 + } else { /* negative x */
1.480 + z = x + pio2_1;
1.481 + if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
1.482 + y[0] = z + pio2_1t;
1.483 + y[1] = (z-y[0])+pio2_1t;
1.484 + } else { /* near pi/2, use 33+33+53 bit pi */
1.485 + z += pio2_2;
1.486 + y[0] = z + pio2_2t;
1.487 + y[1] = (z-y[0])+pio2_2t;
1.488 + }
1.489 + return -1;
1.490 + }
1.491 + }
1.492 + if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
1.493 + t = fabsd(x);
1.494 + n = (int) (t*invpio2+half);
1.495 + fn = (double)n;
1.496 + r = t-fn*pio2_1;
1.497 + w = fn*pio2_1t; /* 1st round good to 85 bit */
1.498 + if(n<32&&ix!=npio2_hw[n-1]) {
1.499 + y[0] = r-w; /* quick check no cancellation */
1.500 + } else {
1.501 + j = ix>>20;
1.502 + y[0] = r-w;
1.503 + i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff);
1.504 + if(i>16) { /* 2nd iteration needed, good to 118 */
1.505 + t = r;
1.506 + w = fn*pio2_2;
1.507 + r = t-w;
1.508 + w = fn*pio2_2t-((t-r)-w);
1.509 + y[0] = r-w;
1.510 + i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff);
1.511 + if(i>49) { /* 3rd iteration need, 151 bits acc */
1.512 + t = r; /* will cover all possible cases */
1.513 + w = fn*pio2_3;
1.514 + r = t-w;
1.515 + w = fn*pio2_3t-((t-r)-w);
1.516 + y[0] = r-w;
1.517 + }
1.518 + }
1.519 + }
1.520 + y[1] = (r-y[0])-w;
1.521 + if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
1.522 + else return n;
1.523 + }
1.524 + /*
1.525 + * all other (large) arguments
1.526 + */
1.527 + if(ix>=0x7ff00000) { /* x is inf or NaN */
1.528 + y[0]=y[1]=x-x; return 0;
1.529 + }
1.530 + /* set z = scalbn(|x|,ilogb(x)-23) */
1.531 + *(1-i0+(int*)&z) = *(1-i0+(int*)&x);
1.532 + e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */
1.533 + *(i0+(int*)&z) = ix - (e0<<20);
1.534 + for(i=0;i<2;i++) {
1.535 + tx[i] = (double)((int)(z));
1.536 + z = (z-tx[i])*two24A;
1.537 + }
1.538 + tx[2] = z;
1.539 + nx = 3;
1.540 + while(tx[nx-1]==zeroA) nx--; /* skip zero term */
1.541 + n = __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi);
1.542 + if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
1.543 + return n;
1.544 +}
1.545 +
1.546 +
1.547 +/* __kernel_sin( x, y, iy)
1.548 + * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854
1.549 + * Input x is assumed to be bounded by ~pi/4 in magnitude.
1.550 + * Input y is the tail of x.
1.551 + * Input iy indicates whether y is 0. (if iy=0, y assume to be 0).
1.552 + *
1.553 + * Algorithm
1.554 + * 1. Since sin(-x) = -sin(x), we need only to consider positive x.
1.555 + * 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0.
1.556 + * 3. sin(x) is approximated by a polynomial of degree 13 on
1.557 + * [0,pi/4]
1.558 + * 3 13
1.559 + * sin(x) ~ x + S1*x + ... + S6*x
1.560 + * where
1.561 + *
1.562 + * |sin(x) 2 4 6 8 10 12 | -58
1.563 + * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2
1.564 + * | x |
1.565 + *
1.566 + * 4. sin(x+y) = sin(x) + sin'(x')*y
1.567 + * ~ sin(x) + (1-x*x/2)*y
1.568 + * For better accuracy, let
1.569 + * 3 2 2 2 2
1.570 + * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6))))
1.571 + * then 3 2
1.572 + * sin(x) = x + (S1*x + (x *(r-y/2)+y))
1.573 + */
1.574 +
1.575 +static const double
1.576 +S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */
1.577 +S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */
1.578 +S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */
1.579 +S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */
1.580 +S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */
1.581 +S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */
1.582 +
1.583 +static double __kernel_sin(double x, double y, int iy)
1.584 +{
1.585 + double z,r,v;
1.586 + int ix;
1.587 + ix = __HI(x)&0x7fffffff; /* high word of x */
1.588 + if(ix<0x3e400000) /* |x| < 2**-27 */
1.589 + {if((int)x==0) return x;} /* generate inexact */
1.590 + z = x*x;
1.591 + v = z*x;
1.592 + r = S2+z*(S3+z*(S4+z*(S5+z*S6)));
1.593 + if(iy==0) return x+v*(S1+z*r);
1.594 + else return x-((z*(half*y-v*r)-y)-v*S1);
1.595 +}
1.596 +
1.597 +/*
1.598 + * __kernel_cos( x, y )
1.599 + * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
1.600 + * Input x is assumed to be bounded by ~pi/4 in magnitude.
1.601 + * Input y is the tail of x.
1.602 + *
1.603 + * Algorithm
1.604 + * 1. Since cos(-x) = cos(x), we need only to consider positive x.
1.605 + * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
1.606 + * 3. cos(x) is approximated by a polynomial of degree 14 on
1.607 + * [0,pi/4]
1.608 + * 4 14
1.609 + * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
1.610 + * where the remez error is
1.611 + *
1.612 + * | 2 4 6 8 10 12 14 | -58
1.613 + * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
1.614 + * | |
1.615 + *
1.616 + * 4 6 8 10 12 14
1.617 + * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
1.618 + * cos(x) = 1 - x*x/2 + r
1.619 + * since cos(x+y) ~ cos(x) - sin(x)*y
1.620 + * ~ cos(x) - x*y,
1.621 + * a correction term is necessary in cos(x) and hence
1.622 + * cos(x+y) = 1 - (x*x/2 - (r - x*y))
1.623 + * For better accuracy when x > 0.3, let qx = |x|/4 with
1.624 + * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
1.625 + * Then
1.626 + * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
1.627 + * Note that 1-qx and (x*x/2-qx) is EXACT here, and the
1.628 + * magnitude of the latter is at least a quarter of x*x/2,
1.629 + * thus, reducing the rounding error in the subtraction.
1.630 + */
1.631 +
1.632 +static const double
1.633 +C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
1.634 +C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
1.635 +C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
1.636 +C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
1.637 +C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
1.638 +C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
1.639 +
1.640 +static double __kernel_cos(double x, double y)
1.641 +{
1.642 + double a,hz,z,r,qx;
1.643 + int ix;
1.644 + ix = __HI(x)&0x7fffffff; /* ix = |x|'s high word*/
1.645 + if(ix<0x3e400000) { /* if x < 2**27 */
1.646 + if(((int)x)==0) return one; /* generate inexact */
1.647 + }
1.648 + z = x*x;
1.649 + r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
1.650 + if(ix < 0x3FD33333) /* if |x| < 0.3 */
1.651 + return one - (0.5*z - (z*r - x*y));
1.652 + else {
1.653 + if(ix > 0x3fe90000) { /* x > 0.78125 */
1.654 + qx = 0.28125;
1.655 + } else {
1.656 + __HI(qx) = ix-0x00200000; /* x/4 */
1.657 + __LO(qx) = 0;
1.658 + }
1.659 + hz = 0.5*z-qx;
1.660 + a = one-qx;
1.661 + return a - (hz - (z*r-x*y));
1.662 + }
1.663 +}
1.664 +
1.665 +/* __kernel_tan( x, y, k )
1.666 + * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
1.667 + * Input x is assumed to be bounded by ~pi/4 in magnitude.
1.668 + * Input y is the tail of x.
1.669 + * Input k indicates whether tan (if k=1) or
1.670 + * -1/tan (if k= -1) is returned.
1.671 + *
1.672 + * Algorithm
1.673 + * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
1.674 + * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
1.675 + * 3. tan(x) is approximated by a odd polynomial of degree 27 on
1.676 + * [0,0.67434]
1.677 + * 3 27
1.678 + * tan(x) ~ x + T1*x + ... + T13*x
1.679 + * where
1.680 + *
1.681 + * |tan(x) 2 4 26 | -59.2
1.682 + * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
1.683 + * | x |
1.684 + *
1.685 + * Note: tan(x+y) = tan(x) + tan'(x)*y
1.686 + * ~ tan(x) + (1+x*x)*y
1.687 + * Therefore, for better accuracy in computing tan(x+y), let
1.688 + * 3 2 2 2 2
1.689 + * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
1.690 + * then
1.691 + * 3 2
1.692 + * tan(x+y) = x + (T1*x + (x *(r+y)+y))
1.693 + *
1.694 + * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
1.695 + * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
1.696 + * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
1.697 + */
1.698 +
1.699 +static const double
1.700 +pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
1.701 +pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
1.702 +T[] = {
1.703 + 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
1.704 + 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
1.705 + 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
1.706 + 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
1.707 + 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
1.708 + 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
1.709 + 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
1.710 + 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
1.711 + 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
1.712 + 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
1.713 + 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
1.714 + -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
1.715 + 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
1.716 +};
1.717 +
1.718 +static double __kernel_tan(double x, double y, int iy)
1.719 +{
1.720 + double z,r,v,w,s;
1.721 + int ix,hx;
1.722 + hx = __HI(x); /* high word of x */
1.723 + ix = hx&0x7fffffff; /* high word of |x| */
1.724 + if(ix<0x3e300000) { /* x < 2**-28 */
1.725 + if((int)x==0) { /* generate inexact */
1.726 + if (((ix | __LO(x)) | (iy + 1)) == 0)
1.727 + return one / fabsd(x);
1.728 + else {
1.729 + if (iy == 1)
1.730 + return x;
1.731 + else { /* compute -1 / (x+y) carefully */
1.732 + double a, t;
1.733 +
1.734 + z = w = x + y;
1.735 + __LO(z) = 0;
1.736 + v = y - (z - x);
1.737 + t = a = -one / w;
1.738 + __LO(t) = 0;
1.739 + s = one + t * z;
1.740 + return t + a * (s + t * v);
1.741 + }
1.742 + }
1.743 + }
1.744 + }
1.745 + if(ix>=0x3FE59428) { /* |x|>=0.6744 */
1.746 + if(hx<0) {x = -x; y = -y;}
1.747 + z = pio4-x;
1.748 + w = pio4lo-y;
1.749 + x = z+w; y = 0.0;
1.750 + }
1.751 + z = x*x;
1.752 + w = z*z;
1.753 + /* Break x^5*(T[1]+x^2*T[2]+...) into
1.754 + * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
1.755 + * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
1.756 + */
1.757 + r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
1.758 + v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
1.759 + s = z*x;
1.760 + r = y + z*(s*(r+v)+y);
1.761 + r += T[0]*s;
1.762 + w = x+r;
1.763 + if(ix>=0x3FE59428) {
1.764 + v = (double)iy;
1.765 + return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
1.766 + }
1.767 + if(iy==1) return w;
1.768 + else { /* if allow error up to 2 ulp,
1.769 + simply return -1.0/(x+r) here */
1.770 + /* compute -1.0/(x+r) accurately */
1.771 + double a,t;
1.772 + z = w;
1.773 + __LO(z) = 0;
1.774 + v = r-(z - x); /* z+v = r+x */
1.775 + t = a = -1.0/w; /* a = -1.0/w */
1.776 + __LO(t) = 0;
1.777 + s = 1.0+t*z;
1.778 + return t+a*(s+t*v);
1.779 + }
1.780 +}
1.781 +
1.782 +
1.783 +//----------------------------------------------------------------------
1.784 +//
1.785 +// Routines for new sin/cos implementation
1.786 +//
1.787 +//----------------------------------------------------------------------
1.788 +
1.789 +/* sin(x)
1.790 + * Return sine function of x.
1.791 + *
1.792 + * kernel function:
1.793 + * __kernel_sin ... sine function on [-pi/4,pi/4]
1.794 + * __kernel_cos ... cose function on [-pi/4,pi/4]
1.795 + * __ieee754_rem_pio2 ... argument reduction routine
1.796 + *
1.797 + * Method.
1.798 + * Let S,C and T denote the sin, cos and tan respectively on
1.799 + * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
1.800 + * in [-pi/4 , +pi/4], and let n = k mod 4.
1.801 + * We have
1.802 + *
1.803 + * n sin(x) cos(x) tan(x)
1.804 + * ----------------------------------------------------------
1.805 + * 0 S C T
1.806 + * 1 C -S -1/T
1.807 + * 2 -S -C T
1.808 + * 3 -C S -1/T
1.809 + * ----------------------------------------------------------
1.810 + *
1.811 + * Special cases:
1.812 + * Let trig be any of sin, cos, or tan.
1.813 + * trig(+-INF) is NaN, with signals;
1.814 + * trig(NaN) is that NaN;
1.815 + *
1.816 + * Accuracy:
1.817 + * TRIG(x) returns trig(x) nearly rounded
1.818 + */
1.819 +
1.820 +JRT_LEAF(jdouble, SharedRuntime::dsin(jdouble x))
1.821 + double y[2],z=0.0;
1.822 + int n, ix;
1.823 +
1.824 + /* High word of x. */
1.825 + ix = __HI(x);
1.826 +
1.827 + /* |x| ~< pi/4 */
1.828 + ix &= 0x7fffffff;
1.829 + if(ix <= 0x3fe921fb) return __kernel_sin(x,z,0);
1.830 +
1.831 + /* sin(Inf or NaN) is NaN */
1.832 + else if (ix>=0x7ff00000) return x-x;
1.833 +
1.834 + /* argument reduction needed */
1.835 + else {
1.836 + n = __ieee754_rem_pio2(x,y);
1.837 + switch(n&3) {
1.838 + case 0: return __kernel_sin(y[0],y[1],1);
1.839 + case 1: return __kernel_cos(y[0],y[1]);
1.840 + case 2: return -__kernel_sin(y[0],y[1],1);
1.841 + default:
1.842 + return -__kernel_cos(y[0],y[1]);
1.843 + }
1.844 + }
1.845 +JRT_END
1.846 +
1.847 +/* cos(x)
1.848 + * Return cosine function of x.
1.849 + *
1.850 + * kernel function:
1.851 + * __kernel_sin ... sine function on [-pi/4,pi/4]
1.852 + * __kernel_cos ... cosine function on [-pi/4,pi/4]
1.853 + * __ieee754_rem_pio2 ... argument reduction routine
1.854 + *
1.855 + * Method.
1.856 + * Let S,C and T denote the sin, cos and tan respectively on
1.857 + * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
1.858 + * in [-pi/4 , +pi/4], and let n = k mod 4.
1.859 + * We have
1.860 + *
1.861 + * n sin(x) cos(x) tan(x)
1.862 + * ----------------------------------------------------------
1.863 + * 0 S C T
1.864 + * 1 C -S -1/T
1.865 + * 2 -S -C T
1.866 + * 3 -C S -1/T
1.867 + * ----------------------------------------------------------
1.868 + *
1.869 + * Special cases:
1.870 + * Let trig be any of sin, cos, or tan.
1.871 + * trig(+-INF) is NaN, with signals;
1.872 + * trig(NaN) is that NaN;
1.873 + *
1.874 + * Accuracy:
1.875 + * TRIG(x) returns trig(x) nearly rounded
1.876 + */
1.877 +
1.878 +JRT_LEAF(jdouble, SharedRuntime::dcos(jdouble x))
1.879 + double y[2],z=0.0;
1.880 + int n, ix;
1.881 +
1.882 + /* High word of x. */
1.883 + ix = __HI(x);
1.884 +
1.885 + /* |x| ~< pi/4 */
1.886 + ix &= 0x7fffffff;
1.887 + if(ix <= 0x3fe921fb) return __kernel_cos(x,z);
1.888 +
1.889 + /* cos(Inf or NaN) is NaN */
1.890 + else if (ix>=0x7ff00000) return x-x;
1.891 +
1.892 + /* argument reduction needed */
1.893 + else {
1.894 + n = __ieee754_rem_pio2(x,y);
1.895 + switch(n&3) {
1.896 + case 0: return __kernel_cos(y[0],y[1]);
1.897 + case 1: return -__kernel_sin(y[0],y[1],1);
1.898 + case 2: return -__kernel_cos(y[0],y[1]);
1.899 + default:
1.900 + return __kernel_sin(y[0],y[1],1);
1.901 + }
1.902 + }
1.903 +JRT_END
1.904 +
1.905 +/* tan(x)
1.906 + * Return tangent function of x.
1.907 + *
1.908 + * kernel function:
1.909 + * __kernel_tan ... tangent function on [-pi/4,pi/4]
1.910 + * __ieee754_rem_pio2 ... argument reduction routine
1.911 + *
1.912 + * Method.
1.913 + * Let S,C and T denote the sin, cos and tan respectively on
1.914 + * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
1.915 + * in [-pi/4 , +pi/4], and let n = k mod 4.
1.916 + * We have
1.917 + *
1.918 + * n sin(x) cos(x) tan(x)
1.919 + * ----------------------------------------------------------
1.920 + * 0 S C T
1.921 + * 1 C -S -1/T
1.922 + * 2 -S -C T
1.923 + * 3 -C S -1/T
1.924 + * ----------------------------------------------------------
1.925 + *
1.926 + * Special cases:
1.927 + * Let trig be any of sin, cos, or tan.
1.928 + * trig(+-INF) is NaN, with signals;
1.929 + * trig(NaN) is that NaN;
1.930 + *
1.931 + * Accuracy:
1.932 + * TRIG(x) returns trig(x) nearly rounded
1.933 + */
1.934 +
1.935 +JRT_LEAF(jdouble, SharedRuntime::dtan(jdouble x))
1.936 + double y[2],z=0.0;
1.937 + int n, ix;
1.938 +
1.939 + /* High word of x. */
1.940 + ix = __HI(x);
1.941 +
1.942 + /* |x| ~< pi/4 */
1.943 + ix &= 0x7fffffff;
1.944 + if(ix <= 0x3fe921fb) return __kernel_tan(x,z,1);
1.945 +
1.946 + /* tan(Inf or NaN) is NaN */
1.947 + else if (ix>=0x7ff00000) return x-x; /* NaN */
1.948 +
1.949 + /* argument reduction needed */
1.950 + else {
1.951 + n = __ieee754_rem_pio2(x,y);
1.952 + return __kernel_tan(y[0],y[1],1-((n&1)<<1)); /* 1 -- n even
1.953 + -1 -- n odd */
1.954 + }
1.955 +JRT_END
1.956 +
1.957 +
1.958 +#ifdef WIN32
1.959 +# pragma optimize ( "", on )
1.960 +#endif
