Tue, 08 Aug 2017 15:57:29 +0800
merge
aoqi@0 | 1 | /* |
aoqi@0 | 2 | * Copyright (c) 2001, 2010, Oracle and/or its affiliates. All rights reserved. |
aoqi@0 | 3 | * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. |
aoqi@0 | 4 | * |
aoqi@0 | 5 | * This code is free software; you can redistribute it and/or modify it |
aoqi@0 | 6 | * under the terms of the GNU General Public License version 2 only, as |
aoqi@0 | 7 | * published by the Free Software Foundation. |
aoqi@0 | 8 | * |
aoqi@0 | 9 | * This code is distributed in the hope that it will be useful, but WITHOUT |
aoqi@0 | 10 | * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
aoqi@0 | 11 | * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
aoqi@0 | 12 | * version 2 for more details (a copy is included in the LICENSE file that |
aoqi@0 | 13 | * accompanied this code). |
aoqi@0 | 14 | * |
aoqi@0 | 15 | * You should have received a copy of the GNU General Public License version |
aoqi@0 | 16 | * 2 along with this work; if not, write to the Free Software Foundation, |
aoqi@0 | 17 | * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. |
aoqi@0 | 18 | * |
aoqi@0 | 19 | * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA |
aoqi@0 | 20 | * or visit www.oracle.com if you need additional information or have any |
aoqi@0 | 21 | * questions. |
aoqi@0 | 22 | * |
aoqi@0 | 23 | */ |
aoqi@0 | 24 | |
aoqi@1 | 25 | /* |
aoqi@1 | 26 | * This file has been modified by Loongson Technology in 2015. These |
aoqi@1 | 27 | * modifications are Copyright (c) 2015 Loongson Technology, and are made |
aoqi@1 | 28 | * available on the same license terms set forth above. |
aoqi@1 | 29 | */ |
aoqi@1 | 30 | |
aoqi@0 | 31 | #include "precompiled.hpp" |
aoqi@0 | 32 | #include "prims/jni.h" |
aoqi@0 | 33 | #include "runtime/interfaceSupport.hpp" |
aoqi@0 | 34 | #include "runtime/sharedRuntime.hpp" |
aoqi@0 | 35 | |
aoqi@0 | 36 | // This file contains copies of the fdlibm routines used by |
aoqi@0 | 37 | // StrictMath. It turns out that it is almost always required to use |
aoqi@0 | 38 | // these runtime routines; the Intel CPU doesn't meet the Java |
aoqi@0 | 39 | // specification for sin/cos outside a certain limited argument range, |
aoqi@0 | 40 | // and the SPARC CPU doesn't appear to have sin/cos instructions. It |
aoqi@0 | 41 | // also turns out that avoiding the indirect call through function |
aoqi@0 | 42 | // pointer out to libjava.so in SharedRuntime speeds these routines up |
aoqi@0 | 43 | // by roughly 15% on both Win32/x86 and Solaris/SPARC. |
aoqi@0 | 44 | |
aoqi@0 | 45 | // Enabling optimizations in this file causes incorrect code to be |
aoqi@0 | 46 | // generated; can not figure out how to turn down optimization for one |
aoqi@0 | 47 | // file in the IDE on Windows |
aoqi@0 | 48 | #ifdef WIN32 |
aoqi@0 | 49 | # pragma optimize ( "", off ) |
aoqi@0 | 50 | #endif |
aoqi@0 | 51 | |
aoqi@0 | 52 | /* The above workaround now causes more problems with the latest MS compiler. |
aoqi@0 | 53 | * Visual Studio 2010's /GS option tries to guard against buffer overruns. |
aoqi@0 | 54 | * /GS is on by default if you specify optimizations, which we do globally |
aoqi@0 | 55 | * via /W3 /O2. However the above selective turning off of optimizations means |
aoqi@0 | 56 | * that /GS issues a warning "4748". And since we treat warnings as errors (/WX) |
aoqi@0 | 57 | * then the compilation fails. There are several possible solutions |
aoqi@0 | 58 | * (1) Remove that pragma above as obsolete with VS2010 - requires testing. |
aoqi@0 | 59 | * (2) Stop treating warnings as errors - would be a backward step |
aoqi@0 | 60 | * (3) Disable /GS - may help performance but you lose the security checks |
aoqi@0 | 61 | * (4) Disable the warning with "#pragma warning( disable : 4748 )" |
aoqi@0 | 62 | * (5) Disable planting the code with __declspec(safebuffers) |
aoqi@0 | 63 | * I've opted for (5) although we should investigate the local performance |
aoqi@0 | 64 | * benefits of (1) and global performance benefit of (3). |
aoqi@0 | 65 | */ |
aoqi@0 | 66 | #if defined(WIN32) && (defined(_MSC_VER) && (_MSC_VER >= 1600)) |
aoqi@0 | 67 | #define SAFEBUF __declspec(safebuffers) |
aoqi@0 | 68 | #else |
aoqi@0 | 69 | #define SAFEBUF |
aoqi@0 | 70 | #endif |
aoqi@0 | 71 | |
aoqi@0 | 72 | #include <math.h> |
aoqi@0 | 73 | |
aoqi@0 | 74 | // VM_LITTLE_ENDIAN is #defined appropriately in the Makefiles |
aoqi@0 | 75 | // [jk] this is not 100% correct because the float word order may different |
aoqi@0 | 76 | // from the byte order (e.g. on ARM) |
aoqi@0 | 77 | #ifdef VM_LITTLE_ENDIAN |
aoqi@0 | 78 | # define __HI(x) *(1+(int*)&x) |
aoqi@0 | 79 | # define __LO(x) *(int*)&x |
aoqi@0 | 80 | #else |
aoqi@0 | 81 | # define __HI(x) *(int*)&x |
aoqi@0 | 82 | # define __LO(x) *(1+(int*)&x) |
aoqi@0 | 83 | #endif |
aoqi@0 | 84 | |
aoqi@0 | 85 | static double copysignA(double x, double y) { |
aoqi@0 | 86 | __HI(x) = (__HI(x)&0x7fffffff)|(__HI(y)&0x80000000); |
aoqi@0 | 87 | return x; |
aoqi@0 | 88 | } |
aoqi@0 | 89 | |
aoqi@0 | 90 | /* |
aoqi@0 | 91 | * scalbn (double x, int n) |
aoqi@0 | 92 | * scalbn(x,n) returns x* 2**n computed by exponent |
aoqi@0 | 93 | * manipulation rather than by actually performing an |
aoqi@0 | 94 | * exponentiation or a multiplication. |
aoqi@0 | 95 | */ |
aoqi@0 | 96 | |
aoqi@0 | 97 | static const double |
aoqi@0 | 98 | two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ |
aoqi@0 | 99 | twom54 = 5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */ |
aoqi@0 | 100 | hugeX = 1.0e+300, |
aoqi@0 | 101 | tiny = 1.0e-300; |
aoqi@0 | 102 | |
aoqi@0 | 103 | static double scalbnA (double x, int n) { |
aoqi@0 | 104 | int k,hx,lx; |
aoqi@0 | 105 | hx = __HI(x); |
aoqi@0 | 106 | lx = __LO(x); |
aoqi@0 | 107 | k = (hx&0x7ff00000)>>20; /* extract exponent */ |
aoqi@0 | 108 | if (k==0) { /* 0 or subnormal x */ |
aoqi@0 | 109 | if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */ |
aoqi@0 | 110 | x *= two54; |
aoqi@0 | 111 | hx = __HI(x); |
aoqi@0 | 112 | k = ((hx&0x7ff00000)>>20) - 54; |
aoqi@0 | 113 | if (n< -50000) return tiny*x; /*underflow*/ |
aoqi@0 | 114 | } |
aoqi@0 | 115 | if (k==0x7ff) return x+x; /* NaN or Inf */ |
aoqi@0 | 116 | k = k+n; |
aoqi@0 | 117 | if (k > 0x7fe) return hugeX*copysignA(hugeX,x); /* overflow */ |
aoqi@0 | 118 | if (k > 0) /* normal result */ |
aoqi@0 | 119 | {__HI(x) = (hx&0x800fffff)|(k<<20); return x;} |
aoqi@0 | 120 | if (k <= -54) { |
aoqi@0 | 121 | if (n > 50000) /* in case integer overflow in n+k */ |
aoqi@0 | 122 | return hugeX*copysignA(hugeX,x); /*overflow*/ |
aoqi@0 | 123 | else return tiny*copysignA(tiny,x); /*underflow*/ |
aoqi@0 | 124 | } |
aoqi@0 | 125 | k += 54; /* subnormal result */ |
aoqi@0 | 126 | __HI(x) = (hx&0x800fffff)|(k<<20); |
aoqi@0 | 127 | return x*twom54; |
aoqi@0 | 128 | } |
aoqi@0 | 129 | |
aoqi@0 | 130 | /* |
aoqi@0 | 131 | * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) |
aoqi@0 | 132 | * double x[],y[]; int e0,nx,prec; int ipio2[]; |
aoqi@0 | 133 | * |
aoqi@0 | 134 | * __kernel_rem_pio2 return the last three digits of N with |
aoqi@0 | 135 | * y = x - N*pi/2 |
aoqi@0 | 136 | * so that |y| < pi/2. |
aoqi@0 | 137 | * |
aoqi@0 | 138 | * The method is to compute the integer (mod 8) and fraction parts of |
aoqi@0 | 139 | * (2/pi)*x without doing the full multiplication. In general we |
aoqi@0 | 140 | * skip the part of the product that are known to be a huge integer ( |
aoqi@0 | 141 | * more accurately, = 0 mod 8 ). Thus the number of operations are |
aoqi@0 | 142 | * independent of the exponent of the input. |
aoqi@0 | 143 | * |
aoqi@0 | 144 | * (2/pi) is represented by an array of 24-bit integers in ipio2[]. |
aoqi@0 | 145 | * |
aoqi@0 | 146 | * Input parameters: |
aoqi@0 | 147 | * x[] The input value (must be positive) is broken into nx |
aoqi@0 | 148 | * pieces of 24-bit integers in double precision format. |
aoqi@0 | 149 | * x[i] will be the i-th 24 bit of x. The scaled exponent |
aoqi@0 | 150 | * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 |
aoqi@0 | 151 | * match x's up to 24 bits. |
aoqi@0 | 152 | * |
aoqi@0 | 153 | * Example of breaking a double positive z into x[0]+x[1]+x[2]: |
aoqi@0 | 154 | * e0 = ilogb(z)-23 |
aoqi@0 | 155 | * z = scalbn(z,-e0) |
aoqi@0 | 156 | * for i = 0,1,2 |
aoqi@0 | 157 | * x[i] = floor(z) |
aoqi@0 | 158 | * z = (z-x[i])*2**24 |
aoqi@0 | 159 | * |
aoqi@0 | 160 | * |
aoqi@0 | 161 | * y[] ouput result in an array of double precision numbers. |
aoqi@0 | 162 | * The dimension of y[] is: |
aoqi@0 | 163 | * 24-bit precision 1 |
aoqi@0 | 164 | * 53-bit precision 2 |
aoqi@0 | 165 | * 64-bit precision 2 |
aoqi@0 | 166 | * 113-bit precision 3 |
aoqi@0 | 167 | * The actual value is the sum of them. Thus for 113-bit |
aoqi@0 | 168 | * precsion, one may have to do something like: |
aoqi@0 | 169 | * |
aoqi@0 | 170 | * long double t,w,r_head, r_tail; |
aoqi@0 | 171 | * t = (long double)y[2] + (long double)y[1]; |
aoqi@0 | 172 | * w = (long double)y[0]; |
aoqi@0 | 173 | * r_head = t+w; |
aoqi@0 | 174 | * r_tail = w - (r_head - t); |
aoqi@0 | 175 | * |
aoqi@0 | 176 | * e0 The exponent of x[0] |
aoqi@0 | 177 | * |
aoqi@0 | 178 | * nx dimension of x[] |
aoqi@0 | 179 | * |
aoqi@0 | 180 | * prec an interger indicating the precision: |
aoqi@0 | 181 | * 0 24 bits (single) |
aoqi@0 | 182 | * 1 53 bits (double) |
aoqi@0 | 183 | * 2 64 bits (extended) |
aoqi@0 | 184 | * 3 113 bits (quad) |
aoqi@0 | 185 | * |
aoqi@0 | 186 | * ipio2[] |
aoqi@0 | 187 | * integer array, contains the (24*i)-th to (24*i+23)-th |
aoqi@0 | 188 | * bit of 2/pi after binary point. The corresponding |
aoqi@0 | 189 | * floating value is |
aoqi@0 | 190 | * |
aoqi@0 | 191 | * ipio2[i] * 2^(-24(i+1)). |
aoqi@0 | 192 | * |
aoqi@0 | 193 | * External function: |
aoqi@0 | 194 | * double scalbn(), floor(); |
aoqi@0 | 195 | * |
aoqi@0 | 196 | * |
aoqi@0 | 197 | * Here is the description of some local variables: |
aoqi@0 | 198 | * |
aoqi@0 | 199 | * jk jk+1 is the initial number of terms of ipio2[] needed |
aoqi@0 | 200 | * in the computation. The recommended value is 2,3,4, |
aoqi@0 | 201 | * 6 for single, double, extended,and quad. |
aoqi@0 | 202 | * |
aoqi@0 | 203 | * jz local integer variable indicating the number of |
aoqi@0 | 204 | * terms of ipio2[] used. |
aoqi@0 | 205 | * |
aoqi@0 | 206 | * jx nx - 1 |
aoqi@0 | 207 | * |
aoqi@0 | 208 | * jv index for pointing to the suitable ipio2[] for the |
aoqi@0 | 209 | * computation. In general, we want |
aoqi@0 | 210 | * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 |
aoqi@0 | 211 | * is an integer. Thus |
aoqi@0 | 212 | * e0-3-24*jv >= 0 or (e0-3)/24 >= jv |
aoqi@0 | 213 | * Hence jv = max(0,(e0-3)/24). |
aoqi@0 | 214 | * |
aoqi@0 | 215 | * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. |
aoqi@0 | 216 | * |
aoqi@0 | 217 | * q[] double array with integral value, representing the |
aoqi@0 | 218 | * 24-bits chunk of the product of x and 2/pi. |
aoqi@0 | 219 | * |
aoqi@0 | 220 | * q0 the corresponding exponent of q[0]. Note that the |
aoqi@0 | 221 | * exponent for q[i] would be q0-24*i. |
aoqi@0 | 222 | * |
aoqi@0 | 223 | * PIo2[] double precision array, obtained by cutting pi/2 |
aoqi@0 | 224 | * into 24 bits chunks. |
aoqi@0 | 225 | * |
aoqi@0 | 226 | * f[] ipio2[] in floating point |
aoqi@0 | 227 | * |
aoqi@0 | 228 | * iq[] integer array by breaking up q[] in 24-bits chunk. |
aoqi@0 | 229 | * |
aoqi@0 | 230 | * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] |
aoqi@0 | 231 | * |
aoqi@0 | 232 | * ih integer. If >0 it indicats q[] is >= 0.5, hence |
aoqi@0 | 233 | * it also indicates the *sign* of the result. |
aoqi@0 | 234 | * |
aoqi@0 | 235 | */ |
aoqi@0 | 236 | |
aoqi@0 | 237 | |
aoqi@0 | 238 | /* |
aoqi@0 | 239 | * Constants: |
aoqi@0 | 240 | * The hexadecimal values are the intended ones for the following |
aoqi@0 | 241 | * constants. The decimal values may be used, provided that the |
aoqi@0 | 242 | * compiler will convert from decimal to binary accurately enough |
aoqi@0 | 243 | * to produce the hexadecimal values shown. |
aoqi@0 | 244 | */ |
aoqi@0 | 245 | |
aoqi@0 | 246 | |
aoqi@0 | 247 | static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ |
aoqi@0 | 248 | |
aoqi@0 | 249 | static const double PIo2[] = { |
aoqi@0 | 250 | 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ |
aoqi@0 | 251 | 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ |
aoqi@0 | 252 | 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ |
aoqi@0 | 253 | 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ |
aoqi@0 | 254 | 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ |
aoqi@0 | 255 | 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ |
aoqi@0 | 256 | 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ |
aoqi@0 | 257 | 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ |
aoqi@0 | 258 | }; |
aoqi@0 | 259 | |
aoqi@0 | 260 | static const double |
aoqi@0 | 261 | zeroB = 0.0, |
aoqi@0 | 262 | one = 1.0, |
aoqi@0 | 263 | two24B = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ |
aoqi@0 | 264 | twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ |
aoqi@0 | 265 | |
aoqi@0 | 266 | static SAFEBUF int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2) { |
aoqi@0 | 267 | int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; |
aoqi@0 | 268 | double z,fw,f[20],fq[20],q[20]; |
aoqi@0 | 269 | |
aoqi@0 | 270 | /* initialize jk*/ |
aoqi@0 | 271 | jk = init_jk[prec]; |
aoqi@0 | 272 | jp = jk; |
aoqi@0 | 273 | |
aoqi@0 | 274 | /* determine jx,jv,q0, note that 3>q0 */ |
aoqi@0 | 275 | jx = nx-1; |
aoqi@0 | 276 | jv = (e0-3)/24; if(jv<0) jv=0; |
aoqi@0 | 277 | q0 = e0-24*(jv+1); |
aoqi@0 | 278 | |
aoqi@0 | 279 | /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ |
aoqi@0 | 280 | j = jv-jx; m = jx+jk; |
aoqi@0 | 281 | for(i=0;i<=m;i++,j++) f[i] = (j<0)? zeroB : (double) ipio2[j]; |
aoqi@0 | 282 | |
aoqi@0 | 283 | /* compute q[0],q[1],...q[jk] */ |
aoqi@0 | 284 | for (i=0;i<=jk;i++) { |
aoqi@0 | 285 | for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; |
aoqi@0 | 286 | } |
aoqi@0 | 287 | |
aoqi@0 | 288 | jz = jk; |
aoqi@0 | 289 | recompute: |
aoqi@0 | 290 | /* distill q[] into iq[] reversingly */ |
aoqi@0 | 291 | for(i=0,j=jz,z=q[jz];j>0;i++,j--) { |
aoqi@0 | 292 | fw = (double)((int)(twon24* z)); |
aoqi@0 | 293 | iq[i] = (int)(z-two24B*fw); |
aoqi@0 | 294 | z = q[j-1]+fw; |
aoqi@0 | 295 | } |
aoqi@0 | 296 | |
aoqi@0 | 297 | /* compute n */ |
aoqi@0 | 298 | z = scalbnA(z,q0); /* actual value of z */ |
aoqi@0 | 299 | z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ |
aoqi@0 | 300 | n = (int) z; |
aoqi@0 | 301 | z -= (double)n; |
aoqi@0 | 302 | ih = 0; |
aoqi@0 | 303 | if(q0>0) { /* need iq[jz-1] to determine n */ |
aoqi@0 | 304 | i = (iq[jz-1]>>(24-q0)); n += i; |
aoqi@0 | 305 | iq[jz-1] -= i<<(24-q0); |
aoqi@0 | 306 | ih = iq[jz-1]>>(23-q0); |
aoqi@0 | 307 | } |
aoqi@0 | 308 | else if(q0==0) ih = iq[jz-1]>>23; |
aoqi@0 | 309 | else if(z>=0.5) ih=2; |
aoqi@0 | 310 | |
aoqi@0 | 311 | if(ih>0) { /* q > 0.5 */ |
aoqi@0 | 312 | n += 1; carry = 0; |
aoqi@0 | 313 | for(i=0;i<jz ;i++) { /* compute 1-q */ |
aoqi@0 | 314 | j = iq[i]; |
aoqi@0 | 315 | if(carry==0) { |
aoqi@0 | 316 | if(j!=0) { |
aoqi@0 | 317 | carry = 1; iq[i] = 0x1000000- j; |
aoqi@0 | 318 | } |
aoqi@0 | 319 | } else iq[i] = 0xffffff - j; |
aoqi@0 | 320 | } |
aoqi@0 | 321 | if(q0>0) { /* rare case: chance is 1 in 12 */ |
aoqi@0 | 322 | switch(q0) { |
aoqi@0 | 323 | case 1: |
aoqi@0 | 324 | iq[jz-1] &= 0x7fffff; break; |
aoqi@0 | 325 | case 2: |
aoqi@0 | 326 | iq[jz-1] &= 0x3fffff; break; |
aoqi@0 | 327 | } |
aoqi@0 | 328 | } |
aoqi@0 | 329 | if(ih==2) { |
aoqi@0 | 330 | z = one - z; |
aoqi@0 | 331 | if(carry!=0) z -= scalbnA(one,q0); |
aoqi@0 | 332 | } |
aoqi@0 | 333 | } |
aoqi@0 | 334 | |
aoqi@0 | 335 | /* check if recomputation is needed */ |
aoqi@0 | 336 | if(z==zeroB) { |
aoqi@0 | 337 | j = 0; |
aoqi@0 | 338 | for (i=jz-1;i>=jk;i--) j |= iq[i]; |
aoqi@0 | 339 | if(j==0) { /* need recomputation */ |
aoqi@0 | 340 | for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ |
aoqi@0 | 341 | |
aoqi@0 | 342 | for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ |
aoqi@0 | 343 | f[jx+i] = (double) ipio2[jv+i]; |
aoqi@0 | 344 | for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; |
aoqi@0 | 345 | q[i] = fw; |
aoqi@0 | 346 | } |
aoqi@0 | 347 | jz += k; |
aoqi@0 | 348 | goto recompute; |
aoqi@0 | 349 | } |
aoqi@0 | 350 | } |
aoqi@0 | 351 | |
aoqi@0 | 352 | /* chop off zero terms */ |
aoqi@0 | 353 | if(z==0.0) { |
aoqi@0 | 354 | jz -= 1; q0 -= 24; |
aoqi@0 | 355 | while(iq[jz]==0) { jz--; q0-=24;} |
aoqi@0 | 356 | } else { /* break z into 24-bit if neccessary */ |
aoqi@0 | 357 | z = scalbnA(z,-q0); |
aoqi@0 | 358 | if(z>=two24B) { |
aoqi@0 | 359 | fw = (double)((int)(twon24*z)); |
aoqi@0 | 360 | iq[jz] = (int)(z-two24B*fw); |
aoqi@0 | 361 | jz += 1; q0 += 24; |
aoqi@0 | 362 | iq[jz] = (int) fw; |
aoqi@0 | 363 | } else iq[jz] = (int) z ; |
aoqi@0 | 364 | } |
aoqi@0 | 365 | |
aoqi@0 | 366 | /* convert integer "bit" chunk to floating-point value */ |
aoqi@0 | 367 | fw = scalbnA(one,q0); |
aoqi@0 | 368 | for(i=jz;i>=0;i--) { |
aoqi@0 | 369 | q[i] = fw*(double)iq[i]; fw*=twon24; |
aoqi@0 | 370 | } |
aoqi@0 | 371 | |
aoqi@0 | 372 | /* compute PIo2[0,...,jp]*q[jz,...,0] */ |
aoqi@0 | 373 | for(i=jz;i>=0;i--) { |
aoqi@0 | 374 | for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; |
aoqi@0 | 375 | fq[jz-i] = fw; |
aoqi@0 | 376 | } |
aoqi@0 | 377 | |
aoqi@0 | 378 | /* compress fq[] into y[] */ |
aoqi@0 | 379 | switch(prec) { |
aoqi@0 | 380 | case 0: |
aoqi@0 | 381 | fw = 0.0; |
aoqi@0 | 382 | for (i=jz;i>=0;i--) fw += fq[i]; |
aoqi@0 | 383 | y[0] = (ih==0)? fw: -fw; |
aoqi@0 | 384 | break; |
aoqi@0 | 385 | case 1: |
aoqi@0 | 386 | case 2: |
aoqi@0 | 387 | fw = 0.0; |
aoqi@0 | 388 | for (i=jz;i>=0;i--) fw += fq[i]; |
aoqi@0 | 389 | y[0] = (ih==0)? fw: -fw; |
aoqi@0 | 390 | fw = fq[0]-fw; |
aoqi@0 | 391 | for (i=1;i<=jz;i++) fw += fq[i]; |
aoqi@0 | 392 | y[1] = (ih==0)? fw: -fw; |
aoqi@0 | 393 | break; |
aoqi@0 | 394 | case 3: /* painful */ |
aoqi@0 | 395 | for (i=jz;i>0;i--) { |
aoqi@0 | 396 | fw = fq[i-1]+fq[i]; |
aoqi@0 | 397 | fq[i] += fq[i-1]-fw; |
aoqi@0 | 398 | fq[i-1] = fw; |
aoqi@0 | 399 | } |
aoqi@0 | 400 | for (i=jz;i>1;i--) { |
aoqi@0 | 401 | fw = fq[i-1]+fq[i]; |
aoqi@0 | 402 | fq[i] += fq[i-1]-fw; |
aoqi@0 | 403 | fq[i-1] = fw; |
aoqi@0 | 404 | } |
aoqi@0 | 405 | for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; |
aoqi@0 | 406 | if(ih==0) { |
aoqi@0 | 407 | y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; |
aoqi@0 | 408 | } else { |
aoqi@0 | 409 | y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; |
aoqi@0 | 410 | } |
aoqi@0 | 411 | } |
aoqi@0 | 412 | return n&7; |
aoqi@0 | 413 | } |
aoqi@0 | 414 | |
aoqi@0 | 415 | |
aoqi@0 | 416 | /* |
aoqi@0 | 417 | * ==================================================== |
aoqi@0 | 418 | * Copyright (c) 1993 Oracle and/or its affilates. All rights reserved. |
aoqi@0 | 419 | * |
aoqi@0 | 420 | * Developed at SunPro, a Sun Microsystems, Inc. business. |
aoqi@0 | 421 | * Permission to use, copy, modify, and distribute this |
aoqi@0 | 422 | * software is freely granted, provided that this notice |
aoqi@0 | 423 | * is preserved. |
aoqi@0 | 424 | * ==================================================== |
aoqi@0 | 425 | * |
aoqi@0 | 426 | */ |
aoqi@0 | 427 | |
aoqi@0 | 428 | /* __ieee754_rem_pio2(x,y) |
aoqi@0 | 429 | * |
aoqi@0 | 430 | * return the remainder of x rem pi/2 in y[0]+y[1] |
aoqi@0 | 431 | * use __kernel_rem_pio2() |
aoqi@0 | 432 | */ |
aoqi@0 | 433 | |
aoqi@0 | 434 | /* |
aoqi@0 | 435 | * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi |
aoqi@0 | 436 | */ |
aoqi@0 | 437 | static const int two_over_pi[] = { |
aoqi@0 | 438 | 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, |
aoqi@0 | 439 | 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, |
aoqi@0 | 440 | 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, |
aoqi@0 | 441 | 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, |
aoqi@0 | 442 | 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, |
aoqi@0 | 443 | 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF, |
aoqi@0 | 444 | 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, |
aoqi@0 | 445 | 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, |
aoqi@0 | 446 | 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, |
aoqi@0 | 447 | 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, |
aoqi@0 | 448 | 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B, |
aoqi@0 | 449 | }; |
aoqi@0 | 450 | |
aoqi@0 | 451 | static const int npio2_hw[] = { |
aoqi@0 | 452 | 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, |
aoqi@0 | 453 | 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C, |
aoqi@0 | 454 | 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A, |
aoqi@0 | 455 | 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C, |
aoqi@0 | 456 | 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB, |
aoqi@0 | 457 | 0x404858EB, 0x404921FB, |
aoqi@0 | 458 | }; |
aoqi@0 | 459 | |
aoqi@0 | 460 | /* |
aoqi@0 | 461 | * invpio2: 53 bits of 2/pi |
aoqi@0 | 462 | * pio2_1: first 33 bit of pi/2 |
aoqi@0 | 463 | * pio2_1t: pi/2 - pio2_1 |
aoqi@0 | 464 | * pio2_2: second 33 bit of pi/2 |
aoqi@0 | 465 | * pio2_2t: pi/2 - (pio2_1+pio2_2) |
aoqi@0 | 466 | * pio2_3: third 33 bit of pi/2 |
aoqi@0 | 467 | * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) |
aoqi@0 | 468 | */ |
aoqi@0 | 469 | |
aoqi@0 | 470 | static const double |
aoqi@0 | 471 | zeroA = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ |
aoqi@0 | 472 | half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ |
aoqi@0 | 473 | two24A = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ |
aoqi@0 | 474 | invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ |
aoqi@0 | 475 | pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */ |
aoqi@0 | 476 | pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */ |
aoqi@0 | 477 | pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */ |
aoqi@0 | 478 | pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */ |
aoqi@0 | 479 | pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */ |
aoqi@0 | 480 | pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */ |
aoqi@0 | 481 | |
aoqi@0 | 482 | static SAFEBUF int __ieee754_rem_pio2(double x, double *y) { |
aoqi@0 | 483 | double z,w,t,r,fn; |
aoqi@0 | 484 | double tx[3]; |
aoqi@0 | 485 | int e0,i,j,nx,n,ix,hx,i0; |
aoqi@0 | 486 | |
aoqi@0 | 487 | i0 = ((*(int*)&two24A)>>30)^1; /* high word index */ |
aoqi@0 | 488 | hx = *(i0+(int*)&x); /* high word of x */ |
aoqi@0 | 489 | ix = hx&0x7fffffff; |
aoqi@0 | 490 | if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */ |
aoqi@0 | 491 | {y[0] = x; y[1] = 0; return 0;} |
aoqi@0 | 492 | if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */ |
aoqi@0 | 493 | if(hx>0) { |
aoqi@0 | 494 | z = x - pio2_1; |
aoqi@0 | 495 | if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ |
aoqi@0 | 496 | y[0] = z - pio2_1t; |
aoqi@0 | 497 | y[1] = (z-y[0])-pio2_1t; |
aoqi@0 | 498 | } else { /* near pi/2, use 33+33+53 bit pi */ |
aoqi@0 | 499 | z -= pio2_2; |
aoqi@0 | 500 | y[0] = z - pio2_2t; |
aoqi@0 | 501 | y[1] = (z-y[0])-pio2_2t; |
aoqi@0 | 502 | } |
aoqi@0 | 503 | return 1; |
aoqi@0 | 504 | } else { /* negative x */ |
aoqi@0 | 505 | z = x + pio2_1; |
aoqi@0 | 506 | if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ |
aoqi@0 | 507 | y[0] = z + pio2_1t; |
aoqi@0 | 508 | y[1] = (z-y[0])+pio2_1t; |
aoqi@0 | 509 | } else { /* near pi/2, use 33+33+53 bit pi */ |
aoqi@0 | 510 | z += pio2_2; |
aoqi@0 | 511 | y[0] = z + pio2_2t; |
aoqi@0 | 512 | y[1] = (z-y[0])+pio2_2t; |
aoqi@0 | 513 | } |
aoqi@0 | 514 | return -1; |
aoqi@0 | 515 | } |
aoqi@0 | 516 | } |
aoqi@0 | 517 | if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */ |
aoqi@0 | 518 | t = fabsd(x); |
aoqi@0 | 519 | n = (int) (t*invpio2+half); |
aoqi@0 | 520 | fn = (double)n; |
aoqi@0 | 521 | r = t-fn*pio2_1; |
aoqi@0 | 522 | w = fn*pio2_1t; /* 1st round good to 85 bit */ |
aoqi@0 | 523 | if(n<32&&ix!=npio2_hw[n-1]) { |
aoqi@0 | 524 | y[0] = r-w; /* quick check no cancellation */ |
aoqi@0 | 525 | } else { |
aoqi@0 | 526 | j = ix>>20; |
aoqi@0 | 527 | y[0] = r-w; |
aoqi@0 | 528 | i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff); |
aoqi@0 | 529 | if(i>16) { /* 2nd iteration needed, good to 118 */ |
aoqi@0 | 530 | t = r; |
aoqi@0 | 531 | w = fn*pio2_2; |
aoqi@0 | 532 | r = t-w; |
aoqi@0 | 533 | w = fn*pio2_2t-((t-r)-w); |
aoqi@0 | 534 | y[0] = r-w; |
aoqi@0 | 535 | i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff); |
aoqi@0 | 536 | if(i>49) { /* 3rd iteration need, 151 bits acc */ |
aoqi@0 | 537 | t = r; /* will cover all possible cases */ |
aoqi@0 | 538 | w = fn*pio2_3; |
aoqi@0 | 539 | r = t-w; |
aoqi@0 | 540 | w = fn*pio2_3t-((t-r)-w); |
aoqi@0 | 541 | y[0] = r-w; |
aoqi@0 | 542 | } |
aoqi@0 | 543 | } |
aoqi@0 | 544 | } |
aoqi@0 | 545 | y[1] = (r-y[0])-w; |
aoqi@0 | 546 | if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} |
aoqi@0 | 547 | else return n; |
aoqi@0 | 548 | } |
aoqi@0 | 549 | /* |
aoqi@0 | 550 | * all other (large) arguments |
aoqi@0 | 551 | */ |
aoqi@0 | 552 | if(ix>=0x7ff00000) { /* x is inf or NaN */ |
aoqi@0 | 553 | y[0]=y[1]=x-x; return 0; |
aoqi@0 | 554 | } |
aoqi@0 | 555 | /* set z = scalbn(|x|,ilogb(x)-23) */ |
aoqi@0 | 556 | *(1-i0+(int*)&z) = *(1-i0+(int*)&x); |
aoqi@0 | 557 | e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */ |
aoqi@0 | 558 | *(i0+(int*)&z) = ix - (e0<<20); |
aoqi@0 | 559 | for(i=0;i<2;i++) { |
aoqi@0 | 560 | tx[i] = (double)((int)(z)); |
aoqi@0 | 561 | z = (z-tx[i])*two24A; |
aoqi@0 | 562 | } |
aoqi@0 | 563 | tx[2] = z; |
aoqi@0 | 564 | nx = 3; |
aoqi@0 | 565 | while(tx[nx-1]==zeroA) nx--; /* skip zero term */ |
aoqi@0 | 566 | n = __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi); |
aoqi@0 | 567 | if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} |
aoqi@0 | 568 | return n; |
aoqi@0 | 569 | } |
aoqi@0 | 570 | |
aoqi@0 | 571 | |
aoqi@0 | 572 | /* __kernel_sin( x, y, iy) |
aoqi@0 | 573 | * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 |
aoqi@0 | 574 | * Input x is assumed to be bounded by ~pi/4 in magnitude. |
aoqi@0 | 575 | * Input y is the tail of x. |
aoqi@0 | 576 | * Input iy indicates whether y is 0. (if iy=0, y assume to be 0). |
aoqi@0 | 577 | * |
aoqi@0 | 578 | * Algorithm |
aoqi@0 | 579 | * 1. Since sin(-x) = -sin(x), we need only to consider positive x. |
aoqi@0 | 580 | * 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0. |
aoqi@0 | 581 | * 3. sin(x) is approximated by a polynomial of degree 13 on |
aoqi@0 | 582 | * [0,pi/4] |
aoqi@0 | 583 | * 3 13 |
aoqi@0 | 584 | * sin(x) ~ x + S1*x + ... + S6*x |
aoqi@0 | 585 | * where |
aoqi@0 | 586 | * |
aoqi@0 | 587 | * |sin(x) 2 4 6 8 10 12 | -58 |
aoqi@0 | 588 | * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 |
aoqi@0 | 589 | * | x | |
aoqi@0 | 590 | * |
aoqi@0 | 591 | * 4. sin(x+y) = sin(x) + sin'(x')*y |
aoqi@0 | 592 | * ~ sin(x) + (1-x*x/2)*y |
aoqi@0 | 593 | * For better accuracy, let |
aoqi@0 | 594 | * 3 2 2 2 2 |
aoqi@0 | 595 | * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) |
aoqi@0 | 596 | * then 3 2 |
aoqi@0 | 597 | * sin(x) = x + (S1*x + (x *(r-y/2)+y)) |
aoqi@0 | 598 | */ |
aoqi@1 | 599 | #ifdef MIPS64 |
aoqi@1 | 600 | #undef S1 |
aoqi@1 | 601 | #undef S2 |
aoqi@1 | 602 | #undef S3 |
aoqi@1 | 603 | #undef S4 |
aoqi@1 | 604 | #undef S5 |
aoqi@1 | 605 | #undef S6 |
aoqi@1 | 606 | #endif |
aoqi@0 | 607 | |
aoqi@0 | 608 | static const double |
aoqi@0 | 609 | S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */ |
aoqi@0 | 610 | S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */ |
aoqi@0 | 611 | S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */ |
aoqi@0 | 612 | S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */ |
aoqi@0 | 613 | S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */ |
aoqi@0 | 614 | S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */ |
aoqi@0 | 615 | |
aoqi@0 | 616 | static double __kernel_sin(double x, double y, int iy) |
aoqi@0 | 617 | { |
aoqi@0 | 618 | double z,r,v; |
aoqi@0 | 619 | int ix; |
aoqi@0 | 620 | ix = __HI(x)&0x7fffffff; /* high word of x */ |
aoqi@0 | 621 | if(ix<0x3e400000) /* |x| < 2**-27 */ |
aoqi@0 | 622 | {if((int)x==0) return x;} /* generate inexact */ |
aoqi@0 | 623 | z = x*x; |
aoqi@0 | 624 | v = z*x; |
aoqi@0 | 625 | r = S2+z*(S3+z*(S4+z*(S5+z*S6))); |
aoqi@0 | 626 | if(iy==0) return x+v*(S1+z*r); |
aoqi@0 | 627 | else return x-((z*(half*y-v*r)-y)-v*S1); |
aoqi@0 | 628 | } |
aoqi@0 | 629 | |
aoqi@0 | 630 | /* |
aoqi@0 | 631 | * __kernel_cos( x, y ) |
aoqi@0 | 632 | * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 |
aoqi@0 | 633 | * Input x is assumed to be bounded by ~pi/4 in magnitude. |
aoqi@0 | 634 | * Input y is the tail of x. |
aoqi@0 | 635 | * |
aoqi@0 | 636 | * Algorithm |
aoqi@0 | 637 | * 1. Since cos(-x) = cos(x), we need only to consider positive x. |
aoqi@0 | 638 | * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. |
aoqi@0 | 639 | * 3. cos(x) is approximated by a polynomial of degree 14 on |
aoqi@0 | 640 | * [0,pi/4] |
aoqi@0 | 641 | * 4 14 |
aoqi@0 | 642 | * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x |
aoqi@0 | 643 | * where the remez error is |
aoqi@0 | 644 | * |
aoqi@0 | 645 | * | 2 4 6 8 10 12 14 | -58 |
aoqi@0 | 646 | * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 |
aoqi@0 | 647 | * | | |
aoqi@0 | 648 | * |
aoqi@0 | 649 | * 4 6 8 10 12 14 |
aoqi@0 | 650 | * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then |
aoqi@0 | 651 | * cos(x) = 1 - x*x/2 + r |
aoqi@0 | 652 | * since cos(x+y) ~ cos(x) - sin(x)*y |
aoqi@0 | 653 | * ~ cos(x) - x*y, |
aoqi@0 | 654 | * a correction term is necessary in cos(x) and hence |
aoqi@0 | 655 | * cos(x+y) = 1 - (x*x/2 - (r - x*y)) |
aoqi@0 | 656 | * For better accuracy when x > 0.3, let qx = |x|/4 with |
aoqi@0 | 657 | * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. |
aoqi@0 | 658 | * Then |
aoqi@0 | 659 | * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). |
aoqi@0 | 660 | * Note that 1-qx and (x*x/2-qx) is EXACT here, and the |
aoqi@0 | 661 | * magnitude of the latter is at least a quarter of x*x/2, |
aoqi@0 | 662 | * thus, reducing the rounding error in the subtraction. |
aoqi@0 | 663 | */ |
aoqi@0 | 664 | |
aoqi@0 | 665 | static const double |
aoqi@0 | 666 | C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ |
aoqi@0 | 667 | C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ |
aoqi@0 | 668 | C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ |
aoqi@0 | 669 | C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ |
aoqi@0 | 670 | C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ |
aoqi@0 | 671 | C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ |
aoqi@0 | 672 | |
aoqi@0 | 673 | static double __kernel_cos(double x, double y) |
aoqi@0 | 674 | { |
aoqi@0 | 675 | double a,h,z,r,qx; |
aoqi@0 | 676 | int ix; |
aoqi@0 | 677 | ix = __HI(x)&0x7fffffff; /* ix = |x|'s high word*/ |
aoqi@0 | 678 | if(ix<0x3e400000) { /* if x < 2**27 */ |
aoqi@0 | 679 | if(((int)x)==0) return one; /* generate inexact */ |
aoqi@0 | 680 | } |
aoqi@0 | 681 | z = x*x; |
aoqi@0 | 682 | r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6))))); |
aoqi@0 | 683 | if(ix < 0x3FD33333) /* if |x| < 0.3 */ |
aoqi@0 | 684 | return one - (0.5*z - (z*r - x*y)); |
aoqi@0 | 685 | else { |
aoqi@0 | 686 | if(ix > 0x3fe90000) { /* x > 0.78125 */ |
aoqi@0 | 687 | qx = 0.28125; |
aoqi@0 | 688 | } else { |
aoqi@0 | 689 | __HI(qx) = ix-0x00200000; /* x/4 */ |
aoqi@0 | 690 | __LO(qx) = 0; |
aoqi@0 | 691 | } |
aoqi@0 | 692 | h = 0.5*z-qx; |
aoqi@0 | 693 | a = one-qx; |
aoqi@0 | 694 | return a - (h - (z*r-x*y)); |
aoqi@0 | 695 | } |
aoqi@0 | 696 | } |
aoqi@0 | 697 | |
aoqi@0 | 698 | /* __kernel_tan( x, y, k ) |
aoqi@0 | 699 | * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 |
aoqi@0 | 700 | * Input x is assumed to be bounded by ~pi/4 in magnitude. |
aoqi@0 | 701 | * Input y is the tail of x. |
aoqi@0 | 702 | * Input k indicates whether tan (if k=1) or |
aoqi@0 | 703 | * -1/tan (if k= -1) is returned. |
aoqi@0 | 704 | * |
aoqi@0 | 705 | * Algorithm |
aoqi@0 | 706 | * 1. Since tan(-x) = -tan(x), we need only to consider positive x. |
aoqi@0 | 707 | * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. |
aoqi@0 | 708 | * 3. tan(x) is approximated by a odd polynomial of degree 27 on |
aoqi@0 | 709 | * [0,0.67434] |
aoqi@0 | 710 | * 3 27 |
aoqi@0 | 711 | * tan(x) ~ x + T1*x + ... + T13*x |
aoqi@0 | 712 | * where |
aoqi@0 | 713 | * |
aoqi@0 | 714 | * |tan(x) 2 4 26 | -59.2 |
aoqi@0 | 715 | * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 |
aoqi@0 | 716 | * | x | |
aoqi@0 | 717 | * |
aoqi@0 | 718 | * Note: tan(x+y) = tan(x) + tan'(x)*y |
aoqi@0 | 719 | * ~ tan(x) + (1+x*x)*y |
aoqi@0 | 720 | * Therefore, for better accuracy in computing tan(x+y), let |
aoqi@0 | 721 | * 3 2 2 2 2 |
aoqi@0 | 722 | * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) |
aoqi@0 | 723 | * then |
aoqi@0 | 724 | * 3 2 |
aoqi@0 | 725 | * tan(x+y) = x + (T1*x + (x *(r+y)+y)) |
aoqi@0 | 726 | * |
aoqi@0 | 727 | * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then |
aoqi@0 | 728 | * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) |
aoqi@0 | 729 | * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) |
aoqi@0 | 730 | */ |
aoqi@0 | 731 | |
aoqi@0 | 732 | static const double |
aoqi@0 | 733 | pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ |
aoqi@0 | 734 | pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */ |
aoqi@0 | 735 | T[] = { |
aoqi@0 | 736 | 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */ |
aoqi@0 | 737 | 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */ |
aoqi@0 | 738 | 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */ |
aoqi@0 | 739 | 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */ |
aoqi@0 | 740 | 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */ |
aoqi@0 | 741 | 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */ |
aoqi@0 | 742 | 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */ |
aoqi@0 | 743 | 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */ |
aoqi@0 | 744 | 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */ |
aoqi@0 | 745 | 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */ |
aoqi@0 | 746 | 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */ |
aoqi@0 | 747 | -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */ |
aoqi@0 | 748 | 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */ |
aoqi@0 | 749 | }; |
aoqi@0 | 750 | |
aoqi@0 | 751 | static double __kernel_tan(double x, double y, int iy) |
aoqi@0 | 752 | { |
aoqi@0 | 753 | double z,r,v,w,s; |
aoqi@0 | 754 | int ix,hx; |
aoqi@0 | 755 | hx = __HI(x); /* high word of x */ |
aoqi@0 | 756 | ix = hx&0x7fffffff; /* high word of |x| */ |
aoqi@0 | 757 | if(ix<0x3e300000) { /* x < 2**-28 */ |
aoqi@0 | 758 | if((int)x==0) { /* generate inexact */ |
aoqi@0 | 759 | if (((ix | __LO(x)) | (iy + 1)) == 0) |
aoqi@0 | 760 | return one / fabsd(x); |
aoqi@0 | 761 | else { |
aoqi@0 | 762 | if (iy == 1) |
aoqi@0 | 763 | return x; |
aoqi@0 | 764 | else { /* compute -1 / (x+y) carefully */ |
aoqi@0 | 765 | double a, t; |
aoqi@0 | 766 | |
aoqi@0 | 767 | z = w = x + y; |
aoqi@0 | 768 | __LO(z) = 0; |
aoqi@0 | 769 | v = y - (z - x); |
aoqi@0 | 770 | t = a = -one / w; |
aoqi@0 | 771 | __LO(t) = 0; |
aoqi@0 | 772 | s = one + t * z; |
aoqi@0 | 773 | return t + a * (s + t * v); |
aoqi@0 | 774 | } |
aoqi@0 | 775 | } |
aoqi@0 | 776 | } |
aoqi@0 | 777 | } |
aoqi@0 | 778 | if(ix>=0x3FE59428) { /* |x|>=0.6744 */ |
aoqi@0 | 779 | if(hx<0) {x = -x; y = -y;} |
aoqi@0 | 780 | z = pio4-x; |
aoqi@0 | 781 | w = pio4lo-y; |
aoqi@0 | 782 | x = z+w; y = 0.0; |
aoqi@0 | 783 | } |
aoqi@0 | 784 | z = x*x; |
aoqi@0 | 785 | w = z*z; |
aoqi@0 | 786 | /* Break x^5*(T[1]+x^2*T[2]+...) into |
aoqi@0 | 787 | * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + |
aoqi@0 | 788 | * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) |
aoqi@0 | 789 | */ |
aoqi@0 | 790 | r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11])))); |
aoqi@0 | 791 | v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12]))))); |
aoqi@0 | 792 | s = z*x; |
aoqi@0 | 793 | r = y + z*(s*(r+v)+y); |
aoqi@0 | 794 | r += T[0]*s; |
aoqi@0 | 795 | w = x+r; |
aoqi@0 | 796 | if(ix>=0x3FE59428) { |
aoqi@0 | 797 | v = (double)iy; |
aoqi@0 | 798 | return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r))); |
aoqi@0 | 799 | } |
aoqi@0 | 800 | if(iy==1) return w; |
aoqi@0 | 801 | else { /* if allow error up to 2 ulp, |
aoqi@0 | 802 | simply return -1.0/(x+r) here */ |
aoqi@0 | 803 | /* compute -1.0/(x+r) accurately */ |
aoqi@0 | 804 | double a,t; |
aoqi@0 | 805 | z = w; |
aoqi@0 | 806 | __LO(z) = 0; |
aoqi@0 | 807 | v = r-(z - x); /* z+v = r+x */ |
aoqi@0 | 808 | t = a = -1.0/w; /* a = -1.0/w */ |
aoqi@0 | 809 | __LO(t) = 0; |
aoqi@0 | 810 | s = 1.0+t*z; |
aoqi@0 | 811 | return t+a*(s+t*v); |
aoqi@0 | 812 | } |
aoqi@0 | 813 | } |
aoqi@0 | 814 | |
aoqi@0 | 815 | |
aoqi@0 | 816 | //---------------------------------------------------------------------- |
aoqi@0 | 817 | // |
aoqi@0 | 818 | // Routines for new sin/cos implementation |
aoqi@0 | 819 | // |
aoqi@0 | 820 | //---------------------------------------------------------------------- |
aoqi@0 | 821 | |
aoqi@0 | 822 | /* sin(x) |
aoqi@0 | 823 | * Return sine function of x. |
aoqi@0 | 824 | * |
aoqi@0 | 825 | * kernel function: |
aoqi@0 | 826 | * __kernel_sin ... sine function on [-pi/4,pi/4] |
aoqi@0 | 827 | * __kernel_cos ... cose function on [-pi/4,pi/4] |
aoqi@0 | 828 | * __ieee754_rem_pio2 ... argument reduction routine |
aoqi@0 | 829 | * |
aoqi@0 | 830 | * Method. |
aoqi@0 | 831 | * Let S,C and T denote the sin, cos and tan respectively on |
aoqi@0 | 832 | * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 |
aoqi@0 | 833 | * in [-pi/4 , +pi/4], and let n = k mod 4. |
aoqi@0 | 834 | * We have |
aoqi@0 | 835 | * |
aoqi@0 | 836 | * n sin(x) cos(x) tan(x) |
aoqi@0 | 837 | * ---------------------------------------------------------- |
aoqi@0 | 838 | * 0 S C T |
aoqi@0 | 839 | * 1 C -S -1/T |
aoqi@0 | 840 | * 2 -S -C T |
aoqi@0 | 841 | * 3 -C S -1/T |
aoqi@0 | 842 | * ---------------------------------------------------------- |
aoqi@0 | 843 | * |
aoqi@0 | 844 | * Special cases: |
aoqi@0 | 845 | * Let trig be any of sin, cos, or tan. |
aoqi@0 | 846 | * trig(+-INF) is NaN, with signals; |
aoqi@0 | 847 | * trig(NaN) is that NaN; |
aoqi@0 | 848 | * |
aoqi@0 | 849 | * Accuracy: |
aoqi@0 | 850 | * TRIG(x) returns trig(x) nearly rounded |
aoqi@0 | 851 | */ |
aoqi@0 | 852 | |
aoqi@0 | 853 | JRT_LEAF(jdouble, SharedRuntime::dsin(jdouble x)) |
aoqi@0 | 854 | double y[2],z=0.0; |
aoqi@0 | 855 | int n, ix; |
aoqi@0 | 856 | |
aoqi@0 | 857 | /* High word of x. */ |
aoqi@0 | 858 | ix = __HI(x); |
aoqi@0 | 859 | |
aoqi@0 | 860 | /* |x| ~< pi/4 */ |
aoqi@0 | 861 | ix &= 0x7fffffff; |
aoqi@0 | 862 | if(ix <= 0x3fe921fb) return __kernel_sin(x,z,0); |
aoqi@0 | 863 | |
aoqi@0 | 864 | /* sin(Inf or NaN) is NaN */ |
aoqi@0 | 865 | else if (ix>=0x7ff00000) return x-x; |
aoqi@0 | 866 | |
aoqi@0 | 867 | /* argument reduction needed */ |
aoqi@0 | 868 | else { |
aoqi@0 | 869 | n = __ieee754_rem_pio2(x,y); |
aoqi@0 | 870 | switch(n&3) { |
aoqi@0 | 871 | case 0: return __kernel_sin(y[0],y[1],1); |
aoqi@0 | 872 | case 1: return __kernel_cos(y[0],y[1]); |
aoqi@0 | 873 | case 2: return -__kernel_sin(y[0],y[1],1); |
aoqi@0 | 874 | default: |
aoqi@0 | 875 | return -__kernel_cos(y[0],y[1]); |
aoqi@0 | 876 | } |
aoqi@0 | 877 | } |
aoqi@0 | 878 | JRT_END |
aoqi@0 | 879 | |
aoqi@0 | 880 | /* cos(x) |
aoqi@0 | 881 | * Return cosine function of x. |
aoqi@0 | 882 | * |
aoqi@0 | 883 | * kernel function: |
aoqi@0 | 884 | * __kernel_sin ... sine function on [-pi/4,pi/4] |
aoqi@0 | 885 | * __kernel_cos ... cosine function on [-pi/4,pi/4] |
aoqi@0 | 886 | * __ieee754_rem_pio2 ... argument reduction routine |
aoqi@0 | 887 | * |
aoqi@0 | 888 | * Method. |
aoqi@0 | 889 | * Let S,C and T denote the sin, cos and tan respectively on |
aoqi@0 | 890 | * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 |
aoqi@0 | 891 | * in [-pi/4 , +pi/4], and let n = k mod 4. |
aoqi@0 | 892 | * We have |
aoqi@0 | 893 | * |
aoqi@0 | 894 | * n sin(x) cos(x) tan(x) |
aoqi@0 | 895 | * ---------------------------------------------------------- |
aoqi@0 | 896 | * 0 S C T |
aoqi@0 | 897 | * 1 C -S -1/T |
aoqi@0 | 898 | * 2 -S -C T |
aoqi@0 | 899 | * 3 -C S -1/T |
aoqi@0 | 900 | * ---------------------------------------------------------- |
aoqi@0 | 901 | * |
aoqi@0 | 902 | * Special cases: |
aoqi@0 | 903 | * Let trig be any of sin, cos, or tan. |
aoqi@0 | 904 | * trig(+-INF) is NaN, with signals; |
aoqi@0 | 905 | * trig(NaN) is that NaN; |
aoqi@0 | 906 | * |
aoqi@0 | 907 | * Accuracy: |
aoqi@0 | 908 | * TRIG(x) returns trig(x) nearly rounded |
aoqi@0 | 909 | */ |
aoqi@0 | 910 | |
aoqi@0 | 911 | JRT_LEAF(jdouble, SharedRuntime::dcos(jdouble x)) |
aoqi@0 | 912 | double y[2],z=0.0; |
aoqi@0 | 913 | int n, ix; |
aoqi@0 | 914 | |
aoqi@0 | 915 | /* High word of x. */ |
aoqi@0 | 916 | ix = __HI(x); |
aoqi@0 | 917 | |
aoqi@0 | 918 | /* |x| ~< pi/4 */ |
aoqi@0 | 919 | ix &= 0x7fffffff; |
aoqi@0 | 920 | if(ix <= 0x3fe921fb) return __kernel_cos(x,z); |
aoqi@0 | 921 | |
aoqi@0 | 922 | /* cos(Inf or NaN) is NaN */ |
aoqi@0 | 923 | else if (ix>=0x7ff00000) return x-x; |
aoqi@0 | 924 | |
aoqi@0 | 925 | /* argument reduction needed */ |
aoqi@0 | 926 | else { |
aoqi@0 | 927 | n = __ieee754_rem_pio2(x,y); |
aoqi@0 | 928 | switch(n&3) { |
aoqi@0 | 929 | case 0: return __kernel_cos(y[0],y[1]); |
aoqi@0 | 930 | case 1: return -__kernel_sin(y[0],y[1],1); |
aoqi@0 | 931 | case 2: return -__kernel_cos(y[0],y[1]); |
aoqi@0 | 932 | default: |
aoqi@0 | 933 | return __kernel_sin(y[0],y[1],1); |
aoqi@0 | 934 | } |
aoqi@0 | 935 | } |
aoqi@0 | 936 | JRT_END |
aoqi@0 | 937 | |
aoqi@0 | 938 | /* tan(x) |
aoqi@0 | 939 | * Return tangent function of x. |
aoqi@0 | 940 | * |
aoqi@0 | 941 | * kernel function: |
aoqi@0 | 942 | * __kernel_tan ... tangent function on [-pi/4,pi/4] |
aoqi@0 | 943 | * __ieee754_rem_pio2 ... argument reduction routine |
aoqi@0 | 944 | * |
aoqi@0 | 945 | * Method. |
aoqi@0 | 946 | * Let S,C and T denote the sin, cos and tan respectively on |
aoqi@0 | 947 | * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 |
aoqi@0 | 948 | * in [-pi/4 , +pi/4], and let n = k mod 4. |
aoqi@0 | 949 | * We have |
aoqi@0 | 950 | * |
aoqi@0 | 951 | * n sin(x) cos(x) tan(x) |
aoqi@0 | 952 | * ---------------------------------------------------------- |
aoqi@0 | 953 | * 0 S C T |
aoqi@0 | 954 | * 1 C -S -1/T |
aoqi@0 | 955 | * 2 -S -C T |
aoqi@0 | 956 | * 3 -C S -1/T |
aoqi@0 | 957 | * ---------------------------------------------------------- |
aoqi@0 | 958 | * |
aoqi@0 | 959 | * Special cases: |
aoqi@0 | 960 | * Let trig be any of sin, cos, or tan. |
aoqi@0 | 961 | * trig(+-INF) is NaN, with signals; |
aoqi@0 | 962 | * trig(NaN) is that NaN; |
aoqi@0 | 963 | * |
aoqi@0 | 964 | * Accuracy: |
aoqi@0 | 965 | * TRIG(x) returns trig(x) nearly rounded |
aoqi@0 | 966 | */ |
aoqi@0 | 967 | |
aoqi@0 | 968 | JRT_LEAF(jdouble, SharedRuntime::dtan(jdouble x)) |
aoqi@0 | 969 | double y[2],z=0.0; |
aoqi@0 | 970 | int n, ix; |
aoqi@0 | 971 | |
aoqi@0 | 972 | /* High word of x. */ |
aoqi@0 | 973 | ix = __HI(x); |
aoqi@0 | 974 | |
aoqi@0 | 975 | /* |x| ~< pi/4 */ |
aoqi@0 | 976 | ix &= 0x7fffffff; |
aoqi@0 | 977 | if(ix <= 0x3fe921fb) return __kernel_tan(x,z,1); |
aoqi@0 | 978 | |
aoqi@0 | 979 | /* tan(Inf or NaN) is NaN */ |
aoqi@0 | 980 | else if (ix>=0x7ff00000) return x-x; /* NaN */ |
aoqi@0 | 981 | |
aoqi@0 | 982 | /* argument reduction needed */ |
aoqi@0 | 983 | else { |
aoqi@0 | 984 | n = __ieee754_rem_pio2(x,y); |
aoqi@0 | 985 | return __kernel_tan(y[0],y[1],1-((n&1)<<1)); /* 1 -- n even |
aoqi@0 | 986 | -1 -- n odd */ |
aoqi@0 | 987 | } |
aoqi@0 | 988 | JRT_END |
aoqi@0 | 989 | |
aoqi@0 | 990 | |
aoqi@0 | 991 | #ifdef WIN32 |
aoqi@0 | 992 | # pragma optimize ( "", on ) |
aoqi@0 | 993 | #endif |