Mercurial > hg > graal-compiler
annotate src/share/vm/runtime/sharedRuntimeTrig.cpp @ 1749:b4099f5786da
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author | never |
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date | Wed, 25 Aug 2010 10:31:45 -0700 |
parents | c18cbe5936b8 |
children | f95d63e2154a |
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0 | 1 /* |
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2 * Copyright (c) 2001, 2005, Oracle and/or its affiliates. All rights reserved. |
0 | 3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. |
4 * | |
5 * This code is free software; you can redistribute it and/or modify it | |
6 * under the terms of the GNU General Public License version 2 only, as | |
7 * published by the Free Software Foundation. | |
8 * | |
9 * This code is distributed in the hope that it will be useful, but WITHOUT | |
10 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
11 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
12 * version 2 for more details (a copy is included in the LICENSE file that | |
13 * accompanied this code). | |
14 * | |
15 * You should have received a copy of the GNU General Public License version | |
16 * 2 along with this work; if not, write to the Free Software Foundation, | |
17 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. | |
18 * | |
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19 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA |
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20 * or visit www.oracle.com if you need additional information or have any |
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21 * questions. |
0 | 22 * |
23 */ | |
24 | |
25 #include "incls/_precompiled.incl" | |
26 #include "incls/_sharedRuntimeTrig.cpp.incl" | |
27 | |
28 // This file contains copies of the fdlibm routines used by | |
29 // StrictMath. It turns out that it is almost always required to use | |
30 // these runtime routines; the Intel CPU doesn't meet the Java | |
31 // specification for sin/cos outside a certain limited argument range, | |
32 // and the SPARC CPU doesn't appear to have sin/cos instructions. It | |
33 // also turns out that avoiding the indirect call through function | |
34 // pointer out to libjava.so in SharedRuntime speeds these routines up | |
35 // by roughly 15% on both Win32/x86 and Solaris/SPARC. | |
36 | |
37 // Enabling optimizations in this file causes incorrect code to be | |
38 // generated; can not figure out how to turn down optimization for one | |
39 // file in the IDE on Windows | |
40 #ifdef WIN32 | |
41 # pragma optimize ( "", off ) | |
42 #endif | |
43 | |
1485 | 44 /* The above workaround now causes more problems with the latest MS compiler. |
45 * Visual Studio 2010's /GS option tries to guard against buffer overruns. | |
46 * /GS is on by default if you specify optimizations, which we do globally | |
47 * via /W3 /O2. However the above selective turning off of optimizations means | |
48 * that /GS issues a warning "4748". And since we treat warnings as errors (/WX) | |
49 * then the compilation fails. There are several possible solutions | |
50 * (1) Remove that pragma above as obsolete with VS2010 - requires testing. | |
51 * (2) Stop treating warnings as errors - would be a backward step | |
52 * (3) Disable /GS - may help performance but you lose the security checks | |
53 * (4) Disable the warning with "#pragma warning( disable : 4748 )" | |
54 * (5) Disable planting the code with __declspec(safebuffers) | |
55 * I've opted for (5) although we should investigate the local performance | |
56 * benefits of (1) and global performance benefit of (3). | |
57 */ | |
58 #if defined(WIN32) && (defined(_MSC_VER) && (_MSC_VER >= 1600)) | |
59 #define SAFEBUF __declspec(safebuffers) | |
60 #else | |
61 #define SAFEBUF | |
62 #endif | |
63 | |
0 | 64 #include <math.h> |
65 | |
66 // VM_LITTLE_ENDIAN is #defined appropriately in the Makefiles | |
67 // [jk] this is not 100% correct because the float word order may different | |
68 // from the byte order (e.g. on ARM) | |
69 #ifdef VM_LITTLE_ENDIAN | |
70 # define __HI(x) *(1+(int*)&x) | |
71 # define __LO(x) *(int*)&x | |
72 #else | |
73 # define __HI(x) *(int*)&x | |
74 # define __LO(x) *(1+(int*)&x) | |
75 #endif | |
76 | |
77 static double copysignA(double x, double y) { | |
78 __HI(x) = (__HI(x)&0x7fffffff)|(__HI(y)&0x80000000); | |
79 return x; | |
80 } | |
81 | |
82 /* | |
83 * scalbn (double x, int n) | |
84 * scalbn(x,n) returns x* 2**n computed by exponent | |
85 * manipulation rather than by actually performing an | |
86 * exponentiation or a multiplication. | |
87 */ | |
88 | |
89 static const double | |
90 two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ | |
91 twom54 = 5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */ | |
92 hugeX = 1.0e+300, | |
93 tiny = 1.0e-300; | |
94 | |
95 static double scalbnA (double x, int n) { | |
96 int k,hx,lx; | |
97 hx = __HI(x); | |
98 lx = __LO(x); | |
99 k = (hx&0x7ff00000)>>20; /* extract exponent */ | |
100 if (k==0) { /* 0 or subnormal x */ | |
101 if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */ | |
102 x *= two54; | |
103 hx = __HI(x); | |
104 k = ((hx&0x7ff00000)>>20) - 54; | |
105 if (n< -50000) return tiny*x; /*underflow*/ | |
106 } | |
107 if (k==0x7ff) return x+x; /* NaN or Inf */ | |
108 k = k+n; | |
109 if (k > 0x7fe) return hugeX*copysignA(hugeX,x); /* overflow */ | |
110 if (k > 0) /* normal result */ | |
111 {__HI(x) = (hx&0x800fffff)|(k<<20); return x;} | |
112 if (k <= -54) { | |
113 if (n > 50000) /* in case integer overflow in n+k */ | |
114 return hugeX*copysignA(hugeX,x); /*overflow*/ | |
115 else return tiny*copysignA(tiny,x); /*underflow*/ | |
116 } | |
117 k += 54; /* subnormal result */ | |
118 __HI(x) = (hx&0x800fffff)|(k<<20); | |
119 return x*twom54; | |
120 } | |
121 | |
122 /* | |
123 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) | |
124 * double x[],y[]; int e0,nx,prec; int ipio2[]; | |
125 * | |
126 * __kernel_rem_pio2 return the last three digits of N with | |
127 * y = x - N*pi/2 | |
128 * so that |y| < pi/2. | |
129 * | |
130 * The method is to compute the integer (mod 8) and fraction parts of | |
131 * (2/pi)*x without doing the full multiplication. In general we | |
132 * skip the part of the product that are known to be a huge integer ( | |
133 * more accurately, = 0 mod 8 ). Thus the number of operations are | |
134 * independent of the exponent of the input. | |
135 * | |
136 * (2/pi) is represented by an array of 24-bit integers in ipio2[]. | |
137 * | |
138 * Input parameters: | |
139 * x[] The input value (must be positive) is broken into nx | |
140 * pieces of 24-bit integers in double precision format. | |
141 * x[i] will be the i-th 24 bit of x. The scaled exponent | |
142 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 | |
143 * match x's up to 24 bits. | |
144 * | |
145 * Example of breaking a double positive z into x[0]+x[1]+x[2]: | |
146 * e0 = ilogb(z)-23 | |
147 * z = scalbn(z,-e0) | |
148 * for i = 0,1,2 | |
149 * x[i] = floor(z) | |
150 * z = (z-x[i])*2**24 | |
151 * | |
152 * | |
153 * y[] ouput result in an array of double precision numbers. | |
154 * The dimension of y[] is: | |
155 * 24-bit precision 1 | |
156 * 53-bit precision 2 | |
157 * 64-bit precision 2 | |
158 * 113-bit precision 3 | |
159 * The actual value is the sum of them. Thus for 113-bit | |
160 * precsion, one may have to do something like: | |
161 * | |
162 * long double t,w,r_head, r_tail; | |
163 * t = (long double)y[2] + (long double)y[1]; | |
164 * w = (long double)y[0]; | |
165 * r_head = t+w; | |
166 * r_tail = w - (r_head - t); | |
167 * | |
168 * e0 The exponent of x[0] | |
169 * | |
170 * nx dimension of x[] | |
171 * | |
172 * prec an interger indicating the precision: | |
173 * 0 24 bits (single) | |
174 * 1 53 bits (double) | |
175 * 2 64 bits (extended) | |
176 * 3 113 bits (quad) | |
177 * | |
178 * ipio2[] | |
179 * integer array, contains the (24*i)-th to (24*i+23)-th | |
180 * bit of 2/pi after binary point. The corresponding | |
181 * floating value is | |
182 * | |
183 * ipio2[i] * 2^(-24(i+1)). | |
184 * | |
185 * External function: | |
186 * double scalbn(), floor(); | |
187 * | |
188 * | |
189 * Here is the description of some local variables: | |
190 * | |
191 * jk jk+1 is the initial number of terms of ipio2[] needed | |
192 * in the computation. The recommended value is 2,3,4, | |
193 * 6 for single, double, extended,and quad. | |
194 * | |
195 * jz local integer variable indicating the number of | |
196 * terms of ipio2[] used. | |
197 * | |
198 * jx nx - 1 | |
199 * | |
200 * jv index for pointing to the suitable ipio2[] for the | |
201 * computation. In general, we want | |
202 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 | |
203 * is an integer. Thus | |
204 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv | |
205 * Hence jv = max(0,(e0-3)/24). | |
206 * | |
207 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. | |
208 * | |
209 * q[] double array with integral value, representing the | |
210 * 24-bits chunk of the product of x and 2/pi. | |
211 * | |
212 * q0 the corresponding exponent of q[0]. Note that the | |
213 * exponent for q[i] would be q0-24*i. | |
214 * | |
215 * PIo2[] double precision array, obtained by cutting pi/2 | |
216 * into 24 bits chunks. | |
217 * | |
218 * f[] ipio2[] in floating point | |
219 * | |
220 * iq[] integer array by breaking up q[] in 24-bits chunk. | |
221 * | |
222 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] | |
223 * | |
224 * ih integer. If >0 it indicats q[] is >= 0.5, hence | |
225 * it also indicates the *sign* of the result. | |
226 * | |
227 */ | |
228 | |
229 | |
230 /* | |
231 * Constants: | |
232 * The hexadecimal values are the intended ones for the following | |
233 * constants. The decimal values may be used, provided that the | |
234 * compiler will convert from decimal to binary accurately enough | |
235 * to produce the hexadecimal values shown. | |
236 */ | |
237 | |
238 | |
239 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ | |
240 | |
241 static const double PIo2[] = { | |
242 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ | |
243 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ | |
244 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ | |
245 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ | |
246 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ | |
247 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ | |
248 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ | |
249 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ | |
250 }; | |
251 | |
252 static const double | |
253 zeroB = 0.0, | |
254 one = 1.0, | |
255 two24B = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ | |
256 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ | |
257 | |
1485 | 258 static SAFEBUF int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2) { |
0 | 259 int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; |
260 double z,fw,f[20],fq[20],q[20]; | |
261 | |
262 /* initialize jk*/ | |
263 jk = init_jk[prec]; | |
264 jp = jk; | |
265 | |
266 /* determine jx,jv,q0, note that 3>q0 */ | |
267 jx = nx-1; | |
268 jv = (e0-3)/24; if(jv<0) jv=0; | |
269 q0 = e0-24*(jv+1); | |
270 | |
271 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ | |
272 j = jv-jx; m = jx+jk; | |
273 for(i=0;i<=m;i++,j++) f[i] = (j<0)? zeroB : (double) ipio2[j]; | |
274 | |
275 /* compute q[0],q[1],...q[jk] */ | |
276 for (i=0;i<=jk;i++) { | |
277 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; | |
278 } | |
279 | |
280 jz = jk; | |
281 recompute: | |
282 /* distill q[] into iq[] reversingly */ | |
283 for(i=0,j=jz,z=q[jz];j>0;i++,j--) { | |
284 fw = (double)((int)(twon24* z)); | |
285 iq[i] = (int)(z-two24B*fw); | |
286 z = q[j-1]+fw; | |
287 } | |
288 | |
289 /* compute n */ | |
290 z = scalbnA(z,q0); /* actual value of z */ | |
291 z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ | |
292 n = (int) z; | |
293 z -= (double)n; | |
294 ih = 0; | |
295 if(q0>0) { /* need iq[jz-1] to determine n */ | |
296 i = (iq[jz-1]>>(24-q0)); n += i; | |
297 iq[jz-1] -= i<<(24-q0); | |
298 ih = iq[jz-1]>>(23-q0); | |
299 } | |
300 else if(q0==0) ih = iq[jz-1]>>23; | |
301 else if(z>=0.5) ih=2; | |
302 | |
303 if(ih>0) { /* q > 0.5 */ | |
304 n += 1; carry = 0; | |
305 for(i=0;i<jz ;i++) { /* compute 1-q */ | |
306 j = iq[i]; | |
307 if(carry==0) { | |
308 if(j!=0) { | |
309 carry = 1; iq[i] = 0x1000000- j; | |
310 } | |
311 } else iq[i] = 0xffffff - j; | |
312 } | |
313 if(q0>0) { /* rare case: chance is 1 in 12 */ | |
314 switch(q0) { | |
315 case 1: | |
316 iq[jz-1] &= 0x7fffff; break; | |
317 case 2: | |
318 iq[jz-1] &= 0x3fffff; break; | |
319 } | |
320 } | |
321 if(ih==2) { | |
322 z = one - z; | |
323 if(carry!=0) z -= scalbnA(one,q0); | |
324 } | |
325 } | |
326 | |
327 /* check if recomputation is needed */ | |
328 if(z==zeroB) { | |
329 j = 0; | |
330 for (i=jz-1;i>=jk;i--) j |= iq[i]; | |
331 if(j==0) { /* need recomputation */ | |
332 for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ | |
333 | |
334 for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ | |
335 f[jx+i] = (double) ipio2[jv+i]; | |
336 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; | |
337 q[i] = fw; | |
338 } | |
339 jz += k; | |
340 goto recompute; | |
341 } | |
342 } | |
343 | |
344 /* chop off zero terms */ | |
345 if(z==0.0) { | |
346 jz -= 1; q0 -= 24; | |
347 while(iq[jz]==0) { jz--; q0-=24;} | |
348 } else { /* break z into 24-bit if neccessary */ | |
349 z = scalbnA(z,-q0); | |
350 if(z>=two24B) { | |
351 fw = (double)((int)(twon24*z)); | |
352 iq[jz] = (int)(z-two24B*fw); | |
353 jz += 1; q0 += 24; | |
354 iq[jz] = (int) fw; | |
355 } else iq[jz] = (int) z ; | |
356 } | |
357 | |
358 /* convert integer "bit" chunk to floating-point value */ | |
359 fw = scalbnA(one,q0); | |
360 for(i=jz;i>=0;i--) { | |
361 q[i] = fw*(double)iq[i]; fw*=twon24; | |
362 } | |
363 | |
364 /* compute PIo2[0,...,jp]*q[jz,...,0] */ | |
365 for(i=jz;i>=0;i--) { | |
366 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; | |
367 fq[jz-i] = fw; | |
368 } | |
369 | |
370 /* compress fq[] into y[] */ | |
371 switch(prec) { | |
372 case 0: | |
373 fw = 0.0; | |
374 for (i=jz;i>=0;i--) fw += fq[i]; | |
375 y[0] = (ih==0)? fw: -fw; | |
376 break; | |
377 case 1: | |
378 case 2: | |
379 fw = 0.0; | |
380 for (i=jz;i>=0;i--) fw += fq[i]; | |
381 y[0] = (ih==0)? fw: -fw; | |
382 fw = fq[0]-fw; | |
383 for (i=1;i<=jz;i++) fw += fq[i]; | |
384 y[1] = (ih==0)? fw: -fw; | |
385 break; | |
386 case 3: /* painful */ | |
387 for (i=jz;i>0;i--) { | |
388 fw = fq[i-1]+fq[i]; | |
389 fq[i] += fq[i-1]-fw; | |
390 fq[i-1] = fw; | |
391 } | |
392 for (i=jz;i>1;i--) { | |
393 fw = fq[i-1]+fq[i]; | |
394 fq[i] += fq[i-1]-fw; | |
395 fq[i-1] = fw; | |
396 } | |
397 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; | |
398 if(ih==0) { | |
399 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; | |
400 } else { | |
401 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; | |
402 } | |
403 } | |
404 return n&7; | |
405 } | |
406 | |
407 | |
408 /* | |
409 * ==================================================== | |
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410 * Copyright (c) 1993 Oracle and/or its affilates. All rights reserved. |
0 | 411 * |
412 * Developed at SunPro, a Sun Microsystems, Inc. business. | |
413 * Permission to use, copy, modify, and distribute this | |
414 * software is freely granted, provided that this notice | |
415 * is preserved. | |
416 * ==================================================== | |
417 * | |
418 */ | |
419 | |
420 /* __ieee754_rem_pio2(x,y) | |
421 * | |
422 * return the remainder of x rem pi/2 in y[0]+y[1] | |
423 * use __kernel_rem_pio2() | |
424 */ | |
425 | |
426 /* | |
427 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi | |
428 */ | |
429 static const int two_over_pi[] = { | |
430 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, | |
431 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, | |
432 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, | |
433 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, | |
434 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, | |
435 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF, | |
436 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, | |
437 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, | |
438 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, | |
439 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, | |
440 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B, | |
441 }; | |
442 | |
443 static const int npio2_hw[] = { | |
444 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, | |
445 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C, | |
446 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A, | |
447 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C, | |
448 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB, | |
449 0x404858EB, 0x404921FB, | |
450 }; | |
451 | |
452 /* | |
453 * invpio2: 53 bits of 2/pi | |
454 * pio2_1: first 33 bit of pi/2 | |
455 * pio2_1t: pi/2 - pio2_1 | |
456 * pio2_2: second 33 bit of pi/2 | |
457 * pio2_2t: pi/2 - (pio2_1+pio2_2) | |
458 * pio2_3: third 33 bit of pi/2 | |
459 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) | |
460 */ | |
461 | |
462 static const double | |
463 zeroA = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ | |
464 half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ | |
465 two24A = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ | |
466 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ | |
467 pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */ | |
468 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */ | |
469 pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */ | |
470 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */ | |
471 pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */ | |
472 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */ | |
473 | |
1485 | 474 static SAFEBUF int __ieee754_rem_pio2(double x, double *y) { |
0 | 475 double z,w,t,r,fn; |
476 double tx[3]; | |
477 int e0,i,j,nx,n,ix,hx,i0; | |
478 | |
479 i0 = ((*(int*)&two24A)>>30)^1; /* high word index */ | |
480 hx = *(i0+(int*)&x); /* high word of x */ | |
481 ix = hx&0x7fffffff; | |
482 if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */ | |
483 {y[0] = x; y[1] = 0; return 0;} | |
484 if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */ | |
485 if(hx>0) { | |
486 z = x - pio2_1; | |
487 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ | |
488 y[0] = z - pio2_1t; | |
489 y[1] = (z-y[0])-pio2_1t; | |
490 } else { /* near pi/2, use 33+33+53 bit pi */ | |
491 z -= pio2_2; | |
492 y[0] = z - pio2_2t; | |
493 y[1] = (z-y[0])-pio2_2t; | |
494 } | |
495 return 1; | |
496 } else { /* negative x */ | |
497 z = x + pio2_1; | |
498 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ | |
499 y[0] = z + pio2_1t; | |
500 y[1] = (z-y[0])+pio2_1t; | |
501 } else { /* near pi/2, use 33+33+53 bit pi */ | |
502 z += pio2_2; | |
503 y[0] = z + pio2_2t; | |
504 y[1] = (z-y[0])+pio2_2t; | |
505 } | |
506 return -1; | |
507 } | |
508 } | |
509 if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */ | |
510 t = fabsd(x); | |
511 n = (int) (t*invpio2+half); | |
512 fn = (double)n; | |
513 r = t-fn*pio2_1; | |
514 w = fn*pio2_1t; /* 1st round good to 85 bit */ | |
515 if(n<32&&ix!=npio2_hw[n-1]) { | |
516 y[0] = r-w; /* quick check no cancellation */ | |
517 } else { | |
518 j = ix>>20; | |
519 y[0] = r-w; | |
520 i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff); | |
521 if(i>16) { /* 2nd iteration needed, good to 118 */ | |
522 t = r; | |
523 w = fn*pio2_2; | |
524 r = t-w; | |
525 w = fn*pio2_2t-((t-r)-w); | |
526 y[0] = r-w; | |
527 i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff); | |
528 if(i>49) { /* 3rd iteration need, 151 bits acc */ | |
529 t = r; /* will cover all possible cases */ | |
530 w = fn*pio2_3; | |
531 r = t-w; | |
532 w = fn*pio2_3t-((t-r)-w); | |
533 y[0] = r-w; | |
534 } | |
535 } | |
536 } | |
537 y[1] = (r-y[0])-w; | |
538 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} | |
539 else return n; | |
540 } | |
541 /* | |
542 * all other (large) arguments | |
543 */ | |
544 if(ix>=0x7ff00000) { /* x is inf or NaN */ | |
545 y[0]=y[1]=x-x; return 0; | |
546 } | |
547 /* set z = scalbn(|x|,ilogb(x)-23) */ | |
548 *(1-i0+(int*)&z) = *(1-i0+(int*)&x); | |
549 e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */ | |
550 *(i0+(int*)&z) = ix - (e0<<20); | |
551 for(i=0;i<2;i++) { | |
552 tx[i] = (double)((int)(z)); | |
553 z = (z-tx[i])*two24A; | |
554 } | |
555 tx[2] = z; | |
556 nx = 3; | |
557 while(tx[nx-1]==zeroA) nx--; /* skip zero term */ | |
558 n = __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi); | |
559 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} | |
560 return n; | |
561 } | |
562 | |
563 | |
564 /* __kernel_sin( x, y, iy) | |
565 * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 | |
566 * Input x is assumed to be bounded by ~pi/4 in magnitude. | |
567 * Input y is the tail of x. | |
568 * Input iy indicates whether y is 0. (if iy=0, y assume to be 0). | |
569 * | |
570 * Algorithm | |
571 * 1. Since sin(-x) = -sin(x), we need only to consider positive x. | |
572 * 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0. | |
573 * 3. sin(x) is approximated by a polynomial of degree 13 on | |
574 * [0,pi/4] | |
575 * 3 13 | |
576 * sin(x) ~ x + S1*x + ... + S6*x | |
577 * where | |
578 * | |
579 * |sin(x) 2 4 6 8 10 12 | -58 | |
580 * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 | |
581 * | x | | |
582 * | |
583 * 4. sin(x+y) = sin(x) + sin'(x')*y | |
584 * ~ sin(x) + (1-x*x/2)*y | |
585 * For better accuracy, let | |
586 * 3 2 2 2 2 | |
587 * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) | |
588 * then 3 2 | |
589 * sin(x) = x + (S1*x + (x *(r-y/2)+y)) | |
590 */ | |
591 | |
592 static const double | |
593 S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */ | |
594 S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */ | |
595 S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */ | |
596 S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */ | |
597 S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */ | |
598 S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */ | |
599 | |
600 static double __kernel_sin(double x, double y, int iy) | |
601 { | |
602 double z,r,v; | |
603 int ix; | |
604 ix = __HI(x)&0x7fffffff; /* high word of x */ | |
605 if(ix<0x3e400000) /* |x| < 2**-27 */ | |
606 {if((int)x==0) return x;} /* generate inexact */ | |
607 z = x*x; | |
608 v = z*x; | |
609 r = S2+z*(S3+z*(S4+z*(S5+z*S6))); | |
610 if(iy==0) return x+v*(S1+z*r); | |
611 else return x-((z*(half*y-v*r)-y)-v*S1); | |
612 } | |
613 | |
614 /* | |
615 * __kernel_cos( x, y ) | |
616 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 | |
617 * Input x is assumed to be bounded by ~pi/4 in magnitude. | |
618 * Input y is the tail of x. | |
619 * | |
620 * Algorithm | |
621 * 1. Since cos(-x) = cos(x), we need only to consider positive x. | |
622 * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. | |
623 * 3. cos(x) is approximated by a polynomial of degree 14 on | |
624 * [0,pi/4] | |
625 * 4 14 | |
626 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x | |
627 * where the remez error is | |
628 * | |
629 * | 2 4 6 8 10 12 14 | -58 | |
630 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 | |
631 * | | | |
632 * | |
633 * 4 6 8 10 12 14 | |
634 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then | |
635 * cos(x) = 1 - x*x/2 + r | |
636 * since cos(x+y) ~ cos(x) - sin(x)*y | |
637 * ~ cos(x) - x*y, | |
638 * a correction term is necessary in cos(x) and hence | |
639 * cos(x+y) = 1 - (x*x/2 - (r - x*y)) | |
640 * For better accuracy when x > 0.3, let qx = |x|/4 with | |
641 * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. | |
642 * Then | |
643 * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). | |
644 * Note that 1-qx and (x*x/2-qx) is EXACT here, and the | |
645 * magnitude of the latter is at least a quarter of x*x/2, | |
646 * thus, reducing the rounding error in the subtraction. | |
647 */ | |
648 | |
649 static const double | |
650 C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ | |
651 C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ | |
652 C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ | |
653 C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ | |
654 C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ | |
655 C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ | |
656 | |
657 static double __kernel_cos(double x, double y) | |
658 { | |
659 double a,hz,z,r,qx; | |
660 int ix; | |
661 ix = __HI(x)&0x7fffffff; /* ix = |x|'s high word*/ | |
662 if(ix<0x3e400000) { /* if x < 2**27 */ | |
663 if(((int)x)==0) return one; /* generate inexact */ | |
664 } | |
665 z = x*x; | |
666 r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6))))); | |
667 if(ix < 0x3FD33333) /* if |x| < 0.3 */ | |
668 return one - (0.5*z - (z*r - x*y)); | |
669 else { | |
670 if(ix > 0x3fe90000) { /* x > 0.78125 */ | |
671 qx = 0.28125; | |
672 } else { | |
673 __HI(qx) = ix-0x00200000; /* x/4 */ | |
674 __LO(qx) = 0; | |
675 } | |
676 hz = 0.5*z-qx; | |
677 a = one-qx; | |
678 return a - (hz - (z*r-x*y)); | |
679 } | |
680 } | |
681 | |
682 /* __kernel_tan( x, y, k ) | |
683 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 | |
684 * Input x is assumed to be bounded by ~pi/4 in magnitude. | |
685 * Input y is the tail of x. | |
686 * Input k indicates whether tan (if k=1) or | |
687 * -1/tan (if k= -1) is returned. | |
688 * | |
689 * Algorithm | |
690 * 1. Since tan(-x) = -tan(x), we need only to consider positive x. | |
691 * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. | |
692 * 3. tan(x) is approximated by a odd polynomial of degree 27 on | |
693 * [0,0.67434] | |
694 * 3 27 | |
695 * tan(x) ~ x + T1*x + ... + T13*x | |
696 * where | |
697 * | |
698 * |tan(x) 2 4 26 | -59.2 | |
699 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 | |
700 * | x | | |
701 * | |
702 * Note: tan(x+y) = tan(x) + tan'(x)*y | |
703 * ~ tan(x) + (1+x*x)*y | |
704 * Therefore, for better accuracy in computing tan(x+y), let | |
705 * 3 2 2 2 2 | |
706 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) | |
707 * then | |
708 * 3 2 | |
709 * tan(x+y) = x + (T1*x + (x *(r+y)+y)) | |
710 * | |
711 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then | |
712 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) | |
713 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) | |
714 */ | |
715 | |
716 static const double | |
717 pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ | |
718 pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */ | |
719 T[] = { | |
720 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */ | |
721 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */ | |
722 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */ | |
723 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */ | |
724 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */ | |
725 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */ | |
726 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */ | |
727 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */ | |
728 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */ | |
729 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */ | |
730 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */ | |
731 -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */ | |
732 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */ | |
733 }; | |
734 | |
735 static double __kernel_tan(double x, double y, int iy) | |
736 { | |
737 double z,r,v,w,s; | |
738 int ix,hx; | |
739 hx = __HI(x); /* high word of x */ | |
740 ix = hx&0x7fffffff; /* high word of |x| */ | |
741 if(ix<0x3e300000) { /* x < 2**-28 */ | |
742 if((int)x==0) { /* generate inexact */ | |
743 if (((ix | __LO(x)) | (iy + 1)) == 0) | |
744 return one / fabsd(x); | |
745 else { | |
746 if (iy == 1) | |
747 return x; | |
748 else { /* compute -1 / (x+y) carefully */ | |
749 double a, t; | |
750 | |
751 z = w = x + y; | |
752 __LO(z) = 0; | |
753 v = y - (z - x); | |
754 t = a = -one / w; | |
755 __LO(t) = 0; | |
756 s = one + t * z; | |
757 return t + a * (s + t * v); | |
758 } | |
759 } | |
760 } | |
761 } | |
762 if(ix>=0x3FE59428) { /* |x|>=0.6744 */ | |
763 if(hx<0) {x = -x; y = -y;} | |
764 z = pio4-x; | |
765 w = pio4lo-y; | |
766 x = z+w; y = 0.0; | |
767 } | |
768 z = x*x; | |
769 w = z*z; | |
770 /* Break x^5*(T[1]+x^2*T[2]+...) into | |
771 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + | |
772 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) | |
773 */ | |
774 r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11])))); | |
775 v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12]))))); | |
776 s = z*x; | |
777 r = y + z*(s*(r+v)+y); | |
778 r += T[0]*s; | |
779 w = x+r; | |
780 if(ix>=0x3FE59428) { | |
781 v = (double)iy; | |
782 return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r))); | |
783 } | |
784 if(iy==1) return w; | |
785 else { /* if allow error up to 2 ulp, | |
786 simply return -1.0/(x+r) here */ | |
787 /* compute -1.0/(x+r) accurately */ | |
788 double a,t; | |
789 z = w; | |
790 __LO(z) = 0; | |
791 v = r-(z - x); /* z+v = r+x */ | |
792 t = a = -1.0/w; /* a = -1.0/w */ | |
793 __LO(t) = 0; | |
794 s = 1.0+t*z; | |
795 return t+a*(s+t*v); | |
796 } | |
797 } | |
798 | |
799 | |
800 //---------------------------------------------------------------------- | |
801 // | |
802 // Routines for new sin/cos implementation | |
803 // | |
804 //---------------------------------------------------------------------- | |
805 | |
806 /* sin(x) | |
807 * Return sine function of x. | |
808 * | |
809 * kernel function: | |
810 * __kernel_sin ... sine function on [-pi/4,pi/4] | |
811 * __kernel_cos ... cose function on [-pi/4,pi/4] | |
812 * __ieee754_rem_pio2 ... argument reduction routine | |
813 * | |
814 * Method. | |
815 * Let S,C and T denote the sin, cos and tan respectively on | |
816 * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 | |
817 * in [-pi/4 , +pi/4], and let n = k mod 4. | |
818 * We have | |
819 * | |
820 * n sin(x) cos(x) tan(x) | |
821 * ---------------------------------------------------------- | |
822 * 0 S C T | |
823 * 1 C -S -1/T | |
824 * 2 -S -C T | |
825 * 3 -C S -1/T | |
826 * ---------------------------------------------------------- | |
827 * | |
828 * Special cases: | |
829 * Let trig be any of sin, cos, or tan. | |
830 * trig(+-INF) is NaN, with signals; | |
831 * trig(NaN) is that NaN; | |
832 * | |
833 * Accuracy: | |
834 * TRIG(x) returns trig(x) nearly rounded | |
835 */ | |
836 | |
837 JRT_LEAF(jdouble, SharedRuntime::dsin(jdouble x)) | |
838 double y[2],z=0.0; | |
839 int n, ix; | |
840 | |
841 /* High word of x. */ | |
842 ix = __HI(x); | |
843 | |
844 /* |x| ~< pi/4 */ | |
845 ix &= 0x7fffffff; | |
846 if(ix <= 0x3fe921fb) return __kernel_sin(x,z,0); | |
847 | |
848 /* sin(Inf or NaN) is NaN */ | |
849 else if (ix>=0x7ff00000) return x-x; | |
850 | |
851 /* argument reduction needed */ | |
852 else { | |
853 n = __ieee754_rem_pio2(x,y); | |
854 switch(n&3) { | |
855 case 0: return __kernel_sin(y[0],y[1],1); | |
856 case 1: return __kernel_cos(y[0],y[1]); | |
857 case 2: return -__kernel_sin(y[0],y[1],1); | |
858 default: | |
859 return -__kernel_cos(y[0],y[1]); | |
860 } | |
861 } | |
862 JRT_END | |
863 | |
864 /* cos(x) | |
865 * Return cosine function of x. | |
866 * | |
867 * kernel function: | |
868 * __kernel_sin ... sine function on [-pi/4,pi/4] | |
869 * __kernel_cos ... cosine function on [-pi/4,pi/4] | |
870 * __ieee754_rem_pio2 ... argument reduction routine | |
871 * | |
872 * Method. | |
873 * Let S,C and T denote the sin, cos and tan respectively on | |
874 * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 | |
875 * in [-pi/4 , +pi/4], and let n = k mod 4. | |
876 * We have | |
877 * | |
878 * n sin(x) cos(x) tan(x) | |
879 * ---------------------------------------------------------- | |
880 * 0 S C T | |
881 * 1 C -S -1/T | |
882 * 2 -S -C T | |
883 * 3 -C S -1/T | |
884 * ---------------------------------------------------------- | |
885 * | |
886 * Special cases: | |
887 * Let trig be any of sin, cos, or tan. | |
888 * trig(+-INF) is NaN, with signals; | |
889 * trig(NaN) is that NaN; | |
890 * | |
891 * Accuracy: | |
892 * TRIG(x) returns trig(x) nearly rounded | |
893 */ | |
894 | |
895 JRT_LEAF(jdouble, SharedRuntime::dcos(jdouble x)) | |
896 double y[2],z=0.0; | |
897 int n, ix; | |
898 | |
899 /* High word of x. */ | |
900 ix = __HI(x); | |
901 | |
902 /* |x| ~< pi/4 */ | |
903 ix &= 0x7fffffff; | |
904 if(ix <= 0x3fe921fb) return __kernel_cos(x,z); | |
905 | |
906 /* cos(Inf or NaN) is NaN */ | |
907 else if (ix>=0x7ff00000) return x-x; | |
908 | |
909 /* argument reduction needed */ | |
910 else { | |
911 n = __ieee754_rem_pio2(x,y); | |
912 switch(n&3) { | |
913 case 0: return __kernel_cos(y[0],y[1]); | |
914 case 1: return -__kernel_sin(y[0],y[1],1); | |
915 case 2: return -__kernel_cos(y[0],y[1]); | |
916 default: | |
917 return __kernel_sin(y[0],y[1],1); | |
918 } | |
919 } | |
920 JRT_END | |
921 | |
922 /* tan(x) | |
923 * Return tangent function of x. | |
924 * | |
925 * kernel function: | |
926 * __kernel_tan ... tangent function on [-pi/4,pi/4] | |
927 * __ieee754_rem_pio2 ... argument reduction routine | |
928 * | |
929 * Method. | |
930 * Let S,C and T denote the sin, cos and tan respectively on | |
931 * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 | |
932 * in [-pi/4 , +pi/4], and let n = k mod 4. | |
933 * We have | |
934 * | |
935 * n sin(x) cos(x) tan(x) | |
936 * ---------------------------------------------------------- | |
937 * 0 S C T | |
938 * 1 C -S -1/T | |
939 * 2 -S -C T | |
940 * 3 -C S -1/T | |
941 * ---------------------------------------------------------- | |
942 * | |
943 * Special cases: | |
944 * Let trig be any of sin, cos, or tan. | |
945 * trig(+-INF) is NaN, with signals; | |
946 * trig(NaN) is that NaN; | |
947 * | |
948 * Accuracy: | |
949 * TRIG(x) returns trig(x) nearly rounded | |
950 */ | |
951 | |
952 JRT_LEAF(jdouble, SharedRuntime::dtan(jdouble x)) | |
953 double y[2],z=0.0; | |
954 int n, ix; | |
955 | |
956 /* High word of x. */ | |
957 ix = __HI(x); | |
958 | |
959 /* |x| ~< pi/4 */ | |
960 ix &= 0x7fffffff; | |
961 if(ix <= 0x3fe921fb) return __kernel_tan(x,z,1); | |
962 | |
963 /* tan(Inf or NaN) is NaN */ | |
964 else if (ix>=0x7ff00000) return x-x; /* NaN */ | |
965 | |
966 /* argument reduction needed */ | |
967 else { | |
968 n = __ieee754_rem_pio2(x,y); | |
969 return __kernel_tan(y[0],y[1],1-((n&1)<<1)); /* 1 -- n even | |
970 -1 -- n odd */ | |
971 } | |
972 JRT_END | |
973 | |
974 | |
975 #ifdef WIN32 | |
976 # pragma optimize ( "", on ) | |
977 #endif |