Mercurial > hg > graal-compiler
comparison src/share/vm/runtime/sharedRuntimeTrig.cpp @ 0:a61af66fc99e jdk7-b24
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author | duke |
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date | Sat, 01 Dec 2007 00:00:00 +0000 |
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children | fb57d4cf76c2 |
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1 /* | |
2 * Copyright 2001-2005 Sun Microsystems, Inc. All Rights Reserved. | |
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 * | |
19 * Please contact Sun Microsystems, Inc., 4150 Network Circle, Santa Clara, | |
20 * CA 95054 USA or visit www.sun.com if you need additional information or | |
21 * have any questions. | |
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 | |
44 #include <math.h> | |
45 | |
46 // VM_LITTLE_ENDIAN is #defined appropriately in the Makefiles | |
47 // [jk] this is not 100% correct because the float word order may different | |
48 // from the byte order (e.g. on ARM) | |
49 #ifdef VM_LITTLE_ENDIAN | |
50 # define __HI(x) *(1+(int*)&x) | |
51 # define __LO(x) *(int*)&x | |
52 #else | |
53 # define __HI(x) *(int*)&x | |
54 # define __LO(x) *(1+(int*)&x) | |
55 #endif | |
56 | |
57 static double copysignA(double x, double y) { | |
58 __HI(x) = (__HI(x)&0x7fffffff)|(__HI(y)&0x80000000); | |
59 return x; | |
60 } | |
61 | |
62 /* | |
63 * scalbn (double x, int n) | |
64 * scalbn(x,n) returns x* 2**n computed by exponent | |
65 * manipulation rather than by actually performing an | |
66 * exponentiation or a multiplication. | |
67 */ | |
68 | |
69 static const double | |
70 two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ | |
71 twom54 = 5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */ | |
72 hugeX = 1.0e+300, | |
73 tiny = 1.0e-300; | |
74 | |
75 static double scalbnA (double x, int n) { | |
76 int k,hx,lx; | |
77 hx = __HI(x); | |
78 lx = __LO(x); | |
79 k = (hx&0x7ff00000)>>20; /* extract exponent */ | |
80 if (k==0) { /* 0 or subnormal x */ | |
81 if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */ | |
82 x *= two54; | |
83 hx = __HI(x); | |
84 k = ((hx&0x7ff00000)>>20) - 54; | |
85 if (n< -50000) return tiny*x; /*underflow*/ | |
86 } | |
87 if (k==0x7ff) return x+x; /* NaN or Inf */ | |
88 k = k+n; | |
89 if (k > 0x7fe) return hugeX*copysignA(hugeX,x); /* overflow */ | |
90 if (k > 0) /* normal result */ | |
91 {__HI(x) = (hx&0x800fffff)|(k<<20); return x;} | |
92 if (k <= -54) { | |
93 if (n > 50000) /* in case integer overflow in n+k */ | |
94 return hugeX*copysignA(hugeX,x); /*overflow*/ | |
95 else return tiny*copysignA(tiny,x); /*underflow*/ | |
96 } | |
97 k += 54; /* subnormal result */ | |
98 __HI(x) = (hx&0x800fffff)|(k<<20); | |
99 return x*twom54; | |
100 } | |
101 | |
102 /* | |
103 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) | |
104 * double x[],y[]; int e0,nx,prec; int ipio2[]; | |
105 * | |
106 * __kernel_rem_pio2 return the last three digits of N with | |
107 * y = x - N*pi/2 | |
108 * so that |y| < pi/2. | |
109 * | |
110 * The method is to compute the integer (mod 8) and fraction parts of | |
111 * (2/pi)*x without doing the full multiplication. In general we | |
112 * skip the part of the product that are known to be a huge integer ( | |
113 * more accurately, = 0 mod 8 ). Thus the number of operations are | |
114 * independent of the exponent of the input. | |
115 * | |
116 * (2/pi) is represented by an array of 24-bit integers in ipio2[]. | |
117 * | |
118 * Input parameters: | |
119 * x[] The input value (must be positive) is broken into nx | |
120 * pieces of 24-bit integers in double precision format. | |
121 * x[i] will be the i-th 24 bit of x. The scaled exponent | |
122 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 | |
123 * match x's up to 24 bits. | |
124 * | |
125 * Example of breaking a double positive z into x[0]+x[1]+x[2]: | |
126 * e0 = ilogb(z)-23 | |
127 * z = scalbn(z,-e0) | |
128 * for i = 0,1,2 | |
129 * x[i] = floor(z) | |
130 * z = (z-x[i])*2**24 | |
131 * | |
132 * | |
133 * y[] ouput result in an array of double precision numbers. | |
134 * The dimension of y[] is: | |
135 * 24-bit precision 1 | |
136 * 53-bit precision 2 | |
137 * 64-bit precision 2 | |
138 * 113-bit precision 3 | |
139 * The actual value is the sum of them. Thus for 113-bit | |
140 * precsion, one may have to do something like: | |
141 * | |
142 * long double t,w,r_head, r_tail; | |
143 * t = (long double)y[2] + (long double)y[1]; | |
144 * w = (long double)y[0]; | |
145 * r_head = t+w; | |
146 * r_tail = w - (r_head - t); | |
147 * | |
148 * e0 The exponent of x[0] | |
149 * | |
150 * nx dimension of x[] | |
151 * | |
152 * prec an interger indicating the precision: | |
153 * 0 24 bits (single) | |
154 * 1 53 bits (double) | |
155 * 2 64 bits (extended) | |
156 * 3 113 bits (quad) | |
157 * | |
158 * ipio2[] | |
159 * integer array, contains the (24*i)-th to (24*i+23)-th | |
160 * bit of 2/pi after binary point. The corresponding | |
161 * floating value is | |
162 * | |
163 * ipio2[i] * 2^(-24(i+1)). | |
164 * | |
165 * External function: | |
166 * double scalbn(), floor(); | |
167 * | |
168 * | |
169 * Here is the description of some local variables: | |
170 * | |
171 * jk jk+1 is the initial number of terms of ipio2[] needed | |
172 * in the computation. The recommended value is 2,3,4, | |
173 * 6 for single, double, extended,and quad. | |
174 * | |
175 * jz local integer variable indicating the number of | |
176 * terms of ipio2[] used. | |
177 * | |
178 * jx nx - 1 | |
179 * | |
180 * jv index for pointing to the suitable ipio2[] for the | |
181 * computation. In general, we want | |
182 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 | |
183 * is an integer. Thus | |
184 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv | |
185 * Hence jv = max(0,(e0-3)/24). | |
186 * | |
187 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. | |
188 * | |
189 * q[] double array with integral value, representing the | |
190 * 24-bits chunk of the product of x and 2/pi. | |
191 * | |
192 * q0 the corresponding exponent of q[0]. Note that the | |
193 * exponent for q[i] would be q0-24*i. | |
194 * | |
195 * PIo2[] double precision array, obtained by cutting pi/2 | |
196 * into 24 bits chunks. | |
197 * | |
198 * f[] ipio2[] in floating point | |
199 * | |
200 * iq[] integer array by breaking up q[] in 24-bits chunk. | |
201 * | |
202 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] | |
203 * | |
204 * ih integer. If >0 it indicats q[] is >= 0.5, hence | |
205 * it also indicates the *sign* of the result. | |
206 * | |
207 */ | |
208 | |
209 | |
210 /* | |
211 * Constants: | |
212 * The hexadecimal values are the intended ones for the following | |
213 * constants. The decimal values may be used, provided that the | |
214 * compiler will convert from decimal to binary accurately enough | |
215 * to produce the hexadecimal values shown. | |
216 */ | |
217 | |
218 | |
219 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ | |
220 | |
221 static const double PIo2[] = { | |
222 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ | |
223 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ | |
224 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ | |
225 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ | |
226 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ | |
227 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ | |
228 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ | |
229 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ | |
230 }; | |
231 | |
232 static const double | |
233 zeroB = 0.0, | |
234 one = 1.0, | |
235 two24B = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ | |
236 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ | |
237 | |
238 static int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2) { | |
239 int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; | |
240 double z,fw,f[20],fq[20],q[20]; | |
241 | |
242 /* initialize jk*/ | |
243 jk = init_jk[prec]; | |
244 jp = jk; | |
245 | |
246 /* determine jx,jv,q0, note that 3>q0 */ | |
247 jx = nx-1; | |
248 jv = (e0-3)/24; if(jv<0) jv=0; | |
249 q0 = e0-24*(jv+1); | |
250 | |
251 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ | |
252 j = jv-jx; m = jx+jk; | |
253 for(i=0;i<=m;i++,j++) f[i] = (j<0)? zeroB : (double) ipio2[j]; | |
254 | |
255 /* compute q[0],q[1],...q[jk] */ | |
256 for (i=0;i<=jk;i++) { | |
257 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; | |
258 } | |
259 | |
260 jz = jk; | |
261 recompute: | |
262 /* distill q[] into iq[] reversingly */ | |
263 for(i=0,j=jz,z=q[jz];j>0;i++,j--) { | |
264 fw = (double)((int)(twon24* z)); | |
265 iq[i] = (int)(z-two24B*fw); | |
266 z = q[j-1]+fw; | |
267 } | |
268 | |
269 /* compute n */ | |
270 z = scalbnA(z,q0); /* actual value of z */ | |
271 z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ | |
272 n = (int) z; | |
273 z -= (double)n; | |
274 ih = 0; | |
275 if(q0>0) { /* need iq[jz-1] to determine n */ | |
276 i = (iq[jz-1]>>(24-q0)); n += i; | |
277 iq[jz-1] -= i<<(24-q0); | |
278 ih = iq[jz-1]>>(23-q0); | |
279 } | |
280 else if(q0==0) ih = iq[jz-1]>>23; | |
281 else if(z>=0.5) ih=2; | |
282 | |
283 if(ih>0) { /* q > 0.5 */ | |
284 n += 1; carry = 0; | |
285 for(i=0;i<jz ;i++) { /* compute 1-q */ | |
286 j = iq[i]; | |
287 if(carry==0) { | |
288 if(j!=0) { | |
289 carry = 1; iq[i] = 0x1000000- j; | |
290 } | |
291 } else iq[i] = 0xffffff - j; | |
292 } | |
293 if(q0>0) { /* rare case: chance is 1 in 12 */ | |
294 switch(q0) { | |
295 case 1: | |
296 iq[jz-1] &= 0x7fffff; break; | |
297 case 2: | |
298 iq[jz-1] &= 0x3fffff; break; | |
299 } | |
300 } | |
301 if(ih==2) { | |
302 z = one - z; | |
303 if(carry!=0) z -= scalbnA(one,q0); | |
304 } | |
305 } | |
306 | |
307 /* check if recomputation is needed */ | |
308 if(z==zeroB) { | |
309 j = 0; | |
310 for (i=jz-1;i>=jk;i--) j |= iq[i]; | |
311 if(j==0) { /* need recomputation */ | |
312 for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ | |
313 | |
314 for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ | |
315 f[jx+i] = (double) ipio2[jv+i]; | |
316 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; | |
317 q[i] = fw; | |
318 } | |
319 jz += k; | |
320 goto recompute; | |
321 } | |
322 } | |
323 | |
324 /* chop off zero terms */ | |
325 if(z==0.0) { | |
326 jz -= 1; q0 -= 24; | |
327 while(iq[jz]==0) { jz--; q0-=24;} | |
328 } else { /* break z into 24-bit if neccessary */ | |
329 z = scalbnA(z,-q0); | |
330 if(z>=two24B) { | |
331 fw = (double)((int)(twon24*z)); | |
332 iq[jz] = (int)(z-two24B*fw); | |
333 jz += 1; q0 += 24; | |
334 iq[jz] = (int) fw; | |
335 } else iq[jz] = (int) z ; | |
336 } | |
337 | |
338 /* convert integer "bit" chunk to floating-point value */ | |
339 fw = scalbnA(one,q0); | |
340 for(i=jz;i>=0;i--) { | |
341 q[i] = fw*(double)iq[i]; fw*=twon24; | |
342 } | |
343 | |
344 /* compute PIo2[0,...,jp]*q[jz,...,0] */ | |
345 for(i=jz;i>=0;i--) { | |
346 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; | |
347 fq[jz-i] = fw; | |
348 } | |
349 | |
350 /* compress fq[] into y[] */ | |
351 switch(prec) { | |
352 case 0: | |
353 fw = 0.0; | |
354 for (i=jz;i>=0;i--) fw += fq[i]; | |
355 y[0] = (ih==0)? fw: -fw; | |
356 break; | |
357 case 1: | |
358 case 2: | |
359 fw = 0.0; | |
360 for (i=jz;i>=0;i--) fw += fq[i]; | |
361 y[0] = (ih==0)? fw: -fw; | |
362 fw = fq[0]-fw; | |
363 for (i=1;i<=jz;i++) fw += fq[i]; | |
364 y[1] = (ih==0)? fw: -fw; | |
365 break; | |
366 case 3: /* painful */ | |
367 for (i=jz;i>0;i--) { | |
368 fw = fq[i-1]+fq[i]; | |
369 fq[i] += fq[i-1]-fw; | |
370 fq[i-1] = fw; | |
371 } | |
372 for (i=jz;i>1;i--) { | |
373 fw = fq[i-1]+fq[i]; | |
374 fq[i] += fq[i-1]-fw; | |
375 fq[i-1] = fw; | |
376 } | |
377 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; | |
378 if(ih==0) { | |
379 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; | |
380 } else { | |
381 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; | |
382 } | |
383 } | |
384 return n&7; | |
385 } | |
386 | |
387 | |
388 /* | |
389 * ==================================================== | |
390 * Copyright 13 Dec 1993 Sun Microsystems, Inc. All Rights Reserved. | |
391 * | |
392 * Developed at SunPro, a Sun Microsystems, Inc. business. | |
393 * Permission to use, copy, modify, and distribute this | |
394 * software is freely granted, provided that this notice | |
395 * is preserved. | |
396 * ==================================================== | |
397 * | |
398 */ | |
399 | |
400 /* __ieee754_rem_pio2(x,y) | |
401 * | |
402 * return the remainder of x rem pi/2 in y[0]+y[1] | |
403 * use __kernel_rem_pio2() | |
404 */ | |
405 | |
406 /* | |
407 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi | |
408 */ | |
409 static const int two_over_pi[] = { | |
410 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, | |
411 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, | |
412 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, | |
413 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, | |
414 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, | |
415 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF, | |
416 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, | |
417 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, | |
418 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, | |
419 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, | |
420 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B, | |
421 }; | |
422 | |
423 static const int npio2_hw[] = { | |
424 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, | |
425 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C, | |
426 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A, | |
427 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C, | |
428 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB, | |
429 0x404858EB, 0x404921FB, | |
430 }; | |
431 | |
432 /* | |
433 * invpio2: 53 bits of 2/pi | |
434 * pio2_1: first 33 bit of pi/2 | |
435 * pio2_1t: pi/2 - pio2_1 | |
436 * pio2_2: second 33 bit of pi/2 | |
437 * pio2_2t: pi/2 - (pio2_1+pio2_2) | |
438 * pio2_3: third 33 bit of pi/2 | |
439 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) | |
440 */ | |
441 | |
442 static const double | |
443 zeroA = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ | |
444 half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ | |
445 two24A = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ | |
446 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ | |
447 pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */ | |
448 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */ | |
449 pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */ | |
450 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */ | |
451 pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */ | |
452 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */ | |
453 | |
454 static int __ieee754_rem_pio2(double x, double *y) { | |
455 double z,w,t,r,fn; | |
456 double tx[3]; | |
457 int e0,i,j,nx,n,ix,hx,i0; | |
458 | |
459 i0 = ((*(int*)&two24A)>>30)^1; /* high word index */ | |
460 hx = *(i0+(int*)&x); /* high word of x */ | |
461 ix = hx&0x7fffffff; | |
462 if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */ | |
463 {y[0] = x; y[1] = 0; return 0;} | |
464 if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */ | |
465 if(hx>0) { | |
466 z = x - pio2_1; | |
467 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ | |
468 y[0] = z - pio2_1t; | |
469 y[1] = (z-y[0])-pio2_1t; | |
470 } else { /* near pi/2, use 33+33+53 bit pi */ | |
471 z -= pio2_2; | |
472 y[0] = z - pio2_2t; | |
473 y[1] = (z-y[0])-pio2_2t; | |
474 } | |
475 return 1; | |
476 } else { /* negative x */ | |
477 z = x + pio2_1; | |
478 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ | |
479 y[0] = z + pio2_1t; | |
480 y[1] = (z-y[0])+pio2_1t; | |
481 } else { /* near pi/2, use 33+33+53 bit pi */ | |
482 z += pio2_2; | |
483 y[0] = z + pio2_2t; | |
484 y[1] = (z-y[0])+pio2_2t; | |
485 } | |
486 return -1; | |
487 } | |
488 } | |
489 if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */ | |
490 t = fabsd(x); | |
491 n = (int) (t*invpio2+half); | |
492 fn = (double)n; | |
493 r = t-fn*pio2_1; | |
494 w = fn*pio2_1t; /* 1st round good to 85 bit */ | |
495 if(n<32&&ix!=npio2_hw[n-1]) { | |
496 y[0] = r-w; /* quick check no cancellation */ | |
497 } else { | |
498 j = ix>>20; | |
499 y[0] = r-w; | |
500 i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff); | |
501 if(i>16) { /* 2nd iteration needed, good to 118 */ | |
502 t = r; | |
503 w = fn*pio2_2; | |
504 r = t-w; | |
505 w = fn*pio2_2t-((t-r)-w); | |
506 y[0] = r-w; | |
507 i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff); | |
508 if(i>49) { /* 3rd iteration need, 151 bits acc */ | |
509 t = r; /* will cover all possible cases */ | |
510 w = fn*pio2_3; | |
511 r = t-w; | |
512 w = fn*pio2_3t-((t-r)-w); | |
513 y[0] = r-w; | |
514 } | |
515 } | |
516 } | |
517 y[1] = (r-y[0])-w; | |
518 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} | |
519 else return n; | |
520 } | |
521 /* | |
522 * all other (large) arguments | |
523 */ | |
524 if(ix>=0x7ff00000) { /* x is inf or NaN */ | |
525 y[0]=y[1]=x-x; return 0; | |
526 } | |
527 /* set z = scalbn(|x|,ilogb(x)-23) */ | |
528 *(1-i0+(int*)&z) = *(1-i0+(int*)&x); | |
529 e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */ | |
530 *(i0+(int*)&z) = ix - (e0<<20); | |
531 for(i=0;i<2;i++) { | |
532 tx[i] = (double)((int)(z)); | |
533 z = (z-tx[i])*two24A; | |
534 } | |
535 tx[2] = z; | |
536 nx = 3; | |
537 while(tx[nx-1]==zeroA) nx--; /* skip zero term */ | |
538 n = __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi); | |
539 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} | |
540 return n; | |
541 } | |
542 | |
543 | |
544 /* __kernel_sin( x, y, iy) | |
545 * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 | |
546 * Input x is assumed to be bounded by ~pi/4 in magnitude. | |
547 * Input y is the tail of x. | |
548 * Input iy indicates whether y is 0. (if iy=0, y assume to be 0). | |
549 * | |
550 * Algorithm | |
551 * 1. Since sin(-x) = -sin(x), we need only to consider positive x. | |
552 * 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0. | |
553 * 3. sin(x) is approximated by a polynomial of degree 13 on | |
554 * [0,pi/4] | |
555 * 3 13 | |
556 * sin(x) ~ x + S1*x + ... + S6*x | |
557 * where | |
558 * | |
559 * |sin(x) 2 4 6 8 10 12 | -58 | |
560 * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 | |
561 * | x | | |
562 * | |
563 * 4. sin(x+y) = sin(x) + sin'(x')*y | |
564 * ~ sin(x) + (1-x*x/2)*y | |
565 * For better accuracy, let | |
566 * 3 2 2 2 2 | |
567 * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) | |
568 * then 3 2 | |
569 * sin(x) = x + (S1*x + (x *(r-y/2)+y)) | |
570 */ | |
571 | |
572 static const double | |
573 S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */ | |
574 S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */ | |
575 S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */ | |
576 S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */ | |
577 S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */ | |
578 S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */ | |
579 | |
580 static double __kernel_sin(double x, double y, int iy) | |
581 { | |
582 double z,r,v; | |
583 int ix; | |
584 ix = __HI(x)&0x7fffffff; /* high word of x */ | |
585 if(ix<0x3e400000) /* |x| < 2**-27 */ | |
586 {if((int)x==0) return x;} /* generate inexact */ | |
587 z = x*x; | |
588 v = z*x; | |
589 r = S2+z*(S3+z*(S4+z*(S5+z*S6))); | |
590 if(iy==0) return x+v*(S1+z*r); | |
591 else return x-((z*(half*y-v*r)-y)-v*S1); | |
592 } | |
593 | |
594 /* | |
595 * __kernel_cos( x, y ) | |
596 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 | |
597 * Input x is assumed to be bounded by ~pi/4 in magnitude. | |
598 * Input y is the tail of x. | |
599 * | |
600 * Algorithm | |
601 * 1. Since cos(-x) = cos(x), we need only to consider positive x. | |
602 * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. | |
603 * 3. cos(x) is approximated by a polynomial of degree 14 on | |
604 * [0,pi/4] | |
605 * 4 14 | |
606 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x | |
607 * where the remez error is | |
608 * | |
609 * | 2 4 6 8 10 12 14 | -58 | |
610 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 | |
611 * | | | |
612 * | |
613 * 4 6 8 10 12 14 | |
614 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then | |
615 * cos(x) = 1 - x*x/2 + r | |
616 * since cos(x+y) ~ cos(x) - sin(x)*y | |
617 * ~ cos(x) - x*y, | |
618 * a correction term is necessary in cos(x) and hence | |
619 * cos(x+y) = 1 - (x*x/2 - (r - x*y)) | |
620 * For better accuracy when x > 0.3, let qx = |x|/4 with | |
621 * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. | |
622 * Then | |
623 * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). | |
624 * Note that 1-qx and (x*x/2-qx) is EXACT here, and the | |
625 * magnitude of the latter is at least a quarter of x*x/2, | |
626 * thus, reducing the rounding error in the subtraction. | |
627 */ | |
628 | |
629 static const double | |
630 C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ | |
631 C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ | |
632 C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ | |
633 C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ | |
634 C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ | |
635 C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ | |
636 | |
637 static double __kernel_cos(double x, double y) | |
638 { | |
639 double a,hz,z,r,qx; | |
640 int ix; | |
641 ix = __HI(x)&0x7fffffff; /* ix = |x|'s high word*/ | |
642 if(ix<0x3e400000) { /* if x < 2**27 */ | |
643 if(((int)x)==0) return one; /* generate inexact */ | |
644 } | |
645 z = x*x; | |
646 r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6))))); | |
647 if(ix < 0x3FD33333) /* if |x| < 0.3 */ | |
648 return one - (0.5*z - (z*r - x*y)); | |
649 else { | |
650 if(ix > 0x3fe90000) { /* x > 0.78125 */ | |
651 qx = 0.28125; | |
652 } else { | |
653 __HI(qx) = ix-0x00200000; /* x/4 */ | |
654 __LO(qx) = 0; | |
655 } | |
656 hz = 0.5*z-qx; | |
657 a = one-qx; | |
658 return a - (hz - (z*r-x*y)); | |
659 } | |
660 } | |
661 | |
662 /* __kernel_tan( x, y, k ) | |
663 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 | |
664 * Input x is assumed to be bounded by ~pi/4 in magnitude. | |
665 * Input y is the tail of x. | |
666 * Input k indicates whether tan (if k=1) or | |
667 * -1/tan (if k= -1) is returned. | |
668 * | |
669 * Algorithm | |
670 * 1. Since tan(-x) = -tan(x), we need only to consider positive x. | |
671 * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. | |
672 * 3. tan(x) is approximated by a odd polynomial of degree 27 on | |
673 * [0,0.67434] | |
674 * 3 27 | |
675 * tan(x) ~ x + T1*x + ... + T13*x | |
676 * where | |
677 * | |
678 * |tan(x) 2 4 26 | -59.2 | |
679 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 | |
680 * | x | | |
681 * | |
682 * Note: tan(x+y) = tan(x) + tan'(x)*y | |
683 * ~ tan(x) + (1+x*x)*y | |
684 * Therefore, for better accuracy in computing tan(x+y), let | |
685 * 3 2 2 2 2 | |
686 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) | |
687 * then | |
688 * 3 2 | |
689 * tan(x+y) = x + (T1*x + (x *(r+y)+y)) | |
690 * | |
691 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then | |
692 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) | |
693 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) | |
694 */ | |
695 | |
696 static const double | |
697 pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ | |
698 pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */ | |
699 T[] = { | |
700 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */ | |
701 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */ | |
702 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */ | |
703 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */ | |
704 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */ | |
705 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */ | |
706 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */ | |
707 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */ | |
708 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */ | |
709 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */ | |
710 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */ | |
711 -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */ | |
712 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */ | |
713 }; | |
714 | |
715 static double __kernel_tan(double x, double y, int iy) | |
716 { | |
717 double z,r,v,w,s; | |
718 int ix,hx; | |
719 hx = __HI(x); /* high word of x */ | |
720 ix = hx&0x7fffffff; /* high word of |x| */ | |
721 if(ix<0x3e300000) { /* x < 2**-28 */ | |
722 if((int)x==0) { /* generate inexact */ | |
723 if (((ix | __LO(x)) | (iy + 1)) == 0) | |
724 return one / fabsd(x); | |
725 else { | |
726 if (iy == 1) | |
727 return x; | |
728 else { /* compute -1 / (x+y) carefully */ | |
729 double a, t; | |
730 | |
731 z = w = x + y; | |
732 __LO(z) = 0; | |
733 v = y - (z - x); | |
734 t = a = -one / w; | |
735 __LO(t) = 0; | |
736 s = one + t * z; | |
737 return t + a * (s + t * v); | |
738 } | |
739 } | |
740 } | |
741 } | |
742 if(ix>=0x3FE59428) { /* |x|>=0.6744 */ | |
743 if(hx<0) {x = -x; y = -y;} | |
744 z = pio4-x; | |
745 w = pio4lo-y; | |
746 x = z+w; y = 0.0; | |
747 } | |
748 z = x*x; | |
749 w = z*z; | |
750 /* Break x^5*(T[1]+x^2*T[2]+...) into | |
751 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + | |
752 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) | |
753 */ | |
754 r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11])))); | |
755 v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12]))))); | |
756 s = z*x; | |
757 r = y + z*(s*(r+v)+y); | |
758 r += T[0]*s; | |
759 w = x+r; | |
760 if(ix>=0x3FE59428) { | |
761 v = (double)iy; | |
762 return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r))); | |
763 } | |
764 if(iy==1) return w; | |
765 else { /* if allow error up to 2 ulp, | |
766 simply return -1.0/(x+r) here */ | |
767 /* compute -1.0/(x+r) accurately */ | |
768 double a,t; | |
769 z = w; | |
770 __LO(z) = 0; | |
771 v = r-(z - x); /* z+v = r+x */ | |
772 t = a = -1.0/w; /* a = -1.0/w */ | |
773 __LO(t) = 0; | |
774 s = 1.0+t*z; | |
775 return t+a*(s+t*v); | |
776 } | |
777 } | |
778 | |
779 | |
780 //---------------------------------------------------------------------- | |
781 // | |
782 // Routines for new sin/cos implementation | |
783 // | |
784 //---------------------------------------------------------------------- | |
785 | |
786 /* sin(x) | |
787 * Return sine function of x. | |
788 * | |
789 * kernel function: | |
790 * __kernel_sin ... sine function on [-pi/4,pi/4] | |
791 * __kernel_cos ... cose function on [-pi/4,pi/4] | |
792 * __ieee754_rem_pio2 ... argument reduction routine | |
793 * | |
794 * Method. | |
795 * Let S,C and T denote the sin, cos and tan respectively on | |
796 * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 | |
797 * in [-pi/4 , +pi/4], and let n = k mod 4. | |
798 * We have | |
799 * | |
800 * n sin(x) cos(x) tan(x) | |
801 * ---------------------------------------------------------- | |
802 * 0 S C T | |
803 * 1 C -S -1/T | |
804 * 2 -S -C T | |
805 * 3 -C S -1/T | |
806 * ---------------------------------------------------------- | |
807 * | |
808 * Special cases: | |
809 * Let trig be any of sin, cos, or tan. | |
810 * trig(+-INF) is NaN, with signals; | |
811 * trig(NaN) is that NaN; | |
812 * | |
813 * Accuracy: | |
814 * TRIG(x) returns trig(x) nearly rounded | |
815 */ | |
816 | |
817 JRT_LEAF(jdouble, SharedRuntime::dsin(jdouble x)) | |
818 double y[2],z=0.0; | |
819 int n, ix; | |
820 | |
821 /* High word of x. */ | |
822 ix = __HI(x); | |
823 | |
824 /* |x| ~< pi/4 */ | |
825 ix &= 0x7fffffff; | |
826 if(ix <= 0x3fe921fb) return __kernel_sin(x,z,0); | |
827 | |
828 /* sin(Inf or NaN) is NaN */ | |
829 else if (ix>=0x7ff00000) return x-x; | |
830 | |
831 /* argument reduction needed */ | |
832 else { | |
833 n = __ieee754_rem_pio2(x,y); | |
834 switch(n&3) { | |
835 case 0: return __kernel_sin(y[0],y[1],1); | |
836 case 1: return __kernel_cos(y[0],y[1]); | |
837 case 2: return -__kernel_sin(y[0],y[1],1); | |
838 default: | |
839 return -__kernel_cos(y[0],y[1]); | |
840 } | |
841 } | |
842 JRT_END | |
843 | |
844 /* cos(x) | |
845 * Return cosine function of x. | |
846 * | |
847 * kernel function: | |
848 * __kernel_sin ... sine function on [-pi/4,pi/4] | |
849 * __kernel_cos ... cosine function on [-pi/4,pi/4] | |
850 * __ieee754_rem_pio2 ... argument reduction routine | |
851 * | |
852 * Method. | |
853 * Let S,C and T denote the sin, cos and tan respectively on | |
854 * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 | |
855 * in [-pi/4 , +pi/4], and let n = k mod 4. | |
856 * We have | |
857 * | |
858 * n sin(x) cos(x) tan(x) | |
859 * ---------------------------------------------------------- | |
860 * 0 S C T | |
861 * 1 C -S -1/T | |
862 * 2 -S -C T | |
863 * 3 -C S -1/T | |
864 * ---------------------------------------------------------- | |
865 * | |
866 * Special cases: | |
867 * Let trig be any of sin, cos, or tan. | |
868 * trig(+-INF) is NaN, with signals; | |
869 * trig(NaN) is that NaN; | |
870 * | |
871 * Accuracy: | |
872 * TRIG(x) returns trig(x) nearly rounded | |
873 */ | |
874 | |
875 JRT_LEAF(jdouble, SharedRuntime::dcos(jdouble x)) | |
876 double y[2],z=0.0; | |
877 int n, ix; | |
878 | |
879 /* High word of x. */ | |
880 ix = __HI(x); | |
881 | |
882 /* |x| ~< pi/4 */ | |
883 ix &= 0x7fffffff; | |
884 if(ix <= 0x3fe921fb) return __kernel_cos(x,z); | |
885 | |
886 /* cos(Inf or NaN) is NaN */ | |
887 else if (ix>=0x7ff00000) return x-x; | |
888 | |
889 /* argument reduction needed */ | |
890 else { | |
891 n = __ieee754_rem_pio2(x,y); | |
892 switch(n&3) { | |
893 case 0: return __kernel_cos(y[0],y[1]); | |
894 case 1: return -__kernel_sin(y[0],y[1],1); | |
895 case 2: return -__kernel_cos(y[0],y[1]); | |
896 default: | |
897 return __kernel_sin(y[0],y[1],1); | |
898 } | |
899 } | |
900 JRT_END | |
901 | |
902 /* tan(x) | |
903 * Return tangent function of x. | |
904 * | |
905 * kernel function: | |
906 * __kernel_tan ... tangent function on [-pi/4,pi/4] | |
907 * __ieee754_rem_pio2 ... argument reduction routine | |
908 * | |
909 * Method. | |
910 * Let S,C and T denote the sin, cos and tan respectively on | |
911 * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 | |
912 * in [-pi/4 , +pi/4], and let n = k mod 4. | |
913 * We have | |
914 * | |
915 * n sin(x) cos(x) tan(x) | |
916 * ---------------------------------------------------------- | |
917 * 0 S C T | |
918 * 1 C -S -1/T | |
919 * 2 -S -C T | |
920 * 3 -C S -1/T | |
921 * ---------------------------------------------------------- | |
922 * | |
923 * Special cases: | |
924 * Let trig be any of sin, cos, or tan. | |
925 * trig(+-INF) is NaN, with signals; | |
926 * trig(NaN) is that NaN; | |
927 * | |
928 * Accuracy: | |
929 * TRIG(x) returns trig(x) nearly rounded | |
930 */ | |
931 | |
932 JRT_LEAF(jdouble, SharedRuntime::dtan(jdouble x)) | |
933 double y[2],z=0.0; | |
934 int n, ix; | |
935 | |
936 /* High word of x. */ | |
937 ix = __HI(x); | |
938 | |
939 /* |x| ~< pi/4 */ | |
940 ix &= 0x7fffffff; | |
941 if(ix <= 0x3fe921fb) return __kernel_tan(x,z,1); | |
942 | |
943 /* tan(Inf or NaN) is NaN */ | |
944 else if (ix>=0x7ff00000) return x-x; /* NaN */ | |
945 | |
946 /* argument reduction needed */ | |
947 else { | |
948 n = __ieee754_rem_pio2(x,y); | |
949 return __kernel_tan(y[0],y[1],1-((n&1)<<1)); /* 1 -- n even | |
950 -1 -- n odd */ | |
951 } | |
952 JRT_END | |
953 | |
954 | |
955 #ifdef WIN32 | |
956 # pragma optimize ( "", on ) | |
957 #endif |