Mercurial > hg > truffle
annotate src/share/vm/runtime/sharedRuntimeTrig.cpp @ 17458:ddcb2ac2900d
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author | drchase |
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date | Thu, 12 Dec 2013 20:55:36 -0500 |
parents | f95d63e2154a |
children | 63a4eb8bcd23 bdd155477289 |
rev | line source |
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0 | 1 /* |
1972 | 2 * Copyright (c) 2001, 2010, 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 | |
1972 | 25 #include "precompiled.hpp" |
26 #include "prims/jni.h" | |
27 #include "runtime/interfaceSupport.hpp" | |
28 #include "runtime/sharedRuntime.hpp" | |
0 | 29 |
30 // This file contains copies of the fdlibm routines used by | |
31 // StrictMath. It turns out that it is almost always required to use | |
32 // these runtime routines; the Intel CPU doesn't meet the Java | |
33 // specification for sin/cos outside a certain limited argument range, | |
34 // and the SPARC CPU doesn't appear to have sin/cos instructions. It | |
35 // also turns out that avoiding the indirect call through function | |
36 // pointer out to libjava.so in SharedRuntime speeds these routines up | |
37 // by roughly 15% on both Win32/x86 and Solaris/SPARC. | |
38 | |
39 // Enabling optimizations in this file causes incorrect code to be | |
40 // generated; can not figure out how to turn down optimization for one | |
41 // file in the IDE on Windows | |
42 #ifdef WIN32 | |
43 # pragma optimize ( "", off ) | |
44 #endif | |
45 | |
1485 | 46 /* The above workaround now causes more problems with the latest MS compiler. |
47 * Visual Studio 2010's /GS option tries to guard against buffer overruns. | |
48 * /GS is on by default if you specify optimizations, which we do globally | |
49 * via /W3 /O2. However the above selective turning off of optimizations means | |
50 * that /GS issues a warning "4748". And since we treat warnings as errors (/WX) | |
51 * then the compilation fails. There are several possible solutions | |
52 * (1) Remove that pragma above as obsolete with VS2010 - requires testing. | |
53 * (2) Stop treating warnings as errors - would be a backward step | |
54 * (3) Disable /GS - may help performance but you lose the security checks | |
55 * (4) Disable the warning with "#pragma warning( disable : 4748 )" | |
56 * (5) Disable planting the code with __declspec(safebuffers) | |
57 * I've opted for (5) although we should investigate the local performance | |
58 * benefits of (1) and global performance benefit of (3). | |
59 */ | |
60 #if defined(WIN32) && (defined(_MSC_VER) && (_MSC_VER >= 1600)) | |
61 #define SAFEBUF __declspec(safebuffers) | |
62 #else | |
63 #define SAFEBUF | |
64 #endif | |
65 | |
0 | 66 #include <math.h> |
67 | |
68 // VM_LITTLE_ENDIAN is #defined appropriately in the Makefiles | |
69 // [jk] this is not 100% correct because the float word order may different | |
70 // from the byte order (e.g. on ARM) | |
71 #ifdef VM_LITTLE_ENDIAN | |
72 # define __HI(x) *(1+(int*)&x) | |
73 # define __LO(x) *(int*)&x | |
74 #else | |
75 # define __HI(x) *(int*)&x | |
76 # define __LO(x) *(1+(int*)&x) | |
77 #endif | |
78 | |
79 static double copysignA(double x, double y) { | |
80 __HI(x) = (__HI(x)&0x7fffffff)|(__HI(y)&0x80000000); | |
81 return x; | |
82 } | |
83 | |
84 /* | |
85 * scalbn (double x, int n) | |
86 * scalbn(x,n) returns x* 2**n computed by exponent | |
87 * manipulation rather than by actually performing an | |
88 * exponentiation or a multiplication. | |
89 */ | |
90 | |
91 static const double | |
92 two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ | |
93 twom54 = 5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */ | |
94 hugeX = 1.0e+300, | |
95 tiny = 1.0e-300; | |
96 | |
97 static double scalbnA (double x, int n) { | |
98 int k,hx,lx; | |
99 hx = __HI(x); | |
100 lx = __LO(x); | |
101 k = (hx&0x7ff00000)>>20; /* extract exponent */ | |
102 if (k==0) { /* 0 or subnormal x */ | |
103 if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */ | |
104 x *= two54; | |
105 hx = __HI(x); | |
106 k = ((hx&0x7ff00000)>>20) - 54; | |
107 if (n< -50000) return tiny*x; /*underflow*/ | |
108 } | |
109 if (k==0x7ff) return x+x; /* NaN or Inf */ | |
110 k = k+n; | |
111 if (k > 0x7fe) return hugeX*copysignA(hugeX,x); /* overflow */ | |
112 if (k > 0) /* normal result */ | |
113 {__HI(x) = (hx&0x800fffff)|(k<<20); return x;} | |
114 if (k <= -54) { | |
115 if (n > 50000) /* in case integer overflow in n+k */ | |
116 return hugeX*copysignA(hugeX,x); /*overflow*/ | |
117 else return tiny*copysignA(tiny,x); /*underflow*/ | |
118 } | |
119 k += 54; /* subnormal result */ | |
120 __HI(x) = (hx&0x800fffff)|(k<<20); | |
121 return x*twom54; | |
122 } | |
123 | |
124 /* | |
125 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) | |
126 * double x[],y[]; int e0,nx,prec; int ipio2[]; | |
127 * | |
128 * __kernel_rem_pio2 return the last three digits of N with | |
129 * y = x - N*pi/2 | |
130 * so that |y| < pi/2. | |
131 * | |
132 * The method is to compute the integer (mod 8) and fraction parts of | |
133 * (2/pi)*x without doing the full multiplication. In general we | |
134 * skip the part of the product that are known to be a huge integer ( | |
135 * more accurately, = 0 mod 8 ). Thus the number of operations are | |
136 * independent of the exponent of the input. | |
137 * | |
138 * (2/pi) is represented by an array of 24-bit integers in ipio2[]. | |
139 * | |
140 * Input parameters: | |
141 * x[] The input value (must be positive) is broken into nx | |
142 * pieces of 24-bit integers in double precision format. | |
143 * x[i] will be the i-th 24 bit of x. The scaled exponent | |
144 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 | |
145 * match x's up to 24 bits. | |
146 * | |
147 * Example of breaking a double positive z into x[0]+x[1]+x[2]: | |
148 * e0 = ilogb(z)-23 | |
149 * z = scalbn(z,-e0) | |
150 * for i = 0,1,2 | |
151 * x[i] = floor(z) | |
152 * z = (z-x[i])*2**24 | |
153 * | |
154 * | |
155 * y[] ouput result in an array of double precision numbers. | |
156 * The dimension of y[] is: | |
157 * 24-bit precision 1 | |
158 * 53-bit precision 2 | |
159 * 64-bit precision 2 | |
160 * 113-bit precision 3 | |
161 * The actual value is the sum of them. Thus for 113-bit | |
162 * precsion, one may have to do something like: | |
163 * | |
164 * long double t,w,r_head, r_tail; | |
165 * t = (long double)y[2] + (long double)y[1]; | |
166 * w = (long double)y[0]; | |
167 * r_head = t+w; | |
168 * r_tail = w - (r_head - t); | |
169 * | |
170 * e0 The exponent of x[0] | |
171 * | |
172 * nx dimension of x[] | |
173 * | |
174 * prec an interger indicating the precision: | |
175 * 0 24 bits (single) | |
176 * 1 53 bits (double) | |
177 * 2 64 bits (extended) | |
178 * 3 113 bits (quad) | |
179 * | |
180 * ipio2[] | |
181 * integer array, contains the (24*i)-th to (24*i+23)-th | |
182 * bit of 2/pi after binary point. The corresponding | |
183 * floating value is | |
184 * | |
185 * ipio2[i] * 2^(-24(i+1)). | |
186 * | |
187 * External function: | |
188 * double scalbn(), floor(); | |
189 * | |
190 * | |
191 * Here is the description of some local variables: | |
192 * | |
193 * jk jk+1 is the initial number of terms of ipio2[] needed | |
194 * in the computation. The recommended value is 2,3,4, | |
195 * 6 for single, double, extended,and quad. | |
196 * | |
197 * jz local integer variable indicating the number of | |
198 * terms of ipio2[] used. | |
199 * | |
200 * jx nx - 1 | |
201 * | |
202 * jv index for pointing to the suitable ipio2[] for the | |
203 * computation. In general, we want | |
204 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 | |
205 * is an integer. Thus | |
206 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv | |
207 * Hence jv = max(0,(e0-3)/24). | |
208 * | |
209 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. | |
210 * | |
211 * q[] double array with integral value, representing the | |
212 * 24-bits chunk of the product of x and 2/pi. | |
213 * | |
214 * q0 the corresponding exponent of q[0]. Note that the | |
215 * exponent for q[i] would be q0-24*i. | |
216 * | |
217 * PIo2[] double precision array, obtained by cutting pi/2 | |
218 * into 24 bits chunks. | |
219 * | |
220 * f[] ipio2[] in floating point | |
221 * | |
222 * iq[] integer array by breaking up q[] in 24-bits chunk. | |
223 * | |
224 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] | |
225 * | |
226 * ih integer. If >0 it indicats q[] is >= 0.5, hence | |
227 * it also indicates the *sign* of the result. | |
228 * | |
229 */ | |
230 | |
231 | |
232 /* | |
233 * Constants: | |
234 * The hexadecimal values are the intended ones for the following | |
235 * constants. The decimal values may be used, provided that the | |
236 * compiler will convert from decimal to binary accurately enough | |
237 * to produce the hexadecimal values shown. | |
238 */ | |
239 | |
240 | |
241 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ | |
242 | |
243 static const double PIo2[] = { | |
244 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ | |
245 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ | |
246 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ | |
247 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ | |
248 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ | |
249 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ | |
250 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ | |
251 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ | |
252 }; | |
253 | |
254 static const double | |
255 zeroB = 0.0, | |
256 one = 1.0, | |
257 two24B = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ | |
258 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ | |
259 | |
1485 | 260 static SAFEBUF int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2) { |
0 | 261 int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; |
262 double z,fw,f[20],fq[20],q[20]; | |
263 | |
264 /* initialize jk*/ | |
265 jk = init_jk[prec]; | |
266 jp = jk; | |
267 | |
268 /* determine jx,jv,q0, note that 3>q0 */ | |
269 jx = nx-1; | |
270 jv = (e0-3)/24; if(jv<0) jv=0; | |
271 q0 = e0-24*(jv+1); | |
272 | |
273 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ | |
274 j = jv-jx; m = jx+jk; | |
275 for(i=0;i<=m;i++,j++) f[i] = (j<0)? zeroB : (double) ipio2[j]; | |
276 | |
277 /* compute q[0],q[1],...q[jk] */ | |
278 for (i=0;i<=jk;i++) { | |
279 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; | |
280 } | |
281 | |
282 jz = jk; | |
283 recompute: | |
284 /* distill q[] into iq[] reversingly */ | |
285 for(i=0,j=jz,z=q[jz];j>0;i++,j--) { | |
286 fw = (double)((int)(twon24* z)); | |
287 iq[i] = (int)(z-two24B*fw); | |
288 z = q[j-1]+fw; | |
289 } | |
290 | |
291 /* compute n */ | |
292 z = scalbnA(z,q0); /* actual value of z */ | |
293 z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ | |
294 n = (int) z; | |
295 z -= (double)n; | |
296 ih = 0; | |
297 if(q0>0) { /* need iq[jz-1] to determine n */ | |
298 i = (iq[jz-1]>>(24-q0)); n += i; | |
299 iq[jz-1] -= i<<(24-q0); | |
300 ih = iq[jz-1]>>(23-q0); | |
301 } | |
302 else if(q0==0) ih = iq[jz-1]>>23; | |
303 else if(z>=0.5) ih=2; | |
304 | |
305 if(ih>0) { /* q > 0.5 */ | |
306 n += 1; carry = 0; | |
307 for(i=0;i<jz ;i++) { /* compute 1-q */ | |
308 j = iq[i]; | |
309 if(carry==0) { | |
310 if(j!=0) { | |
311 carry = 1; iq[i] = 0x1000000- j; | |
312 } | |
313 } else iq[i] = 0xffffff - j; | |
314 } | |
315 if(q0>0) { /* rare case: chance is 1 in 12 */ | |
316 switch(q0) { | |
317 case 1: | |
318 iq[jz-1] &= 0x7fffff; break; | |
319 case 2: | |
320 iq[jz-1] &= 0x3fffff; break; | |
321 } | |
322 } | |
323 if(ih==2) { | |
324 z = one - z; | |
325 if(carry!=0) z -= scalbnA(one,q0); | |
326 } | |
327 } | |
328 | |
329 /* check if recomputation is needed */ | |
330 if(z==zeroB) { | |
331 j = 0; | |
332 for (i=jz-1;i>=jk;i--) j |= iq[i]; | |
333 if(j==0) { /* need recomputation */ | |
334 for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ | |
335 | |
336 for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ | |
337 f[jx+i] = (double) ipio2[jv+i]; | |
338 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; | |
339 q[i] = fw; | |
340 } | |
341 jz += k; | |
342 goto recompute; | |
343 } | |
344 } | |
345 | |
346 /* chop off zero terms */ | |
347 if(z==0.0) { | |
348 jz -= 1; q0 -= 24; | |
349 while(iq[jz]==0) { jz--; q0-=24;} | |
350 } else { /* break z into 24-bit if neccessary */ | |
351 z = scalbnA(z,-q0); | |
352 if(z>=two24B) { | |
353 fw = (double)((int)(twon24*z)); | |
354 iq[jz] = (int)(z-two24B*fw); | |
355 jz += 1; q0 += 24; | |
356 iq[jz] = (int) fw; | |
357 } else iq[jz] = (int) z ; | |
358 } | |
359 | |
360 /* convert integer "bit" chunk to floating-point value */ | |
361 fw = scalbnA(one,q0); | |
362 for(i=jz;i>=0;i--) { | |
363 q[i] = fw*(double)iq[i]; fw*=twon24; | |
364 } | |
365 | |
366 /* compute PIo2[0,...,jp]*q[jz,...,0] */ | |
367 for(i=jz;i>=0;i--) { | |
368 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; | |
369 fq[jz-i] = fw; | |
370 } | |
371 | |
372 /* compress fq[] into y[] */ | |
373 switch(prec) { | |
374 case 0: | |
375 fw = 0.0; | |
376 for (i=jz;i>=0;i--) fw += fq[i]; | |
377 y[0] = (ih==0)? fw: -fw; | |
378 break; | |
379 case 1: | |
380 case 2: | |
381 fw = 0.0; | |
382 for (i=jz;i>=0;i--) fw += fq[i]; | |
383 y[0] = (ih==0)? fw: -fw; | |
384 fw = fq[0]-fw; | |
385 for (i=1;i<=jz;i++) fw += fq[i]; | |
386 y[1] = (ih==0)? fw: -fw; | |
387 break; | |
388 case 3: /* painful */ | |
389 for (i=jz;i>0;i--) { | |
390 fw = fq[i-1]+fq[i]; | |
391 fq[i] += fq[i-1]-fw; | |
392 fq[i-1] = fw; | |
393 } | |
394 for (i=jz;i>1;i--) { | |
395 fw = fq[i-1]+fq[i]; | |
396 fq[i] += fq[i-1]-fw; | |
397 fq[i-1] = fw; | |
398 } | |
399 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; | |
400 if(ih==0) { | |
401 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; | |
402 } else { | |
403 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; | |
404 } | |
405 } | |
406 return n&7; | |
407 } | |
408 | |
409 | |
410 /* | |
411 * ==================================================== | |
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412 * Copyright (c) 1993 Oracle and/or its affilates. All rights reserved. |
0 | 413 * |
414 * Developed at SunPro, a Sun Microsystems, Inc. business. | |
415 * Permission to use, copy, modify, and distribute this | |
416 * software is freely granted, provided that this notice | |
417 * is preserved. | |
418 * ==================================================== | |
419 * | |
420 */ | |
421 | |
422 /* __ieee754_rem_pio2(x,y) | |
423 * | |
424 * return the remainder of x rem pi/2 in y[0]+y[1] | |
425 * use __kernel_rem_pio2() | |
426 */ | |
427 | |
428 /* | |
429 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi | |
430 */ | |
431 static const int two_over_pi[] = { | |
432 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, | |
433 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, | |
434 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, | |
435 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, | |
436 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, | |
437 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF, | |
438 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, | |
439 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, | |
440 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, | |
441 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, | |
442 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B, | |
443 }; | |
444 | |
445 static const int npio2_hw[] = { | |
446 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, | |
447 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C, | |
448 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A, | |
449 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C, | |
450 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB, | |
451 0x404858EB, 0x404921FB, | |
452 }; | |
453 | |
454 /* | |
455 * invpio2: 53 bits of 2/pi | |
456 * pio2_1: first 33 bit of pi/2 | |
457 * pio2_1t: pi/2 - pio2_1 | |
458 * pio2_2: second 33 bit of pi/2 | |
459 * pio2_2t: pi/2 - (pio2_1+pio2_2) | |
460 * pio2_3: third 33 bit of pi/2 | |
461 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) | |
462 */ | |
463 | |
464 static const double | |
465 zeroA = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ | |
466 half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ | |
467 two24A = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ | |
468 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ | |
469 pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */ | |
470 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */ | |
471 pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */ | |
472 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */ | |
473 pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */ | |
474 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */ | |
475 | |
1485 | 476 static SAFEBUF int __ieee754_rem_pio2(double x, double *y) { |
0 | 477 double z,w,t,r,fn; |
478 double tx[3]; | |
479 int e0,i,j,nx,n,ix,hx,i0; | |
480 | |
481 i0 = ((*(int*)&two24A)>>30)^1; /* high word index */ | |
482 hx = *(i0+(int*)&x); /* high word of x */ | |
483 ix = hx&0x7fffffff; | |
484 if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */ | |
485 {y[0] = x; y[1] = 0; return 0;} | |
486 if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */ | |
487 if(hx>0) { | |
488 z = x - pio2_1; | |
489 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ | |
490 y[0] = z - pio2_1t; | |
491 y[1] = (z-y[0])-pio2_1t; | |
492 } else { /* near pi/2, use 33+33+53 bit pi */ | |
493 z -= pio2_2; | |
494 y[0] = z - pio2_2t; | |
495 y[1] = (z-y[0])-pio2_2t; | |
496 } | |
497 return 1; | |
498 } else { /* negative x */ | |
499 z = x + pio2_1; | |
500 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ | |
501 y[0] = z + pio2_1t; | |
502 y[1] = (z-y[0])+pio2_1t; | |
503 } else { /* near pi/2, use 33+33+53 bit pi */ | |
504 z += pio2_2; | |
505 y[0] = z + pio2_2t; | |
506 y[1] = (z-y[0])+pio2_2t; | |
507 } | |
508 return -1; | |
509 } | |
510 } | |
511 if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */ | |
512 t = fabsd(x); | |
513 n = (int) (t*invpio2+half); | |
514 fn = (double)n; | |
515 r = t-fn*pio2_1; | |
516 w = fn*pio2_1t; /* 1st round good to 85 bit */ | |
517 if(n<32&&ix!=npio2_hw[n-1]) { | |
518 y[0] = r-w; /* quick check no cancellation */ | |
519 } else { | |
520 j = ix>>20; | |
521 y[0] = r-w; | |
522 i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff); | |
523 if(i>16) { /* 2nd iteration needed, good to 118 */ | |
524 t = r; | |
525 w = fn*pio2_2; | |
526 r = t-w; | |
527 w = fn*pio2_2t-((t-r)-w); | |
528 y[0] = r-w; | |
529 i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff); | |
530 if(i>49) { /* 3rd iteration need, 151 bits acc */ | |
531 t = r; /* will cover all possible cases */ | |
532 w = fn*pio2_3; | |
533 r = t-w; | |
534 w = fn*pio2_3t-((t-r)-w); | |
535 y[0] = r-w; | |
536 } | |
537 } | |
538 } | |
539 y[1] = (r-y[0])-w; | |
540 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} | |
541 else return n; | |
542 } | |
543 /* | |
544 * all other (large) arguments | |
545 */ | |
546 if(ix>=0x7ff00000) { /* x is inf or NaN */ | |
547 y[0]=y[1]=x-x; return 0; | |
548 } | |
549 /* set z = scalbn(|x|,ilogb(x)-23) */ | |
550 *(1-i0+(int*)&z) = *(1-i0+(int*)&x); | |
551 e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */ | |
552 *(i0+(int*)&z) = ix - (e0<<20); | |
553 for(i=0;i<2;i++) { | |
554 tx[i] = (double)((int)(z)); | |
555 z = (z-tx[i])*two24A; | |
556 } | |
557 tx[2] = z; | |
558 nx = 3; | |
559 while(tx[nx-1]==zeroA) nx--; /* skip zero term */ | |
560 n = __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi); | |
561 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} | |
562 return n; | |
563 } | |
564 | |
565 | |
566 /* __kernel_sin( x, y, iy) | |
567 * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 | |
568 * Input x is assumed to be bounded by ~pi/4 in magnitude. | |
569 * Input y is the tail of x. | |
570 * Input iy indicates whether y is 0. (if iy=0, y assume to be 0). | |
571 * | |
572 * Algorithm | |
573 * 1. Since sin(-x) = -sin(x), we need only to consider positive x. | |
574 * 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0. | |
575 * 3. sin(x) is approximated by a polynomial of degree 13 on | |
576 * [0,pi/4] | |
577 * 3 13 | |
578 * sin(x) ~ x + S1*x + ... + S6*x | |
579 * where | |
580 * | |
581 * |sin(x) 2 4 6 8 10 12 | -58 | |
582 * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 | |
583 * | x | | |
584 * | |
585 * 4. sin(x+y) = sin(x) + sin'(x')*y | |
586 * ~ sin(x) + (1-x*x/2)*y | |
587 * For better accuracy, let | |
588 * 3 2 2 2 2 | |
589 * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) | |
590 * then 3 2 | |
591 * sin(x) = x + (S1*x + (x *(r-y/2)+y)) | |
592 */ | |
593 | |
594 static const double | |
595 S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */ | |
596 S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */ | |
597 S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */ | |
598 S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */ | |
599 S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */ | |
600 S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */ | |
601 | |
602 static double __kernel_sin(double x, double y, int iy) | |
603 { | |
604 double z,r,v; | |
605 int ix; | |
606 ix = __HI(x)&0x7fffffff; /* high word of x */ | |
607 if(ix<0x3e400000) /* |x| < 2**-27 */ | |
608 {if((int)x==0) return x;} /* generate inexact */ | |
609 z = x*x; | |
610 v = z*x; | |
611 r = S2+z*(S3+z*(S4+z*(S5+z*S6))); | |
612 if(iy==0) return x+v*(S1+z*r); | |
613 else return x-((z*(half*y-v*r)-y)-v*S1); | |
614 } | |
615 | |
616 /* | |
617 * __kernel_cos( x, y ) | |
618 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 | |
619 * Input x is assumed to be bounded by ~pi/4 in magnitude. | |
620 * Input y is the tail of x. | |
621 * | |
622 * Algorithm | |
623 * 1. Since cos(-x) = cos(x), we need only to consider positive x. | |
624 * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. | |
625 * 3. cos(x) is approximated by a polynomial of degree 14 on | |
626 * [0,pi/4] | |
627 * 4 14 | |
628 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x | |
629 * where the remez error is | |
630 * | |
631 * | 2 4 6 8 10 12 14 | -58 | |
632 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 | |
633 * | | | |
634 * | |
635 * 4 6 8 10 12 14 | |
636 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then | |
637 * cos(x) = 1 - x*x/2 + r | |
638 * since cos(x+y) ~ cos(x) - sin(x)*y | |
639 * ~ cos(x) - x*y, | |
640 * a correction term is necessary in cos(x) and hence | |
641 * cos(x+y) = 1 - (x*x/2 - (r - x*y)) | |
642 * For better accuracy when x > 0.3, let qx = |x|/4 with | |
643 * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. | |
644 * Then | |
645 * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). | |
646 * Note that 1-qx and (x*x/2-qx) is EXACT here, and the | |
647 * magnitude of the latter is at least a quarter of x*x/2, | |
648 * thus, reducing the rounding error in the subtraction. | |
649 */ | |
650 | |
651 static const double | |
652 C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ | |
653 C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ | |
654 C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ | |
655 C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ | |
656 C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ | |
657 C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ | |
658 | |
659 static double __kernel_cos(double x, double y) | |
660 { | |
661 double a,hz,z,r,qx; | |
662 int ix; | |
663 ix = __HI(x)&0x7fffffff; /* ix = |x|'s high word*/ | |
664 if(ix<0x3e400000) { /* if x < 2**27 */ | |
665 if(((int)x)==0) return one; /* generate inexact */ | |
666 } | |
667 z = x*x; | |
668 r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6))))); | |
669 if(ix < 0x3FD33333) /* if |x| < 0.3 */ | |
670 return one - (0.5*z - (z*r - x*y)); | |
671 else { | |
672 if(ix > 0x3fe90000) { /* x > 0.78125 */ | |
673 qx = 0.28125; | |
674 } else { | |
675 __HI(qx) = ix-0x00200000; /* x/4 */ | |
676 __LO(qx) = 0; | |
677 } | |
678 hz = 0.5*z-qx; | |
679 a = one-qx; | |
680 return a - (hz - (z*r-x*y)); | |
681 } | |
682 } | |
683 | |
684 /* __kernel_tan( x, y, k ) | |
685 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 | |
686 * Input x is assumed to be bounded by ~pi/4 in magnitude. | |
687 * Input y is the tail of x. | |
688 * Input k indicates whether tan (if k=1) or | |
689 * -1/tan (if k= -1) is returned. | |
690 * | |
691 * Algorithm | |
692 * 1. Since tan(-x) = -tan(x), we need only to consider positive x. | |
693 * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. | |
694 * 3. tan(x) is approximated by a odd polynomial of degree 27 on | |
695 * [0,0.67434] | |
696 * 3 27 | |
697 * tan(x) ~ x + T1*x + ... + T13*x | |
698 * where | |
699 * | |
700 * |tan(x) 2 4 26 | -59.2 | |
701 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 | |
702 * | x | | |
703 * | |
704 * Note: tan(x+y) = tan(x) + tan'(x)*y | |
705 * ~ tan(x) + (1+x*x)*y | |
706 * Therefore, for better accuracy in computing tan(x+y), let | |
707 * 3 2 2 2 2 | |
708 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) | |
709 * then | |
710 * 3 2 | |
711 * tan(x+y) = x + (T1*x + (x *(r+y)+y)) | |
712 * | |
713 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then | |
714 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) | |
715 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) | |
716 */ | |
717 | |
718 static const double | |
719 pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ | |
720 pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */ | |
721 T[] = { | |
722 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */ | |
723 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */ | |
724 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */ | |
725 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */ | |
726 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */ | |
727 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */ | |
728 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */ | |
729 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */ | |
730 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */ | |
731 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */ | |
732 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */ | |
733 -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */ | |
734 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */ | |
735 }; | |
736 | |
737 static double __kernel_tan(double x, double y, int iy) | |
738 { | |
739 double z,r,v,w,s; | |
740 int ix,hx; | |
741 hx = __HI(x); /* high word of x */ | |
742 ix = hx&0x7fffffff; /* high word of |x| */ | |
743 if(ix<0x3e300000) { /* x < 2**-28 */ | |
744 if((int)x==0) { /* generate inexact */ | |
745 if (((ix | __LO(x)) | (iy + 1)) == 0) | |
746 return one / fabsd(x); | |
747 else { | |
748 if (iy == 1) | |
749 return x; | |
750 else { /* compute -1 / (x+y) carefully */ | |
751 double a, t; | |
752 | |
753 z = w = x + y; | |
754 __LO(z) = 0; | |
755 v = y - (z - x); | |
756 t = a = -one / w; | |
757 __LO(t) = 0; | |
758 s = one + t * z; | |
759 return t + a * (s + t * v); | |
760 } | |
761 } | |
762 } | |
763 } | |
764 if(ix>=0x3FE59428) { /* |x|>=0.6744 */ | |
765 if(hx<0) {x = -x; y = -y;} | |
766 z = pio4-x; | |
767 w = pio4lo-y; | |
768 x = z+w; y = 0.0; | |
769 } | |
770 z = x*x; | |
771 w = z*z; | |
772 /* Break x^5*(T[1]+x^2*T[2]+...) into | |
773 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + | |
774 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) | |
775 */ | |
776 r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11])))); | |
777 v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12]))))); | |
778 s = z*x; | |
779 r = y + z*(s*(r+v)+y); | |
780 r += T[0]*s; | |
781 w = x+r; | |
782 if(ix>=0x3FE59428) { | |
783 v = (double)iy; | |
784 return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r))); | |
785 } | |
786 if(iy==1) return w; | |
787 else { /* if allow error up to 2 ulp, | |
788 simply return -1.0/(x+r) here */ | |
789 /* compute -1.0/(x+r) accurately */ | |
790 double a,t; | |
791 z = w; | |
792 __LO(z) = 0; | |
793 v = r-(z - x); /* z+v = r+x */ | |
794 t = a = -1.0/w; /* a = -1.0/w */ | |
795 __LO(t) = 0; | |
796 s = 1.0+t*z; | |
797 return t+a*(s+t*v); | |
798 } | |
799 } | |
800 | |
801 | |
802 //---------------------------------------------------------------------- | |
803 // | |
804 // Routines for new sin/cos implementation | |
805 // | |
806 //---------------------------------------------------------------------- | |
807 | |
808 /* sin(x) | |
809 * Return sine function of x. | |
810 * | |
811 * kernel function: | |
812 * __kernel_sin ... sine function on [-pi/4,pi/4] | |
813 * __kernel_cos ... cose function on [-pi/4,pi/4] | |
814 * __ieee754_rem_pio2 ... argument reduction routine | |
815 * | |
816 * Method. | |
817 * Let S,C and T denote the sin, cos and tan respectively on | |
818 * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 | |
819 * in [-pi/4 , +pi/4], and let n = k mod 4. | |
820 * We have | |
821 * | |
822 * n sin(x) cos(x) tan(x) | |
823 * ---------------------------------------------------------- | |
824 * 0 S C T | |
825 * 1 C -S -1/T | |
826 * 2 -S -C T | |
827 * 3 -C S -1/T | |
828 * ---------------------------------------------------------- | |
829 * | |
830 * Special cases: | |
831 * Let trig be any of sin, cos, or tan. | |
832 * trig(+-INF) is NaN, with signals; | |
833 * trig(NaN) is that NaN; | |
834 * | |
835 * Accuracy: | |
836 * TRIG(x) returns trig(x) nearly rounded | |
837 */ | |
838 | |
839 JRT_LEAF(jdouble, SharedRuntime::dsin(jdouble x)) | |
840 double y[2],z=0.0; | |
841 int n, ix; | |
842 | |
843 /* High word of x. */ | |
844 ix = __HI(x); | |
845 | |
846 /* |x| ~< pi/4 */ | |
847 ix &= 0x7fffffff; | |
848 if(ix <= 0x3fe921fb) return __kernel_sin(x,z,0); | |
849 | |
850 /* sin(Inf or NaN) is NaN */ | |
851 else if (ix>=0x7ff00000) return x-x; | |
852 | |
853 /* argument reduction needed */ | |
854 else { | |
855 n = __ieee754_rem_pio2(x,y); | |
856 switch(n&3) { | |
857 case 0: return __kernel_sin(y[0],y[1],1); | |
858 case 1: return __kernel_cos(y[0],y[1]); | |
859 case 2: return -__kernel_sin(y[0],y[1],1); | |
860 default: | |
861 return -__kernel_cos(y[0],y[1]); | |
862 } | |
863 } | |
864 JRT_END | |
865 | |
866 /* cos(x) | |
867 * Return cosine function of x. | |
868 * | |
869 * kernel function: | |
870 * __kernel_sin ... sine function on [-pi/4,pi/4] | |
871 * __kernel_cos ... cosine function on [-pi/4,pi/4] | |
872 * __ieee754_rem_pio2 ... argument reduction routine | |
873 * | |
874 * Method. | |
875 * Let S,C and T denote the sin, cos and tan respectively on | |
876 * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 | |
877 * in [-pi/4 , +pi/4], and let n = k mod 4. | |
878 * We have | |
879 * | |
880 * n sin(x) cos(x) tan(x) | |
881 * ---------------------------------------------------------- | |
882 * 0 S C T | |
883 * 1 C -S -1/T | |
884 * 2 -S -C T | |
885 * 3 -C S -1/T | |
886 * ---------------------------------------------------------- | |
887 * | |
888 * Special cases: | |
889 * Let trig be any of sin, cos, or tan. | |
890 * trig(+-INF) is NaN, with signals; | |
891 * trig(NaN) is that NaN; | |
892 * | |
893 * Accuracy: | |
894 * TRIG(x) returns trig(x) nearly rounded | |
895 */ | |
896 | |
897 JRT_LEAF(jdouble, SharedRuntime::dcos(jdouble x)) | |
898 double y[2],z=0.0; | |
899 int n, ix; | |
900 | |
901 /* High word of x. */ | |
902 ix = __HI(x); | |
903 | |
904 /* |x| ~< pi/4 */ | |
905 ix &= 0x7fffffff; | |
906 if(ix <= 0x3fe921fb) return __kernel_cos(x,z); | |
907 | |
908 /* cos(Inf or NaN) is NaN */ | |
909 else if (ix>=0x7ff00000) return x-x; | |
910 | |
911 /* argument reduction needed */ | |
912 else { | |
913 n = __ieee754_rem_pio2(x,y); | |
914 switch(n&3) { | |
915 case 0: return __kernel_cos(y[0],y[1]); | |
916 case 1: return -__kernel_sin(y[0],y[1],1); | |
917 case 2: return -__kernel_cos(y[0],y[1]); | |
918 default: | |
919 return __kernel_sin(y[0],y[1],1); | |
920 } | |
921 } | |
922 JRT_END | |
923 | |
924 /* tan(x) | |
925 * Return tangent function of x. | |
926 * | |
927 * kernel function: | |
928 * __kernel_tan ... tangent function on [-pi/4,pi/4] | |
929 * __ieee754_rem_pio2 ... argument reduction routine | |
930 * | |
931 * Method. | |
932 * Let S,C and T denote the sin, cos and tan respectively on | |
933 * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 | |
934 * in [-pi/4 , +pi/4], and let n = k mod 4. | |
935 * We have | |
936 * | |
937 * n sin(x) cos(x) tan(x) | |
938 * ---------------------------------------------------------- | |
939 * 0 S C T | |
940 * 1 C -S -1/T | |
941 * 2 -S -C T | |
942 * 3 -C S -1/T | |
943 * ---------------------------------------------------------- | |
944 * | |
945 * Special cases: | |
946 * Let trig be any of sin, cos, or tan. | |
947 * trig(+-INF) is NaN, with signals; | |
948 * trig(NaN) is that NaN; | |
949 * | |
950 * Accuracy: | |
951 * TRIG(x) returns trig(x) nearly rounded | |
952 */ | |
953 | |
954 JRT_LEAF(jdouble, SharedRuntime::dtan(jdouble x)) | |
955 double y[2],z=0.0; | |
956 int n, ix; | |
957 | |
958 /* High word of x. */ | |
959 ix = __HI(x); | |
960 | |
961 /* |x| ~< pi/4 */ | |
962 ix &= 0x7fffffff; | |
963 if(ix <= 0x3fe921fb) return __kernel_tan(x,z,1); | |
964 | |
965 /* tan(Inf or NaN) is NaN */ | |
966 else if (ix>=0x7ff00000) return x-x; /* NaN */ | |
967 | |
968 /* argument reduction needed */ | |
969 else { | |
970 n = __ieee754_rem_pio2(x,y); | |
971 return __kernel_tan(y[0],y[1],1-((n&1)<<1)); /* 1 -- n even | |
972 -1 -- n odd */ | |
973 } | |
974 JRT_END | |
975 | |
976 | |
977 #ifdef WIN32 | |
978 # pragma optimize ( "", on ) | |
979 #endif |