1 /*
2 * Copyright (c) 2005, 2017, Oracle and/or its affiliates. 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 Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
20 * or visit www.oracle.com if you need additional information or have any
21 * questions.
22 *
23 */
24
25 #include "precompiled.hpp"
26 #include "jni.h"
27 #include "runtime/interfaceSupport.inline.hpp"
28 #include "runtime/sharedRuntime.hpp"
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 warning( disable: 4748 ) // /GS can not protect parameters and local variables from local buffer overrun because optimizations are disabled in function
44 # pragma optimize ( "", off )
45 #endif
46
47 #include "runtime/sharedRuntimeMath.hpp"
48
49 /* __ieee754_log(x)
50 * Return the logarithm of x
51 *
52 * Method :
53 * 1. Argument Reduction: find k and f such that
54 * x = 2^k * (1+f),
55 * where sqrt(2)/2 < 1+f < sqrt(2) .
56 *
57 * 2. Approximation of log(1+f).
58 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
59 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
60 * = 2s + s*R
61 * We use a special Reme algorithm on [0,0.1716] to generate
62 * a polynomial of degree 14 to approximate R The maximum error
63 * of this polynomial approximation is bounded by 2**-58.45. In
64 * other words,
65 * 2 4 6 8 10 12 14
66 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
67 * (the values of Lg1 to Lg7 are listed in the program)
68 * and
69 * | 2 14 | -58.45
70 * | Lg1*s +...+Lg7*s - R(z) | <= 2
71 * | |
72 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
73 * In order to guarantee error in log below 1ulp, we compute log
74 * by
75 * log(1+f) = f - s*(f - R) (if f is not too large)
76 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
77 *
78 * 3. Finally, log(x) = k*ln2 + log(1+f).
79 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
80 * Here ln2 is split into two floating point number:
81 * ln2_hi + ln2_lo,
82 * where n*ln2_hi is always exact for |n| < 2000.
83 *
84 * Special cases:
85 * log(x) is NaN with signal if x < 0 (including -INF) ;
86 * log(+INF) is +INF; log(0) is -INF with signal;
87 * log(NaN) is that NaN with no signal.
88 *
89 * Accuracy:
90 * according to an error analysis, the error is always less than
91 * 1 ulp (unit in the last place).
92 *
93 * Constants:
94 * The hexadecimal values are the intended ones for the following
95 * constants. The decimal values may be used, provided that the
96 * compiler will convert from decimal to binary accurately enough
97 * to produce the hexadecimal values shown.
98 */
99
100 static const double
101 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
102 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
103 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
104 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
105 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
106 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
107 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
108 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
109 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
110
111 static double zero = 0.0;
112
113 static double __ieee754_log(double x) {
114 double hfsq,f,s,z,R,w,t1,t2,dk;
115 int k,hx,i,j;
116 unsigned lx;
117
118 hx = high(x); /* high word of x */
119 lx = low(x); /* low word of x */
120
121 k=0;
122 if (hx < 0x00100000) { /* x < 2**-1022 */
123 if (((hx&0x7fffffff)|lx)==0)
124 return -two54/zero; /* log(+-0)=-inf */
125 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
126 k -= 54; x *= two54; /* subnormal number, scale up x */
127 hx = high(x); /* high word of x */
128 }
129 if (hx >= 0x7ff00000) return x+x;
130 k += (hx>>20)-1023;
131 hx &= 0x000fffff;
132 i = (hx+0x95f64)&0x100000;
133 set_high(&x, hx|(i^0x3ff00000)); /* normalize x or x/2 */
134 k += (i>>20);
135 f = x-1.0;
136 if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
137 if(f==zero) {
138 if (k==0) return zero;
139 else {dk=(double)k; return dk*ln2_hi+dk*ln2_lo;}
140 }
141 R = f*f*(0.5-0.33333333333333333*f);
142 if(k==0) return f-R; else {dk=(double)k;
143 return dk*ln2_hi-((R-dk*ln2_lo)-f);}
144 }
145 s = f/(2.0+f);
146 dk = (double)k;
147 z = s*s;
148 i = hx-0x6147a;
149 w = z*z;
150 j = 0x6b851-hx;
151 t1= w*(Lg2+w*(Lg4+w*Lg6));
152 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
153 i |= j;
154 R = t2+t1;
155 if(i>0) {
156 hfsq=0.5*f*f;
157 if(k==0) return f-(hfsq-s*(hfsq+R)); else
158 return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
159 } else {
160 if(k==0) return f-s*(f-R); else
161 return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
162 }
163 }
164
165 JRT_LEAF(jdouble, SharedRuntime::dlog(jdouble x))
166 return __ieee754_log(x);
167 JRT_END
168
169 /* __ieee754_log10(x)
170 * Return the base 10 logarithm of x
171 *
172 * Method :
173 * Let log10_2hi = leading 40 bits of log10(2) and
174 * log10_2lo = log10(2) - log10_2hi,
175 * ivln10 = 1/log(10) rounded.
176 * Then
177 * n = ilogb(x),
178 * if(n<0) n = n+1;
179 * x = scalbn(x,-n);
180 * log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))
181 *
182 * Note 1:
183 * To guarantee log10(10**n)=n, where 10**n is normal, the rounding
184 * mode must set to Round-to-Nearest.
185 * Note 2:
186 * [1/log(10)] rounded to 53 bits has error .198 ulps;
187 * log10 is monotonic at all binary break points.
188 *
189 * Special cases:
190 * log10(x) is NaN with signal if x < 0;
191 * log10(+INF) is +INF with no signal; log10(0) is -INF with signal;
192 * log10(NaN) is that NaN with no signal;
193 * log10(10**N) = N for N=0,1,...,22.
194 *
195 * Constants:
196 * The hexadecimal values are the intended ones for the following constants.
197 * The decimal values may be used, provided that the compiler will convert
198 * from decimal to binary accurately enough to produce the hexadecimal values
199 * shown.
200 */
201
202 static const double
203 ivln10 = 4.34294481903251816668e-01, /* 0x3FDBCB7B, 0x1526E50E */
204 log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
205 log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */
206
207 static double __ieee754_log10(double x) {
208 double y,z;
209 int i,k,hx;
210 unsigned lx;
211
212 hx = high(x); /* high word of x */
213 lx = low(x); /* low word of x */
214
215 k=0;
216 if (hx < 0x00100000) { /* x < 2**-1022 */
217 if (((hx&0x7fffffff)|lx)==0)
218 return -two54/zero; /* log(+-0)=-inf */
219 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
220 k -= 54; x *= two54; /* subnormal number, scale up x */
221 hx = high(x); /* high word of x */
222 }
223 if (hx >= 0x7ff00000) return x+x;
224 k += (hx>>20)-1023;
225 i = ((unsigned)k&0x80000000)>>31;
226 hx = (hx&0x000fffff)|((0x3ff-i)<<20);
227 y = (double)(k+i);
228 set_high(&x, hx);
229 z = y*log10_2lo + ivln10*__ieee754_log(x);
230 return z+y*log10_2hi;
231 }
232
233 JRT_LEAF(jdouble, SharedRuntime::dlog10(jdouble x))
234 return __ieee754_log10(x);
235 JRT_END
236
237
238 /* __ieee754_exp(x)
239 * Returns the exponential of x.
240 *
241 * Method
242 * 1. Argument reduction:
243 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
244 * Given x, find r and integer k such that
245 *
246 * x = k*ln2 + r, |r| <= 0.5*ln2.
247 *
248 * Here r will be represented as r = hi-lo for better
249 * accuracy.
250 *
251 * 2. Approximation of exp(r) by a special rational function on
252 * the interval [0,0.34658]:
253 * Write
254 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
255 * We use a special Reme algorithm on [0,0.34658] to generate
256 * a polynomial of degree 5 to approximate R. The maximum error
257 * of this polynomial approximation is bounded by 2**-59. In
258 * other words,
259 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
260 * (where z=r*r, and the values of P1 to P5 are listed below)
261 * and
262 * | 5 | -59
263 * | 2.0+P1*z+...+P5*z - R(z) | <= 2
264 * | |
265 * The computation of exp(r) thus becomes
266 * 2*r
267 * exp(r) = 1 + -------
268 * R - r
269 * r*R1(r)
270 * = 1 + r + ----------- (for better accuracy)
271 * 2 - R1(r)
272 * where
273 * 2 4 10
274 * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
275 *
276 * 3. Scale back to obtain exp(x):
277 * From step 1, we have
278 * exp(x) = 2^k * exp(r)
279 *
280 * Special cases:
281 * exp(INF) is INF, exp(NaN) is NaN;
282 * exp(-INF) is 0, and
283 * for finite argument, only exp(0)=1 is exact.
284 *
285 * Accuracy:
286 * according to an error analysis, the error is always less than
287 * 1 ulp (unit in the last place).
288 *
289 * Misc. info.
290 * For IEEE double
291 * if x > 7.09782712893383973096e+02 then exp(x) overflow
292 * if x < -7.45133219101941108420e+02 then exp(x) underflow
293 *
294 * Constants:
295 * The hexadecimal values are the intended ones for the following
296 * constants. The decimal values may be used, provided that the
297 * compiler will convert from decimal to binary accurately enough
298 * to produce the hexadecimal values shown.
299 */
300
301 static const double
302 one = 1.0,
303 halF[2] = {0.5,-0.5,},
304 twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
305 o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
306 u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
307 ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
308 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
309 ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
310 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
311 invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
312 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
313 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
314 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
315 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
316 P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
317
318 static double __ieee754_exp(double x) {
319 double y,hi=0,lo=0,c,t;
320 int k=0,xsb;
321 unsigned hx;
322
323 hx = high(x); /* high word of x */
324 xsb = (hx>>31)&1; /* sign bit of x */
325 hx &= 0x7fffffff; /* high word of |x| */
326
327 /* filter out non-finite argument */
328 if(hx >= 0x40862E42) { /* if |x|>=709.78... */
329 if(hx>=0x7ff00000) {
330 if(((hx&0xfffff)|low(x))!=0)
331 return x+x; /* NaN */
332 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
333 }
334 if(x > o_threshold) return hugeX*hugeX; /* overflow */
335 if(x < u_threshold) return twom1000*twom1000; /* underflow */
336 }
337
338 /* argument reduction */
339 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
340 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
341 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
342 } else {
343 k = (int)(invln2*x+halF[xsb]);
344 t = k;
345 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
346 lo = t*ln2LO[0];
347 }
348 x = hi - lo;
349 }
350 else if(hx < 0x3e300000) { /* when |x|<2**-28 */
351 if(hugeX+x>one) return one+x;/* trigger inexact */
352 }
353 else k = 0;
354
355 /* x is now in primary range */
356 t = x*x;
357 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
358 if(k==0) return one-((x*c)/(c-2.0)-x);
359 else y = one-((lo-(x*c)/(2.0-c))-hi);
360 if(k >= -1021) {
361 set_high(&y, high(y) + (k<<20)); /* add k to y's exponent */
362 return y;
363 } else {
364 set_high(&y, high(y) + ((k+1000)<<20)); /* add k to y's exponent */
365 return y*twom1000;
366 }
367 }
368
369 JRT_LEAF(jdouble, SharedRuntime::dexp(jdouble x))
370 return __ieee754_exp(x);
371 JRT_END
372
373 /* __ieee754_pow(x,y) return x**y
374 *
375 * n
376 * Method: Let x = 2 * (1+f)
377 * 1. Compute and return log2(x) in two pieces:
378 * log2(x) = w1 + w2,
379 * where w1 has 53-24 = 29 bit trailing zeros.
380 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
381 * arithmetic, where |y'|<=0.5.
382 * 3. Return x**y = 2**n*exp(y'*log2)
383 *
384 * Special cases:
385 * 1. (anything) ** 0 is 1
386 * 2. (anything) ** 1 is itself
387 * 3. (anything) ** NAN is NAN
388 * 4. NAN ** (anything except 0) is NAN
389 * 5. +-(|x| > 1) ** +INF is +INF
390 * 6. +-(|x| > 1) ** -INF is +0
391 * 7. +-(|x| < 1) ** +INF is +0
392 * 8. +-(|x| < 1) ** -INF is +INF
393 * 9. +-1 ** +-INF is NAN
394 * 10. +0 ** (+anything except 0, NAN) is +0
395 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
396 * 12. +0 ** (-anything except 0, NAN) is +INF
397 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
398 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
399 * 15. +INF ** (+anything except 0,NAN) is +INF
400 * 16. +INF ** (-anything except 0,NAN) is +0
401 * 17. -INF ** (anything) = -0 ** (-anything)
402 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
403 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
404 *
405 * Accuracy:
406 * pow(x,y) returns x**y nearly rounded. In particular
407 * pow(integer,integer)
408 * always returns the correct integer provided it is
409 * representable.
410 *
411 * Constants :
412 * The hexadecimal values are the intended ones for the following
413 * constants. The decimal values may be used, provided that the
414 * compiler will convert from decimal to binary accurately enough
415 * to produce the hexadecimal values shown.
416 */
417
418 static const double
419 bp[] = {1.0, 1.5,},
420 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
421 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
422 zeroX = 0.0,
423 two = 2.0,
424 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
425 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
426 L1X = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
427 L2X = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
428 L3X = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
429 L4X = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
430 L5X = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
431 L6X = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
432 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
433 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
434 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
435 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
436 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
437 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
438 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
439 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
440 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
441 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
442
443 double __ieee754_pow(double x, double y) {
444 double z,ax,z_h,z_l,p_h,p_l;
445 double y1,t1,t2,r,s,t,u,v,w;
446 int i0,i1,i,j,k,yisint,n;
447 int hx,hy,ix,iy;
448 unsigned lx,ly;
449
450 i0 = ((*(int*)&one)>>29)^1; i1=1-i0;
451 hx = high(x); lx = low(x);
452 hy = high(y); ly = low(y);
453 ix = hx&0x7fffffff; iy = hy&0x7fffffff;
454
455 /* y==zero: x**0 = 1 */
456 if((iy|ly)==0) return one;
457
458 /* +-NaN return x+y */
459 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
460 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
461 return x+y;
462
463 /* determine if y is an odd int when x < 0
464 * yisint = 0 ... y is not an integer
465 * yisint = 1 ... y is an odd int
466 * yisint = 2 ... y is an even int
467 */
468 yisint = 0;
469 if(hx<0) {
470 if(iy>=0x43400000) yisint = 2; /* even integer y */
471 else if(iy>=0x3ff00000) {
472 k = (iy>>20)-0x3ff; /* exponent */
473 if(k>20) {
474 j = ly>>(52-k);
475 if((unsigned)(j<<(52-k))==ly) yisint = 2-(j&1);
476 } else if(ly==0) {
477 j = iy>>(20-k);
478 if((j<<(20-k))==iy) yisint = 2-(j&1);
479 }
480 }
481 }
482
483 /* special value of y */
484 if(ly==0) {
485 if (iy==0x7ff00000) { /* y is +-inf */
486 if(((ix-0x3ff00000)|lx)==0)
487 return y - y; /* inf**+-1 is NaN */
488 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
489 return (hy>=0)? y: zeroX;
490 else /* (|x|<1)**-,+inf = inf,0 */
491 return (hy<0)?-y: zeroX;
492 }
493 if(iy==0x3ff00000) { /* y is +-1 */
494 if(hy<0) return one/x; else return x;
495 }
496 if(hy==0x40000000) return x*x; /* y is 2 */
497 if(hy==0x3fe00000) { /* y is 0.5 */
498 if(hx>=0) /* x >= +0 */
499 return sqrt(x);
500 }
501 }
502
503 ax = fabsd(x);
504 /* special value of x */
505 if(lx==0) {
506 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
507 z = ax; /*x is +-0,+-inf,+-1*/
508 if(hy<0) z = one/z; /* z = (1/|x|) */
509 if(hx<0) {
510 if(((ix-0x3ff00000)|yisint)==0) {
511 #ifdef CAN_USE_NAN_DEFINE
512 z = NAN;
513 #else
514 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
515 #endif
516 } else if(yisint==1)
517 z = -1.0*z; /* (x<0)**odd = -(|x|**odd) */
518 }
519 return z;
520 }
521 }
522
523 n = (hx>>31)+1;
524
525 /* (x<0)**(non-int) is NaN */
526 if((n|yisint)==0)
527 #ifdef CAN_USE_NAN_DEFINE
528 return NAN;
529 #else
530 return (x-x)/(x-x);
531 #endif
532
533 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
534 if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
535
536 /* |y| is huge */
537 if(iy>0x41e00000) { /* if |y| > 2**31 */
538 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
539 if(ix<=0x3fefffff) return (hy<0)? hugeX*hugeX:tiny*tiny;
540 if(ix>=0x3ff00000) return (hy>0)? hugeX*hugeX:tiny*tiny;
541 }
542 /* over/underflow if x is not close to one */
543 if(ix<0x3fefffff) return (hy<0)? s*hugeX*hugeX:s*tiny*tiny;
544 if(ix>0x3ff00000) return (hy>0)? s*hugeX*hugeX:s*tiny*tiny;
545 /* now |1-x| is tiny <= 2**-20, suffice to compute
546 log(x) by x-x^2/2+x^3/3-x^4/4 */
547 t = ax-one; /* t has 20 trailing zeros */
548 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
549 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
550 v = t*ivln2_l-w*ivln2;
551 t1 = u+v;
552 set_low(&t1, 0);
553 t2 = v-(t1-u);
554 } else {
555 double ss,s2,s_h,s_l,t_h,t_l;
556 n = 0;
557 /* take care subnormal number */
558 if(ix<0x00100000)
559 {ax *= two53; n -= 53; ix = high(ax); }
560 n += ((ix)>>20)-0x3ff;
561 j = ix&0x000fffff;
562 /* determine interval */
563 ix = j|0x3ff00000; /* normalize ix */
564 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
565 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
566 else {k=0;n+=1;ix -= 0x00100000;}
567 set_high(&ax, ix);
568
569 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
570 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
571 v = one/(ax+bp[k]);
572 ss = u*v;
573 s_h = ss;
574 set_low(&s_h, 0);
575 /* t_h=ax+bp[k] High */
576 t_h = zeroX;
577 set_high(&t_h, ((ix>>1)|0x20000000)+0x00080000+(k<<18));
578 t_l = ax - (t_h-bp[k]);
579 s_l = v*((u-s_h*t_h)-s_h*t_l);
580 /* compute log(ax) */
581 s2 = ss*ss;
582 r = s2*s2*(L1X+s2*(L2X+s2*(L3X+s2*(L4X+s2*(L5X+s2*L6X)))));
583 r += s_l*(s_h+ss);
584 s2 = s_h*s_h;
585 t_h = 3.0+s2+r;
586 set_low(&t_h, 0);
587 t_l = r-((t_h-3.0)-s2);
588 /* u+v = ss*(1+...) */
589 u = s_h*t_h;
590 v = s_l*t_h+t_l*ss;
591 /* 2/(3log2)*(ss+...) */
592 p_h = u+v;
593 set_low(&p_h, 0);
594 p_l = v-(p_h-u);
595 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
596 z_l = cp_l*p_h+p_l*cp+dp_l[k];
597 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
598 t = (double)n;
599 t1 = (((z_h+z_l)+dp_h[k])+t);
600 set_low(&t1, 0);
601 t2 = z_l-(((t1-t)-dp_h[k])-z_h);
602 }
603
604 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
605 y1 = y;
606 set_low(&y1, 0);
607 p_l = (y-y1)*t1+y*t2;
608 p_h = y1*t1;
609 z = p_l+p_h;
610 j = high(z);
611 i = low(z);
612 if (j>=0x40900000) { /* z >= 1024 */
613 if(((j-0x40900000)|i)!=0) /* if z > 1024 */
614 return s*hugeX*hugeX; /* overflow */
615 else {
616 if(p_l+ovt>z-p_h) return s*hugeX*hugeX; /* overflow */
617 }
618 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
619 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
620 return s*tiny*tiny; /* underflow */
621 else {
622 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
623 }
624 }
625 /*
626 * compute 2**(p_h+p_l)
627 */
628 i = j&0x7fffffff;
629 k = (i>>20)-0x3ff;
630 n = 0;
631 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
632 n = j+(0x00100000>>(k+1));
633 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
634 t = zeroX;
635 set_high(&t, (n&~(0x000fffff>>k)));
636 n = ((n&0x000fffff)|0x00100000)>>(20-k);
637 if(j<0) n = -n;
638 p_h -= t;
639 }
640 t = p_l+p_h;
641 set_low(&t, 0);
642 u = t*lg2_h;
643 v = (p_l-(t-p_h))*lg2+t*lg2_l;
644 z = u+v;
645 w = v-(z-u);
646 t = z*z;
647 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
648 r = (z*t1)/(t1-two)-(w+z*w);
649 z = one-(r-z);
650 j = high(z);
651 j += (n<<20);
652 if((j>>20)<=0) z = scalbnA(z,n); /* subnormal output */
653 else set_high(&z, high(z) + (n<<20));
654 return s*z;
655 }
656
657
658 JRT_LEAF(jdouble, SharedRuntime::dpow(jdouble x, jdouble y))
659 return __ieee754_pow(x, y);
660 JRT_END
661
662 #ifdef WIN32
663 # pragma optimize ( "", on )
664 #endif