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.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