1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/share/vm/runtime/sharedRuntimeTrans.cpp Sat Dec 01 00:00:00 2007 +0000
1.3 @@ -0,0 +1,719 @@
1.4 +/*
1.5 + * Copyright 2005 Sun Microsystems, Inc. All Rights Reserved.
1.6 + * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
1.7 + *
1.8 + * This code is free software; you can redistribute it and/or modify it
1.9 + * under the terms of the GNU General Public License version 2 only, as
1.10 + * published by the Free Software Foundation.
1.11 + *
1.12 + * This code is distributed in the hope that it will be useful, but WITHOUT
1.13 + * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
1.14 + * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
1.15 + * version 2 for more details (a copy is included in the LICENSE file that
1.16 + * accompanied this code).
1.17 + *
1.18 + * You should have received a copy of the GNU General Public License version
1.19 + * 2 along with this work; if not, write to the Free Software Foundation,
1.20 + * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
1.21 + *
1.22 + * Please contact Sun Microsystems, Inc., 4150 Network Circle, Santa Clara,
1.23 + * CA 95054 USA or visit www.sun.com if you need additional information or
1.24 + * have any questions.
1.25 + *
1.26 + */
1.27 +
1.28 +#include "incls/_precompiled.incl"
1.29 +#include "incls/_sharedRuntimeTrans.cpp.incl"
1.30 +
1.31 +// This file contains copies of the fdlibm routines used by
1.32 +// StrictMath. It turns out that it is almost always required to use
1.33 +// these runtime routines; the Intel CPU doesn't meet the Java
1.34 +// specification for sin/cos outside a certain limited argument range,
1.35 +// and the SPARC CPU doesn't appear to have sin/cos instructions. It
1.36 +// also turns out that avoiding the indirect call through function
1.37 +// pointer out to libjava.so in SharedRuntime speeds these routines up
1.38 +// by roughly 15% on both Win32/x86 and Solaris/SPARC.
1.39 +
1.40 +// Enabling optimizations in this file causes incorrect code to be
1.41 +// generated; can not figure out how to turn down optimization for one
1.42 +// file in the IDE on Windows
1.43 +#ifdef WIN32
1.44 +# pragma optimize ( "", off )
1.45 +#endif
1.46 +
1.47 +#include <math.h>
1.48 +
1.49 +// VM_LITTLE_ENDIAN is #defined appropriately in the Makefiles
1.50 +// [jk] this is not 100% correct because the float word order may different
1.51 +// from the byte order (e.g. on ARM)
1.52 +#ifdef VM_LITTLE_ENDIAN
1.53 +# define __HI(x) *(1+(int*)&x)
1.54 +# define __LO(x) *(int*)&x
1.55 +#else
1.56 +# define __HI(x) *(int*)&x
1.57 +# define __LO(x) *(1+(int*)&x)
1.58 +#endif
1.59 +
1.60 +double copysign(double x, double y) {
1.61 + __HI(x) = (__HI(x)&0x7fffffff)|(__HI(y)&0x80000000);
1.62 + return x;
1.63 +}
1.64 +
1.65 +/*
1.66 + * ====================================================
1.67 + * Copyright (C) 1998 by Sun Microsystems, Inc. All rights reserved.
1.68 + *
1.69 + * Developed at SunSoft, a Sun Microsystems, Inc. business.
1.70 + * Permission to use, copy, modify, and distribute this
1.71 + * software is freely granted, provided that this notice
1.72 + * is preserved.
1.73 + * ====================================================
1.74 + */
1.75 +
1.76 +/*
1.77 + * scalbn (double x, int n)
1.78 + * scalbn(x,n) returns x* 2**n computed by exponent
1.79 + * manipulation rather than by actually performing an
1.80 + * exponentiation or a multiplication.
1.81 + */
1.82 +
1.83 +static const double
1.84 +two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
1.85 + twom54 = 5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */
1.86 + hugeX = 1.0e+300,
1.87 + tiny = 1.0e-300;
1.88 +
1.89 +double scalbn (double x, int n) {
1.90 + int k,hx,lx;
1.91 + hx = __HI(x);
1.92 + lx = __LO(x);
1.93 + k = (hx&0x7ff00000)>>20; /* extract exponent */
1.94 + if (k==0) { /* 0 or subnormal x */
1.95 + if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */
1.96 + x *= two54;
1.97 + hx = __HI(x);
1.98 + k = ((hx&0x7ff00000)>>20) - 54;
1.99 + if (n< -50000) return tiny*x; /*underflow*/
1.100 + }
1.101 + if (k==0x7ff) return x+x; /* NaN or Inf */
1.102 + k = k+n;
1.103 + if (k > 0x7fe) return hugeX*copysign(hugeX,x); /* overflow */
1.104 + if (k > 0) /* normal result */
1.105 + {__HI(x) = (hx&0x800fffff)|(k<<20); return x;}
1.106 + if (k <= -54) {
1.107 + if (n > 50000) /* in case integer overflow in n+k */
1.108 + return hugeX*copysign(hugeX,x); /*overflow*/
1.109 + else return tiny*copysign(tiny,x); /*underflow*/
1.110 + }
1.111 + k += 54; /* subnormal result */
1.112 + __HI(x) = (hx&0x800fffff)|(k<<20);
1.113 + return x*twom54;
1.114 +}
1.115 +
1.116 +/* __ieee754_log(x)
1.117 + * Return the logrithm of x
1.118 + *
1.119 + * Method :
1.120 + * 1. Argument Reduction: find k and f such that
1.121 + * x = 2^k * (1+f),
1.122 + * where sqrt(2)/2 < 1+f < sqrt(2) .
1.123 + *
1.124 + * 2. Approximation of log(1+f).
1.125 + * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
1.126 + * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
1.127 + * = 2s + s*R
1.128 + * We use a special Reme algorithm on [0,0.1716] to generate
1.129 + * a polynomial of degree 14 to approximate R The maximum error
1.130 + * of this polynomial approximation is bounded by 2**-58.45. In
1.131 + * other words,
1.132 + * 2 4 6 8 10 12 14
1.133 + * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
1.134 + * (the values of Lg1 to Lg7 are listed in the program)
1.135 + * and
1.136 + * | 2 14 | -58.45
1.137 + * | Lg1*s +...+Lg7*s - R(z) | <= 2
1.138 + * | |
1.139 + * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
1.140 + * In order to guarantee error in log below 1ulp, we compute log
1.141 + * by
1.142 + * log(1+f) = f - s*(f - R) (if f is not too large)
1.143 + * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
1.144 + *
1.145 + * 3. Finally, log(x) = k*ln2 + log(1+f).
1.146 + * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
1.147 + * Here ln2 is split into two floating point number:
1.148 + * ln2_hi + ln2_lo,
1.149 + * where n*ln2_hi is always exact for |n| < 2000.
1.150 + *
1.151 + * Special cases:
1.152 + * log(x) is NaN with signal if x < 0 (including -INF) ;
1.153 + * log(+INF) is +INF; log(0) is -INF with signal;
1.154 + * log(NaN) is that NaN with no signal.
1.155 + *
1.156 + * Accuracy:
1.157 + * according to an error analysis, the error is always less than
1.158 + * 1 ulp (unit in the last place).
1.159 + *
1.160 + * Constants:
1.161 + * The hexadecimal values are the intended ones for the following
1.162 + * constants. The decimal values may be used, provided that the
1.163 + * compiler will convert from decimal to binary accurately enough
1.164 + * to produce the hexadecimal values shown.
1.165 + */
1.166 +
1.167 +static const double
1.168 +ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
1.169 + ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
1.170 + Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
1.171 + Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
1.172 + Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
1.173 + Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
1.174 + Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
1.175 + Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
1.176 + Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
1.177 +
1.178 +static double zero = 0.0;
1.179 +
1.180 +static double __ieee754_log(double x) {
1.181 + double hfsq,f,s,z,R,w,t1,t2,dk;
1.182 + int k,hx,i,j;
1.183 + unsigned lx;
1.184 +
1.185 + hx = __HI(x); /* high word of x */
1.186 + lx = __LO(x); /* low word of x */
1.187 +
1.188 + k=0;
1.189 + if (hx < 0x00100000) { /* x < 2**-1022 */
1.190 + if (((hx&0x7fffffff)|lx)==0)
1.191 + return -two54/zero; /* log(+-0)=-inf */
1.192 + if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
1.193 + k -= 54; x *= two54; /* subnormal number, scale up x */
1.194 + hx = __HI(x); /* high word of x */
1.195 + }
1.196 + if (hx >= 0x7ff00000) return x+x;
1.197 + k += (hx>>20)-1023;
1.198 + hx &= 0x000fffff;
1.199 + i = (hx+0x95f64)&0x100000;
1.200 + __HI(x) = hx|(i^0x3ff00000); /* normalize x or x/2 */
1.201 + k += (i>>20);
1.202 + f = x-1.0;
1.203 + if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
1.204 + if(f==zero) {
1.205 + if (k==0) return zero;
1.206 + else {dk=(double)k; return dk*ln2_hi+dk*ln2_lo;}
1.207 + }
1.208 + R = f*f*(0.5-0.33333333333333333*f);
1.209 + if(k==0) return f-R; else {dk=(double)k;
1.210 + return dk*ln2_hi-((R-dk*ln2_lo)-f);}
1.211 + }
1.212 + s = f/(2.0+f);
1.213 + dk = (double)k;
1.214 + z = s*s;
1.215 + i = hx-0x6147a;
1.216 + w = z*z;
1.217 + j = 0x6b851-hx;
1.218 + t1= w*(Lg2+w*(Lg4+w*Lg6));
1.219 + t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
1.220 + i |= j;
1.221 + R = t2+t1;
1.222 + if(i>0) {
1.223 + hfsq=0.5*f*f;
1.224 + if(k==0) return f-(hfsq-s*(hfsq+R)); else
1.225 + return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
1.226 + } else {
1.227 + if(k==0) return f-s*(f-R); else
1.228 + return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
1.229 + }
1.230 +}
1.231 +
1.232 +JRT_LEAF(jdouble, SharedRuntime::dlog(jdouble x))
1.233 + return __ieee754_log(x);
1.234 +JRT_END
1.235 +
1.236 +/* __ieee754_log10(x)
1.237 + * Return the base 10 logarithm of x
1.238 + *
1.239 + * Method :
1.240 + * Let log10_2hi = leading 40 bits of log10(2) and
1.241 + * log10_2lo = log10(2) - log10_2hi,
1.242 + * ivln10 = 1/log(10) rounded.
1.243 + * Then
1.244 + * n = ilogb(x),
1.245 + * if(n<0) n = n+1;
1.246 + * x = scalbn(x,-n);
1.247 + * log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))
1.248 + *
1.249 + * Note 1:
1.250 + * To guarantee log10(10**n)=n, where 10**n is normal, the rounding
1.251 + * mode must set to Round-to-Nearest.
1.252 + * Note 2:
1.253 + * [1/log(10)] rounded to 53 bits has error .198 ulps;
1.254 + * log10 is monotonic at all binary break points.
1.255 + *
1.256 + * Special cases:
1.257 + * log10(x) is NaN with signal if x < 0;
1.258 + * log10(+INF) is +INF with no signal; log10(0) is -INF with signal;
1.259 + * log10(NaN) is that NaN with no signal;
1.260 + * log10(10**N) = N for N=0,1,...,22.
1.261 + *
1.262 + * Constants:
1.263 + * The hexadecimal values are the intended ones for the following constants.
1.264 + * The decimal values may be used, provided that the compiler will convert
1.265 + * from decimal to binary accurately enough to produce the hexadecimal values
1.266 + * shown.
1.267 + */
1.268 +
1.269 +static const double
1.270 +ivln10 = 4.34294481903251816668e-01, /* 0x3FDBCB7B, 0x1526E50E */
1.271 + log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
1.272 + log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */
1.273 +
1.274 +static double __ieee754_log10(double x) {
1.275 + double y,z;
1.276 + int i,k,hx;
1.277 + unsigned lx;
1.278 +
1.279 + hx = __HI(x); /* high word of x */
1.280 + lx = __LO(x); /* low word of x */
1.281 +
1.282 + k=0;
1.283 + if (hx < 0x00100000) { /* x < 2**-1022 */
1.284 + if (((hx&0x7fffffff)|lx)==0)
1.285 + return -two54/zero; /* log(+-0)=-inf */
1.286 + if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
1.287 + k -= 54; x *= two54; /* subnormal number, scale up x */
1.288 + hx = __HI(x); /* high word of x */
1.289 + }
1.290 + if (hx >= 0x7ff00000) return x+x;
1.291 + k += (hx>>20)-1023;
1.292 + i = ((unsigned)k&0x80000000)>>31;
1.293 + hx = (hx&0x000fffff)|((0x3ff-i)<<20);
1.294 + y = (double)(k+i);
1.295 + __HI(x) = hx;
1.296 + z = y*log10_2lo + ivln10*__ieee754_log(x);
1.297 + return z+y*log10_2hi;
1.298 +}
1.299 +
1.300 +JRT_LEAF(jdouble, SharedRuntime::dlog10(jdouble x))
1.301 + return __ieee754_log10(x);
1.302 +JRT_END
1.303 +
1.304 +
1.305 +/* __ieee754_exp(x)
1.306 + * Returns the exponential of x.
1.307 + *
1.308 + * Method
1.309 + * 1. Argument reduction:
1.310 + * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
1.311 + * Given x, find r and integer k such that
1.312 + *
1.313 + * x = k*ln2 + r, |r| <= 0.5*ln2.
1.314 + *
1.315 + * Here r will be represented as r = hi-lo for better
1.316 + * accuracy.
1.317 + *
1.318 + * 2. Approximation of exp(r) by a special rational function on
1.319 + * the interval [0,0.34658]:
1.320 + * Write
1.321 + * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
1.322 + * We use a special Reme algorithm on [0,0.34658] to generate
1.323 + * a polynomial of degree 5 to approximate R. The maximum error
1.324 + * of this polynomial approximation is bounded by 2**-59. In
1.325 + * other words,
1.326 + * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
1.327 + * (where z=r*r, and the values of P1 to P5 are listed below)
1.328 + * and
1.329 + * | 5 | -59
1.330 + * | 2.0+P1*z+...+P5*z - R(z) | <= 2
1.331 + * | |
1.332 + * The computation of exp(r) thus becomes
1.333 + * 2*r
1.334 + * exp(r) = 1 + -------
1.335 + * R - r
1.336 + * r*R1(r)
1.337 + * = 1 + r + ----------- (for better accuracy)
1.338 + * 2 - R1(r)
1.339 + * where
1.340 + * 2 4 10
1.341 + * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
1.342 + *
1.343 + * 3. Scale back to obtain exp(x):
1.344 + * From step 1, we have
1.345 + * exp(x) = 2^k * exp(r)
1.346 + *
1.347 + * Special cases:
1.348 + * exp(INF) is INF, exp(NaN) is NaN;
1.349 + * exp(-INF) is 0, and
1.350 + * for finite argument, only exp(0)=1 is exact.
1.351 + *
1.352 + * Accuracy:
1.353 + * according to an error analysis, the error is always less than
1.354 + * 1 ulp (unit in the last place).
1.355 + *
1.356 + * Misc. info.
1.357 + * For IEEE double
1.358 + * if x > 7.09782712893383973096e+02 then exp(x) overflow
1.359 + * if x < -7.45133219101941108420e+02 then exp(x) underflow
1.360 + *
1.361 + * Constants:
1.362 + * The hexadecimal values are the intended ones for the following
1.363 + * constants. The decimal values may be used, provided that the
1.364 + * compiler will convert from decimal to binary accurately enough
1.365 + * to produce the hexadecimal values shown.
1.366 + */
1.367 +
1.368 +static const double
1.369 +one = 1.0,
1.370 + halF[2] = {0.5,-0.5,},
1.371 + twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
1.372 + o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
1.373 + u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
1.374 + ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
1.375 + -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
1.376 + ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
1.377 + -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
1.378 + invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
1.379 + P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
1.380 + P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
1.381 + P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
1.382 + P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
1.383 + P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
1.384 +
1.385 +static double __ieee754_exp(double x) {
1.386 + double y,hi=0,lo=0,c,t;
1.387 + int k=0,xsb;
1.388 + unsigned hx;
1.389 +
1.390 + hx = __HI(x); /* high word of x */
1.391 + xsb = (hx>>31)&1; /* sign bit of x */
1.392 + hx &= 0x7fffffff; /* high word of |x| */
1.393 +
1.394 + /* filter out non-finite argument */
1.395 + if(hx >= 0x40862E42) { /* if |x|>=709.78... */
1.396 + if(hx>=0x7ff00000) {
1.397 + if(((hx&0xfffff)|__LO(x))!=0)
1.398 + return x+x; /* NaN */
1.399 + else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
1.400 + }
1.401 + if(x > o_threshold) return hugeX*hugeX; /* overflow */
1.402 + if(x < u_threshold) return twom1000*twom1000; /* underflow */
1.403 + }
1.404 +
1.405 + /* argument reduction */
1.406 + if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
1.407 + if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
1.408 + hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
1.409 + } else {
1.410 + k = (int)(invln2*x+halF[xsb]);
1.411 + t = k;
1.412 + hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
1.413 + lo = t*ln2LO[0];
1.414 + }
1.415 + x = hi - lo;
1.416 + }
1.417 + else if(hx < 0x3e300000) { /* when |x|<2**-28 */
1.418 + if(hugeX+x>one) return one+x;/* trigger inexact */
1.419 + }
1.420 + else k = 0;
1.421 +
1.422 + /* x is now in primary range */
1.423 + t = x*x;
1.424 + c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
1.425 + if(k==0) return one-((x*c)/(c-2.0)-x);
1.426 + else y = one-((lo-(x*c)/(2.0-c))-hi);
1.427 + if(k >= -1021) {
1.428 + __HI(y) += (k<<20); /* add k to y's exponent */
1.429 + return y;
1.430 + } else {
1.431 + __HI(y) += ((k+1000)<<20);/* add k to y's exponent */
1.432 + return y*twom1000;
1.433 + }
1.434 +}
1.435 +
1.436 +JRT_LEAF(jdouble, SharedRuntime::dexp(jdouble x))
1.437 + return __ieee754_exp(x);
1.438 +JRT_END
1.439 +
1.440 +/* __ieee754_pow(x,y) return x**y
1.441 + *
1.442 + * n
1.443 + * Method: Let x = 2 * (1+f)
1.444 + * 1. Compute and return log2(x) in two pieces:
1.445 + * log2(x) = w1 + w2,
1.446 + * where w1 has 53-24 = 29 bit trailing zeros.
1.447 + * 2. Perform y*log2(x) = n+y' by simulating muti-precision
1.448 + * arithmetic, where |y'|<=0.5.
1.449 + * 3. Return x**y = 2**n*exp(y'*log2)
1.450 + *
1.451 + * Special cases:
1.452 + * 1. (anything) ** 0 is 1
1.453 + * 2. (anything) ** 1 is itself
1.454 + * 3. (anything) ** NAN is NAN
1.455 + * 4. NAN ** (anything except 0) is NAN
1.456 + * 5. +-(|x| > 1) ** +INF is +INF
1.457 + * 6. +-(|x| > 1) ** -INF is +0
1.458 + * 7. +-(|x| < 1) ** +INF is +0
1.459 + * 8. +-(|x| < 1) ** -INF is +INF
1.460 + * 9. +-1 ** +-INF is NAN
1.461 + * 10. +0 ** (+anything except 0, NAN) is +0
1.462 + * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
1.463 + * 12. +0 ** (-anything except 0, NAN) is +INF
1.464 + * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
1.465 + * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
1.466 + * 15. +INF ** (+anything except 0,NAN) is +INF
1.467 + * 16. +INF ** (-anything except 0,NAN) is +0
1.468 + * 17. -INF ** (anything) = -0 ** (-anything)
1.469 + * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
1.470 + * 19. (-anything except 0 and inf) ** (non-integer) is NAN
1.471 + *
1.472 + * Accuracy:
1.473 + * pow(x,y) returns x**y nearly rounded. In particular
1.474 + * pow(integer,integer)
1.475 + * always returns the correct integer provided it is
1.476 + * representable.
1.477 + *
1.478 + * Constants :
1.479 + * The hexadecimal values are the intended ones for the following
1.480 + * constants. The decimal values may be used, provided that the
1.481 + * compiler will convert from decimal to binary accurately enough
1.482 + * to produce the hexadecimal values shown.
1.483 + */
1.484 +
1.485 +static const double
1.486 +bp[] = {1.0, 1.5,},
1.487 + dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
1.488 + dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
1.489 + zeroX = 0.0,
1.490 + two = 2.0,
1.491 + two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
1.492 + /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
1.493 + L1X = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
1.494 + L2X = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
1.495 + L3X = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
1.496 + L4X = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
1.497 + L5X = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
1.498 + L6X = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
1.499 + lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
1.500 + lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
1.501 + lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
1.502 + ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
1.503 + cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
1.504 + cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
1.505 + cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
1.506 + ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
1.507 + ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
1.508 + ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
1.509 +
1.510 +double __ieee754_pow(double x, double y) {
1.511 + double z,ax,z_h,z_l,p_h,p_l;
1.512 + double y1,t1,t2,r,s,t,u,v,w;
1.513 + int i0,i1,i,j,k,yisint,n;
1.514 + int hx,hy,ix,iy;
1.515 + unsigned lx,ly;
1.516 +
1.517 + i0 = ((*(int*)&one)>>29)^1; i1=1-i0;
1.518 + hx = __HI(x); lx = __LO(x);
1.519 + hy = __HI(y); ly = __LO(y);
1.520 + ix = hx&0x7fffffff; iy = hy&0x7fffffff;
1.521 +
1.522 + /* y==zero: x**0 = 1 */
1.523 + if((iy|ly)==0) return one;
1.524 +
1.525 + /* +-NaN return x+y */
1.526 + if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
1.527 + iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
1.528 + return x+y;
1.529 +
1.530 + /* determine if y is an odd int when x < 0
1.531 + * yisint = 0 ... y is not an integer
1.532 + * yisint = 1 ... y is an odd int
1.533 + * yisint = 2 ... y is an even int
1.534 + */
1.535 + yisint = 0;
1.536 + if(hx<0) {
1.537 + if(iy>=0x43400000) yisint = 2; /* even integer y */
1.538 + else if(iy>=0x3ff00000) {
1.539 + k = (iy>>20)-0x3ff; /* exponent */
1.540 + if(k>20) {
1.541 + j = ly>>(52-k);
1.542 + if((unsigned)(j<<(52-k))==ly) yisint = 2-(j&1);
1.543 + } else if(ly==0) {
1.544 + j = iy>>(20-k);
1.545 + if((j<<(20-k))==iy) yisint = 2-(j&1);
1.546 + }
1.547 + }
1.548 + }
1.549 +
1.550 + /* special value of y */
1.551 + if(ly==0) {
1.552 + if (iy==0x7ff00000) { /* y is +-inf */
1.553 + if(((ix-0x3ff00000)|lx)==0)
1.554 + return y - y; /* inf**+-1 is NaN */
1.555 + else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
1.556 + return (hy>=0)? y: zeroX;
1.557 + else /* (|x|<1)**-,+inf = inf,0 */
1.558 + return (hy<0)?-y: zeroX;
1.559 + }
1.560 + if(iy==0x3ff00000) { /* y is +-1 */
1.561 + if(hy<0) return one/x; else return x;
1.562 + }
1.563 + if(hy==0x40000000) return x*x; /* y is 2 */
1.564 + if(hy==0x3fe00000) { /* y is 0.5 */
1.565 + if(hx>=0) /* x >= +0 */
1.566 + return sqrt(x);
1.567 + }
1.568 + }
1.569 +
1.570 + ax = fabsd(x);
1.571 + /* special value of x */
1.572 + if(lx==0) {
1.573 + if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
1.574 + z = ax; /*x is +-0,+-inf,+-1*/
1.575 + if(hy<0) z = one/z; /* z = (1/|x|) */
1.576 + if(hx<0) {
1.577 + if(((ix-0x3ff00000)|yisint)==0) {
1.578 + z = (z-z)/(z-z); /* (-1)**non-int is NaN */
1.579 + } else if(yisint==1)
1.580 + z = -1.0*z; /* (x<0)**odd = -(|x|**odd) */
1.581 + }
1.582 + return z;
1.583 + }
1.584 + }
1.585 +
1.586 + n = (hx>>31)+1;
1.587 +
1.588 + /* (x<0)**(non-int) is NaN */
1.589 + if((n|yisint)==0) return (x-x)/(x-x);
1.590 +
1.591 + s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
1.592 + if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
1.593 +
1.594 + /* |y| is huge */
1.595 + if(iy>0x41e00000) { /* if |y| > 2**31 */
1.596 + if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
1.597 + if(ix<=0x3fefffff) return (hy<0)? hugeX*hugeX:tiny*tiny;
1.598 + if(ix>=0x3ff00000) return (hy>0)? hugeX*hugeX:tiny*tiny;
1.599 + }
1.600 + /* over/underflow if x is not close to one */
1.601 + if(ix<0x3fefffff) return (hy<0)? s*hugeX*hugeX:s*tiny*tiny;
1.602 + if(ix>0x3ff00000) return (hy>0)? s*hugeX*hugeX:s*tiny*tiny;
1.603 + /* now |1-x| is tiny <= 2**-20, suffice to compute
1.604 + log(x) by x-x^2/2+x^3/3-x^4/4 */
1.605 + t = ax-one; /* t has 20 trailing zeros */
1.606 + w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
1.607 + u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
1.608 + v = t*ivln2_l-w*ivln2;
1.609 + t1 = u+v;
1.610 + __LO(t1) = 0;
1.611 + t2 = v-(t1-u);
1.612 + } else {
1.613 + double ss,s2,s_h,s_l,t_h,t_l;
1.614 + n = 0;
1.615 + /* take care subnormal number */
1.616 + if(ix<0x00100000)
1.617 + {ax *= two53; n -= 53; ix = __HI(ax); }
1.618 + n += ((ix)>>20)-0x3ff;
1.619 + j = ix&0x000fffff;
1.620 + /* determine interval */
1.621 + ix = j|0x3ff00000; /* normalize ix */
1.622 + if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
1.623 + else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
1.624 + else {k=0;n+=1;ix -= 0x00100000;}
1.625 + __HI(ax) = ix;
1.626 +
1.627 + /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
1.628 + u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
1.629 + v = one/(ax+bp[k]);
1.630 + ss = u*v;
1.631 + s_h = ss;
1.632 + __LO(s_h) = 0;
1.633 + /* t_h=ax+bp[k] High */
1.634 + t_h = zeroX;
1.635 + __HI(t_h)=((ix>>1)|0x20000000)+0x00080000+(k<<18);
1.636 + t_l = ax - (t_h-bp[k]);
1.637 + s_l = v*((u-s_h*t_h)-s_h*t_l);
1.638 + /* compute log(ax) */
1.639 + s2 = ss*ss;
1.640 + r = s2*s2*(L1X+s2*(L2X+s2*(L3X+s2*(L4X+s2*(L5X+s2*L6X)))));
1.641 + r += s_l*(s_h+ss);
1.642 + s2 = s_h*s_h;
1.643 + t_h = 3.0+s2+r;
1.644 + __LO(t_h) = 0;
1.645 + t_l = r-((t_h-3.0)-s2);
1.646 + /* u+v = ss*(1+...) */
1.647 + u = s_h*t_h;
1.648 + v = s_l*t_h+t_l*ss;
1.649 + /* 2/(3log2)*(ss+...) */
1.650 + p_h = u+v;
1.651 + __LO(p_h) = 0;
1.652 + p_l = v-(p_h-u);
1.653 + z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
1.654 + z_l = cp_l*p_h+p_l*cp+dp_l[k];
1.655 + /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
1.656 + t = (double)n;
1.657 + t1 = (((z_h+z_l)+dp_h[k])+t);
1.658 + __LO(t1) = 0;
1.659 + t2 = z_l-(((t1-t)-dp_h[k])-z_h);
1.660 + }
1.661 +
1.662 + /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
1.663 + y1 = y;
1.664 + __LO(y1) = 0;
1.665 + p_l = (y-y1)*t1+y*t2;
1.666 + p_h = y1*t1;
1.667 + z = p_l+p_h;
1.668 + j = __HI(z);
1.669 + i = __LO(z);
1.670 + if (j>=0x40900000) { /* z >= 1024 */
1.671 + if(((j-0x40900000)|i)!=0) /* if z > 1024 */
1.672 + return s*hugeX*hugeX; /* overflow */
1.673 + else {
1.674 + if(p_l+ovt>z-p_h) return s*hugeX*hugeX; /* overflow */
1.675 + }
1.676 + } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
1.677 + if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
1.678 + return s*tiny*tiny; /* underflow */
1.679 + else {
1.680 + if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
1.681 + }
1.682 + }
1.683 + /*
1.684 + * compute 2**(p_h+p_l)
1.685 + */
1.686 + i = j&0x7fffffff;
1.687 + k = (i>>20)-0x3ff;
1.688 + n = 0;
1.689 + if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
1.690 + n = j+(0x00100000>>(k+1));
1.691 + k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
1.692 + t = zeroX;
1.693 + __HI(t) = (n&~(0x000fffff>>k));
1.694 + n = ((n&0x000fffff)|0x00100000)>>(20-k);
1.695 + if(j<0) n = -n;
1.696 + p_h -= t;
1.697 + }
1.698 + t = p_l+p_h;
1.699 + __LO(t) = 0;
1.700 + u = t*lg2_h;
1.701 + v = (p_l-(t-p_h))*lg2+t*lg2_l;
1.702 + z = u+v;
1.703 + w = v-(z-u);
1.704 + t = z*z;
1.705 + t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
1.706 + r = (z*t1)/(t1-two)-(w+z*w);
1.707 + z = one-(r-z);
1.708 + j = __HI(z);
1.709 + j += (n<<20);
1.710 + if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
1.711 + else __HI(z) += (n<<20);
1.712 + return s*z;
1.713 +}
1.714 +
1.715 +
1.716 +JRT_LEAF(jdouble, SharedRuntime::dpow(jdouble x, jdouble y))
1.717 + return __ieee754_pow(x, y);
1.718 +JRT_END
1.719 +
1.720 +#ifdef WIN32
1.721 +# pragma optimize ( "", on )
1.722 +#endif
