aoqi@0: /* aoqi@0: * Copyright (c) 2005, 2010, Oracle and/or its affiliates. All rights reserved. aoqi@0: * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. aoqi@0: * aoqi@0: * This code is free software; you can redistribute it and/or modify it aoqi@0: * under the terms of the GNU General Public License version 2 only, as aoqi@0: * published by the Free Software Foundation. aoqi@0: * aoqi@0: * This code is distributed in the hope that it will be useful, but WITHOUT aoqi@0: * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or aoqi@0: * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License aoqi@0: * version 2 for more details (a copy is included in the LICENSE file that aoqi@0: * accompanied this code). aoqi@0: * aoqi@0: * You should have received a copy of the GNU General Public License version aoqi@0: * 2 along with this work; if not, write to the Free Software Foundation, aoqi@0: * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. aoqi@0: * aoqi@0: * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA aoqi@0: * or visit www.oracle.com if you need additional information or have any aoqi@0: * questions. aoqi@0: * aoqi@0: */ aoqi@0: aoqi@0: #include "precompiled.hpp" aoqi@0: #include "prims/jni.h" aoqi@0: #include "runtime/interfaceSupport.hpp" aoqi@0: #include "runtime/sharedRuntime.hpp" aoqi@0: aoqi@0: // This file contains copies of the fdlibm routines used by aoqi@0: // StrictMath. It turns out that it is almost always required to use aoqi@0: // these runtime routines; the Intel CPU doesn't meet the Java aoqi@0: // specification for sin/cos outside a certain limited argument range, aoqi@0: // and the SPARC CPU doesn't appear to have sin/cos instructions. It aoqi@0: // also turns out that avoiding the indirect call through function aoqi@0: // pointer out to libjava.so in SharedRuntime speeds these routines up aoqi@0: // by roughly 15% on both Win32/x86 and Solaris/SPARC. aoqi@0: aoqi@0: // Enabling optimizations in this file causes incorrect code to be aoqi@0: // generated; can not figure out how to turn down optimization for one aoqi@0: // file in the IDE on Windows aoqi@0: #ifdef WIN32 aoqi@0: # pragma optimize ( "", off ) aoqi@0: #endif aoqi@0: aoqi@0: #include aoqi@0: aoqi@0: // VM_LITTLE_ENDIAN is #defined appropriately in the Makefiles aoqi@0: // [jk] this is not 100% correct because the float word order may different aoqi@0: // from the byte order (e.g. on ARM) aoqi@0: #ifdef VM_LITTLE_ENDIAN aoqi@0: # define __HI(x) *(1+(int*)&x) aoqi@0: # define __LO(x) *(int*)&x aoqi@0: #else aoqi@0: # define __HI(x) *(int*)&x aoqi@0: # define __LO(x) *(1+(int*)&x) aoqi@0: #endif aoqi@0: aoqi@0: #if !defined(AIX) aoqi@0: double copysign(double x, double y) { aoqi@0: __HI(x) = (__HI(x)&0x7fffffff)|(__HI(y)&0x80000000); aoqi@0: return x; aoqi@0: } aoqi@0: #endif aoqi@0: aoqi@0: /* aoqi@0: * ==================================================== aoqi@0: * Copyright (c) 1998 Oracle and/or its affiliates. All rights reserved. aoqi@0: * aoqi@0: * Developed at SunSoft, a Sun Microsystems, Inc. business. aoqi@0: * Permission to use, copy, modify, and distribute this aoqi@0: * software is freely granted, provided that this notice aoqi@0: * is preserved. aoqi@0: * ==================================================== aoqi@0: */ aoqi@0: aoqi@0: /* aoqi@0: * scalbn (double x, int n) aoqi@0: * scalbn(x,n) returns x* 2**n computed by exponent aoqi@0: * manipulation rather than by actually performing an aoqi@0: * exponentiation or a multiplication. aoqi@0: */ aoqi@0: aoqi@0: static const double aoqi@0: two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ aoqi@0: twom54 = 5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */ aoqi@0: hugeX = 1.0e+300, aoqi@0: tiny = 1.0e-300; aoqi@0: aoqi@0: #if !defined(AIX) aoqi@0: double scalbn (double x, int n) { aoqi@0: int k,hx,lx; aoqi@0: hx = __HI(x); aoqi@0: lx = __LO(x); aoqi@0: k = (hx&0x7ff00000)>>20; /* extract exponent */ aoqi@0: if (k==0) { /* 0 or subnormal x */ aoqi@0: if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */ aoqi@0: x *= two54; aoqi@0: hx = __HI(x); aoqi@0: k = ((hx&0x7ff00000)>>20) - 54; aoqi@0: if (n< -50000) return tiny*x; /*underflow*/ aoqi@0: } aoqi@0: if (k==0x7ff) return x+x; /* NaN or Inf */ aoqi@0: k = k+n; aoqi@0: if (k > 0x7fe) return hugeX*copysign(hugeX,x); /* overflow */ aoqi@0: if (k > 0) /* normal result */ aoqi@0: {__HI(x) = (hx&0x800fffff)|(k<<20); return x;} aoqi@0: if (k <= -54) { aoqi@0: if (n > 50000) /* in case integer overflow in n+k */ aoqi@0: return hugeX*copysign(hugeX,x); /*overflow*/ aoqi@0: else return tiny*copysign(tiny,x); /*underflow*/ aoqi@0: } aoqi@0: k += 54; /* subnormal result */ aoqi@0: __HI(x) = (hx&0x800fffff)|(k<<20); aoqi@0: return x*twom54; aoqi@0: } aoqi@0: #endif aoqi@0: aoqi@0: /* __ieee754_log(x) aoqi@0: * Return the logrithm of x aoqi@0: * aoqi@0: * Method : aoqi@0: * 1. Argument Reduction: find k and f such that aoqi@0: * x = 2^k * (1+f), aoqi@0: * where sqrt(2)/2 < 1+f < sqrt(2) . aoqi@0: * aoqi@0: * 2. Approximation of log(1+f). aoqi@0: * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) aoqi@0: * = 2s + 2/3 s**3 + 2/5 s**5 + ....., aoqi@0: * = 2s + s*R aoqi@0: * We use a special Reme algorithm on [0,0.1716] to generate aoqi@0: * a polynomial of degree 14 to approximate R The maximum error aoqi@0: * of this polynomial approximation is bounded by 2**-58.45. In aoqi@0: * other words, aoqi@0: * 2 4 6 8 10 12 14 aoqi@0: * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s aoqi@0: * (the values of Lg1 to Lg7 are listed in the program) aoqi@0: * and aoqi@0: * | 2 14 | -58.45 aoqi@0: * | Lg1*s +...+Lg7*s - R(z) | <= 2 aoqi@0: * | | aoqi@0: * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. aoqi@0: * In order to guarantee error in log below 1ulp, we compute log aoqi@0: * by aoqi@0: * log(1+f) = f - s*(f - R) (if f is not too large) aoqi@0: * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) aoqi@0: * aoqi@0: * 3. Finally, log(x) = k*ln2 + log(1+f). aoqi@0: * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) aoqi@0: * Here ln2 is split into two floating point number: aoqi@0: * ln2_hi + ln2_lo, aoqi@0: * where n*ln2_hi is always exact for |n| < 2000. aoqi@0: * aoqi@0: * Special cases: aoqi@0: * log(x) is NaN with signal if x < 0 (including -INF) ; aoqi@0: * log(+INF) is +INF; log(0) is -INF with signal; aoqi@0: * log(NaN) is that NaN with no signal. aoqi@0: * aoqi@0: * Accuracy: aoqi@0: * according to an error analysis, the error is always less than aoqi@0: * 1 ulp (unit in the last place). aoqi@0: * aoqi@0: * Constants: aoqi@0: * The hexadecimal values are the intended ones for the following aoqi@0: * constants. The decimal values may be used, provided that the aoqi@0: * compiler will convert from decimal to binary accurately enough aoqi@0: * to produce the hexadecimal values shown. aoqi@0: */ aoqi@0: aoqi@0: static const double aoqi@0: ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ aoqi@0: ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ aoqi@0: Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ aoqi@0: Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ aoqi@0: Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ aoqi@0: Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ aoqi@0: Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ aoqi@0: Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ aoqi@0: Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ aoqi@0: aoqi@0: static double zero = 0.0; aoqi@0: aoqi@0: static double __ieee754_log(double x) { aoqi@0: double hfsq,f,s,z,R,w,t1,t2,dk; aoqi@0: int k,hx,i,j; aoqi@0: unsigned lx; aoqi@0: aoqi@0: hx = __HI(x); /* high word of x */ aoqi@0: lx = __LO(x); /* low word of x */ aoqi@0: aoqi@0: k=0; aoqi@0: if (hx < 0x00100000) { /* x < 2**-1022 */ aoqi@0: if (((hx&0x7fffffff)|lx)==0) aoqi@0: return -two54/zero; /* log(+-0)=-inf */ aoqi@0: if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ aoqi@0: k -= 54; x *= two54; /* subnormal number, scale up x */ aoqi@0: hx = __HI(x); /* high word of x */ aoqi@0: } aoqi@0: if (hx >= 0x7ff00000) return x+x; aoqi@0: k += (hx>>20)-1023; aoqi@0: hx &= 0x000fffff; aoqi@0: i = (hx+0x95f64)&0x100000; aoqi@0: __HI(x) = hx|(i^0x3ff00000); /* normalize x or x/2 */ aoqi@0: k += (i>>20); aoqi@0: f = x-1.0; aoqi@0: if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */ aoqi@0: if(f==zero) { aoqi@0: if (k==0) return zero; aoqi@0: else {dk=(double)k; return dk*ln2_hi+dk*ln2_lo;} aoqi@0: } aoqi@0: R = f*f*(0.5-0.33333333333333333*f); aoqi@0: if(k==0) return f-R; else {dk=(double)k; aoqi@0: return dk*ln2_hi-((R-dk*ln2_lo)-f);} aoqi@0: } aoqi@0: s = f/(2.0+f); aoqi@0: dk = (double)k; aoqi@0: z = s*s; aoqi@0: i = hx-0x6147a; aoqi@0: w = z*z; aoqi@0: j = 0x6b851-hx; aoqi@0: t1= w*(Lg2+w*(Lg4+w*Lg6)); aoqi@0: t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); aoqi@0: i |= j; aoqi@0: R = t2+t1; aoqi@0: if(i>0) { aoqi@0: hfsq=0.5*f*f; aoqi@0: if(k==0) return f-(hfsq-s*(hfsq+R)); else aoqi@0: return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); aoqi@0: } else { aoqi@0: if(k==0) return f-s*(f-R); else aoqi@0: return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f); aoqi@0: } aoqi@0: } aoqi@0: aoqi@0: JRT_LEAF(jdouble, SharedRuntime::dlog(jdouble x)) aoqi@0: return __ieee754_log(x); aoqi@0: JRT_END aoqi@0: aoqi@0: /* __ieee754_log10(x) aoqi@0: * Return the base 10 logarithm of x aoqi@0: * aoqi@0: * Method : aoqi@0: * Let log10_2hi = leading 40 bits of log10(2) and aoqi@0: * log10_2lo = log10(2) - log10_2hi, aoqi@0: * ivln10 = 1/log(10) rounded. aoqi@0: * Then aoqi@0: * n = ilogb(x), aoqi@0: * if(n<0) n = n+1; aoqi@0: * x = scalbn(x,-n); aoqi@0: * log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x)) aoqi@0: * aoqi@0: * Note 1: aoqi@0: * To guarantee log10(10**n)=n, where 10**n is normal, the rounding aoqi@0: * mode must set to Round-to-Nearest. aoqi@0: * Note 2: aoqi@0: * [1/log(10)] rounded to 53 bits has error .198 ulps; aoqi@0: * log10 is monotonic at all binary break points. aoqi@0: * aoqi@0: * Special cases: aoqi@0: * log10(x) is NaN with signal if x < 0; aoqi@0: * log10(+INF) is +INF with no signal; log10(0) is -INF with signal; aoqi@0: * log10(NaN) is that NaN with no signal; aoqi@0: * log10(10**N) = N for N=0,1,...,22. aoqi@0: * aoqi@0: * Constants: aoqi@0: * The hexadecimal values are the intended ones for the following constants. aoqi@0: * The decimal values may be used, provided that the compiler will convert aoqi@0: * from decimal to binary accurately enough to produce the hexadecimal values aoqi@0: * shown. aoqi@0: */ aoqi@0: aoqi@0: static const double aoqi@0: ivln10 = 4.34294481903251816668e-01, /* 0x3FDBCB7B, 0x1526E50E */ aoqi@0: log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */ aoqi@0: log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */ aoqi@0: aoqi@0: static double __ieee754_log10(double x) { aoqi@0: double y,z; aoqi@0: int i,k,hx; aoqi@0: unsigned lx; aoqi@0: aoqi@0: hx = __HI(x); /* high word of x */ aoqi@0: lx = __LO(x); /* low word of x */ aoqi@0: aoqi@0: k=0; aoqi@0: if (hx < 0x00100000) { /* x < 2**-1022 */ aoqi@0: if (((hx&0x7fffffff)|lx)==0) aoqi@0: return -two54/zero; /* log(+-0)=-inf */ aoqi@0: if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ aoqi@0: k -= 54; x *= two54; /* subnormal number, scale up x */ aoqi@0: hx = __HI(x); /* high word of x */ aoqi@0: } aoqi@0: if (hx >= 0x7ff00000) return x+x; aoqi@0: k += (hx>>20)-1023; aoqi@0: i = ((unsigned)k&0x80000000)>>31; aoqi@0: hx = (hx&0x000fffff)|((0x3ff-i)<<20); aoqi@0: y = (double)(k+i); aoqi@0: __HI(x) = hx; aoqi@0: z = y*log10_2lo + ivln10*__ieee754_log(x); aoqi@0: return z+y*log10_2hi; aoqi@0: } aoqi@0: aoqi@0: JRT_LEAF(jdouble, SharedRuntime::dlog10(jdouble x)) aoqi@0: return __ieee754_log10(x); aoqi@0: JRT_END aoqi@0: aoqi@0: aoqi@0: /* __ieee754_exp(x) aoqi@0: * Returns the exponential of x. aoqi@0: * aoqi@0: * Method aoqi@0: * 1. Argument reduction: aoqi@0: * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. aoqi@0: * Given x, find r and integer k such that aoqi@0: * aoqi@0: * x = k*ln2 + r, |r| <= 0.5*ln2. aoqi@0: * aoqi@0: * Here r will be represented as r = hi-lo for better aoqi@0: * accuracy. aoqi@0: * aoqi@0: * 2. Approximation of exp(r) by a special rational function on aoqi@0: * the interval [0,0.34658]: aoqi@0: * Write aoqi@0: * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... aoqi@0: * We use a special Reme algorithm on [0,0.34658] to generate aoqi@0: * a polynomial of degree 5 to approximate R. The maximum error aoqi@0: * of this polynomial approximation is bounded by 2**-59. In aoqi@0: * other words, aoqi@0: * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 aoqi@0: * (where z=r*r, and the values of P1 to P5 are listed below) aoqi@0: * and aoqi@0: * | 5 | -59 aoqi@0: * | 2.0+P1*z+...+P5*z - R(z) | <= 2 aoqi@0: * | | aoqi@0: * The computation of exp(r) thus becomes aoqi@0: * 2*r aoqi@0: * exp(r) = 1 + ------- aoqi@0: * R - r aoqi@0: * r*R1(r) aoqi@0: * = 1 + r + ----------- (for better accuracy) aoqi@0: * 2 - R1(r) aoqi@0: * where aoqi@0: * 2 4 10 aoqi@0: * R1(r) = r - (P1*r + P2*r + ... + P5*r ). aoqi@0: * aoqi@0: * 3. Scale back to obtain exp(x): aoqi@0: * From step 1, we have aoqi@0: * exp(x) = 2^k * exp(r) aoqi@0: * aoqi@0: * Special cases: aoqi@0: * exp(INF) is INF, exp(NaN) is NaN; aoqi@0: * exp(-INF) is 0, and aoqi@0: * for finite argument, only exp(0)=1 is exact. aoqi@0: * aoqi@0: * Accuracy: aoqi@0: * according to an error analysis, the error is always less than aoqi@0: * 1 ulp (unit in the last place). aoqi@0: * aoqi@0: * Misc. info. aoqi@0: * For IEEE double aoqi@0: * if x > 7.09782712893383973096e+02 then exp(x) overflow aoqi@0: * if x < -7.45133219101941108420e+02 then exp(x) underflow aoqi@0: * aoqi@0: * Constants: aoqi@0: * The hexadecimal values are the intended ones for the following aoqi@0: * constants. The decimal values may be used, provided that the aoqi@0: * compiler will convert from decimal to binary accurately enough aoqi@0: * to produce the hexadecimal values shown. aoqi@0: */ aoqi@0: aoqi@0: static const double aoqi@0: one = 1.0, aoqi@0: halF[2] = {0.5,-0.5,}, aoqi@0: twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ aoqi@0: o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ aoqi@0: u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ aoqi@0: ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ aoqi@0: -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ aoqi@0: ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ aoqi@0: -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ aoqi@0: invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ aoqi@0: P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ aoqi@0: P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ aoqi@0: P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ aoqi@0: P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ aoqi@0: P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ aoqi@0: aoqi@0: static double __ieee754_exp(double x) { aoqi@0: double y,hi=0,lo=0,c,t; aoqi@0: int k=0,xsb; aoqi@0: unsigned hx; aoqi@0: aoqi@0: hx = __HI(x); /* high word of x */ aoqi@0: xsb = (hx>>31)&1; /* sign bit of x */ aoqi@0: hx &= 0x7fffffff; /* high word of |x| */ aoqi@0: aoqi@0: /* filter out non-finite argument */ aoqi@0: if(hx >= 0x40862E42) { /* if |x|>=709.78... */ aoqi@0: if(hx>=0x7ff00000) { aoqi@0: if(((hx&0xfffff)|__LO(x))!=0) aoqi@0: return x+x; /* NaN */ aoqi@0: else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ aoqi@0: } aoqi@0: if(x > o_threshold) return hugeX*hugeX; /* overflow */ aoqi@0: if(x < u_threshold) return twom1000*twom1000; /* underflow */ aoqi@0: } aoqi@0: aoqi@0: /* argument reduction */ aoqi@0: if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ aoqi@0: if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ aoqi@0: hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; aoqi@0: } else { aoqi@0: k = (int)(invln2*x+halF[xsb]); aoqi@0: t = k; aoqi@0: hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ aoqi@0: lo = t*ln2LO[0]; aoqi@0: } aoqi@0: x = hi - lo; aoqi@0: } aoqi@0: else if(hx < 0x3e300000) { /* when |x|<2**-28 */ aoqi@0: if(hugeX+x>one) return one+x;/* trigger inexact */ aoqi@0: } aoqi@0: else k = 0; aoqi@0: aoqi@0: /* x is now in primary range */ aoqi@0: t = x*x; aoqi@0: c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); aoqi@0: if(k==0) return one-((x*c)/(c-2.0)-x); aoqi@0: else y = one-((lo-(x*c)/(2.0-c))-hi); aoqi@0: if(k >= -1021) { aoqi@0: __HI(y) += (k<<20); /* add k to y's exponent */ aoqi@0: return y; aoqi@0: } else { aoqi@0: __HI(y) += ((k+1000)<<20);/* add k to y's exponent */ aoqi@0: return y*twom1000; aoqi@0: } aoqi@0: } aoqi@0: aoqi@0: JRT_LEAF(jdouble, SharedRuntime::dexp(jdouble x)) aoqi@0: return __ieee754_exp(x); aoqi@0: JRT_END aoqi@0: aoqi@0: /* __ieee754_pow(x,y) return x**y aoqi@0: * aoqi@0: * n aoqi@0: * Method: Let x = 2 * (1+f) aoqi@0: * 1. Compute and return log2(x) in two pieces: aoqi@0: * log2(x) = w1 + w2, aoqi@0: * where w1 has 53-24 = 29 bit trailing zeros. aoqi@0: * 2. Perform y*log2(x) = n+y' by simulating muti-precision aoqi@0: * arithmetic, where |y'|<=0.5. aoqi@0: * 3. Return x**y = 2**n*exp(y'*log2) aoqi@0: * aoqi@0: * Special cases: aoqi@0: * 1. (anything) ** 0 is 1 aoqi@0: * 2. (anything) ** 1 is itself aoqi@0: * 3. (anything) ** NAN is NAN aoqi@0: * 4. NAN ** (anything except 0) is NAN aoqi@0: * 5. +-(|x| > 1) ** +INF is +INF aoqi@0: * 6. +-(|x| > 1) ** -INF is +0 aoqi@0: * 7. +-(|x| < 1) ** +INF is +0 aoqi@0: * 8. +-(|x| < 1) ** -INF is +INF aoqi@0: * 9. +-1 ** +-INF is NAN aoqi@0: * 10. +0 ** (+anything except 0, NAN) is +0 aoqi@0: * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 aoqi@0: * 12. +0 ** (-anything except 0, NAN) is +INF aoqi@0: * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF aoqi@0: * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) aoqi@0: * 15. +INF ** (+anything except 0,NAN) is +INF aoqi@0: * 16. +INF ** (-anything except 0,NAN) is +0 aoqi@0: * 17. -INF ** (anything) = -0 ** (-anything) aoqi@0: * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) aoqi@0: * 19. (-anything except 0 and inf) ** (non-integer) is NAN aoqi@0: * aoqi@0: * Accuracy: aoqi@0: * pow(x,y) returns x**y nearly rounded. In particular aoqi@0: * pow(integer,integer) aoqi@0: * always returns the correct integer provided it is aoqi@0: * representable. aoqi@0: * aoqi@0: * Constants : aoqi@0: * The hexadecimal values are the intended ones for the following aoqi@0: * constants. The decimal values may be used, provided that the aoqi@0: * compiler will convert from decimal to binary accurately enough aoqi@0: * to produce the hexadecimal values shown. aoqi@0: */ aoqi@0: aoqi@0: static const double aoqi@0: bp[] = {1.0, 1.5,}, aoqi@0: dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ aoqi@0: dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ aoqi@0: zeroX = 0.0, aoqi@0: two = 2.0, aoqi@0: two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ aoqi@0: /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ aoqi@0: L1X = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ aoqi@0: L2X = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ aoqi@0: L3X = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ aoqi@0: L4X = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ aoqi@0: L5X = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ aoqi@0: L6X = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ aoqi@0: lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ aoqi@0: lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ aoqi@0: lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ aoqi@0: ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ aoqi@0: cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ aoqi@0: cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ aoqi@0: cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ aoqi@0: ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ aoqi@0: ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ aoqi@0: ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ aoqi@0: aoqi@0: double __ieee754_pow(double x, double y) { aoqi@0: double z,ax,z_h,z_l,p_h,p_l; aoqi@0: double y1,t1,t2,r,s,t,u,v,w; aoqi@0: int i0,i1,i,j,k,yisint,n; aoqi@0: int hx,hy,ix,iy; aoqi@0: unsigned lx,ly; aoqi@0: aoqi@0: i0 = ((*(int*)&one)>>29)^1; i1=1-i0; aoqi@0: hx = __HI(x); lx = __LO(x); aoqi@0: hy = __HI(y); ly = __LO(y); aoqi@0: ix = hx&0x7fffffff; iy = hy&0x7fffffff; aoqi@0: aoqi@0: /* y==zero: x**0 = 1 */ aoqi@0: if((iy|ly)==0) return one; aoqi@0: aoqi@0: /* +-NaN return x+y */ aoqi@0: if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || aoqi@0: iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) aoqi@0: return x+y; aoqi@0: aoqi@0: /* determine if y is an odd int when x < 0 aoqi@0: * yisint = 0 ... y is not an integer aoqi@0: * yisint = 1 ... y is an odd int aoqi@0: * yisint = 2 ... y is an even int aoqi@0: */ aoqi@0: yisint = 0; aoqi@0: if(hx<0) { aoqi@0: if(iy>=0x43400000) yisint = 2; /* even integer y */ aoqi@0: else if(iy>=0x3ff00000) { aoqi@0: k = (iy>>20)-0x3ff; /* exponent */ aoqi@0: if(k>20) { aoqi@0: j = ly>>(52-k); aoqi@0: if((unsigned)(j<<(52-k))==ly) yisint = 2-(j&1); aoqi@0: } else if(ly==0) { aoqi@0: j = iy>>(20-k); aoqi@0: if((j<<(20-k))==iy) yisint = 2-(j&1); aoqi@0: } aoqi@0: } aoqi@0: } aoqi@0: aoqi@0: /* special value of y */ aoqi@0: if(ly==0) { aoqi@0: if (iy==0x7ff00000) { /* y is +-inf */ aoqi@0: if(((ix-0x3ff00000)|lx)==0) aoqi@0: return y - y; /* inf**+-1 is NaN */ aoqi@0: else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ aoqi@0: return (hy>=0)? y: zeroX; aoqi@0: else /* (|x|<1)**-,+inf = inf,0 */ aoqi@0: return (hy<0)?-y: zeroX; aoqi@0: } aoqi@0: if(iy==0x3ff00000) { /* y is +-1 */ aoqi@0: if(hy<0) return one/x; else return x; aoqi@0: } aoqi@0: if(hy==0x40000000) return x*x; /* y is 2 */ aoqi@0: if(hy==0x3fe00000) { /* y is 0.5 */ aoqi@0: if(hx>=0) /* x >= +0 */ aoqi@0: return sqrt(x); aoqi@0: } aoqi@0: } aoqi@0: aoqi@0: ax = fabsd(x); aoqi@0: /* special value of x */ aoqi@0: if(lx==0) { aoqi@0: if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ aoqi@0: z = ax; /*x is +-0,+-inf,+-1*/ aoqi@0: if(hy<0) z = one/z; /* z = (1/|x|) */ aoqi@0: if(hx<0) { aoqi@0: if(((ix-0x3ff00000)|yisint)==0) { aoqi@0: #ifdef CAN_USE_NAN_DEFINE aoqi@0: z = NAN; aoqi@0: #else aoqi@0: z = (z-z)/(z-z); /* (-1)**non-int is NaN */ aoqi@0: #endif aoqi@0: } else if(yisint==1) aoqi@0: z = -1.0*z; /* (x<0)**odd = -(|x|**odd) */ aoqi@0: } aoqi@0: return z; aoqi@0: } aoqi@0: } aoqi@0: aoqi@0: n = (hx>>31)+1; aoqi@0: aoqi@0: /* (x<0)**(non-int) is NaN */ aoqi@0: if((n|yisint)==0) aoqi@0: #ifdef CAN_USE_NAN_DEFINE aoqi@0: return NAN; aoqi@0: #else aoqi@0: return (x-x)/(x-x); aoqi@0: #endif aoqi@0: aoqi@0: s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ aoqi@0: if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ aoqi@0: aoqi@0: /* |y| is huge */ aoqi@0: if(iy>0x41e00000) { /* if |y| > 2**31 */ aoqi@0: if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ aoqi@0: if(ix<=0x3fefffff) return (hy<0)? hugeX*hugeX:tiny*tiny; aoqi@0: if(ix>=0x3ff00000) return (hy>0)? hugeX*hugeX:tiny*tiny; aoqi@0: } aoqi@0: /* over/underflow if x is not close to one */ aoqi@0: if(ix<0x3fefffff) return (hy<0)? s*hugeX*hugeX:s*tiny*tiny; aoqi@0: if(ix>0x3ff00000) return (hy>0)? s*hugeX*hugeX:s*tiny*tiny; aoqi@0: /* now |1-x| is tiny <= 2**-20, suffice to compute aoqi@0: log(x) by x-x^2/2+x^3/3-x^4/4 */ aoqi@0: t = ax-one; /* t has 20 trailing zeros */ aoqi@0: w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); aoqi@0: u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ aoqi@0: v = t*ivln2_l-w*ivln2; aoqi@0: t1 = u+v; aoqi@0: __LO(t1) = 0; aoqi@0: t2 = v-(t1-u); aoqi@0: } else { aoqi@0: double ss,s2,s_h,s_l,t_h,t_l; aoqi@0: n = 0; aoqi@0: /* take care subnormal number */ aoqi@0: if(ix<0x00100000) aoqi@0: {ax *= two53; n -= 53; ix = __HI(ax); } aoqi@0: n += ((ix)>>20)-0x3ff; aoqi@0: j = ix&0x000fffff; aoqi@0: /* determine interval */ aoqi@0: ix = j|0x3ff00000; /* normalize ix */ aoqi@0: if(j<=0x3988E) k=0; /* |x|>1)|0x20000000)+0x00080000+(k<<18); aoqi@0: t_l = ax - (t_h-bp[k]); aoqi@0: s_l = v*((u-s_h*t_h)-s_h*t_l); aoqi@0: /* compute log(ax) */ aoqi@0: s2 = ss*ss; aoqi@0: r = s2*s2*(L1X+s2*(L2X+s2*(L3X+s2*(L4X+s2*(L5X+s2*L6X))))); aoqi@0: r += s_l*(s_h+ss); aoqi@0: s2 = s_h*s_h; aoqi@0: t_h = 3.0+s2+r; aoqi@0: __LO(t_h) = 0; aoqi@0: t_l = r-((t_h-3.0)-s2); aoqi@0: /* u+v = ss*(1+...) */ aoqi@0: u = s_h*t_h; aoqi@0: v = s_l*t_h+t_l*ss; aoqi@0: /* 2/(3log2)*(ss+...) */ aoqi@0: p_h = u+v; aoqi@0: __LO(p_h) = 0; aoqi@0: p_l = v-(p_h-u); aoqi@0: z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ aoqi@0: z_l = cp_l*p_h+p_l*cp+dp_l[k]; aoqi@0: /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */ aoqi@0: t = (double)n; aoqi@0: t1 = (((z_h+z_l)+dp_h[k])+t); aoqi@0: __LO(t1) = 0; aoqi@0: t2 = z_l-(((t1-t)-dp_h[k])-z_h); aoqi@0: } aoqi@0: aoqi@0: /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ aoqi@0: y1 = y; aoqi@0: __LO(y1) = 0; aoqi@0: p_l = (y-y1)*t1+y*t2; aoqi@0: p_h = y1*t1; aoqi@0: z = p_l+p_h; aoqi@0: j = __HI(z); aoqi@0: i = __LO(z); aoqi@0: if (j>=0x40900000) { /* z >= 1024 */ aoqi@0: if(((j-0x40900000)|i)!=0) /* if z > 1024 */ aoqi@0: return s*hugeX*hugeX; /* overflow */ aoqi@0: else { aoqi@0: if(p_l+ovt>z-p_h) return s*hugeX*hugeX; /* overflow */ aoqi@0: } aoqi@0: } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ aoqi@0: if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ aoqi@0: return s*tiny*tiny; /* underflow */ aoqi@0: else { aoqi@0: if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ aoqi@0: } aoqi@0: } aoqi@0: /* aoqi@0: * compute 2**(p_h+p_l) aoqi@0: */ aoqi@0: i = j&0x7fffffff; aoqi@0: k = (i>>20)-0x3ff; aoqi@0: n = 0; aoqi@0: if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ aoqi@0: n = j+(0x00100000>>(k+1)); aoqi@0: k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ aoqi@0: t = zeroX; aoqi@0: __HI(t) = (n&~(0x000fffff>>k)); aoqi@0: n = ((n&0x000fffff)|0x00100000)>>(20-k); aoqi@0: if(j<0) n = -n; aoqi@0: p_h -= t; aoqi@0: } aoqi@0: t = p_l+p_h; aoqi@0: __LO(t) = 0; aoqi@0: u = t*lg2_h; aoqi@0: v = (p_l-(t-p_h))*lg2+t*lg2_l; aoqi@0: z = u+v; aoqi@0: w = v-(z-u); aoqi@0: t = z*z; aoqi@0: t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); aoqi@0: r = (z*t1)/(t1-two)-(w+z*w); aoqi@0: z = one-(r-z); aoqi@0: j = __HI(z); aoqi@0: j += (n<<20); aoqi@0: if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ aoqi@0: else __HI(z) += (n<<20); aoqi@0: return s*z; aoqi@0: } aoqi@0: aoqi@0: aoqi@0: JRT_LEAF(jdouble, SharedRuntime::dpow(jdouble x, jdouble y)) aoqi@0: return __ieee754_pow(x, y); aoqi@0: JRT_END aoqi@0: aoqi@0: #ifdef WIN32 aoqi@0: # pragma optimize ( "", on ) aoqi@0: #endif