/* * Copyright (c) 1998, 2016, Oracle and/or its affiliates. All rights reserved. * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. * * This code is free software; you can redistribute it and/or modify it * under the terms of the GNU General Public License version 2 only, as * published by the Free Software Foundation. Oracle designates this * particular file as subject to the "Classpath" exception as provided * by Oracle in the LICENSE file that accompanied this code. * * This code is distributed in the hope that it will be useful, but WITHOUT * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License * version 2 for more details (a copy is included in the LICENSE file that * accompanied this code). * * You should have received a copy of the GNU General Public License version * 2 along with this work; if not, write to the Free Software Foundation, * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. * * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA * or visit www.oracle.com if you need additional information or have any * questions. */ /** * A transliteration of the "Freely Distributable Math Library" * algorithms from C into Java. That is, this port of the algorithms * is as close to the C originals as possible while still being * readable legal Java. */ public class FdlibmTranslit { private FdlibmTranslit() { throw new UnsupportedOperationException("No FdLibmTranslit instances for you."); } /** * Return the low-order 32 bits of the double argument as an int. */ private static int __LO(double x) { long transducer = Double.doubleToRawLongBits(x); return (int)transducer; } /** * Return a double with its low-order bits of the second argument * and the high-order bits of the first argument.. */ private static double __LO(double x, int low) { long transX = Double.doubleToRawLongBits(x); return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L) | (low & 0x0000_0000_FFFF_FFFFL)); } /** * Return the high-order 32 bits of the double argument as an int. */ private static int __HI(double x) { long transducer = Double.doubleToRawLongBits(x); return (int)(transducer >> 32); } /** * Return a double with its high-order bits of the second argument * and the low-order bits of the first argument.. */ private static double __HI(double x, int high) { long transX = Double.doubleToRawLongBits(x); return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL) | ( ((long)high)) << 32 ); } public static double hypot(double x, double y) { return Hypot.compute(x, y); } /** * cbrt(x) * Return cube root of x */ public static class Cbrt { // unsigned private static final int B1 = 715094163; /* B1 = (682-0.03306235651)*2**20 */ private static final int B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */ private static final double C = 5.42857142857142815906e-01; /* 19/35 = 0x3FE15F15, 0xF15F15F1 */ private static final double D = -7.05306122448979611050e-01; /* -864/1225 = 0xBFE691DE, 0x2532C834 */ private static final double E = 1.41428571428571436819e+00; /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */ private static final double F = 1.60714285714285720630e+00; /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */ private static final double G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */ public static strictfp double compute(double x) { int hx; double r, s, t=0.0, w; int sign; // unsigned hx = __HI(x); // high word of x sign = hx & 0x80000000; // sign= sign(x) hx ^= sign; if (hx >= 0x7ff00000) return (x+x); // cbrt(NaN,INF) is itself if ((hx | __LO(x)) == 0) return(x); // cbrt(0) is itself x = __HI(x, hx); // x <- |x| // rough cbrt to 5 bits if (hx < 0x00100000) { // subnormal number t = __HI(t, 0x43500000); // set t= 2**54 t *= x; t = __HI(t, __HI(t)/3+B2); } else { t = __HI(t, hx/3+B1); } // new cbrt to 23 bits, may be implemented in single precision r = t * t/x; s = C + r*t; t *= G + F/(s + E + D/s); // chopped to 20 bits and make it larger than cbrt(x) t = __LO(t, 0); t = __HI(t, __HI(t)+0x00000001); // one step newton iteration to 53 bits with error less than 0.667 ulps s = t * t; // t*t is exact r = x / s; w = t + t; r= (r - t)/(w + r); // r-s is exact t= t + t*r; // retore the sign bit t = __HI(t, __HI(t) | sign); return(t); } } /** * hypot(x,y) * * Method : * If (assume round-to-nearest) z = x*x + y*y * has error less than sqrt(2)/2 ulp, than * sqrt(z) has error less than 1 ulp (exercise). * * So, compute sqrt(x*x + y*y) with some care as * follows to get the error below 1 ulp: * * Assume x > y > 0; * (if possible, set rounding to round-to-nearest) * 1. if x > 2y use * x1*x1 + (y*y + (x2*(x + x1))) for x*x + y*y * where x1 = x with lower 32 bits cleared, x2 = x - x1; else * 2. if x <= 2y use * t1*y1 + ((x-y) * (x-y) + (t1*y2 + t2*y)) * where t1 = 2x with lower 32 bits cleared, t2 = 2x - t1, * y1= y with lower 32 bits chopped, y2 = y - y1. * * NOTE: scaling may be necessary if some argument is too * large or too tiny * * Special cases: * hypot(x,y) is INF if x or y is +INF or -INF; else * hypot(x,y) is NAN if x or y is NAN. * * Accuracy: * hypot(x,y) returns sqrt(x^2 + y^2) with error less * than 1 ulps (units in the last place) */ static class Hypot { public static double compute(double x, double y) { double a = x; double b = y; double t1, t2, y1, y2, w; int j, k, ha, hb; ha = __HI(x) & 0x7fffffff; // high word of x hb = __HI(y) & 0x7fffffff; // high word of y if(hb > ha) { a = y; b = x; j = ha; ha = hb; hb = j; } else { a = x; b = y; } a = __HI(a, ha); // a <- |a| b = __HI(b, hb); // b <- |b| if ((ha - hb) > 0x3c00000) { return a + b; // x / y > 2**60 } k=0; if (ha > 0x5f300000) { // a>2**500 if (ha >= 0x7ff00000) { // Inf or NaN w = a + b; // for sNaN if (((ha & 0xfffff) | __LO(a)) == 0) w = a; if (((hb ^ 0x7ff00000) | __LO(b)) == 0) w = b; return w; } // scale a and b by 2**-600 ha -= 0x25800000; hb -= 0x25800000; k += 600; a = __HI(a, ha); b = __HI(b, hb); } if (hb < 0x20b00000) { // b < 2**-500 if (hb <= 0x000fffff) { // subnormal b or 0 */ if ((hb | (__LO(b))) == 0) return a; t1 = 0; t1 = __HI(t1, 0x7fd00000); // t1=2^1022 b *= t1; a *= t1; k -= 1022; } else { // scale a and b by 2^600 ha += 0x25800000; // a *= 2^600 hb += 0x25800000; // b *= 2^600 k -= 600; a = __HI(a, ha); b = __HI(b, hb); } } // medium size a and b w = a - b; if (w > b) { t1 = 0; t1 = __HI(t1, ha); t2 = a - t1; w = Math.sqrt(t1*t1 - (b*(-b) - t2 * (a + t1))); } else { a = a + a; y1 = 0; y1 = __HI(y1, hb); y2 = b - y1; t1 = 0; t1 = __HI(t1, ha + 0x00100000); t2 = a - t1; w = Math.sqrt(t1*y1 - (w*(-w) - (t1*y2 + t2*b))); } if (k != 0) { t1 = 1.0; int t1_hi = __HI(t1); t1_hi += (k << 20); t1 = __HI(t1, t1_hi); return t1 * w; } else return w; } } /** * Returns the exponential of x. * * Method * 1. Argument reduction: * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. * Given x, find r and integer k such that * * x = k*ln2 + r, |r| <= 0.5*ln2. * * Here r will be represented as r = hi-lo for better * accuracy. * * 2. Approximation of exp(r) by a special rational function on * the interval [0,0.34658]: * Write * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... * We use a special Reme algorithm on [0,0.34658] to generate * a polynomial of degree 5 to approximate R. The maximum error * of this polynomial approximation is bounded by 2**-59. In * other words, * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 * (where z=r*r, and the values of P1 to P5 are listed below) * and * | 5 | -59 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 * | | * The computation of exp(r) thus becomes * 2*r * exp(r) = 1 + ------- * R - r * r*R1(r) * = 1 + r + ----------- (for better accuracy) * 2 - R1(r) * where * 2 4 10 * R1(r) = r - (P1*r + P2*r + ... + P5*r ). * * 3. Scale back to obtain exp(x): * From step 1, we have * exp(x) = 2^k * exp(r) * * Special cases: * exp(INF) is INF, exp(NaN) is NaN; * exp(-INF) is 0, and * for finite argument, only exp(0)=1 is exact. * * Accuracy: * according to an error analysis, the error is always less than * 1 ulp (unit in the last place). * * Misc. info. * For IEEE double * if x > 7.09782712893383973096e+02 then exp(x) overflow * if x < -7.45133219101941108420e+02 then exp(x) underflow * * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ static class Exp { private static final double one = 1.0; private static final double[] halF = {0.5,-0.5,}; private static final double huge = 1.0e+300; private static final double twom1000= 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0*/ private static final double o_threshold= 7.09782712893383973096e+02; /* 0x40862E42, 0xFEFA39EF */ private static final double u_threshold= -7.45133219101941108420e+02; /* 0xc0874910, 0xD52D3051 */ private static final double[] ln2HI ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ -6.93147180369123816490e-01}; /* 0xbfe62e42, 0xfee00000 */ private static final double[] ln2LO ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ -1.90821492927058770002e-10,}; /* 0xbdea39ef, 0x35793c76 */ private static final double invln2 = 1.44269504088896338700e+00; /* 0x3ff71547, 0x652b82fe */ private static final double P1 = 1.66666666666666019037e-01; /* 0x3FC55555, 0x5555553E */ private static final double P2 = -2.77777777770155933842e-03; /* 0xBF66C16C, 0x16BEBD93 */ private static final double P3 = 6.61375632143793436117e-05; /* 0x3F11566A, 0xAF25DE2C */ private static final double P4 = -1.65339022054652515390e-06; /* 0xBEBBBD41, 0xC5D26BF1 */ private static final double P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ public static strictfp double compute(double x) { double y,hi=0,lo=0,c,t; int k=0,xsb; /*unsigned*/ int hx; hx = __HI(x); /* high word of x */ xsb = (hx>>31)&1; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out non-finite argument */ if(hx >= 0x40862E42) { /* if |x|>=709.78... */ if(hx>=0x7ff00000) { if(((hx&0xfffff)|__LO(x))!=0) return x+x; /* NaN */ else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ } if(x > o_threshold) return huge*huge; /* overflow */ if(x < u_threshold) return twom1000*twom1000; /* underflow */ } /* argument reduction */ if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; } else { k = (int)(invln2*x+halF[xsb]); t = k; hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ lo = t*ln2LO[0]; } x = hi - lo; } else if(hx < 0x3e300000) { /* when |x|<2**-28 */ if(huge+x>one) return one+x;/* trigger inexact */ } else k = 0; /* x is now in primary range */ t = x*x; c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); if(k==0) return one-((x*c)/(c-2.0)-x); else y = one-((lo-(x*c)/(2.0-c))-hi); if(k >= -1021) { y = __HI(y, __HI(y) + (k<<20)); /* add k to y's exponent */ return y; } else { y = __HI(y, __HI(y) + ((k+1000)<<20));/* add k to y's exponent */ return y*twom1000; } } } }