| /* |
| * 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; |
| } |
| } |
| } |
| } |