/*
 * Copyright (c) 1998, 2023, 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.
 *
 * 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 sin(double x) {
        return Sin.compute(x);
    }

    public static double cos(double x) {
        return Cos.compute(x);
    }

    public static double tan(double x) {
        return Tan.compute(x);
    }

    public static double asin(double x) {
        return Asin.compute(x);
    }

    public static double acos(double x) {
        return Acos.compute(x);
    }

    public static double atan(double x) {
        return Atan.compute(x);
    }

    public static double atan2(double y, double x) {
        return Atan2.compute(y, x);
    }

    public static double hypot(double x, double y) {
        return Hypot.compute(x, y);
    }

    public static double sqrt(double x) {
        return Sqrt.compute(x);
    }

    public static double cbrt(double x) {
        return Cbrt.compute(x);
    }

    public static double log(double x) {
        return Log.compute(x);
    }

    public static double log10(double x) {
        return Log10.compute(x);
    }

    public static double log1p(double x) {
        return Log1p.compute(x);
    }

    public static double exp(double x) {
        return Exp.compute(x);
    }

    public static double expm1(double x) {
        return Expm1.compute(x);
    }

    public static double sinh(double x) {
        return Sinh.compute(x);
    }

    public static double cosh(double x) {
        return Cosh.compute(x);
    }

    public static double tanh(double x) {
        return Tanh.compute(x);
    }

    public static double IEEEremainder(double f1, double f2) {
        return IEEEremainder.compute(f1, f2);
    }

    // -----------------------------------------------------------------------------------------

    /** sin(x)
     * Return sine function of x.
     *
     * kernel function:
     *      __kernel_sin            ... sine function on [-pi/4,pi/4]
     *      __kernel_cos            ... cose function on [-pi/4,pi/4]
     *      __ieee754_rem_pio2      ... argument reduction routine
     *
     * Method.
     *      Let S,C and T denote the sin, cos and tan respectively on
     *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
     *      in [-pi/4 , +pi/4], and let n = k mod 4.
     *      We have
     *
     *          n        sin(x)      cos(x)        tan(x)
     *     ----------------------------------------------------------
     *          0          S           C             T
     *          1          C          -S            -1/T
     *          2         -S          -C             T
     *          3         -C           S            -1/T
     *     ----------------------------------------------------------
     *
     * Special cases:
     *      Let trig be any of sin, cos, or tan.
     *      trig(+-INF)  is NaN, with signals;
     *      trig(NaN)    is that NaN;
     *
     * Accuracy:
     *      TRIG(x) returns trig(x) nearly rounded
     */
    static class Sin {
        static double compute(double x) {
            double[] y = new double[2];
            double z=0.0;
            int n, ix;

            /* High word of x. */
            ix = __HI(x);

            /* |x| ~< pi/4 */
            ix &= 0x7fffffff;
            if(ix <= 0x3fe921fb) return __kernel_sin(x,z,0);

            /* sin(Inf or NaN) is NaN */
            else if (ix>=0x7ff00000) return x-x;

            /* argument reduction needed */
            else {
                n = RemPio2.__ieee754_rem_pio2(x,y);
                switch(n&3) {
                case 0: return  Sin.__kernel_sin(y[0],y[1],1);
                case 1: return  Cos.__kernel_cos(y[0],y[1]);
                case 2: return -Sin.__kernel_sin(y[0],y[1],1);
                default:
                    return -Cos.__kernel_cos(y[0],y[1]);
                }
            }
        }

        /** __kernel_sin( x, y, iy)
         * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854
         * Input x is assumed to be bounded by ~pi/4 in magnitude.
         * Input y is the tail of x.
         * Input iy indicates whether y is 0. (if iy=0, y assume to be 0).
         *
         * Algorithm
         *      1. Since sin(-x) = -sin(x), we need only to consider positive x.
         *      2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0.
         *      3. sin(x) is approximated by a polynomial of degree 13 on
         *         [0,pi/4]
         *                               3            13
         *              sin(x) ~ x + S1*x + ... + S6*x
         *         where
         *
         *      |sin(x)         2     4     6     8     10     12  |     -58
         *      |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x  +S6*x   )| <= 2
         *      |  x                                               |
         *
         *      4. sin(x+y) = sin(x) + sin'(x')*y
         *                  ~ sin(x) + (1-x*x/2)*y
         *         For better accuracy, let
         *                   3      2      2      2      2
         *              r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6))))
         *         then                   3    2
         *              sin(x) = x + (S1*x + (x *(r-y/2)+y))
         */
        private static final double
            half =  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
            S1  = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */
            S2  =  8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */
            S3  = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */
            S4  =  2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */
            S5  = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */
            S6  =  1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */

        static double __kernel_sin(double x, double y, int iy) {
            double z,r,v;
            int ix;
            ix = __HI(x)&0x7fffffff;        /* high word of x */
            if(ix<0x3e400000)                       /* |x| < 2**-27 */
                {if((int)x==0) return x;}            /* generate inexact */
            z       =  x*x;
            v       =  z*x;
            r       =  S2+z*(S3+z*(S4+z*(S5+z*S6)));
            if(iy==0) return x+v*(S1+z*r);
            else      return x-((z*(half*y-v*r)-y)-v*S1);
        }
    }


    /** cos(x)
     * Return cosine function of x.
     *
     * kernel function:
     *      __kernel_sin            ... sine function on [-pi/4,pi/4]
     *      __kernel_cos            ... cosine function on [-pi/4,pi/4]
     *      __ieee754_rem_pio2      ... argument reduction routine
     *
     * Method.
     *      Let S,C and T denote the sin, cos and tan respectively on
     *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
     *      in [-pi/4 , +pi/4], and let n = k mod 4.
     *      We have
     *
     *          n        sin(x)      cos(x)        tan(x)
     *     ----------------------------------------------------------
     *          0          S           C             T
     *          1          C          -S            -1/T
     *          2         -S          -C             T
     *          3         -C           S            -1/T
     *     ----------------------------------------------------------
     *
     * Special cases:
     *      Let trig be any of sin, cos, or tan.
     *      trig(+-INF)  is NaN, with signals;
     *      trig(NaN)    is that NaN;
     *
     * Accuracy:
     *      TRIG(x) returns trig(x) nearly rounded
     */
    static class Cos {
        static double compute(double x) {
            double[] y = new double[2];
            double z=0.0;
            int n, ix;

            /* High word of x. */
            ix = __HI(x);

            /* |x| ~< pi/4 */
            ix &= 0x7fffffff;
            if(ix <= 0x3fe921fb) return __kernel_cos(x,z);

            /* cos(Inf or NaN) is NaN */
            else if (ix>=0x7ff00000) return x-x;

            /* argument reduction needed */
            else {
                n = RemPio2.__ieee754_rem_pio2(x,y);
                switch(n&3) {
                case 0: return  Cos.__kernel_cos(y[0],y[1]);
                case 1: return -Sin.__kernel_sin(y[0],y[1],1);
                case 2: return -Cos.__kernel_cos(y[0],y[1]);
                default:
                    return  Sin.__kernel_sin(y[0],y[1],1);
                }
            }
        }

        /**
         * __kernel_cos( x,  y )
         * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
         * Input x is assumed to be bounded by ~pi/4 in magnitude.
         * Input y is the tail of x.
         *
         * Algorithm
         *      1. Since cos(-x) = cos(x), we need only to consider positive x.
         *      2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
         *      3. cos(x) is approximated by a polynomial of degree 14 on
         *         [0,pi/4]
         *                                       4            14
         *              cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
         *         where the remez error is
         *
         *      |              2     4     6     8     10    12     14 |     -58
         *      |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
         *      |                                                      |
         *
         *                     4     6     8     10    12     14
         *      4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
         *             cos(x) = 1 - x*x/2 + r
         *         since cos(x+y) ~ cos(x) - sin(x)*y
         *                        ~ cos(x) - x*y,
         *         a correction term is necessary in cos(x) and hence
         *              cos(x+y) = 1 - (x*x/2 - (r - x*y))
         *         For better accuracy when x > 0.3, let qx = |x|/4 with
         *         the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
         *         Then
         *              cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
         *         Note that 1-qx and (x*x/2-qx) is EXACT here, and the
         *         magnitude of the latter is at least a quarter of x*x/2,
         *         thus, reducing the rounding error in the subtraction.
         */
        private static final double
            one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
            C1  =  4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
            C2  = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
            C3  =  2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
            C4  = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
            C5  =  2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
            C6  = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */

        static double __kernel_cos(double x, double y) {
            double a,hz,z,r,qx = 0.0;
            int ix;
            ix = __HI(x)&0x7fffffff;        /* ix = |x|'s high word*/
            if(ix<0x3e400000) {                     /* if x < 2**27 */
                if(((int)x)==0) return one;         /* generate inexact */
            }
            z  = x*x;
            r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
            if(ix < 0x3FD33333)                     /* if |x| < 0.3 */
                return one - (0.5*z - (z*r - x*y));
            else {
                if(ix > 0x3fe90000) {               /* x > 0.78125 */
                    qx = 0.28125;
                } else {
                    //__HI(qx) = ix-0x00200000;       /* x/4 */
                    qx = __HI(qx, ix-0x00200000);
                    // __LO(qx) = 0;
                    qx = __LO(qx, 0);
                }
                hz = 0.5*z-qx;
                a  = one-qx;
                return a - (hz - (z*r-x*y));
            }
        }
    }

    /** tan(x)
     * Return tangent function of x.
     *
     * kernel function:
     *      __kernel_tan            ... tangent function on [-pi/4,pi/4]
     *      __ieee754_rem_pio2      ... argument reduction routine
     *
     * Method.
     *      Let S,C and T denote the sin, cos and tan respectively on
     *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
     *      in [-pi/4 , +pi/4], and let n = k mod 4.
     *      We have
     *
     *          n        sin(x)      cos(x)        tan(x)
     *     ----------------------------------------------------------
     *          0          S           C             T
     *          1          C          -S            -1/T
     *          2         -S          -C             T
     *          3         -C           S            -1/T
     *     ----------------------------------------------------------
     *
     * Special cases:
     *      Let trig be any of sin, cos, or tan.
     *      trig(+-INF)  is NaN, with signals;
     *      trig(NaN)    is that NaN;
     *
     * Accuracy:
     *      TRIG(x) returns trig(x) nearly rounded
     */
    static class Tan {
        static double compute(double x) {
            double[] y= new double[2];
            double z=0.0;
            int n, ix;

            /* High word of x. */
            ix = __HI(x);

            /* |x| ~< pi/4 */
            ix &= 0x7fffffff;
            if(ix <= 0x3fe921fb) return __kernel_tan(x,z,1);

            /* tan(Inf or NaN) is NaN */
            else if (ix>=0x7ff00000) return x-x;            /* NaN */

            /* argument reduction needed */
            else {
                n = RemPio2.__ieee754_rem_pio2(x,y);
                return __kernel_tan(y[0],y[1],1-((n&1)<<1)); /*   1 -- n even
                                                                  -1 -- n odd */
            }
        }

        /** __kernel_tan( x, y, k )
         * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
         * Input x is assumed to be bounded by ~pi/4 in magnitude.
         * Input y is the tail of x.
         * Input k indicates whether tan (if k=1) or
         * -1/tan (if k= -1) is returned.
         *
         * Algorithm
         *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
         *      2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
         *      3. tan(x) is approximated by a odd polynomial of degree 27 on
         *         [0,0.67434]
         *                               3             27
         *              tan(x) ~ x + T1*x + ... + T13*x
         *         where
         *
         *              |tan(x)         2     4            26   |     -59.2
         *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
         *              |  x                                    |
         *
         *         Note: tan(x+y) = tan(x) + tan'(x)*y
         *                        ~ tan(x) + (1+x*x)*y
         *         Therefore, for better accuracy in computing tan(x+y), let
         *                   3      2      2       2       2
         *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
         *         then
         *                                  3    2
         *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
         *
         *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
         *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
         *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
         */
        private static final double
            one   =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
            pio4  =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
            pio4lo=  3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
            T[] =  {
            3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
            1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
            5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
            2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
            8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
            3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
            1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
            5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
            2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
            7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
            7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
            -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
            2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
        };

        static double __kernel_tan(double x, double y, int iy) {
            double z,r,v,w,s;
            int ix,hx;
            hx = __HI(x);   /* high word of x */
            ix = hx&0x7fffffff;     /* high word of |x| */
            if(ix<0x3e300000) {                     /* x < 2**-28 */
                if((int)x==0) {                       /* generate inexact */
                    if (((ix | __LO(x)) | (iy + 1)) == 0)
                        return one / Math.abs(x);
                    else {
                        if (iy == 1)
                            return x;
                        else {    /* compute -1 / (x+y) carefully */
                            double a, t;

                            z = w = x + y;
                            // __LO(z) = 0;
                            z= __LO(z, 0);
                            v = y - (z - x);
                            t = a = -one / w;
                            //__LO(t) = 0;
                            t = __LO(t, 0);
                            s = one + t * z;
                            return t + a * (s + t * v);
                        }
                    }
                }
            }
            if(ix>=0x3FE59428) {                    /* |x|>=0.6744 */
                if(hx<0) {x = -x; y = -y;}
                z = pio4-x;
                w = pio4lo-y;
                x = z+w; y = 0.0;
            }
            z       =  x*x;
            w       =  z*z;
            /* Break x^5*(T[1]+x^2*T[2]+...) into
             *    x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
             *    x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
             */
            r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
            v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
            s = z*x;
            r = y + z*(s*(r+v)+y);
            r += T[0]*s;
            w = x+r;
            if(ix>=0x3FE59428) {
                v = (double)iy;
                return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
            }
            if(iy==1) return w;
            else {          /* if allow error up to 2 ulp,
                               simply return -1.0/(x+r) here */
                /*  compute -1.0/(x+r) accurately */
                double a,t;
                z  = w;
                // __LO(z) = 0;
                z = __LO(z, 0);
                v  = r-(z - x);     /* z+v = r+x */
                t = a  = -1.0/w;    /* a = -1.0/w */
                // __LO(t) = 0;
                t = __LO(t, 0);
                s  = 1.0+t*z;
                return t+a*(s+t*v);
            }
        }
    }

    /** __ieee754_rem_pio2(x,y)
     *
     * return the remainder of x rem pi/2 in y[0]+y[1]
     * use __kernel_rem_pio2()
     */
    static class RemPio2 {
        /*
         * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
         */
        private static final int[] two_over_pi = {
            0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
            0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
            0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
            0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
            0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
            0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
            0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
            0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
            0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
            0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
            0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
        };

        private static final int[] npio2_hw = {
            0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
            0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
            0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
            0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
            0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
            0x404858EB, 0x404921FB,
        };

        /*
         * invpio2:  53 bits of 2/pi
         * pio2_1:   first  33 bit of pi/2
         * pio2_1t:  pi/2 - pio2_1
         * pio2_2:   second 33 bit of pi/2
         * pio2_2t:  pi/2 - (pio2_1+pio2_2)
         * pio2_3:   third  33 bit of pi/2
         * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
         */

        private static final double
            zero =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
            half =  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
            two24 =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
            invpio2 =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
            pio2_1  =  1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
            pio2_1t =  6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
            pio2_2  =  6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
            pio2_2t =  2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
            pio2_3  =  2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
            pio2_3t =  8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */

        static int __ieee754_rem_pio2(double x, double[] y) {
            double z = 0.0,w,t,r,fn;
            double[] tx = new double[3];
            int e0,i,j,nx,n,ix,hx;

            hx = __HI(x);           /* high word of x */
            ix = hx&0x7fffffff;
            if(ix<=0x3fe921fb)   /* |x| ~<= pi/4 , no need for reduction */
                {y[0] = x; y[1] = 0; return 0;}
            if(ix<0x4002d97c) {  /* |x| < 3pi/4, special case with n=+-1 */
                if(hx>0) {
                    z = x - pio2_1;
                    if(ix!=0x3ff921fb) {    /* 33+53 bit pi is good enough */
                        y[0] = z - pio2_1t;
                        y[1] = (z-y[0])-pio2_1t;
                    } else {                /* near pi/2, use 33+33+53 bit pi */
                        z -= pio2_2;
                        y[0] = z - pio2_2t;
                        y[1] = (z-y[0])-pio2_2t;
                    }
                    return 1;
                } else {    /* negative x */
                    z = x + pio2_1;
                    if(ix!=0x3ff921fb) {    /* 33+53 bit pi is good enough */
                        y[0] = z + pio2_1t;
                        y[1] = (z-y[0])+pio2_1t;
                    } else {                /* near pi/2, use 33+33+53 bit pi */
                        z += pio2_2;
                        y[0] = z + pio2_2t;
                        y[1] = (z-y[0])+pio2_2t;
                    }
                    return -1;
                }
            }
            if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
                t  = Math.abs(x);
                n  = (int) (t*invpio2+half);
                fn = (double)n;
                r  = t-fn*pio2_1;
                w  = fn*pio2_1t;    /* 1st round good to 85 bit */
                if(n<32&&ix!=npio2_hw[n-1]) {
                    y[0] = r-w;     /* quick check no cancellation */
                } else {
                    j  = ix>>20;
                    y[0] = r-w;
                    i = j-(((__HI(y[0]))>>20)&0x7ff);
                    if(i>16) {  /* 2nd iteration needed, good to 118 */
                        t  = r;
                        w  = fn*pio2_2;
                        r  = t-w;
                        w  = fn*pio2_2t-((t-r)-w);
                        y[0] = r-w;
                        i = j-(((__HI(y[0]))>>20)&0x7ff);
                        if(i>49)  { /* 3rd iteration need, 151 bits acc */
                            t  = r; /* will cover all possible cases */
                            w  = fn*pio2_3;
                            r  = t-w;
                            w  = fn*pio2_3t-((t-r)-w);
                            y[0] = r-w;
                        }
                    }
                }
                y[1] = (r-y[0])-w;
                if(hx<0)    {y[0] = -y[0]; y[1] = -y[1]; return -n;}
                else         return n;
            }
            /*
             * all other (large) arguments
             */
            if(ix>=0x7ff00000) {            /* x is inf or NaN */
                y[0]=y[1]=x-x; return 0;
            }
            /* set z = scalbn(|x|,ilogb(x)-23) */
            // __LO(z) = __LO(x);
            z = __LO(z, __LO(x));
            e0      = (ix>>20)-1046;        /* e0 = ilogb(z)-23; */
            // __HI(z) = ix - (e0<<20);
            z = __HI(z, ix - (e0<<20));
            for(i=0;i<2;i++) {
                tx[i] = (double)((int)(z));
                z     = (z-tx[i])*two24;
            }
            tx[2] = z;
            nx = 3;
            while(tx[nx-1]==zero) nx--;     /* skip zero term */
            n  =  KernelRemPio2.__kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi);
            if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
            return n;
        }

    }

    /**
     * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
     * double x[],y[]; int e0,nx,prec; int ipio2[];
     *
     * __kernel_rem_pio2 return the last three digits of N with
     *              y = x - N*pi/2
     * so that |y| < pi/2.
     *
     * The method is to compute the integer (mod 8) and fraction parts of
     * (2/pi)*x without doing the full multiplication. In general we
     * skip the part of the product that are known to be a huge integer (
     * more accurately, = 0 mod 8 ). Thus the number of operations are
     * independent of the exponent of the input.
     *
     * (2/pi) is represented by an array of 24-bit integers in ipio2[].
     *
     * Input parameters:
     *      x[]     The input value (must be positive) is broken into nx
     *              pieces of 24-bit integers in double precision format.
     *              x[i] will be the i-th 24 bit of x. The scaled exponent
     *              of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
     *              match x's up to 24 bits.
     *
     *              Example of breaking a double positive z into x[0]+x[1]+x[2]:
     *                      e0 = ilogb(z)-23
     *                      z  = scalbn(z,-e0)
     *              for i = 0,1,2
     *                      x[i] = floor(z)
     *                      z    = (z-x[i])*2**24
     *
     *
     *      y[]     output result in an array of double precision numbers.
     *              The dimension of y[] is:
     *                      24-bit  precision       1
     *                      53-bit  precision       2
     *                      64-bit  precision       2
     *                      113-bit precision       3
     *              The actual value is the sum of them. Thus for 113-bit
     *              precision, one may have to do something like:
     *
     *              long double t,w,r_head, r_tail;
     *              t = (long double)y[2] + (long double)y[1];
     *              w = (long double)y[0];
     *              r_head = t+w;
     *              r_tail = w - (r_head - t);
     *
     *      e0      The exponent of x[0]
     *
     *      nx      dimension of x[]
     *
     *      prec    an integer indicating the precision:
     *                      0       24  bits (single)
     *                      1       53  bits (double)
     *                      2       64  bits (extended)
     *                      3       113 bits (quad)
     *
     *      ipio2[]
     *              integer array, contains the (24*i)-th to (24*i+23)-th
     *              bit of 2/pi after binary point. The corresponding
     *              floating value is
     *
     *                      ipio2[i] * 2^(-24(i+1)).
     *
     * External function:
     *      double scalbn(), floor();
     *
     *
     * Here is the description of some local variables:
     *
     *      jk      jk+1 is the initial number of terms of ipio2[] needed
     *              in the computation. The recommended value is 2,3,4,
     *              6 for single, double, extended,and quad.
     *
     *      jz      local integer variable indicating the number of
     *              terms of ipio2[] used.
     *
     *      jx      nx - 1
     *
     *      jv      index for pointing to the suitable ipio2[] for the
     *              computation. In general, we want
     *                      ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
     *              is an integer. Thus
     *                      e0-3-24*jv >= 0 or (e0-3)/24 >= jv
     *              Hence jv = max(0,(e0-3)/24).
     *
     *      jp      jp+1 is the number of terms in PIo2[] needed, jp = jk.
     *
     *      q[]     double array with integral value, representing the
     *              24-bits chunk of the product of x and 2/pi.
     *
     *      q0      the corresponding exponent of q[0]. Note that the
     *              exponent for q[i] would be q0-24*i.
     *
     *      PIo2[]  double precision array, obtained by cutting pi/2
     *              into 24 bits chunks.
     *
     *      f[]     ipio2[] in floating point
     *
     *      iq[]    integer array by breaking up q[] in 24-bits chunk.
     *
     *      fq[]    final product of x*(2/pi) in fq[0],..,fq[jk]
     *
     *      ih      integer. If >0 it indicates q[] is >= 0.5, hence
     *              it also indicates the *sign* of the result.
     *
     */
    static class KernelRemPio2 {
        /*
         * 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.
         */

        private static final int[] init_jk = {2,3,4,6}; /* initial value for jk */

        private static final double[] PIo2 = {
            1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
            7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
            5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
            3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
            1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
            1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
            2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
            2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
        };
        static final double
            zero   = 0.0,
            one    = 1.0,
            two24   =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
            twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */

        static int __kernel_rem_pio2(double[] x, double[] y, int e0, int nx, int prec, final int[] ipio2) {
            int jz,jx,jv,jp,jk,carry,n,i,j,k,m,q0,ih;
            int[] iq = new int[20];
            double z,fw;
            double [] f = new double[20];
            double [] fq= new double[20];
            double [] q = new double[20];

            /* initialize jk*/
            jk = init_jk[prec];
            jp = jk;

            /* determine jx,jv,q0, note that 3>q0 */
            jx =  nx-1;
            jv = (e0-3)/24; if(jv<0) jv=0;
            q0 =  e0-24*(jv+1);

            /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
            j = jv-jx; m = jx+jk;
            for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];

            /* compute q[0],q[1],...q[jk] */
            for (i=0;i<=jk;i++) {
                for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
                q[i] = fw;
            }

            jz = jk;
            /*
             * Transliteration note: the FDLIBM C sources have a
             * "recompute:" label at this point and a "goto
             * recompute;" later on at the indicated point. This
             * structure was replaced by wrapping the code in the
             * while(true){...} loop below, replacing the goto with
             * the continue to re-execute the loop and by adding
             * breaks to exit the loop on the other control flow
             * paths.
             */
            while(true) {
                /* distill q[] into iq[] reversingly */
                for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
                    fw    =  (double)((int)(twon24* z));
                    iq[i] =  (int)(z-two24*fw);
                    z     =  q[j-1]+fw;
                }

                /* compute n */
                z  = Math.scalb(z,q0);              /* actual value of z */
                z -= 8.0*Math.floor(z*0.125);           /* trim off integer >= 8 */
                n  = (int) z;
                z -= (double)n;
                ih = 0;
                if(q0>0) {      /* need iq[jz-1] to determine n */
                    i  = (iq[jz-1]>>(24-q0)); n += i;
                    iq[jz-1] -= i<<(24-q0);
                    ih = iq[jz-1]>>(23-q0);
                }
                else if(q0==0) ih = iq[jz-1]>>23;
                else if(z>=0.5) ih=2;

                if(ih>0) {      /* q > 0.5 */
                    n += 1; carry = 0;
                    for(i=0;i<jz ;i++) {        /* compute 1-q */
                        j = iq[i];
                        if(carry==0) {
                            if(j!=0) {
                                carry = 1; iq[i] = 0x1000000- j;
                            }
                        } else  iq[i] = 0xffffff - j;
                    }
                    if(q0>0) {          /* rare case: chance is 1 in 12 */
                        switch(q0) {
                        case 1:
                            iq[jz-1] &= 0x7fffff; break;
                        case 2:
                            iq[jz-1] &= 0x3fffff; break;
                        }
                    }
                    if(ih==2) {
                        z = one - z;
                        if(carry!=0) z -= Math.scalb(one,q0);
                    }
                }

                /* check if recomputation is needed */
                if(z==zero) {
                    j = 0;
                    for (i=jz-1;i>=jk;i--) j |= iq[i];
                    if(j==0) { /* need recomputation */
                        for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */

                        for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */
                            f[jx+i] = (double) ipio2[jv+i];
                            for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
                            q[i] = fw;
                        }
                        jz += k;
                        // At this point "goto recompute;" in the original C sources.
                        continue;
                    } else { break;}
                } else {break;}
            }

            /* chop off zero terms */
            if(z==0.0) {
                jz -= 1; q0 -= 24;
                while(iq[jz]==0) { jz--; q0-=24;}
            } else { /* break z into 24-bit if necessary */
                z = Math.scalb(z,-q0);
                if(z>=two24) {
                    fw = (double)((int)(twon24*z));
                    iq[jz] = (int)(z-two24*fw);
                    jz += 1; q0 += 24;
                    iq[jz] = (int) fw;
                } else iq[jz] = (int) z ;
            }

            /* convert integer "bit" chunk to floating-point value */
            fw = Math.scalb(one,q0);
            for(i=jz;i>=0;i--) {
                q[i] = fw*(double)iq[i]; fw*=twon24;
            }

            /* compute PIo2[0,...,jp]*q[jz,...,0] */
            for(i=jz;i>=0;i--) {
                for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
                fq[jz-i] = fw;
            }

            /* compress fq[] into y[] */
            switch(prec) {
            case 0:
                fw = 0.0;
                for (i=jz;i>=0;i--) fw += fq[i];
                y[0] = (ih==0)? fw: -fw;
                break;
            case 1:
            case 2:
                fw = 0.0;
                for (i=jz;i>=0;i--) fw += fq[i];
                y[0] = (ih==0)? fw: -fw;
                fw = fq[0]-fw;
                for (i=1;i<=jz;i++) fw += fq[i];
                y[1] = (ih==0)? fw: -fw;
                break;
            case 3:     /* painful */
                for (i=jz;i>0;i--) {
                    fw      = fq[i-1]+fq[i];
                    fq[i]  += fq[i-1]-fw;
                    fq[i-1] = fw;
                }
                for (i=jz;i>1;i--) {
                    fw      = fq[i-1]+fq[i];
                    fq[i]  += fq[i-1]-fw;
                    fq[i-1] = fw;
                }
                for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
                if(ih==0) {
                    y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;
                } else {
                    y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
                }
            }
            return n&7;
        }
    }

    /** Returns the arcsine of x.
     *
     * Method :
     *      Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
     *      we approximate asin(x) on [0,0.5] by
     *              asin(x) = x + x*x^2*R(x^2)
     *      where
     *              R(x^2) is a rational approximation of (asin(x)-x)/x^3
     *      and its remez error is bounded by
     *              |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
     *
     *      For x in [0.5,1]
     *              asin(x) = pi/2-2*asin(sqrt((1-x)/2))
     *      Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
     *      then for x>0.98
     *              asin(x) = pi/2 - 2*(s+s*z*R(z))
     *                      = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
     *      For x<=0.98, let pio4_hi = pio2_hi/2, then
     *              f = hi part of s;
     *              c = sqrt(z) - f = (z-f*f)/(s+f)         ...f+c=sqrt(z)
     *      and
     *              asin(x) = pi/2 - 2*(s+s*z*R(z))
     *                      = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
     *                      = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
     *
     * Special cases:
     *      if x is NaN, return x itself;
     *      if |x|>1, return NaN with invalid signal.
     *
     */
    static class Asin {
        private static final double
            one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
            huge =  1.000e+300,
            pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
            pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
            pio4_hi =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
        /* coefficient for R(x^2) */
            pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
            pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
            pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
            pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
            pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
            pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
            qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
            qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
            qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
            qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */

        static double compute(double x) {
            double t=0,w,p,q,c,r,s;
            int hx,ix;
            hx = __HI(x);
            ix = hx&0x7fffffff;
            if(ix>= 0x3ff00000) {           /* |x|>= 1 */
                if(((ix-0x3ff00000)|__LO(x))==0)
                    /* asin(1)=+-pi/2 with inexact */
                    return x*pio2_hi+x*pio2_lo;
                return (x-x)/(x-x);         /* asin(|x|>1) is NaN */
            } else if (ix<0x3fe00000) {     /* |x|<0.5 */
                if(ix<0x3e400000) {         /* if |x| < 2**-27 */
                    if(huge+x>one) return x;/* return x with inexact if x!=0*/
                } else
                    t = x*x;
                p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
                q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
                w = p/q;
                return x+x*w;
            }
            /* 1> |x|>= 0.5 */
            w = one-Math.abs(x);
            t = w*0.5;
            p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
            q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
            s = Math.sqrt(t);
            if(ix>=0x3FEF3333) {    /* if |x| > 0.975 */
                w = p/q;
                t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
            } else {
                w  = s;
                // __LO(w) = 0;
                w  = __LO(w, 0);
                c  = (t-w*w)/(s+w);
                r  = p/q;
                p  = 2.0*s*r-(pio2_lo-2.0*c);
                q  = pio4_hi-2.0*w;
                t  = pio4_hi-(p-q);
            }
            if(hx>0) return t; else return -t;
        }
    }

    /** Returns the arccosine of x.
     * Method :
     *      acos(x)  = pi/2 - asin(x)
     *      acos(-x) = pi/2 + asin(x)
     * For |x|<=0.5
     *      acos(x) = pi/2 - (x + x*x^2*R(x^2))     (see asin.c)
     * For x>0.5
     *      acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
     *              = 2asin(sqrt((1-x)/2))
     *              = 2s + 2s*z*R(z)        ...z=(1-x)/2, s=sqrt(z)
     *              = 2f + (2c + 2s*z*R(z))
     *     where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term
     *     for f so that f+c ~ sqrt(z).
     * For x<-0.5
     *      acos(x) = pi - 2asin(sqrt((1-|x|)/2))
     *              = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)
     *
     * Special cases:
     *      if x is NaN, return x itself;
     *      if |x|>1, return NaN with invalid signal.
     *
     * Function needed: sqrt
     */
    static class Acos {
        private static final double
            one=  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
            pi =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
            pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
            pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
            pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
            pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
            pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
            pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
            pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
            pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
            qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
            qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
            qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
            qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */

        static double compute(double x) {
            double z,p,q,r,w,s,c,df;
            int hx,ix;
            hx = __HI(x);
            ix = hx&0x7fffffff;
            if(ix>=0x3ff00000) {    /* |x| >= 1 */
                if(((ix-0x3ff00000)|__LO(x))==0) {  /* |x|==1 */
                    if(hx>0) return 0.0;            /* acos(1) = 0  */
                    else return pi+2.0*pio2_lo;     /* acos(-1)= pi */
                }
                return (x-x)/(x-x);         /* acos(|x|>1) is NaN */
            }
            if(ix<0x3fe00000) {     /* |x| < 0.5 */
                if(ix<=0x3c600000) return pio2_hi+pio2_lo;/*if|x|<2**-57*/
                z = x*x;
                p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
                q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
                r = p/q;
                return pio2_hi - (x - (pio2_lo-x*r));
            } else  if (hx<0) {             /* x < -0.5 */
                z = (one+x)*0.5;
                p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
                q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
                s = Math.sqrt(z);
                r = p/q;
                w = r*s-pio2_lo;
                return pi - 2.0*(s+w);
            } else {                        /* x > 0.5 */
                z = (one-x)*0.5;
                s = Math.sqrt(z);
                df = s;
                // __LO(df) = 0;
                df = __LO(df, 0);
                c  = (z-df*df)/(s+df);
                p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
                q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
                r = p/q;
                w = r*s+c;
                return 2.0*(df+w);
            }
        }
    }

    /* Returns the arctangent of x.
     * Method
     *   1. Reduce x to positive by atan(x) = -atan(-x).
     *   2. According to the integer k=4t+0.25 chopped, t=x, the argument
     *      is further reduced to one of the following intervals and the
     *      arctangent of t is evaluated by the corresponding formula:
     *
     *      [0,7/16]      atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
     *      [7/16,11/16]  atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
     *      [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
     *      [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
     *      [39/16,INF]   atan(x) = atan(INF) + atan( -1/t )
     *
     * 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 Atan {
        private static final double atanhi[] = {
            4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
            7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
            9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
            1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
        };

        private static final double atanlo[] = {
            2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
            3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
            1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
            6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
        };

        private static final double aT[] = {
             3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */
            -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
             1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */
            -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
             9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */
            -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
             6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */
            -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
             4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */
            -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
             1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */
        };

        private static final double
            one   = 1.0,
            huge   = 1.0e300;

        static double compute(double x) {
            double w,s1,s2,z;
            int ix,hx,id;

            hx = __HI(x);
            ix = hx&0x7fffffff;
            if(ix>=0x44100000) {    /* if |x| >= 2^66 */
                if(ix>0x7ff00000||
                   (ix==0x7ff00000&&(__LO(x)!=0)))
                    return x+x;             /* NaN */
                if(hx>0) return  atanhi[3]+atanlo[3];
                else     return -atanhi[3]-atanlo[3];
            } if (ix < 0x3fdc0000) {        /* |x| < 0.4375 */
                if (ix < 0x3e200000) {      /* |x| < 2^-29 */
                    if(huge+x>one) return x;        /* raise inexact */
                }
                id = -1;
            } else {
                x = Math.abs(x);
                if (ix < 0x3ff30000) {          /* |x| < 1.1875 */
                    if (ix < 0x3fe60000) {      /* 7/16 <=|x|<11/16 */
                        id = 0; x = (2.0*x-one)/(2.0+x);
                    } else {                    /* 11/16<=|x|< 19/16 */
                        id = 1; x  = (x-one)/(x+one);
                    }
                } else {
                    if (ix < 0x40038000) {      /* |x| < 2.4375 */
                        id = 2; x  = (x-1.5)/(one+1.5*x);
                    } else {                    /* 2.4375 <= |x| < 2^66 */
                        id = 3; x  = -1.0/x;
                    }
                }}
            /* end of argument reduction */
            z = x*x;
            w = z*z;
            /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
            s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10])))));
            s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9]))));
            if (id<0) return x - x*(s1+s2);
            else {
                z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
                return (hx<0)? -z:z;
            }
        }
    }

    /**
     * Returns the angle theta from the conversion of rectangular
     * coordinates (x, y) to polar coordinates (r, theta).
     *
     * Method :
     *      1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
     *      2. Reduce x to positive by (if x and y are unexceptional):
     *              ARG (x+iy) = arctan(y/x)           ... if x > 0,
     *              ARG (x+iy) = pi - arctan[y/(-x)]   ... if x < 0,
     *
     * Special cases:
     *
     *      ATAN2((anything), NaN ) is NaN;
     *      ATAN2(NAN , (anything) ) is NaN;
     *      ATAN2(+-0, +(anything but NaN)) is +-0  ;
     *      ATAN2(+-0, -(anything but NaN)) is +-pi ;
     *      ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2;
     *      ATAN2(+-(anything but INF and NaN), +INF) is +-0 ;
     *      ATAN2(+-(anything but INF and NaN), -INF) is +-pi;
     *      ATAN2(+-INF,+INF ) is +-pi/4 ;
     *      ATAN2(+-INF,-INF ) is +-3pi/4;
     *      ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2;
     *
     * 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 Atan2 {
        private static final double
            tiny  = 1.0e-300,
            zero  = 0.0,
            pi_o_4  = 7.8539816339744827900E-01, /* 0x3FE921FB, 0x54442D18 */
            pi_o_2  = 1.5707963267948965580E+00, /* 0x3FF921FB, 0x54442D18 */
            pi      = 3.1415926535897931160E+00, /* 0x400921FB, 0x54442D18 */
            pi_lo   = 1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */

        static double compute(double y, double x) {
            double z;
            int k,m,hx,hy,ix,iy;
            /*unsigned*/ int lx,ly;

            hx = __HI(x); ix = hx&0x7fffffff;
            lx = __LO(x);
            hy = __HI(y); iy = hy&0x7fffffff;
            ly = __LO(y);
            if(((ix|((lx|-lx)>>>31))>0x7ff00000)|| // Note unsigned shifts
               ((iy|((ly|-ly)>>>31))>0x7ff00000))    /* x or y is NaN */
                return x+y;
            if(((hx-0x3ff00000)|lx)==0) return atan(y);   /* x=1.0 */
            m = ((hy>>31)&1)|((hx>>30)&2);  /* 2*sign(x)+sign(y) */

            /* when y = 0 */
            if((iy|ly)==0) {
                switch(m) {
                case 0:
                case 1: return y;       /* atan(+-0,+anything)=+-0 */
                case 2: return  pi+tiny;/* atan(+0,-anything) = pi */
                case 3: return -pi-tiny;/* atan(-0,-anything) =-pi */
                }
            }
            /* when x = 0 */
            if((ix|lx)==0) return (hy<0)?  -pi_o_2-tiny: pi_o_2+tiny;

            /* when x is INF */
            if(ix==0x7ff00000) {
                if(iy==0x7ff00000) {
                    switch(m) {
                    case 0: return  pi_o_4+tiny;/* atan(+INF,+INF) */
                    case 1: return -pi_o_4-tiny;/* atan(-INF,+INF) */
                    case 2: return  3.0*pi_o_4+tiny;/*atan(+INF,-INF)*/
                    case 3: return -3.0*pi_o_4-tiny;/*atan(-INF,-INF)*/
                    }
                } else {
                    switch(m) {
                    case 0: return  zero  ;     /* atan(+...,+INF) */
                    case 1: return -1.0*zero  ; /* atan(-...,+INF) */
                    case 2: return  pi+tiny  ;  /* atan(+...,-INF) */
                    case 3: return -pi-tiny  ;  /* atan(-...,-INF) */
                    }
                }
            }
            /* when y is INF */
            if(iy==0x7ff00000) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny;

            /* compute y/x */
            k = (iy-ix)>>20;
            if(k > 60) z=pi_o_2+0.5*pi_lo;  /* |y/x| >  2**60 */
            else if(hx<0&&k<-60) z=0.0;     /* |y|/x < -2**60 */
            else z=atan(Math.abs(y/x));         /* safe to do y/x */
            switch (m) {
            case 0: return       z  ;   /* atan(+,+) */
            case 1:
                // original:__HI(z) ^= 0x80000000;
                z = __HI(z, __HI(z) ^ 0x80000000);
                return       z  ;   /* atan(-,+) */
            case 2: return  pi-(z-pi_lo);/* atan(+,-) */
            default: /* case 3 */
                return  (z-pi_lo)-pi;/* atan(-,-) */
            }
        }
    }

    /**
     * Return correctly rounded sqrt.
     *           ------------------------------------------
     *           |  Use the hardware sqrt if you have one |
     *           ------------------------------------------
     * Method:
     *   Bit by bit method using integer arithmetic. (Slow, but portable)
     *   1. Normalization
     *      Scale x to y in [1,4) with even powers of 2:
     *      find an integer k such that  1 <= (y=x*2^(2k)) < 4, then
     *              sqrt(x) = 2^k * sqrt(y)
     *   2. Bit by bit computation
     *      Let q  = sqrt(y) truncated to i bit after binary point (q = 1),
     *           i                                                   0
     *                                     i+1         2
     *          s  = 2*q , and      y  =  2   * ( y - q  ).         (1)
     *           i      i            i                 i
     *
     *      To compute q    from q , one checks whether
     *                  i+1       i
     *
     *                            -(i+1) 2
     *                      (q + 2      ) <= y.                     (2)
     *                        i
     *                                                            -(i+1)
     *      If (2) is false, then q   = q ; otherwise q   = q  + 2      .
     *                             i+1   i             i+1   i
     *
     *      With some algebraic manipulation, it is not difficult to see
     *      that (2) is equivalent to
     *                             -(i+1)
     *                      s  +  2       <= y                      (3)
     *                       i                i
     *
     *      The advantage of (3) is that s  and y  can be computed by
     *                                    i      i
     *      the following recurrence formula:
     *          if (3) is false
     *
     *          s     =  s  ,       y    = y   ;                    (4)
     *           i+1      i          i+1    i
     *
     *          otherwise,
     *                         -i                     -(i+1)
     *          s     =  s  + 2  ,  y    = y  -  s  - 2             (5)
     *           i+1      i          i+1    i     i
     *
     *      One may easily use induction to prove (4) and (5).
     *      Note. Since the left hand side of (3) contain only i+2 bits,
     *            it does not necessary to do a full (53-bit) comparison
     *            in (3).
     *   3. Final rounding
     *      After generating the 53 bits result, we compute one more bit.
     *      Together with the remainder, we can decide whether the
     *      result is exact, bigger than 1/2ulp, or less than 1/2ulp
     *      (it will never equal to 1/2ulp).
     *      The rounding mode can be detected by checking whether
     *      huge + tiny is equal to huge, and whether huge - tiny is
     *      equal to huge for some floating point number "huge" and "tiny".
     *
     * Special cases:
     *      sqrt(+-0) = +-0         ... exact
     *      sqrt(inf) = inf
     *      sqrt(-ve) = NaN         ... with invalid signal
     *      sqrt(NaN) = NaN         ... with invalid signal for signaling NaN
     *
     * Other methods : see the appended file at the end of the program below.
     *---------------
     */
    static class Sqrt {
        private static final double    one     = 1.0, tiny=1.0e-300;

        public static double compute(double x) {
            double z = 0.0;
            int     sign = (int)0x80000000;
            /*unsigned*/ int r,t1,s1,ix1,q1;
            int ix0,s0,q,m,t,i;

            ix0 = __HI(x);                  /* high word of x */
            ix1 = __LO(x);          /* low word of x */

            /* take care of Inf and NaN */
            if((ix0&0x7ff00000)==0x7ff00000) {
                return x*x+x;               /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
                                               sqrt(-inf)=sNaN */
            }
            /* take care of zero */
            if(ix0<=0) {
                if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */
                else if(ix0<0)
                    return (x-x)/(x-x);             /* sqrt(-ve) = sNaN */
            }
            /* normalize x */
            m = (ix0>>20);
            if(m==0) {                              /* subnormal x */
                while(ix0==0) {
                    m -= 21;
                    ix0 |= (ix1>>>11); ix1 <<= 21; // unsigned shift
                }
                for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1;
                m -= i-1;
                ix0 |= (ix1>>>(32-i)); // unsigned shift
                ix1 <<= i;
            }
            m -= 1023;      /* unbias exponent */
            ix0 = (ix0&0x000fffff)|0x00100000;
            if((m&1) != 0){        /* odd m, double x to make it even */
                ix0 += ix0 + ((ix1&sign)>>>31); // unsigned shift
                ix1 += ix1;
            }
            m >>= 1;        /* m = [m/2] */

            /* generate sqrt(x) bit by bit */
            ix0 += ix0 + ((ix1&sign)>>>31); // unsigned shift
            ix1 += ix1;
            q = q1 = s0 = s1 = 0;   /* [q,q1] = sqrt(x) */
            r = 0x00200000;         /* r = moving bit from right to left */

            while(r!=0) {
                t = s0+r;
                if(t<=ix0) {
                    s0   = t+r;
                    ix0 -= t;
                    q   += r;
                }
                ix0 += ix0 + ((ix1&sign)>>>31); // unsigned shift
                ix1 += ix1;
                r>>>=1; // unsigned shift
            }

            r = sign;
            while(r!=0) {
                t1 = s1+r;
                t  = s0;
                if((t<ix0)||((t==ix0)&&(Integer.compareUnsigned(t1, ix1) <= 0 ))) { // t1<=ix1
                    s1  = t1+r;
                    if(((t1&sign)==sign)&&(s1&sign)==0) s0 += 1;
                    ix0 -= t;
                    if (Integer.compareUnsigned(ix1, t1) < 0) ix0 -= 1; // ix1 < t1
                    ix1 -= t1;
                    q1  += r;
                }
                ix0 += ix0 + ((ix1&sign)>>>31); // unsigned shift
                ix1 += ix1;
                r>>>=1; // unsigned shift
            }

            /* use floating add to find out rounding direction */
            if((ix0|ix1)!=0) {
                z = one-tiny; /* trigger inexact flag */
                if (z>=one) {
                    z = one+tiny;
                    if (q1==0xffffffff) { q1=0; q += 1;}
                    else if (z>one) {
                        if (q1==0xfffffffe) q+=1;
                        q1+=2;
                    } else
                        q1 += (q1&1);
                }
            }
            ix0 = (q>>1)+0x3fe00000;
            ix1 =  q1>>>1; // unsigned shift
            if ((q&1)==1) ix1 |= sign;
            ix0 += (m <<20);
            // __HI(z) = ix0;
            z = __HI(z, ix0);
            // __LO(z) = ix1;
            z = __LO(z, ix1);
            return z;
        }
    }

    /**
     * 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 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.
     */
    private static final 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 */

        static 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;
            }
        }
    }

    /**
     * Return the logarithm of x
     *
     * Method :
     *   1. Argument Reduction: find k and f such that
     *                      x = 2^k * (1+f),
     *         where  sqrt(2)/2 < 1+f < sqrt(2) .
     *
     *   2. Approximation of log(1+f).
     *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
     *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
     *               = 2s + s*R
     *      We use a special Reme algorithm on [0,0.1716] to generate
     *      a polynomial of degree 14 to approximate R The maximum error
     *      of this polynomial approximation is bounded by 2**-58.45. In
     *      other words,
     *                      2      4      6      8      10      12      14
     *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
     *      (the values of Lg1 to Lg7 are listed in the program)
     *      and
     *          |      2          14          |     -58.45
     *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2
     *          |                             |
     *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
     *      In order to guarantee error in log below 1ulp, we compute log
     *      by
     *              log(1+f) = f - s*(f - R)        (if f is not too large)
     *              log(1+f) = f - (hfsq - s*(hfsq+R)).     (better accuracy)
     *
     *      3. Finally,  log(x) = k*ln2 + log(1+f).
     *                          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
     *         Here ln2 is split into two floating point number:
     *                      ln2_hi + ln2_lo,
     *         where n*ln2_hi is always exact for |n| < 2000.
     *
     * Special cases:
     *      log(x) is NaN with signal if x < 0 (including -INF) ;
     *      log(+INF) is +INF; log(0) is -INF with signal;
     *      log(NaN) is that NaN with no signal.
     *
     * Accuracy:
     *      according to an error analysis, the error is always less than
     *      1 ulp (unit in the last place).
     *
     * 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.
     */
    private static final class Log {
        private static final  double
            ln2_hi  =  6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
            ln2_lo  =  1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
            two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
            Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
            Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
            Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
            Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
            Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
            Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
            Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */

        private static double zero   =  0.0;

        static double compute(double x) {
            double hfsq,f,s,z,R,w,t1,t2,dk;
            int k,hx,i,j;
            /*unsigned*/ int lx;

            hx = __HI(x);           /* high word of x */
            lx = __LO(x);           /* low  word of x */

            k=0;
            if (hx < 0x00100000) {                  /* x < 2**-1022  */
                if (((hx&0x7fffffff)|lx)==0)
                    return -two54/zero;             /* log(+-0)=-inf */
                if (hx<0) return (x-x)/zero;        /* log(-#) = NaN */
                k -= 54; x *= two54; /* subnormal number, scale up x */
                hx = __HI(x);               /* high word of x */
            }
            if (hx >= 0x7ff00000) return x+x;
            k += (hx>>20)-1023;
            hx &= 0x000fffff;
            i = (hx+0x95f64)&0x100000;
            // __HI(x) = hx|(i^0x3ff00000);    /* normalize x or x/2 */
            x =__HI(x, hx|(i^0x3ff00000));    /* normalize x or x/2 */
            k += (i>>20);
            f = x-1.0;
            if((0x000fffff&(2+hx))<3) {     /* |f| < 2**-20 */
                if(f==zero) {
                    if (k==0) return zero;
                    else {dk=(double)k; return dk*ln2_hi+dk*ln2_lo;}
                }
                R = f*f*(0.5-0.33333333333333333*f);
                if(k==0) return f-R; else {dk=(double)k;
                    return dk*ln2_hi-((R-dk*ln2_lo)-f);}
            }
            s = f/(2.0+f);
            dk = (double)k;
            z = s*s;
            i = hx-0x6147a;
            w = z*z;
            j = 0x6b851-hx;
            t1= w*(Lg2+w*(Lg4+w*Lg6));
            t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
            i |= j;
            R = t2+t1;
            if(i>0) {
                hfsq=0.5*f*f;
                if(k==0) return f-(hfsq-s*(hfsq+R)); else
                    return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
            } else {
                if(k==0) return f-s*(f-R); else
                    return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
            }
        }
    }

    /**
     * Return the base 10 logarithm of x
     *
     * Method :
     *      Let log10_2hi = leading 40 bits of log10(2) and
     *          log10_2lo = log10(2) - log10_2hi,
     *          ivln10   = 1/log(10) rounded.
     *      Then
     *              n = ilogb(x),
     *              if(n<0)  n = n+1;
     *              x = scalbn(x,-n);
     *              log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))
     *
     * Note 1:
     *      To guarantee log10(10**n)=n, where 10**n is normal, the rounding
     *      mode must set to Round-to-Nearest.
     * Note 2:
     *      [1/log(10)] rounded to 53 bits has error  .198   ulps;
     *      log10 is monotonic at all binary break points.
     *
     * Special cases:
     *      log10(x) is NaN with signal if x < 0;
     *      log10(+INF) is +INF with no signal; log10(0) is -INF with signal;
     *      log10(NaN) is that NaN with no signal;
     *      log10(10**N) = N  for N=0,1,...,22.
     *
     * 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 Log10 {
        private static double two54      =  1.80143985094819840000e+16; /* 0x43500000, 0x00000000 */
        private static double ivln10     =  4.34294481903251816668e-01; /* 0x3FDBCB7B, 0x1526E50E */

        private static double log10_2hi  =  3.01029995663611771306e-01; /* 0x3FD34413, 0x509F6000 */
        private static double log10_2lo  =  3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */

        private static double zero   =  0.0;

        public static double compute(double x) {
            double y,z;
            int i,k,hx;
            /*unsigned*/ int lx;

            hx = __HI(x);   /* high word of x */
            lx = __LO(x);   /* low word of x */

            k=0;
            if (hx < 0x00100000) {                  /* x < 2**-1022  */
                if (((hx&0x7fffffff)|lx)==0)
                    return -two54/zero;             /* log(+-0)=-inf */
                if (hx<0) return (x-x)/zero;        /* log(-#) = NaN */
                k -= 54; x *= two54; /* subnormal number, scale up x */
                hx = __HI(x);                /* high word of x */
            }
            if (hx >= 0x7ff00000) return x+x;
            k += (hx>>20)-1023;
            i  = (k&0x80000000)>>>31; // unsigned shift
            hx = (hx&0x000fffff)|((0x3ff-i)<<20);
            y  = (double)(k+i);
            x = __HI(x, hx); //original: __HI(x) = hx;
            z  = y*log10_2lo + ivln10*log(x);
            return  z+y*log10_2hi;
        }
    }

    /**
     * Returns the natural logarithm of the sum of the argument and 1.
     *
     * Method :
     *   1. Argument Reduction: find k and f such that
     *                      1+x = 2^k * (1+f),
     *         where  sqrt(2)/2 < 1+f < sqrt(2) .
     *
     *      Note. If k=0, then f=x is exact. However, if k!=0, then f
     *      may not be representable exactly. In that case, a correction
     *      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
     *      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
     *      and add back the correction term c/u.
     *      (Note: when x > 2**53, one can simply return log(x))
     *
     *   2. Approximation of log1p(f).
     *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
     *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
     *               = 2s + s*R
     *      We use a special Reme algorithm on [0,0.1716] to generate
     *      a polynomial of degree 14 to approximate R The maximum error
     *      of this polynomial approximation is bounded by 2**-58.45. In
     *      other words,
     *                      2      4      6      8      10      12      14
     *          R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
     *      (the values of Lp1 to Lp7 are listed in the program)
     *      and
     *          |      2          14          |     -58.45
     *          | Lp1*s +...+Lp7*s    -  R(z) | <= 2
     *          |                             |
     *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
     *      In order to guarantee error in log below 1ulp, we compute log
     *      by
     *              log1p(f) = f - (hfsq - s*(hfsq+R)).
     *
     *      3. Finally, log1p(x) = k*ln2 + log1p(f).
     *                           = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
     *         Here ln2 is split into two floating point number:
     *                      ln2_hi + ln2_lo,
     *         where n*ln2_hi is always exact for |n| < 2000.
     *
     * Special cases:
     *      log1p(x) is NaN with signal if x < -1 (including -INF) ;
     *      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
     *      log1p(NaN) is that NaN with no signal.
     *
     * Accuracy:
     *      according to an error analysis, the error is always less than
     *      1 ulp (unit in the last place).
     *
     * 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.
     *
     * Note: Assuming log() return accurate answer, the following
     *       algorithm can be used to compute log1p(x) to within a few ULP:
     *
     *              u = 1+x;
     *              if(u==1.0) return x ; else
     *                         return log(u)*(x/(u-1.0));
     *
     *       See HP-15C Advanced Functions Handbook, p.193.
     */
    static class Log1p {
        private static double ln2_hi  =  6.93147180369123816490e-01;  /* 3fe62e42 fee00000 */
        private static double ln2_lo  =  1.90821492927058770002e-10;  /* 3dea39ef 35793c76 */
        private static double two54   =  1.80143985094819840000e+16;  /* 43500000 00000000 */
        private static double Lp1 = 6.666666666666735130e-01;  /* 3FE55555 55555593 */
        private static double Lp2 = 3.999999999940941908e-01;  /* 3FD99999 9997FA04 */
        private static double Lp3 = 2.857142874366239149e-01;  /* 3FD24924 94229359 */
        private static double Lp4 = 2.222219843214978396e-01;  /* 3FCC71C5 1D8E78AF */
        private static double Lp5 = 1.818357216161805012e-01;  /* 3FC74664 96CB03DE */
        private static double Lp6 = 1.531383769920937332e-01;  /* 3FC39A09 D078C69F */
        private static double Lp7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
        private static double zero = 0.0;

        public static double compute(double x) {
            double hfsq,f=0,c=0,s,z,R,u;
            int k,hx,hu=0,ax;

            hx = __HI(x);           /* high word of x */
            ax = hx&0x7fffffff;

            k = 1;
            if (hx < 0x3FDA827A) {                  /* x < 0.41422  */
                if(ax>=0x3ff00000) {                /* x <= -1.0 */
                    /*
                     * Added redundant test against hx to work around VC++
                     * code generation problem.
                     */
                    if(x==-1.0 && (hx==0xbff00000)) /* log1p(-1)=-inf */
                        return -two54/zero;
                    else
                        return (x-x)/(x-x);           /* log1p(x<-1)=NaN */
                }
                if(ax<0x3e200000) {                 /* |x| < 2**-29 */
                    if(two54+x>zero                 /* raise inexact */
                       &&ax<0x3c900000)            /* |x| < 2**-54 */
                        return x;
                    else
                        return x - x*x*0.5;
                }
                if(hx>0||hx<=((int)0xbfd2bec3)) {
                    k=0;f=x;hu=1;}  /* -0.2929<x<0.41422 */
            }
            if (hx >= 0x7ff00000) return x+x;
            if(k!=0) {
                if(hx<0x43400000) {
                    u  = 1.0+x;
                    hu = __HI(u);           /* high word of u */
                    k  = (hu>>20)-1023;
                    c  = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
                    c /= u;
                } else {
                    u  = x;
                    hu = __HI(u);           /* high word of u */
                    k  = (hu>>20)-1023;
                    c  = 0;
                }
                hu &= 0x000fffff;
                if(hu<0x6a09e) {
                    u = __HI(u, hu|0x3ff00000);        /* normalize u */
                } else {
                    k += 1;
                    u = __HI(u, hu|0x3fe00000);        /* normalize u/2 */
                    hu = (0x00100000-hu)>>2;
                }
                f = u-1.0;
            }
            hfsq=0.5*f*f;
            if(hu==0) {     /* |f| < 2**-20 */
                if(f==zero) { if(k==0) return zero;
                    else {c += k*ln2_lo; return k*ln2_hi+c;}}
                R = hfsq*(1.0-0.66666666666666666*f);
                if(k==0) return f-R; else
                    return k*ln2_hi-((R-(k*ln2_lo+c))-f);
            }
            s = f/(2.0+f);
            z = s*s;
            R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
            if(k==0) return f-(hfsq-s*(hfsq+R)); else
                return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
        }
    }

    /* expm1(x)
     * Returns exp(x)-1, the exponential of x minus 1.
     *
     * Method
     *   1. Argument reduction:
     *      Given x, find r and integer k such that
     *
     *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
     *
     *      Here a correction term c will be computed to compensate
     *      the error in r when rounded to a floating-point number.
     *
     *   2. Approximating expm1(r) by a special rational function on
     *      the interval [0,0.34658]:
     *      Since
     *          r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
     *      we define R1(r*r) by
     *          r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
     *      That is,
     *          R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
     *                   = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
     *                   = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
     *      We use a special Reme algorithm on [0,0.347] to generate
     *      a polynomial of degree 5 in r*r to approximate R1. The
     *      maximum error of this polynomial approximation is bounded
     *      by 2**-61. In other words,
     *          R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
     *      where   Q1  =  -1.6666666666666567384E-2,
     *              Q2  =   3.9682539681370365873E-4,
     *              Q3  =  -9.9206344733435987357E-6,
     *              Q4  =   2.5051361420808517002E-7,
     *              Q5  =  -6.2843505682382617102E-9;
     *      (where z=r*r, and the values of Q1 to Q5 are listed below)
     *      with error bounded by
     *          |                  5           |     -61
     *          | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
     *          |                              |
     *
     *      expm1(r) = exp(r)-1 is then computed by the following
     *      specific way which minimize the accumulation rounding error:
     *                             2     3
     *                            r     r    [ 3 - (R1 + R1*r/2)  ]
     *            expm1(r) = r + --- + --- * [--------------------]
     *                            2     2    [ 6 - r*(3 - R1*r/2) ]
     *
     *      To compensate the error in the argument reduction, we use
     *              expm1(r+c) = expm1(r) + c + expm1(r)*c
     *                         ~ expm1(r) + c + r*c
     *      Thus c+r*c will be added in as the correction terms for
     *      expm1(r+c). Now rearrange the term to avoid optimization
     *      screw up:
     *                      (      2                                    2 )
     *                      ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
     *       expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
     *                      ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
     *                      (                                             )
     *
     *                 = r - E
     *   3. Scale back to obtain expm1(x):
     *      From step 1, we have
     *         expm1(x) = either 2^k*[expm1(r)+1] - 1
     *                  = or     2^k*[expm1(r) + (1-2^-k)]
     *   4. Implementation notes:
     *      (A). To save one multiplication, we scale the coefficient Qi
     *           to Qi*2^i, and replace z by (x^2)/2.
     *      (B). To achieve maximum accuracy, we compute expm1(x) by
     *        (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
     *        (ii)  if k=0, return r-E
     *        (iii) if k=-1, return 0.5*(r-E)-0.5
     *        (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)
     *                     else          return  1.0+2.0*(r-E);
     *        (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
     *        (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
     *        (vii) return 2^k(1-((E+2^-k)-r))
     *
     * Special cases:
     *      expm1(INF) is INF, expm1(NaN) is NaN;
     *      expm1(-INF) is -1, and
     *      for finite argument, only expm1(0)=0 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 expm1(x) overflow
     *
     * 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 Expm1 {
        private static final double one             = 1.0;
        private static final double huge            = 1.0e+300;
        private static final double tiny            = 1.0e-300;
        private static final double o_threshold     = 7.09782712893383973096e+02; /* 0x40862E42, 0xFEFA39EF */
        private static final double ln2_hi          = 6.93147180369123816490e-01; /* 0x3fe62e42, 0xfee00000 */
        private static final double ln2_lo          = 1.90821492927058770002e-10; /* 0x3dea39ef, 0x35793c76 */
        private static final double invln2          = 1.44269504088896338700e+00; /* 0x3ff71547, 0x652b82fe */
        /* scaled coefficients related to expm1 */
        private static final double Q1  =  -3.33333333333331316428e-02; /* BFA11111 111110F4 */
        private static final double Q2  =   1.58730158725481460165e-03; /* 3F5A01A0 19FE5585 */
        private static final double Q3  =  -7.93650757867487942473e-05; /* BF14CE19 9EAADBB7 */
        private static final double Q4  =   4.00821782732936239552e-06; /* 3ED0CFCA 86E65239 */
        private static final double Q5  =  -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */

        static double compute(double x) {
            double y,hi,lo,c=0,t,e,hxs,hfx,r1;
            int k,xsb;
            /*unsigned*/ int hx;

            hx  = __HI(x);  /* high word of x */
            xsb = hx&0x80000000;            /* sign bit of x */
            if(xsb==0) y=x; else y= -x;     /* y = |x| */
            hx &= 0x7fffffff;               /* high word of |x| */

            /* filter out huge and non-finite argument */
            if(hx >= 0x4043687A) {                  /* if |x|>=56*ln2 */
                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:-1.0;/* exp(+-inf)={inf,-1} */
                    }
                    if(x > o_threshold) return huge*huge; /* overflow */
                }
                if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
                    if(x+tiny<0.0)          /* raise inexact */
                        return tiny-one;        /* return -1 */
                }
            }

            /* argument reduction */
            if(hx > 0x3fd62e42) {           /* if  |x| > 0.5 ln2 */
                if(hx < 0x3FF0A2B2) {       /* and |x| < 1.5 ln2 */
                    if(xsb==0)
                        {hi = x - ln2_hi; lo =  ln2_lo;  k =  1;}
                    else
                        {hi = x + ln2_hi; lo = -ln2_lo;  k = -1;}
                } else {
                    k  = (int)(invln2*x+((xsb==0)?0.5:-0.5));
                    t  = k;
                    hi = x - t*ln2_hi;      /* t*ln2_hi is exact here */
                    lo = t*ln2_lo;
                }
                x  = hi - lo;
                c  = (hi-x)-lo;
            }
            else if(hx < 0x3c900000) {      /* when |x|<2**-54, return x */
                t = huge+x; /* return x with inexact flags when x!=0 */
                return x - (t-(huge+x));
            }
            else k = 0;

            /* x is now in primary range */
            hfx = 0.5*x;
            hxs = x*hfx;
            r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
            t  = 3.0-r1*hfx;
            e  = hxs*((r1-t)/(6.0 - x*t));
            if(k==0) return x - (x*e-hxs);          /* c is 0 */
            else {
                e  = (x*(e-c)-c);
                e -= hxs;
                if(k== -1) return 0.5*(x-e)-0.5;
                if(k==1) {
                    if(x < -0.25) return -2.0*(e-(x+0.5));
                    else          return  one+2.0*(x-e);
                }
                if (k <= -2 || k>56) {   /* suffice to return exp(x)-1 */
                    y = one-(e-x);
                    y = __HI(y,  __HI(y) + (k<<20));     /* add k to y's exponent */
                    return y-one;
                }
                t = one;
                if(k<20) {
                    t = __HI(t, 0x3ff00000 - (0x200000>>k));  /* t=1-2^-k */
                    y = t-(e-x);
                    y = __HI(y, __HI(y) + (k<<20));     /* add k to y's exponent */
                } else {
                    t = __HI(t, ((0x3ff-k)<<20));     /* 2^-k */
                    y = x-(e+t);
                    y += one;
                    y = __HI(y, __HI(y) + (k<<20));     /* add k to y's exponent */
                }
            }
            return y;
        }
    }

    /**
     * Method :
     * mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2
     *      1. Replace x by |x| (sinh(-x) = -sinh(x)).
     *      2.
     *                                                  E + E/(E+1)
     *          0        <= x <= 22     :  sinh(x) := --------------, E=expm1(x)
     *                                                      2
     *
     *          22       <= x <= lnovft :  sinh(x) := exp(x)/2
     *          lnovft   <= x <= ln2ovft:  sinh(x) := exp(x/2)/2 * exp(x/2)
     *          ln2ovft  <  x           :  sinh(x) := x*shuge (overflow)
     *
     * Special cases:
     *      sinh(x) is |x| if x is +INF, -INF, or NaN.
     *      only sinh(0)=0 is exact for finite x.
     */
    private static final class Sinh {
        private static final double one = 1.0, shuge = 1.0e307;

        static double compute(double x) {
            double t,w,h;
            int ix,jx;
            /* unsigned */ int lx;

            /* High word of |x|. */
            jx = __HI(x);
            ix = jx&0x7fffffff;

            /* x is INF or NaN */
            if(ix>=0x7ff00000) return x+x;

            h = 0.5;
            if (jx<0) h = -h;
            /* |x| in [0,22], return sign(x)*0.5*(E+E/(E+1))) */
            if (ix < 0x40360000) {          /* |x|<22 */
                if (ix<0x3e300000)          /* |x|<2**-28 */
                    if(shuge+x>one) return x;/* sinh(tiny) = tiny with inexact */
                t = FdlibmTranslit.expm1(Math.abs(x));
                if(ix<0x3ff00000) return h*(2.0*t-t*t/(t+one));
                return h*(t+t/(t+one));
            }

            /* |x| in [22, log(maxdouble)] return 0.5*exp(|x|) */
            if (ix < 0x40862E42) return h*FdlibmTranslit.exp(Math.abs(x));

            /* |x| in [log(maxdouble), overflowthresold] */
            // Note: the original FDLIBM sources use
            // lx = *( (((*(unsigned*)&one)>>29)) + (unsigned*)&x);
            // to set lx to the low-order 32 bits of x. The expression
            // in question is an alternate way to implement the
            // functionality of the C FDLIBM __LO macro and the
            // expression is coded to work on both big-edian and
            // little-endian machines. However, this port will instead
            // use the __LO method call to represent this
            // functionality.
            lx = __LO(x);
            if (ix<0x408633CE || ((ix==0x408633ce)&&(Long.compareUnsigned(lx, 0x8fb9f87d) <= 0 ))) {
                w = exp(0.5*Math.abs(x));
                t = h*w;
                return t*w;
            }

            /* |x| > overflowthresold, sinh(x) overflow */
            return x*shuge;
        }
    }

    /**
     * Method :
     * mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2
     *      1. Replace x by |x| (cosh(x) = cosh(-x)).
     *      2.
     *                                                      [ exp(x) - 1 ]^2
     *          0        <= x <= ln2/2  :  cosh(x) := 1 + -------------------
     *                                                         2*exp(x)
     *
     *                                                exp(x) +  1/exp(x)
     *          ln2/2    <= x <= 22     :  cosh(x) := -------------------
     *                                                        2
     *          22       <= x <= lnovft :  cosh(x) := exp(x)/2
     *          lnovft   <= x <= ln2ovft:  cosh(x) := exp(x/2)/2 * exp(x/2)
     *          ln2ovft  <  x           :  cosh(x) := huge*huge (overflow)
     *
     * Special cases:
     *      cosh(x) is |x| if x is +INF, -INF, or NaN.
     *      only cosh(0)=1 is exact for finite x.
     */
    private static final class Cosh {
        private static final double one = 1.0, half=0.5, huge = 1.0e300;
        static double compute(double x) {
            double t,w;
            int ix;
            /*unsigned*/ int lx;

            /* High word of |x|. */
            ix = __HI(x);
            ix &= 0x7fffffff;

            /* x is INF or NaN */
            if(ix>=0x7ff00000) return x*x;

            /* |x| in [0,0.5*ln2], return 1+expm1(|x|)^2/(2*exp(|x|)) */
            if(ix<0x3fd62e43) {
                t = expm1(Math.abs(x));
                w = one+t;
                if (ix<0x3c800000) return w;        /* cosh(tiny) = 1 */
                return one+(t*t)/(w+w);
            }

            /* |x| in [0.5*ln2,22], return (exp(|x|)+1/exp(|x|)/2; */
            if (ix < 0x40360000) {
                t = exp(Math.abs(x));
                return half*t+half/t;
            }

            /* |x| in [22, log(maxdouble)] return half*exp(|x|) */
            if (ix < 0x40862E42) return half*exp(Math.abs(x));

            /* |x| in [log(maxdouble), overflowthresold] */
            // See note above in the sinh implementation for how this
            // transliteration port uses __LO(x) in the line below
            // that differs from the idiom used in the original FDLIBM.
            lx = __LO(x);
            if (ix<0x408633CE ||
                ((ix==0x408633ce)&&(Integer.compareUnsigned(lx, 0x8fb9f87d) <= 0))) {
                w = exp(half*Math.abs(x));
                t = half*w;
                return t*w;
            }

            /* |x| > overflowthresold, cosh(x) overflow */
            return huge*huge;
        }
    }

    /**
     * Return the Hyperbolic Tangent of x
     *
     * Method :
     *                                     x    -x
     *                                    e  - e
     *      0. tanh(x) is defined to be -----------
     *                                     x    -x
     *                                    e  + e
     *      1. reduce x to non-negative by tanh(-x) = -tanh(x).
     *      2.  0      <= x <= 2**-55 : tanh(x) := x*(one+x)
     *                                              -t
     *          2**-55 <  x <=  1     : tanh(x) := -----; t = expm1(-2x)
     *                                             t + 2
     *                                                   2
     *          1      <= x <=  22.0  : tanh(x) := 1-  ----- ; t=expm1(2x)
     *                                                 t + 2
     *          22.0   <  x <= INF    : tanh(x) := 1.
     *
     * Special cases:
     *      tanh(NaN) is NaN;
     *      only tanh(0)=0 is exact for finite argument.
     */
    private static final class Tanh {
        private static final double one=1.0, two=2.0, tiny = 1.0e-300;
        static double compute(double x) {
            double t,z;
            int jx,ix;

            /* High word of |x|. */
            jx = __HI(x);
            ix = jx&0x7fffffff;

            /* x is INF or NaN */
            if(ix>=0x7ff00000) {
                if (jx>=0) return one/x+one;    /* tanh(+-inf)=+-1 */
                else       return one/x-one;    /* tanh(NaN) = NaN */
            }

            /* |x| < 22 */
            if (ix < 0x40360000) {          /* |x|<22 */
                if (ix<0x3c800000)          /* |x|<2**-55 */
                    return x*(one+x);       /* tanh(small) = small */
                if (ix>=0x3ff00000) {       /* |x|>=1  */
                    t = expm1(two*Math.abs(x));
                    z = one - two/(t+two);
                } else {
                    t = expm1(-two*Math.abs(x));
                    z= -t/(t+two);
                }
                /* |x| > 22, return +-1 */
            } else {
                z = one - tiny;             /* raised inexact flag */
            }
            return (jx>=0)? z: -z;
        }
    }

    private static final class IEEEremainder {
        private static final double zero = 0.0;
        private static double one = 1.0;
        private static double[] Zero = {0.0, -0.0,};

        static double compute(double x, double p) {
            int hx,hp;
            /*unsigned*/ int sx,lx,lp;
            double p_half;

            hx = __HI(x);           /* high word of x */
            lx = __LO(x);           /* low  word of x */
            hp = __HI(p);           /* high word of p */
            lp = __LO(p);           /* low  word of p */
            sx = hx&0x80000000;
            hp &= 0x7fffffff;
            hx &= 0x7fffffff;

            /* purge off exception values */
            if((hp|lp)==0) return (x*p)/(x*p);      /* p = 0 */
            if((hx>=0x7ff00000)||                   /* x not finite */
               ((hp>=0x7ff00000)&&                   /* p is NaN */
                (((hp-0x7ff00000)|lp)!=0)))
                return (x*p)/(x*p);


            if (hp<=0x7fdfffff) x = __ieee754_fmod(x,p+p);  /* now x < 2p */
            if (((hx-hp)|(lx-lp))==0) return zero*x;
            x  = Math.abs(x);
            p  = Math.abs(p);
            if (hp<0x00200000) {
                if(x+x>p) {
                    x-=p;
                    if(x+x>=p) x -= p;
                }
            } else {
                p_half = 0.5*p;
                if(x>p_half) {
                    x-=p;
                    if(x>=p_half) x -= p;
                }
            }
            // __HI(x) ^= sx;
            x = __HI(x, __HI(x)^sx);
            return x;
        }

        private static double __ieee754_fmod(double x, double y) {
            int n,hx,hy,hz,ix,iy,sx,i;
            /*unsigned*/ int lx,ly,lz;

            hx = __HI(x);           /* high word of x */
            lx = __LO(x);           /* low  word of x */
            hy = __HI(y);           /* high word of y */
            ly = __LO(y);           /* low  word of y */
            sx = hx&0x80000000;             /* sign of x */
            hx ^=sx;                /* |x| */
            hy &= 0x7fffffff;       /* |y| */

            /* purge off exception values */
            if((hy|ly)==0||(hx>=0x7ff00000)||       /* y=0,or x not finite */
               ((hy|((ly|-ly)>>>31))>0x7ff00000))     /* or y is NaN  */ // unsigned shift
                return (x*y)/(x*y);
            if(hx<=hy) {
                // if((hx<hy)||(lx<ly)) return x;      /* |x|<|y| return x */
                if((hx<hy)||(Integer.compareUnsigned(lx,ly) < 0))  return x;  /* |x|<|y| return x */
                if(lx==ly)
                    return Zero[/*(unsigned)*/sx>>>31];  /* |x|=|y| return x*0*/ // unsigned shift
            }

            /* determine ix = ilogb(x) */
            if(hx<0x00100000) {     /* subnormal x */
                if(hx==0) {
                    for (ix = -1043, i=lx; i>0; i<<=1) ix -=1;
                } else {
                    for (ix = -1022,i=(hx<<11); i>0; i<<=1) ix -=1;
                }
            } else ix = (hx>>20)-1023;

            /* determine iy = ilogb(y) */
            if(hy<0x00100000) {     /* subnormal y */
                if(hy==0) {
                    for (iy = -1043, i=ly; i>0; i<<=1) iy -=1;
                } else {
                    for (iy = -1022,i=(hy<<11); i>0; i<<=1) iy -=1;
                }
            } else iy = (hy>>20)-1023;

            /* set up {hx,lx}, {hy,ly} and align y to x */
            if(ix >= -1022)
                hx = 0x00100000|(0x000fffff&hx);
            else {          /* subnormal x, shift x to normal */
                n = -1022-ix;
                if(n<=31) {
                    hx = (hx<<n)|(lx >>> (32-n)); // unsigned shift
                    lx <<= n;
                } else {
                    hx = lx<<(n-32);
                    lx = 0;
                }
            }
            if(iy >= -1022)
                hy = 0x00100000|(0x000fffff&hy);
            else {          /* subnormal y, shift y to normal */
                n = -1022-iy;
                if(n<=31) {
                    hy = (hy<<n)|(ly >>> (32-n)); // unsigned shift
                    ly <<= n;
                } else {
                    hy = ly<<(n-32);
                    ly = 0;
                }
            }

            /* fix point fmod */
            n = ix - iy;
            while(n-- != 0) {
                hz=hx-hy;lz=lx-ly;
                // if(lx<ly) hz -= 1;
                if(Integer.compareUnsigned(lx, ly) < 0) hz -= 1;
                if(hz<0){hx = hx+hx+(lx >>> 31); lx = lx+lx;} // unsigned shift
                else {
                    if((hz|lz)==0)          /* return sign(x)*0 */
                        return Zero[/*(unsigned)*/sx>>>31]; // unsigned shift
                    hx = hz+hz+(lz >>> 31); // unsigned shift
                    lx = lz+lz;
                }
            }
            hz=hx-hy;lz=lx-ly;
            // if(lx<ly) hz -= 1;
            if(Integer.compareUnsigned(lx, ly) < 0) hz -= 1;
            if(hz>=0) {hx=hz;lx=lz;}

            /* convert back to floating value and restore the sign */
            if((hx|lx)==0)                  /* return sign(x)*0 */
                return Zero[/*(unsigned)*/sx >>> 31]; // unsigned shift
            while(hx<0x00100000) {          /* normalize x */
                hx = hx+hx+(lx >>> 31); lx = lx+lx; // unsigned shift
                iy -= 1;
            }
            if(iy>= -1022) {        /* normalize output */
                hx = ((hx-0x00100000)|((iy+1023)<<20));
                // __HI(x) = hx|sx;
                x = __HI(x, hx|sx);
                // __LO(x) = lx;
                x = __LO(x, lx);
            } else {                /* subnormal output */
                n = -1022 - iy;
                if(n<=20) {
                    lx = (lx >>> n)|(/*(unsigned)*/hx<<(32-n)); // unsigned shift
                    hx >>= n;
                } else if (n<=31) {
                    lx = (hx<<(32-n))|(lx >>> n); // unsigned shift
                    hx = sx;
                } else {
                    lx = hx>>(n-32); hx = sx;
                }
                // __HI(x) = hx|sx;
                x = __HI(x, hx|sx);
                // __LO(x) = lx;
                x = __LO(x, lx);
                x *= one;           /* create necessary signal */
            }
            return x;               /* exact output */
        }
    }
}