xref: /external/apache-commons-math/src/main/java/org/apache/commons/math/util/FastMath.java

/*
 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You under the Apache License, Version 2.0
 * (the "License"); you may not use this file except in compliance with
 * the License.  You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */
package org.apache.commons.math.util;

/**
 * Faster, more accurate, portable alternative to {@link StrictMath}.
 * <p>
 * Additionally implements the following methods not found in StrictMath:
 * <ul>
 * <li>{@link #asinh(double)}</li>
 * <li>{@link #acosh(double)}</li>
 * <li>{@link #atanh(double)}</li>
 * </ul>
 * The following methods are found in StrictMath since 1.6 only
 * <ul>
 * <li>{@link #copySign(double, double)}</li>
 * <li>{@link #getExponent(double)}</li>
 * <li>{@link #nextAfter(double,double)}</li>
 * <li>{@link #nextUp(double)}</li>
 * <li>{@link #scalb(double, int)}</li>
 * <li>{@link #copySign(float, float)}</li>
 * <li>{@link #getExponent(float)}</li>
 * <li>{@link #nextAfter(float,double)}</li>
 * <li>{@link #nextUp(float)}</li>
 * <li>{@link #scalb(float, int)}</li>
 * </ul>
 * @version $Revision: 1074294 $ $Date: 2011-02-24 22:18:59 +0100 (jeu. 24 févr. 2011) $
 * @since 2.2
 */
public class FastMath {

    /** Archimede's constant PI, ratio of circle circumference to diameter. */
    public static final double PI = 105414357.0 / 33554432.0 + 1.984187159361080883e-9;

    /** Napier's constant e, base of the natural logarithm. */
    public static final double E = 2850325.0 / 1048576.0 + 8.254840070411028747e-8;

    /** Exponential evaluated at integer values,
     * exp(x) =  expIntTableA[x + 750] + expIntTableB[x+750].
     */
    private static final double EXP_INT_TABLE_A[] = new double[1500];

    /** Exponential evaluated at integer values,
     * exp(x) =  expIntTableA[x + 750] + expIntTableB[x+750]
     */
    private static final double EXP_INT_TABLE_B[] = new double[1500];

    /** Exponential over the range of 0 - 1 in increments of 2^-10
     * exp(x/1024) =  expFracTableA[x] + expFracTableB[x].
     */
    private static final double EXP_FRAC_TABLE_A[] = new double[1025];

    /** Exponential over the range of 0 - 1 in increments of 2^-10
     * exp(x/1024) =  expFracTableA[x] + expFracTableB[x].
     */
    private static final double EXP_FRAC_TABLE_B[] = new double[1025];

    /** Factorial table, for Taylor series expansions. */
    private static final double FACT[] = new double[20];

    /** Extended precision logarithm table over the range 1 - 2 in increments of 2^-10. */
    private static final double LN_MANT[][] = new double[1024][];

    /** log(2) (high bits). */
    private static final double LN_2_A = 0.693147063255310059;

    /** log(2) (low bits). */
    private static final double LN_2_B = 1.17304635250823482e-7;

    /** Coefficients for slowLog. */
    private static final double LN_SPLIT_COEF[][] = {
        {2.0, 0.0},
        {0.6666666269302368, 3.9736429850260626E-8},
        {0.3999999761581421, 2.3841857910019882E-8},
        {0.2857142686843872, 1.7029898543501842E-8},
        {0.2222222089767456, 1.3245471311735498E-8},
        {0.1818181574344635, 2.4384203044354907E-8},
        {0.1538461446762085, 9.140260083262505E-9},
        {0.13333332538604736, 9.220590270857665E-9},
        {0.11764700710773468, 1.2393345855018391E-8},
        {0.10526403784751892, 8.251545029714408E-9},
        {0.0952233225107193, 1.2675934823758863E-8},
        {0.08713622391223907, 1.1430250008909141E-8},
        {0.07842259109020233, 2.404307984052299E-9},
        {0.08371849358081818, 1.176342548272881E-8},
        {0.030589580535888672, 1.2958646899018938E-9},
        {0.14982303977012634, 1.225743062930824E-8},
    };

    /** Coefficients for log, when input 0.99 < x < 1.01. */
    private static final double LN_QUICK_COEF[][] = {
        {1.0, 5.669184079525E-24},
        {-0.25, -0.25},
        {0.3333333134651184, 1.986821492305628E-8},
        {-0.25, -6.663542893624021E-14},
        {0.19999998807907104, 1.1921056801463227E-8},
        {-0.1666666567325592, -7.800414592973399E-9},
        {0.1428571343421936, 5.650007086920087E-9},
        {-0.12502530217170715, -7.44321345601866E-11},
        {0.11113807559013367, 9.219544613762692E-9},
    };

    /** Coefficients for log in the range of 1.0 < x < 1.0 + 2^-10. */
    private static final double LN_HI_PREC_COEF[][] = {
        {1.0, -6.032174644509064E-23},
        {-0.25, -0.25},
        {0.3333333134651184, 1.9868161777724352E-8},
        {-0.2499999701976776, -2.957007209750105E-8},
        {0.19999954104423523, 1.5830993332061267E-10},
        {-0.16624879837036133, -2.6033824355191673E-8}
    };

    /** Sine table (high bits). */
    private static final double SINE_TABLE_A[] = new double[14];

    /** Sine table (low bits). */
    private static final double SINE_TABLE_B[] = new double[14];

    /** Cosine table (high bits). */
    private static final double COSINE_TABLE_A[] = new double[14];

    /** Cosine table (low bits). */
    private static final double COSINE_TABLE_B[] = new double[14];

    /** Tangent table, used by atan() (high bits). */
    private static final double TANGENT_TABLE_A[] = new double[14];

    /** Tangent table, used by atan() (low bits). */
    private static final double TANGENT_TABLE_B[] = new double[14];

    /** Bits of 1/(2*pi), need for reducePayneHanek(). */
    private static final long RECIP_2PI[] = new long[] {
        (0x28be60dbL << 32) | 0x9391054aL,
        (0x7f09d5f4L << 32) | 0x7d4d3770L,
        (0x36d8a566L << 32) | 0x4f10e410L,
        (0x7f9458eaL << 32) | 0xf7aef158L,
        (0x6dc91b8eL << 32) | 0x909374b8L,
        (0x01924bbaL << 32) | 0x82746487L,
        (0x3f877ac7L << 32) | 0x2c4a69cfL,
        (0xba208d7dL << 32) | 0x4baed121L,
        (0x3a671c09L << 32) | 0xad17df90L,
        (0x4e64758eL << 32) | 0x60d4ce7dL,
        (0x272117e2L << 32) | 0xef7e4a0eL,
        (0xc7fe25ffL << 32) | 0xf7816603L,
        (0xfbcbc462L << 32) | 0xd6829b47L,
        (0xdb4d9fb3L << 32) | 0xc9f2c26dL,
        (0xd3d18fd9L << 32) | 0xa797fa8bL,
        (0x5d49eeb1L << 32) | 0xfaf97c5eL,
        (0xcf41ce7dL << 32) | 0xe294a4baL,
         0x9afed7ecL << 32  };

    /** Bits of pi/4, need for reducePayneHanek(). */
    private static final long PI_O_4_BITS[] = new long[] {
        (0xc90fdaa2L << 32) | 0x2168c234L,
        (0xc4c6628bL << 32) | 0x80dc1cd1L };

    /** Eighths.
     * This is used by sinQ, because its faster to do a table lookup than
     * a multiply in this time-critical routine
     */
    private static final double EIGHTHS[] = {0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1.0, 1.125, 1.25, 1.375, 1.5, 1.625};

    /** Table of 2^((n+2)/3) */
    private static final double CBRTTWO[] = { 0.6299605249474366,
                                            0.7937005259840998,
                                            1.0,
                                            1.2599210498948732,
                                            1.5874010519681994 };

    /*
     *  There are 52 bits in the mantissa of a double.
     *  For additional precision, the code splits double numbers into two parts,
     *  by clearing the low order 30 bits if possible, and then performs the arithmetic
     *  on each half separately.
     */

    /**
     * 0x40000000 - used to split a double into two parts, both with the low order bits cleared.
     * Equivalent to 2^30.
     */
    private static final long HEX_40000000 = 0x40000000L; // 1073741824L

    /** Mask used to clear low order 30 bits */
    private static final long MASK_30BITS = -1L - (HEX_40000000 -1); // 0xFFFFFFFFC0000000L;

    /** 2^52 - double numbers this large must be integral (no fraction) or NaN or Infinite */
    private static final double TWO_POWER_52 = 4503599627370496.0;

    // Initialize tables
    static {
        int i;

        // Generate an array of factorials
        FACT[0] = 1.0;
        for (i = 1; i < FACT.length; i++) {
            FACT[i] = FACT[i-1] * i;
        }

        double tmp[] = new double[2];
        double recip[] = new double[2];

        // Populate expIntTable
        for (i = 0; i < 750; i++) {
            expint(i, tmp);
            EXP_INT_TABLE_A[i+750] = tmp[0];
            EXP_INT_TABLE_B[i+750] = tmp[1];

            if (i != 0) {
                // Negative integer powers
                splitReciprocal(tmp, recip);
                EXP_INT_TABLE_A[750-i] = recip[0];
                EXP_INT_TABLE_B[750-i] = recip[1];
            }
        }

        // Populate expFracTable
        for (i = 0; i < EXP_FRAC_TABLE_A.length; i++) {
            slowexp(i/1024.0, tmp);
            EXP_FRAC_TABLE_A[i] = tmp[0];
            EXP_FRAC_TABLE_B[i] = tmp[1];
        }

        // Populate lnMant table
        for (i = 0; i < LN_MANT.length; i++) {
            double d = Double.longBitsToDouble( (((long) i) << 42) | 0x3ff0000000000000L );
            LN_MANT[i] = slowLog(d);
        }

        // Build the sine and cosine tables
        buildSinCosTables();
    }

    /**
     * Private Constructor
     */
    private FastMath() {
    }

    // Generic helper methods

    /**
     * Get the high order bits from the mantissa.
     * Equivalent to adding and subtracting HEX_40000 but also works for very large numbers
     *
     * @param d the value to split
     * @return the high order part of the mantissa
     */
    private static double doubleHighPart(double d) {
        if (d > -MathUtils.SAFE_MIN && d < MathUtils.SAFE_MIN){
            return d; // These are un-normalised - don't try to convert
        }
        long xl = Double.doubleToLongBits(d);
        xl = xl & MASK_30BITS; // Drop low order bits
        return Double.longBitsToDouble(xl);
    }

    /** Compute the square root of a number.
     * <p><b>Note:</b> this implementation currently delegates to {@link Math#sqrt}
     * @param a number on which evaluation is done
     * @return square root of a
     */
    public static double sqrt(final double a) {
        return Math.sqrt(a);
    }

    /** Compute the hyperbolic cosine of a number.
     * @param x number on which evaluation is done
     * @return hyperbolic cosine of x
     */
    public static double cosh(double x) {
      if (x != x) {
          return x;
      }

      if (x > 20.0) {
          return exp(x)/2.0;
      }

      if (x < -20) {
          return exp(-x)/2.0;
      }

      double hiPrec[] = new double[2];
      if (x < 0.0) {
          x = -x;
      }
      exp(x, 0.0, hiPrec);

      double ya = hiPrec[0] + hiPrec[1];
      double yb = -(ya - hiPrec[0] - hiPrec[1]);

      double temp = ya * HEX_40000000;
      double yaa = ya + temp - temp;
      double yab = ya - yaa;

      // recip = 1/y
      double recip = 1.0/ya;
      temp = recip * HEX_40000000;
      double recipa = recip + temp - temp;
      double recipb = recip - recipa;

      // Correct for rounding in division
      recipb += (1.0 - yaa*recipa - yaa*recipb - yab*recipa - yab*recipb) * recip;
      // Account for yb
      recipb += -yb * recip * recip;

      // y = y + 1/y
      temp = ya + recipa;
      yb += -(temp - ya - recipa);
      ya = temp;
      temp = ya + recipb;
      yb += -(temp - ya - recipb);
      ya = temp;

      double result = ya + yb;
      result *= 0.5;
      return result;
    }

    /** Compute the hyperbolic sine of a number.
     * @param x number on which evaluation is done
     * @return hyperbolic sine of x
     */
    public static double sinh(double x) {
      boolean negate = false;
      if (x != x) {
          return x;
      }

      if (x > 20.0) {
          return exp(x)/2.0;
      }

      if (x < -20) {
          return -exp(-x)/2.0;
      }

      if (x == 0) {
          return x;
      }

      if (x < 0.0) {
          x = -x;
          negate = true;
      }

      double result;

      if (x > 0.25) {
          double hiPrec[] = new double[2];
          exp(x, 0.0, hiPrec);

          double ya = hiPrec[0] + hiPrec[1];
          double yb = -(ya - hiPrec[0] - hiPrec[1]);

          double temp = ya * HEX_40000000;
          double yaa = ya + temp - temp;
          double yab = ya - yaa;

          // recip = 1/y
          double recip = 1.0/ya;
          temp = recip * HEX_40000000;
          double recipa = recip + temp - temp;
          double recipb = recip - recipa;

          // Correct for rounding in division
          recipb += (1.0 - yaa*recipa - yaa*recipb - yab*recipa - yab*recipb) * recip;
          // Account for yb
          recipb += -yb * recip * recip;

          recipa = -recipa;
          recipb = -recipb;

          // y = y + 1/y
          temp = ya + recipa;
          yb += -(temp - ya - recipa);
          ya = temp;
          temp = ya + recipb;
          yb += -(temp - ya - recipb);
          ya = temp;

          result = ya + yb;
          result *= 0.5;
      }
      else {
          double hiPrec[] = new double[2];
          expm1(x, hiPrec);

          double ya = hiPrec[0] + hiPrec[1];
          double yb = -(ya - hiPrec[0] - hiPrec[1]);

          /* Compute expm1(-x) = -expm1(x) / (expm1(x) + 1) */
          double denom = 1.0 + ya;
          double denomr = 1.0 / denom;
          double denomb = -(denom - 1.0 - ya) + yb;
          double ratio = ya * denomr;
          double temp = ratio * HEX_40000000;
          double ra = ratio + temp - temp;
          double rb = ratio - ra;

          temp = denom * HEX_40000000;
          double za = denom + temp - temp;
          double zb = denom - za;

          rb += (ya - za*ra - za*rb - zb*ra - zb*rb) * denomr;

          // Adjust for yb
          rb += yb*denomr;                        // numerator
          rb += -ya * denomb * denomr * denomr;   // denominator

          // y = y - 1/y
          temp = ya + ra;
          yb += -(temp - ya - ra);
          ya = temp;
          temp = ya + rb;
          yb += -(temp - ya - rb);
          ya = temp;

          result = ya + yb;
          result *= 0.5;
      }

      if (negate) {
          result = -result;
      }

      return result;
    }

    /** Compute the hyperbolic tangent of a number.
     * @param x number on which evaluation is done
     * @return hyperbolic tangent of x
     */
    public static double tanh(double x) {
      boolean negate = false;

      if (x != x) {
          return x;
      }

      if (x > 20.0) {
          return 1.0;
      }

      if (x < -20) {
          return -1.0;
      }

      if (x == 0) {
          return x;
      }

      if (x < 0.0) {
          x = -x;
          negate = true;
      }

      double result;
      if (x >= 0.5) {
          double hiPrec[] = new double[2];
          // tanh(x) = (exp(2x) - 1) / (exp(2x) + 1)
          exp(x*2.0, 0.0, hiPrec);

          double ya = hiPrec[0] + hiPrec[1];
          double yb = -(ya - hiPrec[0] - hiPrec[1]);

          /* Numerator */
          double na = -1.0 + ya;
          double nb = -(na + 1.0 - ya);
          double temp = na + yb;
          nb += -(temp - na - yb);
          na = temp;

          /* Denominator */
          double da = 1.0 + ya;
          double db = -(da - 1.0 - ya);
          temp = da + yb;
          db += -(temp - da - yb);
          da = temp;

          temp = da * HEX_40000000;
          double daa = da + temp - temp;
          double dab = da - daa;

          // ratio = na/da
          double ratio = na/da;
          temp = ratio * HEX_40000000;
          double ratioa = ratio + temp - temp;
          double ratiob = ratio - ratioa;

          // Correct for rounding in division
          ratiob += (na - daa*ratioa - daa*ratiob - dab*ratioa - dab*ratiob) / da;

          // Account for nb
          ratiob += nb / da;
          // Account for db
          ratiob += -db * na / da / da;

          result = ratioa + ratiob;
      }
      else {
          double hiPrec[] = new double[2];
          // tanh(x) = expm1(2x) / (expm1(2x) + 2)
          expm1(x*2.0, hiPrec);

          double ya = hiPrec[0] + hiPrec[1];
          double yb = -(ya - hiPrec[0] - hiPrec[1]);

          /* Numerator */
          double na = ya;
          double nb = yb;

          /* Denominator */
          double da = 2.0 + ya;
          double db = -(da - 2.0 - ya);
          double temp = da + yb;
          db += -(temp - da - yb);
          da = temp;

          temp = da * HEX_40000000;
          double daa = da + temp - temp;
          double dab = da - daa;

          // ratio = na/da
          double ratio = na/da;
          temp = ratio * HEX_40000000;
          double ratioa = ratio + temp - temp;
          double ratiob = ratio - ratioa;

          // Correct for rounding in division
          ratiob += (na - daa*ratioa - daa*ratiob - dab*ratioa - dab*ratiob) / da;

          // Account for nb
          ratiob += nb / da;
          // Account for db
          ratiob += -db * na / da / da;

          result = ratioa + ratiob;
      }

      if (negate) {
          result = -result;
      }

      return result;
    }

    /** Compute the inverse hyperbolic cosine of a number.
     * @param a number on which evaluation is done
     * @return inverse hyperbolic cosine of a
     */
    public static double acosh(final double a) {
        return FastMath.log(a + FastMath.sqrt(a * a - 1));
    }

    /** Compute the inverse hyperbolic sine of a number.
     * @param a number on which evaluation is done
     * @return inverse hyperbolic sine of a
     */
    public static double asinh(double a) {

        boolean negative = false;
        if (a < 0) {
            negative = true;
            a = -a;
        }

        double absAsinh;
        if (a > 0.167) {
            absAsinh = FastMath.log(FastMath.sqrt(a * a + 1) + a);
        } else {
            final double a2 = a * a;
            if (a > 0.097) {
                absAsinh = a * (1 - a2 * (1 / 3.0 - a2 * (1 / 5.0 - a2 * (1 / 7.0 - a2 * (1 / 9.0 - a2 * (1.0 / 11.0 - a2 * (1.0 / 13.0 - a2 * (1.0 / 15.0 - a2 * (1.0 / 17.0) * 15.0 / 16.0) * 13.0 / 14.0) * 11.0 / 12.0) * 9.0 / 10.0) * 7.0 / 8.0) * 5.0 / 6.0) * 3.0 / 4.0) / 2.0);
            } else if (a > 0.036) {
                absAsinh = a * (1 - a2 * (1 / 3.0 - a2 * (1 / 5.0 - a2 * (1 / 7.0 - a2 * (1 / 9.0 - a2 * (1.0 / 11.0 - a2 * (1.0 / 13.0) * 11.0 / 12.0) * 9.0 / 10.0) * 7.0 / 8.0) * 5.0 / 6.0) * 3.0 / 4.0) / 2.0);
            } else if (a > 0.0036) {
                absAsinh = a * (1 - a2 * (1 / 3.0 - a2 * (1 / 5.0 - a2 * (1 / 7.0 - a2 * (1 / 9.0) * 7.0 / 8.0) * 5.0 / 6.0) * 3.0 / 4.0) / 2.0);
            } else {
                absAsinh = a * (1 - a2 * (1 / 3.0 - a2 * (1 / 5.0) * 3.0 / 4.0) / 2.0);
            }
        }

        return negative ? -absAsinh : absAsinh;

    }

    /** Compute the inverse hyperbolic tangent of a number.
     * @param a number on which evaluation is done
     * @return inverse hyperbolic tangent of a
     */
    public static double atanh(double a) {

        boolean negative = false;
        if (a < 0) {
            negative = true;
            a = -a;
        }

        double absAtanh;
        if (a > 0.15) {
            absAtanh = 0.5 * FastMath.log((1 + a) / (1 - a));
        } else {
            final double a2 = a * a;
            if (a > 0.087) {
                absAtanh = a * (1 + a2 * (1.0 / 3.0 + a2 * (1.0 / 5.0 + a2 * (1.0 / 7.0 + a2 * (1.0 / 9.0 + a2 * (1.0 / 11.0 + a2 * (1.0 / 13.0 + a2 * (1.0 / 15.0 + a2 * (1.0 / 17.0)))))))));
            } else if (a > 0.031) {
                absAtanh = a * (1 + a2 * (1.0 / 3.0 + a2 * (1.0 / 5.0 + a2 * (1.0 / 7.0 + a2 * (1.0 / 9.0 + a2 * (1.0 / 11.0 + a2 * (1.0 / 13.0)))))));
            } else if (a > 0.003) {
                absAtanh = a * (1 + a2 * (1.0 / 3.0 + a2 * (1.0 / 5.0 + a2 * (1.0 / 7.0 + a2 * (1.0 / 9.0)))));
            } else {
                absAtanh = a * (1 + a2 * (1.0 / 3.0 + a2 * (1.0 / 5.0)));
            }
        }

        return negative ? -absAtanh : absAtanh;

    }

    /** Compute the signum of a number.
     * The signum is -1 for negative numbers, +1 for positive numbers and 0 otherwise
     * @param a number on which evaluation is done
     * @return -1.0, -0.0, +0.0, +1.0 or NaN depending on sign of a
     */
    public static double signum(final double a) {
        return (a < 0.0) ? -1.0 : ((a > 0.0) ? 1.0 : a); // return +0.0/-0.0/NaN depending on a
    }

    /** Compute the signum of a number.
     * The signum is -1 for negative numbers, +1 for positive numbers and 0 otherwise
     * @param a number on which evaluation is done
     * @return -1.0, -0.0, +0.0, +1.0 or NaN depending on sign of a
     */
    public static float signum(final float a) {
        return (a < 0.0f) ? -1.0f : ((a > 0.0f) ? 1.0f : a); // return +0.0/-0.0/NaN depending on a
    }

    /** Compute next number towards positive infinity.
     * @param a number to which neighbor should be computed
     * @return neighbor of a towards positive infinity
     */
    public static double nextUp(final double a) {
        return nextAfter(a, Double.POSITIVE_INFINITY);
    }

    /** Compute next number towards positive infinity.
     * @param a number to which neighbor should be computed
     * @return neighbor of a towards positive infinity
     */
    public static float nextUp(final float a) {
        return nextAfter(a, Float.POSITIVE_INFINITY);
    }

    /** Returns a pseudo-random number between 0.0 and 1.0.
     * <p><b>Note:</b> this implementation currently delegates to {@link Math#random}
     * @return a random number between 0.0 and 1.0
     */
    public static double random() {
        return Math.random();
    }

    /**
     * Exponential function.
     *
     * Computes exp(x), function result is nearly rounded.   It will be correctly
     * rounded to the theoretical value for 99.9% of input values, otherwise it will
     * have a 1 UPL error.
     *
     * Method:
     *    Lookup intVal = exp(int(x))
     *    Lookup fracVal = exp(int(x-int(x) / 1024.0) * 1024.0 );
     *    Compute z as the exponential of the remaining bits by a polynomial minus one
     *    exp(x) = intVal * fracVal * (1 + z)
     *
     * Accuracy:
     *    Calculation is done with 63 bits of precision, so result should be correctly
     *    rounded for 99.9% of input values, with less than 1 ULP error otherwise.
     *
     * @param x   a double
     * @return double e<sup>x</sup>
     */
    public static double exp(double x) {
        return exp(x, 0.0, null);
    }

    /**
     * Internal helper method for exponential function.
     * @param x original argument of the exponential function
     * @param extra extra bits of precision on input (To Be Confirmed)
     * @param hiPrec extra bits of precision on output (To Be Confirmed)
     * @return exp(x)
     */
    private static double exp(double x, double extra, double[] hiPrec) {
        double intPartA;
        double intPartB;
        int intVal;

        /* Lookup exp(floor(x)).
         * intPartA will have the upper 22 bits, intPartB will have the lower
         * 52 bits.
         */
        if (x < 0.0) {
            intVal = (int) -x;

            if (intVal > 746) {
                if (hiPrec != null) {
                    hiPrec[0] = 0.0;
                    hiPrec[1] = 0.0;
                }
                return 0.0;
            }

            if (intVal > 709) {
                /* This will produce a subnormal output */
                final double result = exp(x+40.19140625, extra, hiPrec) / 285040095144011776.0;
                if (hiPrec != null) {
                    hiPrec[0] /= 285040095144011776.0;
                    hiPrec[1] /= 285040095144011776.0;
                }
                return result;
            }

            if (intVal == 709) {
                /* exp(1.494140625) is nearly a machine number... */
                final double result = exp(x+1.494140625, extra, hiPrec) / 4.455505956692756620;
                if (hiPrec != null) {
                    hiPrec[0] /= 4.455505956692756620;
                    hiPrec[1] /= 4.455505956692756620;
                }
                return result;
            }

            intVal++;

            intPartA = EXP_INT_TABLE_A[750-intVal];
            intPartB = EXP_INT_TABLE_B[750-intVal];

            intVal = -intVal;
        } else {
            intVal = (int) x;

            if (intVal > 709) {
                if (hiPrec != null) {
                    hiPrec[0] = Double.POSITIVE_INFINITY;
                    hiPrec[1] = 0.0;
                }
                return Double.POSITIVE_INFINITY;
            }

            intPartA = EXP_INT_TABLE_A[750+intVal];
            intPartB = EXP_INT_TABLE_B[750+intVal];
        }

        /* Get the fractional part of x, find the greatest multiple of 2^-10 less than
         * x and look up the exp function of it.
         * fracPartA will have the upper 22 bits, fracPartB the lower 52 bits.
         */
        final int intFrac = (int) ((x - intVal) * 1024.0);
        final double fracPartA = EXP_FRAC_TABLE_A[intFrac];
        final double fracPartB = EXP_FRAC_TABLE_B[intFrac];

        /* epsilon is the difference in x from the nearest multiple of 2^-10.  It
         * has a value in the range 0 <= epsilon < 2^-10.
         * Do the subtraction from x as the last step to avoid possible loss of percison.
         */
        final double epsilon = x - (intVal + intFrac / 1024.0);

        /* Compute z = exp(epsilon) - 1.0 via a minimax polynomial.  z has
       full double precision (52 bits).  Since z < 2^-10, we will have
       62 bits of precision when combined with the contant 1.  This will be
       used in the last addition below to get proper rounding. */

        /* Remez generated polynomial.  Converges on the interval [0, 2^-10], error
       is less than 0.5 ULP */
        double z = 0.04168701738764507;
        z = z * epsilon + 0.1666666505023083;
        z = z * epsilon + 0.5000000000042687;
        z = z * epsilon + 1.0;
        z = z * epsilon + -3.940510424527919E-20;

        /* Compute (intPartA+intPartB) * (fracPartA+fracPartB) by binomial
       expansion.
       tempA is exact since intPartA and intPartB only have 22 bits each.
       tempB will have 52 bits of precision.
         */
        double tempA = intPartA * fracPartA;
        double tempB = intPartA * fracPartB + intPartB * fracPartA + intPartB * fracPartB;

        /* Compute the result.  (1+z)(tempA+tempB).  Order of operations is
       important.  For accuracy add by increasing size.  tempA is exact and
       much larger than the others.  If there are extra bits specified from the
       pow() function, use them. */
        final double tempC = tempB + tempA;
        final double result;
        if (extra != 0.0) {
            result = tempC*extra*z + tempC*extra + tempC*z + tempB + tempA;
        } else {
            result = tempC*z + tempB + tempA;
        }

        if (hiPrec != null) {
            // If requesting high precision
            hiPrec[0] = tempA;
            hiPrec[1] = tempC*extra*z + tempC*extra + tempC*z + tempB;
        }

        return result;
    }

    /** Compute exp(x) - 1
     * @param x number to compute shifted exponential
     * @return exp(x) - 1
     */
    public static double expm1(double x) {
      return expm1(x, null);
    }

    /** Internal helper method for expm1
     * @param x number to compute shifted exponential
     * @param hiPrecOut receive high precision result for -1.0 < x < 1.0
     * @return exp(x) - 1
     */
    private static double expm1(double x, double hiPrecOut[]) {
        if (x != x || x == 0.0) { // NaN or zero
            return x;
        }

        if (x <= -1.0 || x >= 1.0) {
            // If not between +/- 1.0
            //return exp(x) - 1.0;
            double hiPrec[] = new double[2];
            exp(x, 0.0, hiPrec);
            if (x > 0.0) {
                return -1.0 + hiPrec[0] + hiPrec[1];
            } else {
                final double ra = -1.0 + hiPrec[0];
                double rb = -(ra + 1.0 - hiPrec[0]);
                rb += hiPrec[1];
                return ra + rb;
            }
        }

        double baseA;
        double baseB;
        double epsilon;
        boolean negative = false;

        if (x < 0.0) {
            x = -x;
            negative = true;
        }

        {
            int intFrac = (int) (x * 1024.0);
            double tempA = EXP_FRAC_TABLE_A[intFrac] - 1.0;
            double tempB = EXP_FRAC_TABLE_B[intFrac];

            double temp = tempA + tempB;
            tempB = -(temp - tempA - tempB);
            tempA = temp;

            temp = tempA * HEX_40000000;
            baseA = tempA + temp - temp;
            baseB = tempB + (tempA - baseA);

            epsilon = x - intFrac/1024.0;
        }


        /* Compute expm1(epsilon) */
        double zb = 0.008336750013465571;
        zb = zb * epsilon + 0.041666663879186654;
        zb = zb * epsilon + 0.16666666666745392;
        zb = zb * epsilon + 0.49999999999999994;
        zb = zb * epsilon;
        zb = zb * epsilon;

        double za = epsilon;
        double temp = za + zb;
        zb = -(temp - za - zb);
        za = temp;

        temp = za * HEX_40000000;
        temp = za + temp - temp;
        zb += za - temp;
        za = temp;

        /* Combine the parts.   expm1(a+b) = expm1(a) + expm1(b) + expm1(a)*expm1(b) */
        double ya = za * baseA;
        //double yb = za*baseB + zb*baseA + zb*baseB;
        temp = ya + za * baseB;
        double yb = -(temp - ya - za * baseB);
        ya = temp;

        temp = ya + zb * baseA;
        yb += -(temp - ya - zb * baseA);
        ya = temp;

        temp = ya + zb * baseB;
        yb += -(temp - ya - zb*baseB);
        ya = temp;

        //ya = ya + za + baseA;
        //yb = yb + zb + baseB;
        temp = ya + baseA;
        yb += -(temp - baseA - ya);
        ya = temp;

        temp = ya + za;
        //yb += (ya > za) ? -(temp - ya - za) : -(temp - za - ya);
        yb += -(temp - ya - za);
        ya = temp;

        temp = ya + baseB;
        //yb += (ya > baseB) ? -(temp - ya - baseB) : -(temp - baseB - ya);
        yb += -(temp - ya - baseB);
        ya = temp;

        temp = ya + zb;
        //yb += (ya > zb) ? -(temp - ya - zb) : -(temp - zb - ya);
        yb += -(temp - ya - zb);
        ya = temp;

        if (negative) {
            /* Compute expm1(-x) = -expm1(x) / (expm1(x) + 1) */
            double denom = 1.0 + ya;
            double denomr = 1.0 / denom;
            double denomb = -(denom - 1.0 - ya) + yb;
            double ratio = ya * denomr;
            temp = ratio * HEX_40000000;
            final double ra = ratio + temp - temp;
            double rb = ratio - ra;

            temp = denom * HEX_40000000;
            za = denom + temp - temp;
            zb = denom - za;

            rb += (ya - za * ra - za * rb - zb * ra - zb * rb) * denomr;

            // f(x) = x/1+x
            // Compute f'(x)
            // Product rule:  d(uv) = du*v + u*dv
            // Chain rule:  d(f(g(x)) = f'(g(x))*f(g'(x))
            // d(1/x) = -1/(x*x)
            // d(1/1+x) = -1/( (1+x)^2) *  1 =  -1/((1+x)*(1+x))
            // d(x/1+x) = -x/((1+x)(1+x)) + 1/1+x = 1 / ((1+x)(1+x))

            // Adjust for yb
            rb += yb * denomr;                      // numerator
            rb += -ya * denomb * denomr * denomr;   // denominator

            // negate
            ya = -ra;
            yb = -rb;
        }

        if (hiPrecOut != null) {
            hiPrecOut[0] = ya;
            hiPrecOut[1] = yb;
        }

        return ya + yb;
    }

    /**
     *  For x between 0 and 1, returns exp(x), uses extended precision
     *  @param x argument of exponential
     *  @param result placeholder where to place exp(x) split in two terms
     *  for extra precision (i.e. exp(x) = result[0] ° result[1]
     *  @return exp(x)
     */
    private static double slowexp(final double x, final double result[]) {
        final double xs[] = new double[2];
        final double ys[] = new double[2];
        final double facts[] = new double[2];
        final double as[] = new double[2];
        split(x, xs);
        ys[0] = ys[1] = 0.0;

        for (int i = 19; i >= 0; i--) {
            splitMult(xs, ys, as);
            ys[0] = as[0];
            ys[1] = as[1];

            split(FACT[i], as);
            splitReciprocal(as, facts);

            splitAdd(ys, facts, as);
            ys[0] = as[0];
            ys[1] = as[1];
        }

        if (result != null) {
            result[0] = ys[0];
            result[1] = ys[1];
        }

        return ys[0] + ys[1];
    }

    /** Compute split[0], split[1] such that their sum is equal to d,
     * and split[0] has its 30 least significant bits as zero.
     * @param d number to split
     * @param split placeholder where to place the result
     */
    private static void split(final double d, final double split[]) {
        if (d < 8e298 && d > -8e298) {
            final double a = d * HEX_40000000;
            split[0] = (d + a) - a;
            split[1] = d - split[0];
        } else {
            final double a = d * 9.31322574615478515625E-10;
            split[0] = (d + a - d) * HEX_40000000;
            split[1] = d - split[0];
        }
    }

    /** Recompute a split.
     * @param a input/out array containing the split, changed
     * on output
     */
    private static void resplit(final double a[]) {
        final double c = a[0] + a[1];
        final double d = -(c - a[0] - a[1]);

        if (c < 8e298 && c > -8e298) {
            double z = c * HEX_40000000;
            a[0] = (c + z) - z;
            a[1] = c - a[0] + d;
        } else {
            double z = c * 9.31322574615478515625E-10;
            a[0] = (c + z - c) * HEX_40000000;
            a[1] = c - a[0] + d;
        }
    }

    /** Multiply two numbers in split form.
     * @param a first term of multiplication
     * @param b second term of multiplication
     * @param ans placeholder where to put the result
     */
    private static void splitMult(double a[], double b[], double ans[]) {
        ans[0] = a[0] * b[0];
        ans[1] = a[0] * b[1] + a[1] * b[0] + a[1] * b[1];

        /* Resplit */
        resplit(ans);
    }

    /** Add two numbers in split form.
     * @param a first term of addition
     * @param b second term of addition
     * @param ans placeholder where to put the result
     */
    private static void splitAdd(final double a[], final double b[], final double ans[]) {
        ans[0] = a[0] + b[0];
        ans[1] = a[1] + b[1];

        resplit(ans);
    }

    /** Compute the reciprocal of in.  Use the following algorithm.
     *  in = c + d.
     *  want to find x + y such that x+y = 1/(c+d) and x is much
     *  larger than y and x has several zero bits on the right.
     *
     *  Set b = 1/(2^22),  a = 1 - b.  Thus (a+b) = 1.
     *  Use following identity to compute (a+b)/(c+d)
     *
     *  (a+b)/(c+d)  =   a/c   +    (bc - ad) / (c^2 + cd)
     *  set x = a/c  and y = (bc - ad) / (c^2 + cd)
     *  This will be close to the right answer, but there will be
     *  some rounding in the calculation of X.  So by carefully
     *  computing 1 - (c+d)(x+y) we can compute an error and
     *  add that back in.   This is done carefully so that terms
     *  of similar size are subtracted first.
     *  @param in initial number, in split form
     *  @param result placeholder where to put the result
     */
    private static void splitReciprocal(final double in[], final double result[]) {
        final double b = 1.0/4194304.0;
        final double a = 1.0 - b;

        if (in[0] == 0.0) {
            in[0] = in[1];
            in[1] = 0.0;
        }

        result[0] = a / in[0];
        result[1] = (b*in[0]-a*in[1]) / (in[0]*in[0] + in[0]*in[1]);

        if (result[1] != result[1]) { // can happen if result[1] is NAN
            result[1] = 0.0;
        }

        /* Resplit */
        resplit(result);

        for (int i = 0; i < 2; i++) {
            /* this may be overkill, probably once is enough */
            double err = 1.0 - result[0] * in[0] - result[0] * in[1] -
            result[1] * in[0] - result[1] * in[1];
            /*err = 1.0 - err; */
            err = err * (result[0] + result[1]);
            /*printf("err = %16e\n", err); */
            result[1] += err;
        }
    }

    /** Compute (a[0] + a[1]) * (b[0] + b[1]) in extended precision.
     * @param a first term of the multiplication
     * @param b second term of the multiplication
     * @param result placeholder where to put the result
     */
    private static void quadMult(final double a[], final double b[], final double result[]) {
        final double xs[] = new double[2];
        final double ys[] = new double[2];
        final double zs[] = new double[2];

        /* a[0] * b[0] */
        split(a[0], xs);
        split(b[0], ys);
        splitMult(xs, ys, zs);

        result[0] = zs[0];
        result[1] = zs[1];

        /* a[0] * b[1] */
        split(b[1], ys);
        splitMult(xs, ys, zs);

        double tmp = result[0] + zs[0];
        result[1] = result[1] - (tmp - result[0] - zs[0]);
        result[0] = tmp;
        tmp = result[0] + zs[1];
        result[1] = result[1] - (tmp - result[0] - zs[1]);
        result[0] = tmp;

        /* a[1] * b[0] */
        split(a[1], xs);
        split(b[0], ys);
        splitMult(xs, ys, zs);

        tmp = result[0] + zs[0];
        result[1] = result[1] - (tmp - result[0] - zs[0]);
        result[0] = tmp;
        tmp = result[0] + zs[1];
        result[1] = result[1] - (tmp - result[0] - zs[1]);
        result[0] = tmp;

        /* a[1] * b[0] */
        split(a[1], xs);
        split(b[1], ys);
        splitMult(xs, ys, zs);

        tmp = result[0] + zs[0];
        result[1] = result[1] - (tmp - result[0] - zs[0]);
        result[0] = tmp;
        tmp = result[0] + zs[1];
        result[1] = result[1] - (tmp - result[0] - zs[1]);
        result[0] = tmp;
    }

    /** Compute exp(p) for a integer p in extended precision.
     * @param p integer whose exponential is requested
     * @param result placeholder where to put the result in extended precision
     * @return exp(p) in standard precision (equal to result[0] + result[1])
     */
    private static double expint(int p, final double result[]) {
        //double x = M_E;
        final double xs[] = new double[2];
        final double as[] = new double[2];
        final double ys[] = new double[2];
        //split(x, xs);
        //xs[1] = (double)(2.7182818284590452353602874713526625L - xs[0]);
        //xs[0] = 2.71827697753906250000;
        //xs[1] = 4.85091998273542816811e-06;
        //xs[0] = Double.longBitsToDouble(0x4005bf0800000000L);
        //xs[1] = Double.longBitsToDouble(0x3ed458a2bb4a9b00L);

        /* E */
        xs[0] = 2.718281828459045;
        xs[1] = 1.4456468917292502E-16;

        split(1.0, ys);

        while (p > 0) {
            if ((p & 1) != 0) {
                quadMult(ys, xs, as);
                ys[0] = as[0]; ys[1] = as[1];
            }

            quadMult(xs, xs, as);
            xs[0] = as[0]; xs[1] = as[1];

            p >>= 1;
        }

        if (result != null) {
            result[0] = ys[0];
            result[1] = ys[1];

            resplit(result);
        }

        return ys[0] + ys[1];
    }


    /**
     * Natural logarithm.
     *
     * @param x   a double
     * @return log(x)
     */
    public static double log(final double x) {
        return log(x, null);
    }

    /**
     * Internal helper method for natural logarithm function.
     * @param x original argument of the natural logarithm function
     * @param hiPrec extra bits of precision on output (To Be Confirmed)
     * @return log(x)
     */
    private static double log(final double x, final double[] hiPrec) {
        if (x==0) { // Handle special case of +0/-0
            return Double.NEGATIVE_INFINITY;
        }
        long bits = Double.doubleToLongBits(x);

        /* Handle special cases of negative input, and NaN */
        if ((bits & 0x8000000000000000L) != 0 || x != x) {
            if (x != 0.0) {
                if (hiPrec != null) {
                    hiPrec[0] = Double.NaN;
                }

                return Double.NaN;
            }
        }

        /* Handle special cases of Positive infinity. */
        if (x == Double.POSITIVE_INFINITY) {
            if (hiPrec != null) {
                hiPrec[0] = Double.POSITIVE_INFINITY;
            }

            return Double.POSITIVE_INFINITY;
        }

        /* Extract the exponent */
        int exp = (int)(bits >> 52)-1023;

        if ((bits & 0x7ff0000000000000L) == 0) {
            // Subnormal!
            if (x == 0) {
                // Zero
                if (hiPrec != null) {
                    hiPrec[0] = Double.NEGATIVE_INFINITY;
                }

                return Double.NEGATIVE_INFINITY;
            }

            /* Normalize the subnormal number. */
            bits <<= 1;
            while ( (bits & 0x0010000000000000L) == 0) {
                exp--;
                bits <<= 1;
            }
        }


        if (exp == -1 || exp == 0) {
            if (x < 1.01 && x > 0.99 && hiPrec == null) {
                /* The normal method doesn't work well in the range [0.99, 1.01], so call do a straight
           polynomial expansion in higer precision. */

               /* Compute x - 1.0 and split it */
                double xa = x - 1.0;
                double xb = xa - x + 1.0;
                double tmp = xa * HEX_40000000;
                double aa = xa + tmp - tmp;
                double ab = xa - aa;
                xa = aa;
                xb = ab;

                double ya = LN_QUICK_COEF[LN_QUICK_COEF.length-1][0];
                double yb = LN_QUICK_COEF[LN_QUICK_COEF.length-1][1];

                for (int i = LN_QUICK_COEF.length - 2; i >= 0; i--) {
                    /* Multiply a = y * x */
                    aa = ya * xa;
                    ab = ya * xb + yb * xa + yb * xb;
                    /* split, so now y = a */
                    tmp = aa * HEX_40000000;
                    ya = aa + tmp - tmp;
                    yb = aa - ya + ab;

                    /* Add  a = y + lnQuickCoef */
                    aa = ya + LN_QUICK_COEF[i][0];
                    ab = yb + LN_QUICK_COEF[i][1];
                    /* Split y = a */
                    tmp = aa * HEX_40000000;
                    ya = aa + tmp - tmp;
                    yb = aa - ya + ab;
                }

                /* Multiply a = y * x */
                aa = ya * xa;
                ab = ya * xb + yb * xa + yb * xb;
                /* split, so now y = a */
                tmp = aa * HEX_40000000;
                ya = aa + tmp - tmp;
                yb = aa - ya + ab;

                return ya + yb;
            }
        }

        // lnm is a log of a number in the range of 1.0 - 2.0, so 0 <= lnm < ln(2)
        double lnm[] = LN_MANT[(int)((bits & 0x000ffc0000000000L) >> 42)];

        /*
    double epsilon = x / Double.longBitsToDouble(bits & 0xfffffc0000000000L);

    epsilon -= 1.0;
         */

        // y is the most significant 10 bits of the mantissa
        //double y = Double.longBitsToDouble(bits & 0xfffffc0000000000L);
        //double epsilon = (x - y) / y;
        double epsilon = (bits & 0x3ffffffffffL) / (TWO_POWER_52 + (bits & 0x000ffc0000000000L));

        double lnza = 0.0;
        double lnzb = 0.0;

        if (hiPrec != null) {
            /* split epsilon -> x */
            double tmp = epsilon * HEX_40000000;
            double aa = epsilon + tmp - tmp;
            double ab = epsilon - aa;
            double xa = aa;
            double xb = ab;

            /* Need a more accurate epsilon, so adjust the division. */
            double numer = bits & 0x3ffffffffffL;
            double denom = TWO_POWER_52 + (bits & 0x000ffc0000000000L);
            aa = numer - xa*denom - xb * denom;
            xb += aa / denom;

            /* Remez polynomial evaluation */
            double ya = LN_HI_PREC_COEF[LN_HI_PREC_COEF.length-1][0];
            double yb = LN_HI_PREC_COEF[LN_HI_PREC_COEF.length-1][1];

            for (int i = LN_HI_PREC_COEF.length - 2; i >= 0; i--) {
                /* Multiply a = y * x */
                aa = ya * xa;
                ab = ya * xb + yb * xa + yb * xb;
                /* split, so now y = a */
                tmp = aa * HEX_40000000;
                ya = aa + tmp - tmp;
                yb = aa - ya + ab;

                /* Add  a = y + lnHiPrecCoef */
                aa = ya + LN_HI_PREC_COEF[i][0];
                ab = yb + LN_HI_PREC_COEF[i][1];
                /* Split y = a */
                tmp = aa * HEX_40000000;
                ya = aa + tmp - tmp;
                yb = aa - ya + ab;
            }

            /* Multiply a = y * x */
            aa = ya * xa;
            ab = ya * xb + yb * xa + yb * xb;

            /* split, so now lnz = a */
            /*
      tmp = aa * 1073741824.0;
      lnza = aa + tmp - tmp;
      lnzb = aa - lnza + ab;
             */
            lnza = aa + ab;
            lnzb = -(lnza - aa - ab);
        } else {
            /* High precision not required.  Eval Remez polynomial
         using standard double precision */
            lnza = -0.16624882440418567;
            lnza = lnza * epsilon + 0.19999954120254515;
            lnza = lnza * epsilon + -0.2499999997677497;
            lnza = lnza * epsilon + 0.3333333333332802;
            lnza = lnza * epsilon + -0.5;
            lnza = lnza * epsilon + 1.0;
            lnza = lnza * epsilon;
        }

        /* Relative sizes:
         * lnzb     [0, 2.33E-10]
         * lnm[1]   [0, 1.17E-7]
         * ln2B*exp [0, 1.12E-4]
         * lnza      [0, 9.7E-4]
         * lnm[0]   [0, 0.692]
         * ln2A*exp [0, 709]
         */

        /* Compute the following sum:
         * lnzb + lnm[1] + ln2B*exp + lnza + lnm[0] + ln2A*exp;
         */

        //return lnzb + lnm[1] + ln2B*exp + lnza + lnm[0] + ln2A*exp;
        double a = LN_2_A*exp;
        double b = 0.0;
        double c = a+lnm[0];
        double d = -(c-a-lnm[0]);
        a = c;
        b = b + d;

        c = a + lnza;
        d = -(c - a - lnza);
        a = c;
        b = b + d;

        c = a + LN_2_B*exp;
        d = -(c - a - LN_2_B*exp);
        a = c;
        b = b + d;

        c = a + lnm[1];
        d = -(c - a - lnm[1]);
        a = c;
        b = b + d;

        c = a + lnzb;
        d = -(c - a - lnzb);
        a = c;
        b = b + d;

        if (hiPrec != null) {
            hiPrec[0] = a;
            hiPrec[1] = b;
        }

        return a + b;
    }

    /** Compute log(1 + x).
     * @param x a number
     * @return log(1 + x)
     */
    public static double log1p(final double x) {
        double xpa = 1.0 + x;
        double xpb = -(xpa - 1.0 - x);

        if (x == -1) {
            return x/0.0;   // -Infinity
        }

        if (x > 0 && 1/x == 0) { // x = Infinity
            return x;
        }

        if (x>1e-6 || x<-1e-6) {
            double hiPrec[] = new double[2];

            final double lores = log(xpa, hiPrec);
            if (Double.isInfinite(lores)){ // don't allow this to be converted to NaN
                return lores;
            }

            /* Do a taylor series expansion around xpa */
            /* f(x+y) = f(x) + f'(x)*y + f''(x)/2 y^2 */
            double fx1 = xpb/xpa;

            double epsilon = 0.5 * fx1 + 1.0;
            epsilon = epsilon * fx1;

            return epsilon + hiPrec[1] + hiPrec[0];
        }

        /* Value is small |x| < 1e6, do a Taylor series centered on 1.0 */
        double y = x * 0.333333333333333 - 0.5;
        y = y * x + 1.0;
        y = y * x;

        return y;
    }

    /** Compute the base 10 logarithm.
     * @param x a number
     * @return log10(x)
     */
    public static double log10(final double x) {
        final double hiPrec[] = new double[2];

        final double lores = log(x, hiPrec);
        if (Double.isInfinite(lores)){ // don't allow this to be converted to NaN
            return lores;
        }

        final double tmp = hiPrec[0] * HEX_40000000;
        final double lna = hiPrec[0] + tmp - tmp;
        final double lnb = hiPrec[0] - lna + hiPrec[1];

        final double rln10a = 0.4342944622039795;
        final double rln10b = 1.9699272335463627E-8;

        return rln10b * lnb + rln10b * lna + rln10a * lnb + rln10a * lna;
    }

    /**
     * Power function.  Compute x^y.
     *
     * @param x   a double
     * @param y   a double
     * @return double
     */
    public static double pow(double x, double y) {
        final double lns[] = new double[2];

        if (y == 0.0) {
            return 1.0;
        }

        if (x != x) { // X is NaN
            return x;
        }


        if (x == 0) {
            long bits = Double.doubleToLongBits(x);
            if ((bits & 0x8000000000000000L) != 0) {
                // -zero
                long yi = (long) y;

                if (y < 0 && y == yi && (yi & 1) == 1) {
                    return Double.NEGATIVE_INFINITY;
                }

                if (y < 0 && y == yi && (yi & 1) == 1) {
                    return -0.0;
                }

                if (y > 0 && y == yi && (yi & 1) == 1) {
                    return -0.0;
                }
            }

            if (y < 0) {
                return Double.POSITIVE_INFINITY;
            }
            if (y > 0) {
                return 0.0;
            }

            return Double.NaN;
        }

        if (x == Double.POSITIVE_INFINITY) {
            if (y != y) { // y is NaN
                return y;
            }
            if (y < 0.0) {
                return 0.0;
            } else {
                return Double.POSITIVE_INFINITY;
            }
        }

        if (y == Double.POSITIVE_INFINITY) {
            if (x * x == 1.0)
              return Double.NaN;

            if (x * x > 1.0) {
                return Double.POSITIVE_INFINITY;
            } else {
                return 0.0;
            }
        }

        if (x == Double.NEGATIVE_INFINITY) {
            if (y != y) { // y is NaN
                return y;
            }

            if (y < 0) {
                long yi = (long) y;
                if (y == yi && (yi & 1) == 1) {
                    return -0.0;
                }

                return 0.0;
            }

            if (y > 0)  {
                long yi = (long) y;
                if (y == yi && (yi & 1) == 1) {
                    return Double.NEGATIVE_INFINITY;
                }

                return Double.POSITIVE_INFINITY;
            }
        }

        if (y == Double.NEGATIVE_INFINITY) {

            if (x * x == 1.0) {
                return Double.NaN;
            }

            if (x * x < 1.0) {
                return Double.POSITIVE_INFINITY;
            } else {
                return 0.0;
            }
        }

        /* Handle special case x<0 */
        if (x < 0) {
            // y is an even integer in this case
            if (y >= TWO_POWER_52 || y <= -TWO_POWER_52) {
                return pow(-x, y);
            }

            if (y == (long) y) {
                // If y is an integer
                return ((long)y & 1) == 0 ? pow(-x, y) : -pow(-x, y);
            } else {
                return Double.NaN;
            }
        }

        /* Split y into ya and yb such that y = ya+yb */
        double ya;
        double yb;
        if (y < 8e298 && y > -8e298) {
            double tmp1 = y * HEX_40000000;
            ya = y + tmp1 - tmp1;
            yb = y - ya;
        } else {
            double tmp1 = y * 9.31322574615478515625E-10;
            double tmp2 = tmp1 * 9.31322574615478515625E-10;
            ya = (tmp1 + tmp2 - tmp1) * HEX_40000000 * HEX_40000000;
            yb = y - ya;
        }

        /* Compute ln(x) */
        final double lores = log(x, lns);
        if (Double.isInfinite(lores)){ // don't allow this to be converted to NaN
            return lores;
        }

        double lna = lns[0];
        double lnb = lns[1];

        /* resplit lns */
        double tmp1 = lna * HEX_40000000;
        double tmp2 = lna + tmp1 - tmp1;
        lnb += lna - tmp2;
        lna = tmp2;

        // y*ln(x) = (aa+ab)
        final double aa = lna * ya;
        final double ab = lna * yb + lnb * ya + lnb * yb;

        lna = aa+ab;
        lnb = -(lna - aa - ab);

        double z = 1.0 / 120.0;
        z = z * lnb + (1.0 / 24.0);
        z = z * lnb + (1.0 / 6.0);
        z = z * lnb + 0.5;
        z = z * lnb + 1.0;
        z = z * lnb;

        final double result = exp(lna, z, null);
        //result = result + result * z;
        return result;
    }

    /** xi in the range of [1, 2].
     *                                3        5        7
     *      x+1           /          x        x        x          \
     *  ln ----- =   2 *  |  x  +   ----  +  ----  +  ---- + ...  |
     *      1-x           \          3        5        7          /
     *
     * So, compute a Remez approximation of the following function
     *
     *  ln ((sqrt(x)+1)/(1-sqrt(x)))  /  x
     *
     * This will be an even function with only positive coefficents.
     * x is in the range [0 - 1/3].
     *
     * Transform xi for input to the above function by setting
     * x = (xi-1)/(xi+1).   Input to the polynomial is x^2, then
     * the result is multiplied by x.
     * @param xi number from which log is requested
     * @return log(xi)
     */
    private static double[] slowLog(double xi) {
        double x[] = new double[2];
        double x2[] = new double[2];
        double y[] = new double[2];
        double a[] = new double[2];

        split(xi, x);

        /* Set X = (x-1)/(x+1) */
        x[0] += 1.0;
        resplit(x);
        splitReciprocal(x, a);
        x[0] -= 2.0;
        resplit(x);
        splitMult(x, a, y);
        x[0] = y[0];
        x[1] = y[1];

        /* Square X -> X2*/
        splitMult(x, x, x2);


        //x[0] -= 1.0;
        //resplit(x);

        y[0] = LN_SPLIT_COEF[LN_SPLIT_COEF.length-1][0];
        y[1] = LN_SPLIT_COEF[LN_SPLIT_COEF.length-1][1];

        for (int i = LN_SPLIT_COEF.length-2; i >= 0; i--) {
            splitMult(y, x2, a);
            y[0] = a[0];
            y[1] = a[1];
            splitAdd(y, LN_SPLIT_COEF[i], a);
            y[0] = a[0];
            y[1] = a[1];
        }

        splitMult(y, x, a);
        y[0] = a[0];
        y[1] = a[1];

        return y;
    }

    /**
     * For x between 0 and pi/4 compute sine.
     * @param x number from which sine is requested
     * @param result placeholder where to put the result in extended precision
     * @return sin(x)
     */
    private static double slowSin(final double x, final double result[]) {
        final double xs[] = new double[2];
        final double ys[] = new double[2];
        final double facts[] = new double[2];
        final double as[] = new double[2];
        split(x, xs);
        ys[0] = ys[1] = 0.0;

        for (int i = 19; i >= 0; i--) {
            splitMult(xs, ys, as);
            ys[0] = as[0]; ys[1] = as[1];

            if ( (i & 1) == 0) {
                continue;
            }

            split(FACT[i], as);
            splitReciprocal(as, facts);

            if ( (i & 2) != 0 ) {
                facts[0] = -facts[0];
                facts[1] = -facts[1];
            }

            splitAdd(ys, facts, as);
            ys[0] = as[0]; ys[1] = as[1];
        }

        if (result != null) {
            result[0] = ys[0];
            result[1] = ys[1];
        }

        return ys[0] + ys[1];
    }

    /**
     *  For x between 0 and pi/4 compute cosine
     * @param x number from which cosine is requested
     * @param result placeholder where to put the result in extended precision
     * @return cos(x)
     */
    private static double slowCos(final double x, final double result[]) {

        final double xs[] = new double[2];
        final double ys[] = new double[2];
        final double facts[] = new double[2];
        final double as[] = new double[2];
        split(x, xs);
        ys[0] = ys[1] = 0.0;

        for (int i = 19; i >= 0; i--) {
            splitMult(xs, ys, as);
            ys[0] = as[0]; ys[1] = as[1];

            if ( (i & 1) != 0) {
                continue;
            }

            split(FACT[i], as);
            splitReciprocal(as, facts);

            if ( (i & 2) != 0 ) {
                facts[0] = -facts[0];
                facts[1] = -facts[1];
            }

            splitAdd(ys, facts, as);
            ys[0] = as[0]; ys[1] = as[1];
        }

        if (result != null) {
            result[0] = ys[0];
            result[1] = ys[1];
        }

        return ys[0] + ys[1];
    }

    /** Build the sine and cosine tables.
     */
    private static void buildSinCosTables() {
        final double result[] = new double[2];

        /* Use taylor series for 0 <= x <= 6/8 */
        for (int i = 0; i < 7; i++) {
            double x = i / 8.0;

            slowSin(x, result);
            SINE_TABLE_A[i] = result[0];
            SINE_TABLE_B[i] = result[1];

            slowCos(x, result);
            COSINE_TABLE_A[i] = result[0];
            COSINE_TABLE_B[i] = result[1];
        }

        /* Use angle addition formula to complete table to 13/8, just beyond pi/2 */
        for (int i = 7; i < 14; i++) {
            double xs[] = new double[2];
            double ys[] = new double[2];
            double as[] = new double[2];
            double bs[] = new double[2];
            double temps[] = new double[2];

            if ( (i & 1) == 0) {
                // Even, use double angle
                xs[0] = SINE_TABLE_A[i/2];
                xs[1] = SINE_TABLE_B[i/2];
                ys[0] = COSINE_TABLE_A[i/2];
                ys[1] = COSINE_TABLE_B[i/2];

                /* compute sine */
                splitMult(xs, ys, result);
                SINE_TABLE_A[i] = result[0] * 2.0;
                SINE_TABLE_B[i] = result[1] * 2.0;

                /* Compute cosine */
                splitMult(ys, ys, as);
                splitMult(xs, xs, temps);
                temps[0] = -temps[0];
                temps[1] = -temps[1];
                splitAdd(as, temps, result);
                COSINE_TABLE_A[i] = result[0];
                COSINE_TABLE_B[i] = result[1];
            } else {
                xs[0] = SINE_TABLE_A[i/2];
                xs[1] = SINE_TABLE_B[i/2];
                ys[0] = COSINE_TABLE_A[i/2];
                ys[1] = COSINE_TABLE_B[i/2];
                as[0] = SINE_TABLE_A[i/2+1];
                as[1] = SINE_TABLE_B[i/2+1];
                bs[0] = COSINE_TABLE_A[i/2+1];
                bs[1] = COSINE_TABLE_B[i/2+1];

                /* compute sine */
                splitMult(xs, bs, temps);
                splitMult(ys, as, result);
                splitAdd(result, temps, result);
                SINE_TABLE_A[i] = result[0];
                SINE_TABLE_B[i] = result[1];

                /* Compute cosine */
                splitMult(ys, bs, result);
                splitMult(xs, as, temps);
                temps[0] = -temps[0];
                temps[1] = -temps[1];
                splitAdd(result, temps, result);
                COSINE_TABLE_A[i] = result[0];
                COSINE_TABLE_B[i] = result[1];
            }
        }

        /* Compute tangent = sine/cosine */
        for (int i = 0; i < 14; i++) {
            double xs[] = new double[2];
            double ys[] = new double[2];
            double as[] = new double[2];

            as[0] = COSINE_TABLE_A[i];
            as[1] = COSINE_TABLE_B[i];

            splitReciprocal(as, ys);

            xs[0] = SINE_TABLE_A[i];
            xs[1] = SINE_TABLE_B[i];

            splitMult(xs, ys, as);

            TANGENT_TABLE_A[i] = as[0];
            TANGENT_TABLE_B[i] = as[1];
        }

    }

    /**
     *  Computes sin(x) - x, where |x| < 1/16.
     *  Use a Remez polynomial approximation.
     *  @param x a number smaller than 1/16
     *  @return sin(x) - x
     */
    private static double polySine(final double x)
    {
        double x2 = x*x;

        double p = 2.7553817452272217E-6;
        p = p * x2 + -1.9841269659586505E-4;
        p = p * x2 + 0.008333333333329196;
        p = p * x2 + -0.16666666666666666;
        //p *= x2;
        //p *= x;
        p = p * x2 * x;

        return p;
    }

    /**
     *  Computes cos(x) - 1, where |x| < 1/16.
     *  Use a Remez polynomial approximation.
     *  @param x a number smaller than 1/16
     *  @return cos(x) - 1
     */
    private static double polyCosine(double x) {
        double x2 = x*x;

        double p = 2.479773539153719E-5;
        p = p * x2 + -0.0013888888689039883;
        p = p * x2 + 0.041666666666621166;
        p = p * x2 + -0.49999999999999994;
        p *= x2;

        return p;
    }

    /**
     *  Compute sine over the first quadrant (0 < x < pi/2).
     *  Use combination of table lookup and rational polynomial expansion.
     *  @param xa number from which sine is requested
     *  @param xb extra bits for x (may be 0.0)
     *  @return sin(xa + xb)
     */
    private static double sinQ(double xa, double xb) {
        int idx = (int) ((xa * 8.0) + 0.5);
        final double epsilon = xa - EIGHTHS[idx]; //idx*0.125;

        // Table lookups
        final double sintA = SINE_TABLE_A[idx];
        final double sintB = SINE_TABLE_B[idx];
        final double costA = COSINE_TABLE_A[idx];
        final double costB = COSINE_TABLE_B[idx];

        // Polynomial eval of sin(epsilon), cos(epsilon)
        double sinEpsA = epsilon;
        double sinEpsB = polySine(epsilon);
        final double cosEpsA = 1.0;
        final double cosEpsB = polyCosine(epsilon);

        // Split epsilon   xa + xb = x
        final double temp = sinEpsA * HEX_40000000;
        double temp2 = (sinEpsA + temp) - temp;
        sinEpsB +=  sinEpsA - temp2;
        sinEpsA = temp2;

        /* Compute sin(x) by angle addition formula */
        double result;

        /* Compute the following sum:
         *
         * result = sintA + costA*sinEpsA + sintA*cosEpsB + costA*sinEpsB +
         *          sintB + costB*sinEpsA + sintB*cosEpsB + costB*sinEpsB;
         *
         * Ranges of elements
         *
         * xxxtA   0            PI/2
         * xxxtB   -1.5e-9      1.5e-9
         * sinEpsA -0.0625      0.0625
         * sinEpsB -6e-11       6e-11
         * cosEpsA  1.0
         * cosEpsB  0           -0.0625
         *
         */

        //result = sintA + costA*sinEpsA + sintA*cosEpsB + costA*sinEpsB +
        //          sintB + costB*sinEpsA + sintB*cosEpsB + costB*sinEpsB;

        //result = sintA + sintA*cosEpsB + sintB + sintB * cosEpsB;
        //result += costA*sinEpsA + costA*sinEpsB + costB*sinEpsA + costB * sinEpsB;
        double a = 0;
        double b = 0;

        double t = sintA;
        double c = a + t;
        double d = -(c - a - t);
        a = c;
        b = b + d;

        t = costA * sinEpsA;
        c = a + t;
        d = -(c - a - t);
        a = c;
        b = b + d;

        b = b + sintA * cosEpsB + costA * sinEpsB;
        /*
    t = sintA*cosEpsB;
    c = a + t;
    d = -(c - a - t);
    a = c;
    b = b + d;

    t = costA*sinEpsB;
    c = a + t;
    d = -(c - a - t);
    a = c;
    b = b + d;
         */

        b = b + sintB + costB * sinEpsA + sintB * cosEpsB + costB * sinEpsB;
        /*
    t = sintB;
    c = a + t;
    d = -(c - a - t);
    a = c;
    b = b + d;

    t = costB*sinEpsA;
    c = a + t;
    d = -(c - a - t);
    a = c;
    b = b + d;

    t = sintB*cosEpsB;
    c = a + t;
    d = -(c - a - t);
    a = c;
    b = b + d;

    t = costB*sinEpsB;
    c = a + t;
    d = -(c - a - t);
    a = c;
    b = b + d;
         */

        if (xb != 0.0) {
            t = ((costA + costB) * (cosEpsA + cosEpsB) -
                 (sintA + sintB) * (sinEpsA + sinEpsB)) * xb;  // approximate cosine*xb
            c = a + t;
            d = -(c - a - t);
            a = c;
            b = b + d;
        }

        result = a + b;

        return result;
    }

    /**
     * Compute cosine in the first quadrant by subtracting input from PI/2 and
     * then calling sinQ.  This is more accurate as the input approaches PI/2.
     *  @param xa number from which cosine is requested
     *  @param xb extra bits for x (may be 0.0)
     *  @return cos(xa + xb)
     */
    private static double cosQ(double xa, double xb) {
        final double pi2a = 1.5707963267948966;
        final double pi2b = 6.123233995736766E-17;

        final double a = pi2a - xa;
        double b = -(a - pi2a + xa);
        b += pi2b - xb;

        return sinQ(a, b);
    }

    /**
     *  Compute tangent (or cotangent) over the first quadrant.   0 < x < pi/2
     *  Use combination of table lookup and rational polynomial expansion.
     *  @param xa number from which sine is requested
     *  @param xb extra bits for x (may be 0.0)
     *  @param cotanFlag if true, compute the cotangent instead of the tangent
     *  @return tan(xa+xb) (or cotangent, depending on cotanFlag)
     */
    private static double tanQ(double xa, double xb, boolean cotanFlag) {

        int idx = (int) ((xa * 8.0) + 0.5);
        final double epsilon = xa - EIGHTHS[idx]; //idx*0.125;

        // Table lookups
        final double sintA = SINE_TABLE_A[idx];
        final double sintB = SINE_TABLE_B[idx];
        final double costA = COSINE_TABLE_A[idx];
        final double costB = COSINE_TABLE_B[idx];

        // Polynomial eval of sin(epsilon), cos(epsilon)
        double sinEpsA = epsilon;
        double sinEpsB = polySine(epsilon);
        final double cosEpsA = 1.0;
        final double cosEpsB = polyCosine(epsilon);

        // Split epsilon   xa + xb = x
        double temp = sinEpsA * HEX_40000000;
        double temp2 = (sinEpsA + temp) - temp;
        sinEpsB +=  sinEpsA - temp2;
        sinEpsA = temp2;

        /* Compute sin(x) by angle addition formula */

        /* Compute the following sum:
         *
         * result = sintA + costA*sinEpsA + sintA*cosEpsB + costA*sinEpsB +
         *          sintB + costB*sinEpsA + sintB*cosEpsB + costB*sinEpsB;
         *
         * Ranges of elements
         *
         * xxxtA   0            PI/2
         * xxxtB   -1.5e-9      1.5e-9
         * sinEpsA -0.0625      0.0625
         * sinEpsB -6e-11       6e-11
         * cosEpsA  1.0
         * cosEpsB  0           -0.0625
         *
         */

        //result = sintA + costA*sinEpsA + sintA*cosEpsB + costA*sinEpsB +
        //          sintB + costB*sinEpsA + sintB*cosEpsB + costB*sinEpsB;

        //result = sintA + sintA*cosEpsB + sintB + sintB * cosEpsB;
        //result += costA*sinEpsA + costA*sinEpsB + costB*sinEpsA + costB * sinEpsB;
        double a = 0;
        double b = 0;

        // Compute sine
        double t = sintA;
        double c = a + t;
        double d = -(c - a - t);
        a = c;
        b = b + d;

        t = costA*sinEpsA;
        c = a + t;
        d = -(c - a - t);
        a = c;
        b = b + d;

        b = b + sintA*cosEpsB + costA*sinEpsB;
        b = b + sintB + costB*sinEpsA + sintB*cosEpsB + costB*sinEpsB;

        double sina = a + b;
        double sinb = -(sina - a - b);

        // Compute cosine

        a = b = c = d = 0.0;

        t = costA*cosEpsA;
        c = a + t;
        d = -(c - a - t);
        a = c;
        b = b + d;

        t = -sintA*sinEpsA;
        c = a + t;
        d = -(c - a - t);
        a = c;
        b = b + d;

        b = b + costB*cosEpsA + costA*cosEpsB + costB*cosEpsB;
        b = b - (sintB*sinEpsA + sintA*sinEpsB + sintB*sinEpsB);

        double cosa = a + b;
        double cosb = -(cosa - a - b);

        if (cotanFlag) {
            double tmp;
            tmp = cosa; cosa = sina; sina = tmp;
            tmp = cosb; cosb = sinb; sinb = tmp;
        }


        /* estimate and correct, compute 1.0/(cosa+cosb) */
        /*
    double est = (sina+sinb)/(cosa+cosb);
    double err = (sina - cosa*est) + (sinb - cosb*est);
    est += err/(cosa+cosb);
    err = (sina - cosa*est) + (sinb - cosb*est);
         */

        // f(x) = 1/x,   f'(x) = -1/x^2

        double est = sina/cosa;

        /* Split the estimate to get more accurate read on division rounding */
        temp = est * HEX_40000000;
        double esta = (est + temp) - temp;
        double estb =  est - esta;

        temp = cosa * HEX_40000000;
        double cosaa = (cosa + temp) - temp;
        double cosab =  cosa - cosaa;

        //double err = (sina - est*cosa)/cosa;  // Correction for division rounding
        double err = (sina - esta*cosaa - esta*cosab - estb*cosaa - estb*cosab)/cosa;  // Correction for division rounding
        err += sinb/cosa;                     // Change in est due to sinb
        err += -sina * cosb / cosa / cosa;    // Change in est due to cosb

        if (xb != 0.0) {
            // tan' = 1 + tan^2      cot' = -(1 + cot^2)
            // Approximate impact of xb
            double xbadj = xb + est*est*xb;
            if (cotanFlag) {
                xbadj = -xbadj;
            }

            err += xbadj;
        }

        return est+err;
    }

    /** Reduce the input argument using the Payne and Hanek method.
     *  This is good for all inputs 0.0 < x < inf
     *  Output is remainder after dividing by PI/2
     *  The result array should contain 3 numbers.
     *  result[0] is the integer portion, so mod 4 this gives the quadrant.
     *  result[1] is the upper bits of the remainder
     *  result[2] is the lower bits of the remainder
     *
     * @param x number to reduce
     * @param result placeholder where to put the result
     */
    private static void reducePayneHanek(double x, double result[])
    {
        /* Convert input double to bits */
        long inbits = Double.doubleToLongBits(x);
        int exponent = (int) ((inbits >> 52) & 0x7ff) - 1023;

        /* Convert to fixed point representation */
        inbits &= 0x000fffffffffffffL;
        inbits |= 0x0010000000000000L;

        /* Normalize input to be between 0.5 and 1.0 */
        exponent++;
        inbits <<= 11;

        /* Based on the exponent, get a shifted copy of recip2pi */
        long shpi0;
        long shpiA;
        long shpiB;
        int idx = exponent >> 6;
        int shift = exponent - (idx << 6);

        if (shift != 0) {
            shpi0 = (idx == 0) ? 0 : (RECIP_2PI[idx-1] << shift);
            shpi0 |= RECIP_2PI[idx] >>> (64-shift);
            shpiA = (RECIP_2PI[idx] << shift) | (RECIP_2PI[idx+1] >>> (64-shift));
            shpiB = (RECIP_2PI[idx+1] << shift) | (RECIP_2PI[idx+2] >>> (64-shift));
        } else {
            shpi0 = (idx == 0) ? 0 : RECIP_2PI[idx-1];
            shpiA = RECIP_2PI[idx];
            shpiB = RECIP_2PI[idx+1];
        }

        /* Multiply input by shpiA */
        long a = inbits >>> 32;
        long b = inbits & 0xffffffffL;

        long c = shpiA >>> 32;
        long d = shpiA & 0xffffffffL;

        long ac = a * c;
        long bd = b * d;
        long bc = b * c;
        long ad = a * d;

        long prodB = bd + (ad << 32);
        long prodA = ac + (ad >>> 32);

        boolean bita = (bd & 0x8000000000000000L) != 0;
        boolean bitb = (ad & 0x80000000L ) != 0;
        boolean bitsum = (prodB & 0x8000000000000000L) != 0;

        /* Carry */
        if ( (bita && bitb) ||
                ((bita || bitb) && !bitsum) ) {
            prodA++;
        }

        bita = (prodB & 0x8000000000000000L) != 0;
        bitb = (bc & 0x80000000L ) != 0;

        prodB = prodB + (bc << 32);
        prodA = prodA + (bc >>> 32);

        bitsum = (prodB & 0x8000000000000000L) != 0;

        /* Carry */
        if ( (bita && bitb) ||
                ((bita || bitb) && !bitsum) ) {
            prodA++;
        }

        /* Multiply input by shpiB */
        c = shpiB >>> 32;
        d = shpiB & 0xffffffffL;
        ac = a * c;
        bc = b * c;
        ad = a * d;

        /* Collect terms */
        ac = ac + ((bc + ad) >>> 32);

        bita = (prodB & 0x8000000000000000L) != 0;
        bitb = (ac & 0x8000000000000000L ) != 0;
        prodB += ac;
        bitsum = (prodB & 0x8000000000000000L) != 0;
        /* Carry */
        if ( (bita && bitb) ||
                ((bita || bitb) && !bitsum) ) {
            prodA++;
        }

        /* Multiply by shpi0 */
        c = shpi0 >>> 32;
        d = shpi0 & 0xffffffffL;

        bd = b * d;
        bc = b * c;
        ad = a * d;

        prodA += bd + ((bc + ad) << 32);

        /*
         * prodA, prodB now contain the remainder as a fraction of PI.  We want this as a fraction of
         * PI/2, so use the following steps:
         * 1.) multiply by 4.
         * 2.) do a fixed point muliply by PI/4.
         * 3.) Convert to floating point.
         * 4.) Multiply by 2
         */

        /* This identifies the quadrant */
        int intPart = (int)(prodA >>> 62);

        /* Multiply by 4 */
        prodA <<= 2;
        prodA |= prodB >>> 62;
        prodB <<= 2;

        /* Multiply by PI/4 */
        a = prodA >>> 32;
        b = prodA & 0xffffffffL;

        c = PI_O_4_BITS[0] >>> 32;
        d = PI_O_4_BITS[0] & 0xffffffffL;

        ac = a * c;
        bd = b * d;
        bc = b * c;
        ad = a * d;

        long prod2B = bd + (ad << 32);
        long prod2A = ac + (ad >>> 32);

        bita = (bd & 0x8000000000000000L) != 0;
        bitb = (ad & 0x80000000L ) != 0;
        bitsum = (prod2B & 0x8000000000000000L) != 0;

        /* Carry */
        if ( (bita && bitb) ||
                ((bita || bitb) && !bitsum) ) {
            prod2A++;
        }

        bita = (prod2B & 0x8000000000000000L) != 0;
        bitb = (bc & 0x80000000L ) != 0;

        prod2B = prod2B + (bc << 32);
        prod2A = prod2A + (bc >>> 32);

        bitsum = (prod2B & 0x8000000000000000L) != 0;

        /* Carry */
        if ( (bita && bitb) ||
                ((bita || bitb) && !bitsum) ) {
            prod2A++;
        }

        /* Multiply input by pio4bits[1] */
        c = PI_O_4_BITS[1] >>> 32;
        d = PI_O_4_BITS[1] & 0xffffffffL;
        ac = a * c;
        bc = b * c;
        ad = a * d;

        /* Collect terms */
        ac = ac + ((bc + ad) >>> 32);

        bita = (prod2B & 0x8000000000000000L) != 0;
        bitb = (ac & 0x8000000000000000L ) != 0;
        prod2B += ac;
        bitsum = (prod2B & 0x8000000000000000L) != 0;
        /* Carry */
        if ( (bita && bitb) ||
                ((bita || bitb) && !bitsum) ) {
            prod2A++;
        }

        /* Multiply inputB by pio4bits[0] */
        a = prodB >>> 32;
        b = prodB & 0xffffffffL;
        c = PI_O_4_BITS[0] >>> 32;
        d = PI_O_4_BITS[0] & 0xffffffffL;
        ac = a * c;
        bc = b * c;
        ad = a * d;

        /* Collect terms */
        ac = ac + ((bc + ad) >>> 32);

        bita = (prod2B & 0x8000000000000000L) != 0;
        bitb = (ac & 0x8000000000000000L ) != 0;
        prod2B += ac;
        bitsum = (prod2B & 0x8000000000000000L) != 0;
        /* Carry */
        if ( (bita && bitb) ||
                ((bita || bitb) && !bitsum) ) {
            prod2A++;
        }

        /* Convert to double */
        double tmpA = (prod2A >>> 12) / TWO_POWER_52;  // High order 52 bits
        double tmpB = (((prod2A & 0xfffL) << 40) + (prod2B >>> 24)) / TWO_POWER_52 / TWO_POWER_52; // Low bits

        double sumA = tmpA + tmpB;
        double sumB = -(sumA - tmpA - tmpB);

        /* Multiply by PI/2 and return */
        result[0] = intPart;
        result[1] = sumA * 2.0;
        result[2] = sumB * 2.0;
    }

    /**
     *  Sine function.
     *  @param x a number
     *  @return sin(x)
     */
    public static double sin(double x) {
        boolean negative = false;
        int quadrant = 0;
        double xa;
        double xb = 0.0;

        /* Take absolute value of the input */
        xa = x;
        if (x < 0) {
            negative = true;
            xa = -xa;
        }

        /* Check for zero and negative zero */
        if (xa == 0.0) {
            long bits = Double.doubleToLongBits(x);
            if (bits < 0) {
                return -0.0;
            }
            return 0.0;
        }

        if (xa != xa || xa == Double.POSITIVE_INFINITY) {
            return Double.NaN;
        }

        /* Perform any argument reduction */
        if (xa > 3294198.0) {
            // PI * (2**20)
            // Argument too big for CodyWaite reduction.  Must use
            // PayneHanek.
            double reduceResults[] = new double[3];
            reducePayneHanek(xa, reduceResults);
            quadrant = ((int) reduceResults[0]) & 3;
            xa = reduceResults[1];
            xb = reduceResults[2];
        } else if (xa > 1.5707963267948966) {
            /* Inline the Cody/Waite reduction for performance */

            // Estimate k
            //k = (int)(xa / 1.5707963267948966);
            int k = (int)(xa * 0.6366197723675814);

            // Compute remainder
            double remA;
            double remB;
            while (true) {
                double a = -k * 1.570796251296997;
                remA = xa + a;
                remB = -(remA - xa - a);

                a = -k * 7.549789948768648E-8;
                double b = remA;
                remA = a + b;
                remB += -(remA - b - a);

                a = -k * 6.123233995736766E-17;
                b = remA;
                remA = a + b;
                remB += -(remA - b - a);

                if (remA > 0.0)
                    break;

                // Remainder is negative, so decrement k and try again.
                // This should only happen if the input is very close
                // to an even multiple of pi/2
                k--;
            }
            quadrant = k & 3;
            xa = remA;
            xb = remB;
        }

        if (negative) {
            quadrant ^= 2;  // Flip bit 1
        }

        switch (quadrant) {
            case 0:
                return sinQ(xa, xb);
            case 1:
                return cosQ(xa, xb);
            case 2:
                return -sinQ(xa, xb);
            case 3:
                return -cosQ(xa, xb);
            default:
                return Double.NaN;
        }
    }

    /**
     *  Cosine function
     *  @param x a number
     *  @return cos(x)
     */
    public static double cos(double x) {
        int quadrant = 0;

        /* Take absolute value of the input */
        double xa = x;
        if (x < 0) {
            xa = -xa;
        }

        if (xa != xa || xa == Double.POSITIVE_INFINITY) {
            return Double.NaN;
        }

        /* Perform any argument reduction */
        double xb = 0;
        if (xa > 3294198.0) {
            // PI * (2**20)
            // Argument too big for CodyWaite reduction.  Must use
            // PayneHanek.
            double reduceResults[] = new double[3];
            reducePayneHanek(xa, reduceResults);
            quadrant = ((int) reduceResults[0]) & 3;
            xa = reduceResults[1];
            xb = reduceResults[2];
        } else if (xa > 1.5707963267948966) {
            /* Inline the Cody/Waite reduction for performance */

            // Estimate k
            //k = (int)(xa / 1.5707963267948966);
            int k = (int)(xa * 0.6366197723675814);

            // Compute remainder
            double remA;
            double remB;
            while (true) {
                double a = -k * 1.570796251296997;
                remA = xa + a;
                remB = -(remA - xa - a);

                a = -k * 7.549789948768648E-8;
                double b = remA;
                remA = a + b;
                remB += -(remA - b - a);

                a = -k * 6.123233995736766E-17;
                b = remA;
                remA = a + b;
                remB += -(remA - b - a);

                if (remA > 0.0)
                    break;

                // Remainder is negative, so decrement k and try again.
                // This should only happen if the input is very close
                // to an even multiple of pi/2
                k--;
            }
            quadrant = k & 3;
            xa = remA;
            xb = remB;
        }

        //if (negative)
        //  quadrant = (quadrant + 2) % 4;

        switch (quadrant) {
            case 0:
                return cosQ(xa, xb);
            case 1:
                return -sinQ(xa, xb);
            case 2:
                return -cosQ(xa, xb);
            case 3:
                return sinQ(xa, xb);
            default:
                return Double.NaN;
        }
    }

    /**
     *   Tangent function
     *  @param x a number
     *  @return tan(x)
     */
    public static double tan(double x) {
        boolean negative = false;
        int quadrant = 0;

        /* Take absolute value of the input */
        double xa = x;
        if (x < 0) {
            negative = true;
            xa = -xa;
        }

        /* Check for zero and negative zero */
        if (xa == 0.0) {
            long bits = Double.doubleToLongBits(x);
            if (bits < 0) {
                return -0.0;
            }
            return 0.0;
        }

        if (xa != xa || xa == Double.POSITIVE_INFINITY) {
            return Double.NaN;
        }

        /* Perform any argument reduction */
        double xb = 0;
        if (xa > 3294198.0) {
            // PI * (2**20)
            // Argument too big for CodyWaite reduction.  Must use
            // PayneHanek.
            double reduceResults[] = new double[3];
            reducePayneHanek(xa, reduceResults);
            quadrant = ((int) reduceResults[0]) & 3;
            xa = reduceResults[1];
            xb = reduceResults[2];
        } else if (xa > 1.5707963267948966) {
            /* Inline the Cody/Waite reduction for performance */

            // Estimate k
            //k = (int)(xa / 1.5707963267948966);
            int k = (int)(xa * 0.6366197723675814);

            // Compute remainder
            double remA;
            double remB;
            while (true) {
                double a = -k * 1.570796251296997;
                remA = xa + a;
                remB = -(remA - xa - a);

                a = -k * 7.549789948768648E-8;
                double b = remA;
                remA = a + b;
                remB += -(remA - b - a);

                a = -k * 6.123233995736766E-17;
                b = remA;
                remA = a + b;
                remB += -(remA - b - a);

                if (remA > 0.0)
                    break;

                // Remainder is negative, so decrement k and try again.
                // This should only happen if the input is very close
                // to an even multiple of pi/2
                k--;
            }
            quadrant = k & 3;
            xa = remA;
            xb = remB;
        }

        if (xa > 1.5) {
            // Accurracy suffers between 1.5 and PI/2
            final double pi2a = 1.5707963267948966;
            final double pi2b = 6.123233995736766E-17;

            final double a = pi2a - xa;
            double b = -(a - pi2a + xa);
            b += pi2b - xb;

            xa = a + b;
            xb = -(xa - a - b);
            quadrant ^= 1;
            negative ^= true;
        }

        double result;
        if ((quadrant & 1) == 0) {
            result = tanQ(xa, xb, false);
        } else {
            result = -tanQ(xa, xb, true);
        }

        if (negative) {
            result = -result;
        }

        return result;
    }

    /**
     * Arctangent function
     *  @param x a number
     *  @return atan(x)
     */
    public static double atan(double x) {
        return atan(x, 0.0, false);
    }

    /** Internal helper function to compute arctangent.
     * @param xa number from which arctangent is requested
     * @param xb extra bits for x (may be 0.0)
     * @param leftPlane if true, result angle must be put in the left half plane
     * @return atan(xa + xb) (or angle shifted by {@code PI} if leftPlane is true)
     */
    private static double atan(double xa, double xb, boolean leftPlane) {
        boolean negate = false;
        int idx;

        if (xa == 0.0) { // Matches +/- 0.0; return correct sign
            return leftPlane ? copySign(Math.PI, xa) : xa;
        }

        if (xa < 0) {
            // negative
            xa = -xa;
            xb = -xb;
            negate = true;
        }

        if (xa > 1.633123935319537E16) { // Very large input
            return (negate ^ leftPlane) ? (-Math.PI/2.0) : (Math.PI/2.0);
        }

        /* Estimate the closest tabulated arctan value, compute eps = xa-tangentTable */
        if (xa < 1.0) {
            idx = (int) (((-1.7168146928204136 * xa * xa + 8.0) * xa) + 0.5);
        } else {
            double temp = 1.0/xa;
            idx = (int) (-((-1.7168146928204136 * temp * temp + 8.0) * temp) + 13.07);
        }
        double epsA = xa - TANGENT_TABLE_A[idx];
        double epsB = -(epsA - xa + TANGENT_TABLE_A[idx]);
        epsB += xb - TANGENT_TABLE_B[idx];

        double temp = epsA + epsB;
        epsB = -(temp - epsA - epsB);
        epsA = temp;

        /* Compute eps = eps / (1.0 + xa*tangent) */
        temp = xa * HEX_40000000;
        double ya = xa + temp - temp;
        double yb = xb + xa - ya;
        xa = ya;
        xb += yb;

        //if (idx > 8 || idx == 0)
        if (idx == 0) {
            /* If the slope of the arctan is gentle enough (< 0.45), this approximation will suffice */
            //double denom = 1.0 / (1.0 + xa*tangentTableA[idx] + xb*tangentTableA[idx] + xa*tangentTableB[idx] + xb*tangentTableB[idx]);
            double denom = 1.0 / (1.0 + (xa + xb) * (TANGENT_TABLE_A[idx] + TANGENT_TABLE_B[idx]));
            //double denom = 1.0 / (1.0 + xa*tangentTableA[idx]);
            ya = epsA * denom;
            yb = epsB * denom;
        } else {
            double temp2 = xa * TANGENT_TABLE_A[idx];
            double za = 1.0 + temp2;
            double zb = -(za - 1.0 - temp2);
            temp2 = xb * TANGENT_TABLE_A[idx] + xa * TANGENT_TABLE_B[idx];
            temp = za + temp2;
            zb += -(temp - za - temp2);
            za = temp;

            zb += xb * TANGENT_TABLE_B[idx];
            ya = epsA / za;

            temp = ya * HEX_40000000;
            final double yaa = (ya + temp) - temp;
            final double yab = ya - yaa;

            temp = za * HEX_40000000;
            final double zaa = (za + temp) - temp;
            final double zab = za - zaa;

            /* Correct for rounding in division */
            yb = (epsA - yaa * zaa - yaa * zab - yab * zaa - yab * zab) / za;

            yb += -epsA * zb / za / za;
            yb += epsB / za;
        }


        epsA = ya;
        epsB = yb;

        /* Evaluate polynomial */
        double epsA2 = epsA*epsA;

        /*
    yb = -0.09001346640161823;
    yb = yb * epsA2 + 0.11110718400605211;
    yb = yb * epsA2 + -0.1428571349122913;
    yb = yb * epsA2 + 0.19999999999273194;
    yb = yb * epsA2 + -0.33333333333333093;
    yb = yb * epsA2 * epsA;
         */

        yb = 0.07490822288864472;
        yb = yb * epsA2 + -0.09088450866185192;
        yb = yb * epsA2 + 0.11111095942313305;
        yb = yb * epsA2 + -0.1428571423679182;
        yb = yb * epsA2 + 0.19999999999923582;
        yb = yb * epsA2 + -0.33333333333333287;
        yb = yb * epsA2 * epsA;


        ya = epsA;

        temp = ya + yb;
        yb = -(temp - ya - yb);
        ya = temp;

        /* Add in effect of epsB.   atan'(x) = 1/(1+x^2) */
        yb += epsB / (1.0 + epsA * epsA);

        double result;
        double resultb;

        //result = yb + eighths[idx] + ya;
        double za = EIGHTHS[idx] + ya;
        double zb = -(za - EIGHTHS[idx] - ya);
        temp = za + yb;
        zb += -(temp - za - yb);
        za = temp;

        result = za + zb;
        resultb = -(result - za - zb);

        if (leftPlane) {
            // Result is in the left plane
            final double pia = 1.5707963267948966*2.0;
            final double pib = 6.123233995736766E-17*2.0;

            za = pia - result;
            zb = -(za - pia + result);
            zb += pib - resultb;

            result = za + zb;
            resultb = -(result - za - zb);
        }


        if (negate ^ leftPlane) {
            result = -result;
        }

        return result;
    }

    /**
     * Two arguments arctangent function
     * @param y ordinate
     * @param x abscissa
     * @return phase angle of point (x,y) between {@code -PI} and {@code PI}
     */
    public static double atan2(double y, double x) {
        if (x !=x || y != y) {
            return Double.NaN;
        }

        if (y == 0.0) {
            double result = x*y;
            double invx = 1.0/x;
            double invy = 1.0/y;

            if (invx == 0.0) { // X is infinite
                if (x > 0) {
                    return y; // return +/- 0.0
                } else {
                    return copySign(Math.PI, y);
                }
            }

            if (x < 0.0 || invx < 0.0) {
                if (y < 0.0 || invy < 0.0) {
                    return -Math.PI;
                } else {
                    return Math.PI;
                }
            } else {
                return result;
            }
        }

        // y cannot now be zero

        if (y == Double.POSITIVE_INFINITY) {
            if (x == Double.POSITIVE_INFINITY) {
                return Math.PI/4.0;
            }

            if (x == Double.NEGATIVE_INFINITY) {
                return Math.PI*3.0/4.0;
            }

            return Math.PI/2.0;
        }

        if (y == Double.NEGATIVE_INFINITY) {
            if (x == Double.POSITIVE_INFINITY) {
                return -Math.PI/4.0;
            }

            if (x == Double.NEGATIVE_INFINITY) {
                return -Math.PI*3.0/4.0;
            }

            return -Math.PI/2.0;
        }

        if (x == Double.POSITIVE_INFINITY) {
            if (y > 0.0 || 1/y > 0.0) {
                return 0.0;
            }

            if (y < 0.0 || 1/y < 0.0) {
                return -0.0;
            }
        }

        if (x == Double.NEGATIVE_INFINITY)
        {
            if (y > 0.0 || 1/y > 0.0) {
                return Math.PI;
            }

            if (y < 0.0 || 1/y < 0.0) {
                return -Math.PI;
            }
        }

        // Neither y nor x can be infinite or NAN here

        if (x == 0) {
            if (y > 0.0 || 1/y > 0.0) {
                return Math.PI/2.0;
            }

            if (y < 0.0 || 1/y < 0.0) {
                return -Math.PI/2.0;
            }
        }

        // Compute ratio r = y/x
        final double r = y/x;
        if (Double.isInfinite(r)) { // bypass calculations that can create NaN
            return atan(r, 0, x < 0);
        }

        double ra = doubleHighPart(r);
        double rb = r - ra;

        // Split x
        final double xa = doubleHighPart(x);
        final double xb = x - xa;

        rb += (y - ra * xa - ra * xb - rb * xa - rb * xb) / x;

        double temp = ra + rb;
        rb = -(temp - ra - rb);
        ra = temp;

        if (ra == 0) { // Fix up the sign so atan works correctly
            ra = copySign(0.0, y);
        }

        // Call atan
        double result = atan(ra, rb, x < 0);

        return result;
    }

    /** Compute the arc sine of a number.
     * @param x number on which evaluation is done
     * @return arc sine of x
     */
    public static double asin(double x) {
      if (x != x) {
          return Double.NaN;
      }

      if (x > 1.0 || x < -1.0) {
          return Double.NaN;
      }

      if (x == 1.0) {
          return Math.PI/2.0;
      }

      if (x == -1.0) {
          return -Math.PI/2.0;
      }

      if (x == 0.0) { // Matches +/- 0.0; return correct sign
          return x;
      }

      /* Compute asin(x) = atan(x/sqrt(1-x*x)) */

      /* Split x */
      double temp = x * HEX_40000000;
      final double xa = x + temp - temp;
      final double xb = x - xa;

      /* Square it */
      double ya = xa*xa;
      double yb = xa*xb*2.0 + xb*xb;

      /* Subtract from 1 */
      ya = -ya;
      yb = -yb;

      double za = 1.0 + ya;
      double zb = -(za - 1.0 - ya);

      temp = za + yb;
      zb += -(temp - za - yb);
      za = temp;

      /* Square root */
      double y;
      y = sqrt(za);
      temp = y * HEX_40000000;
      ya = y + temp - temp;
      yb = y - ya;

      /* Extend precision of sqrt */
      yb += (za - ya*ya - 2*ya*yb - yb*yb) / (2.0*y);

      /* Contribution of zb to sqrt */
      double dx = zb / (2.0*y);

      // Compute ratio r = x/y
      double r = x/y;
      temp = r * HEX_40000000;
      double ra = r + temp - temp;
      double rb = r - ra;

      rb += (x - ra*ya - ra*yb - rb*ya - rb*yb) / y;  // Correct for rounding in division
      rb += -x * dx / y / y;  // Add in effect additional bits of sqrt.

      temp = ra + rb;
      rb = -(temp - ra - rb);
      ra = temp;

      return atan(ra, rb, false);
    }

    /** Compute the arc cosine of a number.
     * @param x number on which evaluation is done
     * @return arc cosine of x
     */
    public static double acos(double x) {
      if (x != x) {
          return Double.NaN;
      }

      if (x > 1.0 || x < -1.0) {
          return Double.NaN;
      }

      if (x == -1.0) {
          return Math.PI;
      }

      if (x == 1.0) {
          return 0.0;
      }

      if (x == 0) {
          return Math.PI/2.0;
      }

      /* Compute acos(x) = atan(sqrt(1-x*x)/x) */

      /* Split x */
      double temp = x * HEX_40000000;
      final double xa = x + temp - temp;
      final double xb = x - xa;

      /* Square it */
      double ya = xa*xa;
      double yb = xa*xb*2.0 + xb*xb;

      /* Subtract from 1 */
      ya = -ya;
      yb = -yb;

      double za = 1.0 + ya;
      double zb = -(za - 1.0 - ya);

      temp = za + yb;
      zb += -(temp - za - yb);
      za = temp;

      /* Square root */
      double y = sqrt(za);
      temp = y * HEX_40000000;
      ya = y + temp - temp;
      yb = y - ya;

      /* Extend precision of sqrt */
      yb += (za - ya*ya - 2*ya*yb - yb*yb) / (2.0*y);

      /* Contribution of zb to sqrt */
      yb += zb / (2.0*y);
      y = ya+yb;
      yb = -(y - ya - yb);

      // Compute ratio r = y/x
      double r = y/x;

      // Did r overflow?
      if (Double.isInfinite(r)) { // x is effectively zero
          return Math.PI/2; // so return the appropriate value
      }

      double ra = doubleHighPart(r);
      double rb = r - ra;

      rb += (y - ra*xa - ra*xb - rb*xa - rb*xb) / x;  // Correct for rounding in division
      rb += yb / x;  // Add in effect additional bits of sqrt.

      temp = ra + rb;
      rb = -(temp - ra - rb);
      ra = temp;

      return atan(ra, rb, x<0);
    }

    /** Compute the cubic root of a number.
     * @param x number on which evaluation is done
     * @return cubic root of x
     */
    public static double cbrt(double x) {
      /* Convert input double to bits */
      long inbits = Double.doubleToLongBits(x);
      int exponent = (int) ((inbits >> 52) & 0x7ff) - 1023;
      boolean subnormal = false;

      if (exponent == -1023) {
          if (x == 0) {
              return x;
          }

          /* Subnormal, so normalize */
          subnormal = true;
          x *= 1.8014398509481984E16;  // 2^54
          inbits = Double.doubleToLongBits(x);
          exponent = (int) ((inbits >> 52) & 0x7ff) - 1023;
      }

      if (exponent == 1024) {
          // Nan or infinity.  Don't care which.
          return x;
      }

      /* Divide the exponent by 3 */
      int exp3 = exponent / 3;

      /* p2 will be the nearest power of 2 to x with its exponent divided by 3 */
      double p2 = Double.longBitsToDouble((inbits & 0x8000000000000000L) |
                                          (long)(((exp3 + 1023) & 0x7ff)) << 52);

      /* This will be a number between 1 and 2 */
      final double mant = Double.longBitsToDouble((inbits & 0x000fffffffffffffL) | 0x3ff0000000000000L);

      /* Estimate the cube root of mant by polynomial */
      double est = -0.010714690733195933;
      est = est * mant + 0.0875862700108075;
      est = est * mant + -0.3058015757857271;
      est = est * mant + 0.7249995199969751;
      est = est * mant + 0.5039018405998233;

      est *= CBRTTWO[exponent % 3 + 2];

      // est should now be good to about 15 bits of precision.   Do 2 rounds of
      // Newton's method to get closer,  this should get us full double precision
      // Scale down x for the purpose of doing newtons method.  This avoids over/under flows.
      final double xs = x / (p2*p2*p2);
      est += (xs - est*est*est) / (3*est*est);
      est += (xs - est*est*est) / (3*est*est);

      // Do one round of Newton's method in extended precision to get the last bit right.
      double temp = est * HEX_40000000;
      double ya = est + temp - temp;
      double yb = est - ya;

      double za = ya * ya;
      double zb = ya * yb * 2.0 + yb * yb;
      temp = za * HEX_40000000;
      double temp2 = za + temp - temp;
      zb += za - temp2;
      za = temp2;

      zb = za * yb + ya * zb + zb * yb;
      za = za * ya;

      double na = xs - za;
      double nb = -(na - xs + za);
      nb -= zb;

      est += (na+nb)/(3*est*est);

      /* Scale by a power of two, so this is exact. */
      est *= p2;

      if (subnormal) {
          est *= 3.814697265625E-6;  // 2^-18
      }

      return est;
    }

    /**
     *  Convert degrees to radians, with error of less than 0.5 ULP
     *  @param x angle in degrees
     *  @return x converted into radians
     */
    public static double toRadians(double x)
    {
        if (Double.isInfinite(x) || x == 0.0) { // Matches +/- 0.0; return correct sign
            return x;
        }

        // These are PI/180 split into high and low order bits
        final double facta = 0.01745329052209854;
        final double factb = 1.997844754509471E-9;

        double xa = doubleHighPart(x);
        double xb = x - xa;

        double result = xb * factb + xb * facta + xa * factb + xa * facta;
        if (result == 0) {
            result = result * x; // ensure correct sign if calculation underflows
        }
        return result;
    }

    /**
     *  Convert radians to degrees, with error of less than 0.5 ULP
     *  @param x angle in radians
     *  @return x converted into degrees
     */
    public static double toDegrees(double x)
    {
        if (Double.isInfinite(x) || x == 0.0) { // Matches +/- 0.0; return correct sign
            return x;
        }

        // These are 180/PI split into high and low order bits
        final double facta = 57.2957763671875;
        final double factb = 3.145894820876798E-6;

        double xa = doubleHighPart(x);
        double xb = x - xa;

        return xb * factb + xb * facta + xa * factb + xa * facta;
    }

    /**
     * Absolute value.
     * @param x number from which absolute value is requested
     * @return abs(x)
     */
    public static int abs(final int x) {
        return (x < 0) ? -x : x;
    }

    /**
     * Absolute value.
     * @param x number from which absolute value is requested
     * @return abs(x)
     */
    public static long abs(final long x) {
        return (x < 0l) ? -x : x;
    }

    /**
     * Absolute value.
     * @param x number from which absolute value is requested
     * @return abs(x)
     */
    public static float abs(final float x) {
        return (x < 0.0f) ? -x : (x == 0.0f) ? 0.0f : x; // -0.0 => +0.0
    }

    /**
     * Absolute value.
     * @param x number from which absolute value is requested
     * @return abs(x)
     */
    public static double abs(double x) {
        return (x < 0.0) ? -x : (x == 0.0) ? 0.0 : x; // -0.0 => +0.0
    }

    /**
     * Compute least significant bit (Unit in Last Position) for a number.
     * @param x number from which ulp is requested
     * @return ulp(x)
     */
    public static double ulp(double x) {
        if (Double.isInfinite(x)) {
            return Double.POSITIVE_INFINITY;
        }
        return abs(x - Double.longBitsToDouble(Double.doubleToLongBits(x) ^ 1));
    }

    /**
     * Compute least significant bit (Unit in Last Position) for a number.
     * @param x number from which ulp is requested
     * @return ulp(x)
     */
    public static float ulp(float x) {
        if (Float.isInfinite(x)) {
            return Float.POSITIVE_INFINITY;
        }
        return abs(x - Float.intBitsToFloat(Float.floatToIntBits(x) ^ 1));
    }

    /**
     * Multiply a double number by a power of 2.
     * @param d number to multiply
     * @param n power of 2
     * @return d &times; 2<sup>n</sup>
     */
    public static double scalb(final double d, final int n) {

        // first simple and fast handling when 2^n can be represented using normal numbers
        if ((n > -1023) && (n < 1024)) {
            return d * Double.longBitsToDouble(((long) (n + 1023)) << 52);
        }

        // handle special cases
        if (Double.isNaN(d) || Double.isInfinite(d) || (d == 0)) {
            return d;
        }
        if (n < -2098) {
            return (d > 0) ? 0.0 : -0.0;
        }
        if (n > 2097) {
            return (d > 0) ? Double.POSITIVE_INFINITY : Double.NEGATIVE_INFINITY;
        }

        // decompose d
        final long bits = Double.doubleToLongBits(d);
        final long sign = bits & 0x8000000000000000L;
        int  exponent   = ((int) (bits >>> 52)) & 0x7ff;
        long mantissa   = bits & 0x000fffffffffffffL;

        // compute scaled exponent
        int scaledExponent = exponent + n;

        if (n < 0) {
            // we are really in the case n <= -1023
            if (scaledExponent > 0) {
                // both the input and the result are normal numbers, we only adjust the exponent
                return Double.longBitsToDouble(sign | (((long) scaledExponent) << 52) | mantissa);
            } else if (scaledExponent > -53) {
                // the input is a normal number and the result is a subnormal number

                // recover the hidden mantissa bit
                mantissa = mantissa | (1L << 52);

                // scales down complete mantissa, hence losing least significant bits
                final long mostSignificantLostBit = mantissa & (1L << (-scaledExponent));
                mantissa = mantissa >>> (1 - scaledExponent);
                if (mostSignificantLostBit != 0) {
                    // we need to add 1 bit to round up the result
                    mantissa++;
                }
                return Double.longBitsToDouble(sign | mantissa);

            } else {
                // no need to compute the mantissa, the number scales down to 0
                return (sign == 0L) ? 0.0 : -0.0;
            }
        } else {
            // we are really in the case n >= 1024
            if (exponent == 0) {

                // the input number is subnormal, normalize it
                while ((mantissa >>> 52) != 1) {
                    mantissa = mantissa << 1;
                    --scaledExponent;
                }
                ++scaledExponent;
                mantissa = mantissa & 0x000fffffffffffffL;

                if (scaledExponent < 2047) {
                    return Double.longBitsToDouble(sign | (((long) scaledExponent) << 52) | mantissa);
                } else {
                    return (sign == 0L) ? Double.POSITIVE_INFINITY : Double.NEGATIVE_INFINITY;
                }

            } else if (scaledExponent < 2047) {
                return Double.longBitsToDouble(sign | (((long) scaledExponent) << 52) | mantissa);
            } else {
                return (sign == 0L) ? Double.POSITIVE_INFINITY : Double.NEGATIVE_INFINITY;
            }
        }

    }

    /**
     * Multiply a float number by a power of 2.
     * @param f number to multiply
     * @param n power of 2
     * @return f &times; 2<sup>n</sup>
     */
    public static float scalb(final float f, final int n) {

        // first simple and fast handling when 2^n can be represented using normal numbers
        if ((n > -127) && (n < 128)) {
            return f * Float.intBitsToFloat((n + 127) << 23);
        }

        // handle special cases
        if (Float.isNaN(f) || Float.isInfinite(f) || (f == 0f)) {
            return f;
        }
        if (n < -277) {
            return (f > 0) ? 0.0f : -0.0f;
        }
        if (n > 276) {
            return (f > 0) ? Float.POSITIVE_INFINITY : Float.NEGATIVE_INFINITY;
        }

        // decompose f
        final int bits = Float.floatToIntBits(f);
        final int sign = bits & 0x80000000;
        int  exponent  = (bits >>> 23) & 0xff;
        int mantissa   = bits & 0x007fffff;

        // compute scaled exponent
        int scaledExponent = exponent + n;

        if (n < 0) {
            // we are really in the case n <= -127
            if (scaledExponent > 0) {
                // both the input and the result are normal numbers, we only adjust the exponent
                return Float.intBitsToFloat(sign | (scaledExponent << 23) | mantissa);
            } else if (scaledExponent > -24) {
                // the input is a normal number and the result is a subnormal number

                // recover the hidden mantissa bit
                mantissa = mantissa | (1 << 23);

                // scales down complete mantissa, hence losing least significant bits
                final int mostSignificantLostBit = mantissa & (1 << (-scaledExponent));
                mantissa = mantissa >>> (1 - scaledExponent);
                if (mostSignificantLostBit != 0) {
                    // we need to add 1 bit to round up the result
                    mantissa++;
                }
                return Float.intBitsToFloat(sign | mantissa);

            } else {
                // no need to compute the mantissa, the number scales down to 0
                return (sign == 0) ? 0.0f : -0.0f;
            }
        } else {
            // we are really in the case n >= 128
            if (exponent == 0) {

                // the input number is subnormal, normalize it
                while ((mantissa >>> 23) != 1) {
                    mantissa = mantissa << 1;
                    --scaledExponent;
                }
                ++scaledExponent;
                mantissa = mantissa & 0x007fffff;

                if (scaledExponent < 255) {
                    return Float.intBitsToFloat(sign | (scaledExponent << 23) | mantissa);
                } else {
                    return (sign == 0) ? Float.POSITIVE_INFINITY : Float.NEGATIVE_INFINITY;
                }

            } else if (scaledExponent < 255) {
                return Float.intBitsToFloat(sign | (scaledExponent << 23) | mantissa);
            } else {
                return (sign == 0) ? Float.POSITIVE_INFINITY : Float.NEGATIVE_INFINITY;
            }
        }

    }

    /**
     * Get the next machine representable number after a number, moving
     * in the direction of another number.
     * <p>
     * The ordering is as follows (increasing):
     * <ul>
     * <li>-INFINITY</li>
     * <li>-MAX_VALUE</li>
     * <li>-MIN_VALUE</li>
     * <li>-0.0</li>
     * <li>+0.0</li>
     * <li>+MIN_VALUE</li>
     * <li>+MAX_VALUE</li>
     * <li>+INFINITY</li>
     * <li></li>
     * <p>
     * If arguments compare equal, then the second argument is returned.
     * <p>
     * If {@code direction} is greater than {@code d},
     * the smallest machine representable number strictly greater than
     * {@code d} is returned; if less, then the largest representable number
     * strictly less than {@code d} is returned.</p>
     * <p>
     * If {@code d} is infinite and direction does not
     * bring it back to finite numbers, it is returned unchanged.</p>
     *
     * @param d base number
     * @param direction (the only important thing is whether
     * {@code direction} is greater or smaller than {@code d})
     * @return the next machine representable number in the specified direction
     */
    public static double nextAfter(double d, double direction) {

        // handling of some important special cases
        if (Double.isNaN(d) || Double.isNaN(direction)) {
            return Double.NaN;
        } else if (d == direction) {
            return direction;
        } else if (Double.isInfinite(d)) {
            return (d < 0) ? -Double.MAX_VALUE : Double.MAX_VALUE;
        } else if (d == 0) {
            return (direction < 0) ? -Double.MIN_VALUE : Double.MIN_VALUE;
        }
        // special cases MAX_VALUE to infinity and  MIN_VALUE to 0
        // are handled just as normal numbers

        final long bits = Double.doubleToLongBits(d);
        final long sign = bits & 0x8000000000000000L;
        if ((direction < d) ^ (sign == 0L)) {
            return Double.longBitsToDouble(sign | ((bits & 0x7fffffffffffffffL) + 1));
        } else {
            return Double.longBitsToDouble(sign | ((bits & 0x7fffffffffffffffL) - 1));
        }

    }

    /**
     * Get the next machine representable number after a number, moving
     * in the direction of another number.
     * <p>
     * The ordering is as follows (increasing):
     * <ul>
     * <li>-INFINITY</li>
     * <li>-MAX_VALUE</li>
     * <li>-MIN_VALUE</li>
     * <li>-0.0</li>
     * <li>+0.0</li>
     * <li>+MIN_VALUE</li>
     * <li>+MAX_VALUE</li>
     * <li>+INFINITY</li>
     * <li></li>
     * <p>
     * If arguments compare equal, then the second argument is returned.
     * <p>
     * If {@code direction} is greater than {@code f},
     * the smallest machine representable number strictly greater than
     * {@code f} is returned; if less, then the largest representable number
     * strictly less than {@code f} is returned.</p>
     * <p>
     * If {@code f} is infinite and direction does not
     * bring it back to finite numbers, it is returned unchanged.</p>
     *
     * @param f base number
     * @param direction (the only important thing is whether
     * {@code direction} is greater or smaller than {@code f})
     * @return the next machine representable number in the specified direction
     */
    public static float nextAfter(final float f, final double direction) {

        // handling of some important special cases
        if (Double.isNaN(f) || Double.isNaN(direction)) {
            return Float.NaN;
        } else if (f == direction) {
            return (float) direction;
        } else if (Float.isInfinite(f)) {
            return (f < 0f) ? -Float.MAX_VALUE : Float.MAX_VALUE;
        } else if (f == 0f) {
            return (direction < 0) ? -Float.MIN_VALUE : Float.MIN_VALUE;
        }
        // special cases MAX_VALUE to infinity and  MIN_VALUE to 0
        // are handled just as normal numbers

        final int bits = Float.floatToIntBits(f);
        final int sign = bits & 0x80000000;
        if ((direction < f) ^ (sign == 0)) {
            return Float.intBitsToFloat(sign | ((bits & 0x7fffffff) + 1));
        } else {
            return Float.intBitsToFloat(sign | ((bits & 0x7fffffff) - 1));
        }

    }

    /** Get the largest whole number smaller than x.
     * @param x number from which floor is requested
     * @return a double number f such that f is an integer f <= x < f + 1.0
     */
    public static double floor(double x) {
        long y;

        if (x != x) { // NaN
            return x;
        }

        if (x >= TWO_POWER_52 || x <= -TWO_POWER_52) {
            return x;
        }

        y = (long) x;
        if (x < 0 && y != x) {
            y--;
        }

        if (y == 0) {
            return x*y;
        }

        return y;
    }

    /** Get the smallest whole number larger than x.
     * @param x number from which ceil is requested
     * @return a double number c such that c is an integer c - 1.0 < x <= c
     */
    public static double ceil(double x) {
        double y;

        if (x != x) { // NaN
            return x;
        }

        y = floor(x);
        if (y == x) {
            return y;
        }

        y += 1.0;

        if (y == 0) {
            return x*y;
        }

        return y;
    }

    /** Get the whole number that is the nearest to x, or the even one if x is exactly half way between two integers.
     * @param x number from which nearest whole number is requested
     * @return a double number r such that r is an integer r - 0.5 <= x <= r + 0.5
     */
    public static double rint(double x) {
        double y = floor(x);
        double d = x - y;

        if (d > 0.5) {
            if (y == -1.0) {
                return -0.0; // Preserve sign of operand
            }
            return y+1.0;
        }
        if (d < 0.5) {
            return y;
        }

        /* half way, round to even */
        long z = (long) y;
        return (z & 1) == 0 ? y : y + 1.0;
    }

    /** Get the closest long to x.
     * @param x number from which closest long is requested
     * @return closest long to x
     */
    public static long round(double x) {
        return (long) floor(x + 0.5);
    }

    /** Get the closest int to x.
     * @param x number from which closest int is requested
     * @return closest int to x
     */
    public static int round(final float x) {
        return (int) floor(x + 0.5f);
    }

    /** Compute the minimum of two values
     * @param a first value
     * @param b second value
     * @return a if a is lesser or equal to b, b otherwise
     */
    public static int min(final int a, final int b) {
        return (a <= b) ? a : b;
    }

    /** Compute the minimum of two values
     * @param a first value
     * @param b second value
     * @return a if a is lesser or equal to b, b otherwise
     */
    public static long min(final long a, final long b) {
        return (a <= b) ? a : b;
    }

    /** Compute the minimum of two values
     * @param a first value
     * @param b second value
     * @return a if a is lesser or equal to b, b otherwise
     */
    public static float min(final float a, final float b) {
        if (a > b) {
            return b;
        }
        if (a < b) {
            return a;
        }
        /* if either arg is NaN, return NaN */
        if (a != b) {
            return Float.NaN;
        }
        /* min(+0.0,-0.0) == -0.0 */
        /* 0x80000000 == Float.floatToRawIntBits(-0.0d) */
        int bits = Float.floatToRawIntBits(a);
        if (bits == 0x80000000) {
            return a;
        }
        return b;
    }

    /** Compute the minimum of two values
     * @param a first value
     * @param b second value
     * @return a if a is lesser or equal to b, b otherwise
     */
    public static double min(final double a, final double b) {
        if (a > b) {
            return b;
        }
        if (a < b) {
            return a;
        }
        /* if either arg is NaN, return NaN */
        if (a != b) {
            return Double.NaN;
        }
        /* min(+0.0,-0.0) == -0.0 */
        /* 0x8000000000000000L == Double.doubleToRawLongBits(-0.0d) */
        long bits = Double.doubleToRawLongBits(a);
        if (bits == 0x8000000000000000L) {
            return a;
        }
        return b;
    }

    /** Compute the maximum of two values
     * @param a first value
     * @param b second value
     * @return b if a is lesser or equal to b, a otherwise
     */
    public static int max(final int a, final int b) {
        return (a <= b) ? b : a;
    }

    /** Compute the maximum of two values
     * @param a first value
     * @param b second value
     * @return b if a is lesser or equal to b, a otherwise
     */
    public static long max(final long a, final long b) {
        return (a <= b) ? b : a;
    }

    /** Compute the maximum of two values
     * @param a first value
     * @param b second value
     * @return b if a is lesser or equal to b, a otherwise
     */
    public static float max(final float a, final float b) {
        if (a > b) {
            return a;
        }
        if (a < b) {
            return b;
        }
        /* if either arg is NaN, return NaN */
        if (a != b) {
            return Float.NaN;
        }
        /* min(+0.0,-0.0) == -0.0 */
        /* 0x80000000 == Float.floatToRawIntBits(-0.0d) */
        int bits = Float.floatToRawIntBits(a);
        if (bits == 0x80000000) {
            return b;
        }
        return a;
    }

    /** Compute the maximum of two values
     * @param a first value
     * @param b second value
     * @return b if a is lesser or equal to b, a otherwise
     */
    public static double max(final double a, final double b) {
        if (a > b) {
            return a;
        }
        if (a < b) {
            return b;
        }
        /* if either arg is NaN, return NaN */
        if (a != b) {
            return Double.NaN;
        }
        /* min(+0.0,-0.0) == -0.0 */
        /* 0x8000000000000000L == Double.doubleToRawLongBits(-0.0d) */
        long bits = Double.doubleToRawLongBits(a);
        if (bits == 0x8000000000000000L) {
            return b;
        }
        return a;
    }

    /**
     * Returns the hypotenuse of a triangle with sides {@code x} and {@code y}
     * - sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)<br/>
     * avoiding intermediate overflow or underflow.
     *
     * <ul>
     * <li> If either argument is infinite, then the result is positive infinity.</li>
     * <li> else, if either argument is NaN then the result is NaN.</li>
     * </ul>
     *
     * @param x a value
     * @param y a value
     * @return sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
     */
    public static double hypot(final double x, final double y) {
        if (Double.isInfinite(x) || Double.isInfinite(y)) {
            return Double.POSITIVE_INFINITY;
        } else if (Double.isNaN(x) || Double.isNaN(y)) {
            return Double.NaN;
        } else {

            final int expX = getExponent(x);
            final int expY = getExponent(y);
            if (expX > expY + 27) {
                // y is neglectible with respect to x
                return abs(x);
            } else if (expY > expX + 27) {
                // x is neglectible with respect to y
                return abs(y);
            } else {

                // find an intermediate scale to avoid both overflow and underflow
                final int middleExp = (expX + expY) / 2;

                // scale parameters without losing precision
                final double scaledX = scalb(x, -middleExp);
                final double scaledY = scalb(y, -middleExp);

                // compute scaled hypotenuse
                final double scaledH = sqrt(scaledX * scaledX + scaledY * scaledY);

                // remove scaling
                return scalb(scaledH, middleExp);

            }

        }
    }

    /**
     * Computes the remainder as prescribed by the IEEE 754 standard.
     * The remainder value is mathematically equal to {@code x - y*n}
     * where {@code n} is the mathematical integer closest to the exact mathematical value
     * of the quotient {@code x/y}.
     * If two mathematical integers are equally close to {@code x/y} then
     * {@code n} is the integer that is even.
     * <p>
     * <ul>
     * <li>If either operand is NaN, the result is NaN.</li>
     * <li>If the result is not NaN, the sign of the result equals the sign of the dividend.</li>
     * <li>If the dividend is an infinity, or the divisor is a zero, or both, the result is NaN.</li>
     * <li>If the dividend is finite and the divisor is an infinity, the result equals the dividend.</li>
     * <li>If the dividend is a zero and the divisor is finite, the result equals the dividend.</li>
     * </ul>
     * <p><b>Note:</b> this implementation currently delegates to {@link StrictMath#IEEEremainder}
     * @param dividend the number to be divided
     * @param divisor the number by which to divide
     * @return the remainder, rounded
     */
    public static double IEEEremainder(double dividend, double divisor) {
        return StrictMath.IEEEremainder(dividend, divisor); // TODO provide our own implementation
    }

    /**
     * Returns the first argument with the sign of the second argument.
     * A NaN {@code sign} argument is treated as positive.
     *
     * @param magnitude the value to return
     * @param sign the sign for the returned value
     * @return the magnitude with the same sign as the {@code sign} argument
     */
    public static double copySign(double magnitude, double sign){
        long m = Double.doubleToLongBits(magnitude);
        long s = Double.doubleToLongBits(sign);
        if ((m >= 0 && s >= 0) || (m < 0 && s < 0)) { // Sign is currently OK
            return magnitude;
        }
        return -magnitude; // flip sign
    }

    /**
     * Returns the first argument with the sign of the second argument.
     * A NaN {@code sign} argument is treated as positive.
     *
     * @param magnitude the value to return
     * @param sign the sign for the returned value
     * @return the magnitude with the same sign as the {@code sign} argument
     */
    public static float copySign(float magnitude, float sign){
        int m = Float.floatToIntBits(magnitude);
        int s = Float.floatToIntBits(sign);
        if ((m >= 0 && s >= 0) || (m < 0 && s < 0)) { // Sign is currently OK
            return magnitude;
        }
        return -magnitude; // flip sign
    }

    /**
     * Return the exponent of a double number, removing the bias.
     * <p>
     * For double numbers of the form 2<sup>x</sup>, the unbiased
     * exponent is exactly x.
     * </p>
     * @param d number from which exponent is requested
     * @return exponent for d in IEEE754 representation, without bias
     */
    public static int getExponent(final double d) {
        return (int) ((Double.doubleToLongBits(d) >>> 52) & 0x7ff) - 1023;
    }

    /**
     * Return the exponent of a float number, removing the bias.
     * <p>
     * For float numbers of the form 2<sup>x</sup>, the unbiased
     * exponent is exactly x.
     * </p>
     * @param f number from which exponent is requested
     * @return exponent for d in IEEE754 representation, without bias
     */
    public static int getExponent(final float f) {
        return ((Float.floatToIntBits(f) >>> 23) & 0xff) - 127;
    }

}