Dfp

/*
 * 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.math4.legacy.core.dfp;

import java.util.Arrays;

import org.apache.commons.math4.legacy.core.RealFieldElement;
import org.apache.commons.math4.legacy.exception.DimensionMismatchException;

/**
 *  Decimal floating point library for Java
 *
 *  <p>Another floating point class.  This one is built using radix 10000
 *  which is 10<sup>4</sup>, so its almost decimal.</p>
 *
 *  <p>The design goals here are:
 *  <ol>
 *    <li>Decimal math, or close to it</li>
 *    <li>Settable precision (but no mix between numbers using different settings)</li>
 *    <li>Portability.  Code should be kept as portable as possible.</li>
 *    <li>Performance</li>
 *    <li>Accuracy  - Results should always be +/- 1 ULP for basic
 *         algebraic operation</li>
 *    <li>Comply with IEEE 854-1987 as much as possible.
 *         (See IEEE 854-1987 notes below)</li>
 *  </ol>
 *
 *  <p>Trade offs:
 *  <ol>
 *    <li>Memory foot print.  I'm using more memory than necessary to
 *         represent numbers to get better performance.</li>
 *    <li>Digits are bigger, so rounding is a greater loss.  So, if you
 *         really need 12 decimal digits, better use 4 base 10000 digits
 *         there can be one partially filled.</li>
 *  </ol>
 *
 *  <p>Numbers are represented  in the following form:
 *  <div style="white-space: pre"><code>
 *  n  =  sign &times; mant &times; (radix)<sup>exp</sup>;
 *  </code></div>
 *  where sign is &plusmn;1, mantissa represents a fractional number between
 *  zero and one.  mant[0] is the least significant digit.
 *  exp is in the range of -32767 to 32768
 *
 *  <p>IEEE 854-1987  Notes and differences</p>
 *
 *  <p>IEEE 854 requires the radix to be either 2 or 10.  The radix here is
 *  10000, so that requirement is not met, but  it is possible that a
 *  subclassed can be made to make it behave as a radix 10
 *  number.  It is my opinion that if it looks and behaves as a radix
 *  10 number then it is one and that requirement would be met.</p>
 *
 *  <p>The radix of 10000 was chosen because it should be faster to operate
 *  on 4 decimal digits at once instead of one at a time.  Radix 10 behavior
 *  can be realized by adding an additional rounding step to ensure that
 *  the number of decimal digits represented is constant.</p>
 *
 *  <p>The IEEE standard specifically leaves out internal data encoding,
 *  so it is reasonable to conclude that such a subclass of this radix
 *  10000 system is merely an encoding of a radix 10 system.</p>
 *
 *  <p>IEEE 854 also specifies the existence of "sub-normal" numbers.  This
 *  class does not contain any such entities.  The most significant radix
 *  10000 digit is always non-zero.  Instead, we support "gradual underflow"
 *  by raising the underflow flag for numbers less with exponent less than
 *  expMin, but don't flush to zero until the exponent reaches MIN_EXP-digits.
 *  Thus the smallest number we can represent would be:
 *  1E(-(MIN_EXP-digits-1)*4),  eg, for digits=5, MIN_EXP=-32767, that would
 *  be 1e-131092.</p>
 *
 *  <p>IEEE 854 defines that the implied radix point lies just to the right
 *  of the most significant digit and to the left of the remaining digits.
 *  This implementation puts the implied radix point to the left of all
 *  digits including the most significant one.  The most significant digit
 *  here is the one just to the right of the radix point.  This is a fine
 *  detail and is really only a matter of definition.  Any side effects of
 *  this can be rendered invisible by a subclass.</p>
 * @see DfpField
 * @since 2.2
 */
public class Dfp implements RealFieldElement<Dfp> {

    /** The radix, or base of this system.  Set to 10000 */
    public static final int RADIX = 10000;

    /** The minimum exponent before underflow is signaled.  Flush to zero
     *  occurs at minExp-DIGITS */
    public static final int MIN_EXP = -32767;

    /** The maximum exponent before overflow is signaled and results flushed
     *  to infinity. */
    public static final int MAX_EXP =  32768;

    /** The amount under/overflows are scaled by before going to trap handler. */
    public static final int ERR_SCALE = 32760;

    /** Indicator value for normal finite numbers. */
    public static final byte FINITE = 0;

    /** Indicator value for Infinity. */
    public static final byte INFINITE = 1;

    /** Indicator value for signaling NaN. */
    public static final byte SNAN = 2;

    /** Indicator value for quiet NaN. */
    public static final byte QNAN = 3;

    /** String for NaN representation. */
    private static final String NAN_STRING = "NaN";

    /** String for positive infinity representation. */
    private static final String POS_INFINITY_STRING = "Infinity";

    /** String for negative infinity representation. */
    private static final String NEG_INFINITY_STRING = "-Infinity";

    /** Name for traps triggered by addition. */
    private static final String ADD_TRAP = "add";

    /** Name for traps triggered by multiplication. */
    private static final String MULTIPLY_TRAP = "multiply";

    /** Name for traps triggered by division. */
    private static final String DIVIDE_TRAP = "divide";

    /** Name for traps triggered by square root. */
    private static final String SQRT_TRAP = "sqrt";

    /** Name for traps triggered by alignment. */
    private static final String ALIGN_TRAP = "align";

    /** Name for traps triggered by truncation. */
    private static final String TRUNC_TRAP = "trunc";

    /** Name for traps triggered by nextAfter. */
    private static final String NEXT_AFTER_TRAP = "nextAfter";

    /** Name for traps triggered by lessThan. */
    private static final String LESS_THAN_TRAP = "lessThan";

    /** Name for traps triggered by greaterThan. */
    private static final String GREATER_THAN_TRAP = "greaterThan";

    /** Name for traps triggered by newInstance. */
    private static final String NEW_INSTANCE_TRAP = "newInstance";

    /** Mantissa. */
    protected int[] mant;

    /** Sign bit: 1 for positive, -1 for negative. */
    protected byte sign;

    /** Exponent. */
    protected int exp;

    /** Indicator for non-finite / non-number values. */
    protected byte nans;

    /** Factory building similar Dfp's. */
    private final DfpField field;

    /** Makes an instance with a value of zero.
     * @param field field to which this instance belongs
     */
    protected Dfp(final DfpField field) {
        mant = new int[field.getRadixDigits()];
        sign = 1;
        exp = 0;
        nans = FINITE;
        this.field = field;
    }

    /** Create an instance from a byte value.
     * @param field field to which this instance belongs
     * @param x value to convert to an instance
     */
    protected Dfp(final DfpField field, byte x) {
        this(field, (long) x);
    }

    /** Create an instance from an int value.
     * @param field field to which this instance belongs
     * @param x value to convert to an instance
     */
    protected Dfp(final DfpField field, int x) {
        this(field, (long) x);
    }

    /** Create an instance from a long value.
     * @param field field to which this instance belongs
     * @param x value to convert to an instance
     */
    protected Dfp(final DfpField field, long x) {

        // initialize as if 0
        mant = new int[field.getRadixDigits()];
        nans = FINITE;
        this.field = field;

        boolean isLongMin = false;
        if (x == Long.MIN_VALUE) {
            // special case for Long.MIN_VALUE (-9223372036854775808)
            // we must shift it before taking its absolute value
            isLongMin = true;
            ++x;
        }

        // set the sign
        if (x < 0) {
            sign = -1;
            x = -x;
        } else {
            sign = 1;
        }

        exp = 0;
        while (x != 0) {
            System.arraycopy(mant, mant.length - exp, mant, mant.length - 1 - exp, exp);
            mant[mant.length - 1] = (int) (x % RADIX);
            x /= RADIX;
            exp++;
        }

        if (isLongMin) {
            // remove the shift added for Long.MIN_VALUE
            // we know in this case that fixing the last digit is sufficient
            for (int i = 0; i < mant.length - 1; i++) {
                if (mant[i] != 0) {
                    mant[i]++;
                    break;
                }
            }
        }
    }

    /** Create an instance from a double value.
     * @param field field to which this instance belongs
     * @param x value to convert to an instance
     */
    protected Dfp(final DfpField field, double x) {

        // initialize as if 0
        mant = new int[field.getRadixDigits()];
        sign = 1;
        exp = 0;
        nans = FINITE;
        this.field = field;

        long bits = Double.doubleToLongBits(x);
        long mantissa = bits & 0x000fffffffffffffL;
        int exponent = (int) ((bits & 0x7ff0000000000000L) >> 52) - 1023;

        if (exponent == -1023) {
            // Zero or sub-normal
            if (x == 0) {
                // make sure 0 has the right sign
                if ((bits & 0x8000000000000000L) != 0) {
                    sign = -1;
                }
                return;
            }

            exponent++;

            // Normalize the subnormal number
            while ((mantissa & 0x0010000000000000L) == 0) {
                exponent--;
                mantissa <<= 1;
            }
            mantissa &= 0x000fffffffffffffL;
        }

        if (exponent == 1024) {
            // infinity or NAN
            if (Double.isNaN(x)) {
                sign = (byte) 1;
                nans = QNAN;
            } else if (x < 0) {
                sign = (byte) -1;
                nans = INFINITE;
            } else {
                sign = (byte) 1;
                nans = INFINITE;
            }
            return;
        }

        Dfp xdfp = new Dfp(field, mantissa);
        xdfp = xdfp.divide(new Dfp(field, 4503599627370496L)).add(field.getOne());  // Divide by 2^52, then add one
        xdfp = xdfp.multiply(DfpMath.pow(field.getTwo(), exponent));

        if ((bits & 0x8000000000000000L) != 0) {
            xdfp = xdfp.negate();
        }

        System.arraycopy(xdfp.mant, 0, mant, 0, mant.length);
        sign = xdfp.sign;
        exp  = xdfp.exp;
        nans = xdfp.nans;
    }

    /** Copy constructor.
     * @param d instance to copy
     */
    public Dfp(final Dfp d) {
        mant  = d.mant.clone();
        sign  = d.sign;
        exp   = d.exp;
        nans  = d.nans;
        field = d.field;
    }

    /** Create an instance from a String representation.
     * @param field field to which this instance belongs
     * @param s string representation of the instance
     */
    protected Dfp(final DfpField field, final String s) {

        // initialize as if 0
        mant = new int[field.getRadixDigits()];
        sign = 1;
        exp = 0;
        nans = FINITE;
        this.field = field;

        boolean decimalFound = false;
        final int rsize = 4;   // size of radix in decimal digits
        final int offset = 4;  // Starting offset into Striped
        final char[] striped = new char[getRadixDigits() * rsize + offset * 2];

        // Check some special cases
        if (s.equals(POS_INFINITY_STRING)) {
            sign = (byte) 1;
            nans = INFINITE;
            return;
        }

        if (s.equals(NEG_INFINITY_STRING)) {
            sign = (byte) -1;
            nans = INFINITE;
            return;
        }

        if (s.equals(NAN_STRING)) {
            sign = (byte) 1;
            nans = QNAN;
            return;
        }

        // Check for scientific notation
        int p = s.indexOf("e");
        if (p == -1) { // try upper case?
            p = s.indexOf("E");
        }

        final String fpdecimal;
        int sciexp = 0;
        if (p != -1) {
            // scientific notation
            fpdecimal = s.substring(0, p);
            String fpexp = s.substring(p + 1);
            boolean negative = false;

            for (int i = 0; i < fpexp.length(); i++) {
                if (fpexp.charAt(i) == '-') {
                    negative = true;
                    continue;
                }
                if (fpexp.charAt(i) >= '0' && fpexp.charAt(i) <= '9') {
                    sciexp = sciexp * 10 + fpexp.charAt(i) - '0';
                }
            }

            if (negative) {
                sciexp = -sciexp;
            }
        } else {
            // normal case
            fpdecimal = s;
        }

        // If there is a minus sign in the number then it is negative
        if (fpdecimal.indexOf("-") !=  -1) {
            sign = -1;
        }

        // First off, find all of the leading zeros, trailing zeros, and significant digits
        p = 0;

        // Move p to first significant digit
        int decimalPos = 0;
        for (;;) {
            if (fpdecimal.charAt(p) >= '1' && fpdecimal.charAt(p) <= '9') {
                break;
            }

            if (decimalFound && fpdecimal.charAt(p) == '0') {
                decimalPos--;
            }

            if (fpdecimal.charAt(p) == '.') {
                decimalFound = true;
            }

            p++;

            if (p == fpdecimal.length()) {
                break;
            }
        }

        // Copy the string onto Stripped
        int q = offset;
        striped[0] = '0';
        striped[1] = '0';
        striped[2] = '0';
        striped[3] = '0';
        int significantDigits = 0;
        for (;;) {
            if (p == (fpdecimal.length())) {
                break;
            }

            // Don't want to run pass the end of the array
            if (q == mant.length * rsize + offset + 1) {
                break;
            }

            if (fpdecimal.charAt(p) == '.') {
                decimalFound = true;
                decimalPos = significantDigits;
                p++;
                continue;
            }

            if (fpdecimal.charAt(p) < '0' || fpdecimal.charAt(p) > '9') {
                p++;
                continue;
            }

            striped[q] = fpdecimal.charAt(p);
            q++;
            p++;
            significantDigits++;
        }


        // If the decimal point has been found then get rid of trailing zeros.
        if (decimalFound && q != offset) {
            for (;;) {
                q--;
                if (q == offset) {
                    break;
                }
                if (striped[q] == '0') {
                    significantDigits--;
                } else {
                    break;
                }
            }
        }

        // special case of numbers like "0.00000"
        if (decimalFound && significantDigits == 0) {
            decimalPos = 0;
        }

        // Implicit decimal point at end of number if not present
        if (!decimalFound) {
            decimalPos = q - offset;
        }

        // Find the number of significant trailing zeros
        q = offset; // set q to point to first sig digit
        p = significantDigits - 1 + offset;

        while (p > q) {
            if (striped[p] != '0') {
                break;
            }
            p--;
        }

        // Make sure the decimal is on a mod 10000 boundary
        int i = ((rsize * 100) - decimalPos - sciexp % rsize) % rsize;
        q -= i;
        decimalPos += i;

        // Make the mantissa length right by adding zeros at the end if necessary
        while ((p - q) < (mant.length * rsize)) {
            for (i = 0; i < rsize; i++) {
                striped[++p] = '0';
            }
        }

        // Ok, now we know how many trailing zeros there are,
        // and where the least significant digit is
        for (i = mant.length - 1; i >= 0; i--) {
            mant[i] = (striped[q]     - '0') * 1000 +
                      (striped[q + 1] - '0') * 100  +
                      (striped[q + 2] - '0') * 10   +
                      (striped[q + 3] - '0');
            q += 4;
        }


        exp = (decimalPos + sciexp) / rsize;

        if (q < striped.length) {
            // Is there possible another digit?
            round((striped[q] - '0') * 1000);
        }
    }

    /** Creates an instance with a non-finite value.
     * @param field field to which this instance belongs
     * @param sign sign of the Dfp to create
     * @param nans code of the value, must be one of {@link #INFINITE},
     * {@link #SNAN},  {@link #QNAN}
     */
    protected Dfp(final DfpField field, final byte sign, final byte nans) {
        this.field = field;
        this.mant    = new int[field.getRadixDigits()];
        this.sign    = sign;
        this.exp     = 0;
        this.nans    = nans;
    }

    /** Create an instance with a value of 0.
     * Use this internally in preference to constructors to facilitate subclasses
     * @return a new instance with a value of 0
     */
    public Dfp newInstance() {
        return new Dfp(getField());
    }

    /** Create an instance from a byte value.
     * @param x value to convert to an instance
     * @return a new instance with value x
     */
    public Dfp newInstance(final byte x) {
        return new Dfp(getField(), x);
    }

    /** Create an instance from an int value.
     * @param x value to convert to an instance
     * @return a new instance with value x
     */
    public Dfp newInstance(final int x) {
        return new Dfp(getField(), x);
    }

    /** Create an instance from a long value.
     * @param x value to convert to an instance
     * @return a new instance with value x
     */
    public Dfp newInstance(final long x) {
        return new Dfp(getField(), x);
    }

    /** Create an instance from a double value.
     * @param x value to convert to an instance
     * @return a new instance with value x
     */
    public Dfp newInstance(final double x) {
        return new Dfp(getField(), x);
    }

    /** Create an instance by copying an existing one.
     * Use this internally in preference to constructors to facilitate subclasses.
     * @param d instance to copy
     * @return a new instance with the same value as d
     */
    public Dfp newInstance(final Dfp d) {

        // make sure we don't mix number with different precision
        if (field.getRadixDigits() != d.field.getRadixDigits()) {
            field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
            final Dfp result = newInstance(getZero());
            result.nans = QNAN;
            return dotrap(DfpField.FLAG_INVALID, NEW_INSTANCE_TRAP, d, result);
        }

        return new Dfp(d);
    }

    /** Create an instance from a String representation.
     * Use this internally in preference to constructors to facilitate subclasses.
     * @param s string representation of the instance
     * @return a new instance parsed from specified string
     */
    public Dfp newInstance(final String s) {
        return new Dfp(field, s);
    }

    /** Creates an instance with a non-finite value.
     * @param sig sign of the Dfp to create
     * @param code code of the value, must be one of {@link #INFINITE},
     * {@link #SNAN},  {@link #QNAN}
     * @return a new instance with a non-finite value
     */
    public Dfp newInstance(final byte sig, final byte code) {
        return field.newDfp(sig, code);
    }

    /** Get the {@link org.apache.commons.math4.legacy.core.Field Field} (really a
     * {@link DfpField}) to which the instance belongs.
     * <p>
     * The field is linked to the number of digits and acts as a factory
     * for {@link Dfp} instances.
     * </p>
     * @return {@link org.apache.commons.math4.legacy.core.Field Field} (really a {@link DfpField}) to which the instance belongs
     */
    @Override
    public DfpField getField() {
        return field;
    }

    /** Get the number of radix digits of the instance.
     * @return number of radix digits
     */
    public int getRadixDigits() {
        return field.getRadixDigits();
    }

    /** Get the constant 0.
     * @return a Dfp with value zero
     */
    public Dfp getZero() {
        return field.getZero();
    }

    /** Get the constant 1.
     * @return a Dfp with value one
     */
    public Dfp getOne() {
        return field.getOne();
    }

    /** Get the constant 2.
     * @return a Dfp with value two
     */
    public Dfp getTwo() {
        return field.getTwo();
    }

    /** Shift the mantissa left, and adjust the exponent to compensate.
     */
    protected void shiftLeft() {
        if (mant.length - 1 > 0) {
            System.arraycopy(mant, 0, mant, 1, mant.length - 1);
        }
        mant[0] = 0;
        exp--;
    }

    /* Note that shiftRight() does not call round() as that round() itself
     uses shiftRight() */
    /** Shift the mantissa right, and adjust the exponent to compensate.
     */
    protected void shiftRight() {
        if (mant.length - 1 > 0) {
            System.arraycopy(mant, 1, mant, 0, mant.length - 1);
        }
        mant[mant.length - 1] = 0;
        exp++;
    }

    /** Make our exp equal to the supplied one, this may cause rounding.
     *  Also causes de-normalized numbers.  These numbers are generally
     *  dangerous because most routines assume normalized numbers.
     *  Align doesn't round, so it will return the last digit destroyed
     *  by shifting right.
     *  @param e desired exponent
     *  @return last digit destroyed by shifting right
     */
    protected int align(int e) {
        int lostdigit = 0;
        boolean inexact = false;

        int diff = exp - e;

        int adiff = diff;
        if (adiff < 0) {
            adiff = -adiff;
        }

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

        if (adiff > (mant.length + 1)) {
            // Special case
            Arrays.fill(mant, 0);
            exp = e;

            field.setIEEEFlagsBits(DfpField.FLAG_INEXACT);
            dotrap(DfpField.FLAG_INEXACT, ALIGN_TRAP, this, this);

            return 0;
        }

        for (int i = 0; i < adiff; i++) {
            if (diff < 0) {
                /* Keep track of loss -- only signal inexact after losing 2 digits.
                 * the first lost digit is returned to add() and may be incorporated
                 * into the result.
                 */
                if (lostdigit != 0) {
                    inexact = true;
                }

                lostdigit = mant[0];

                shiftRight();
            } else {
                shiftLeft();
            }
        }

        if (inexact) {
            field.setIEEEFlagsBits(DfpField.FLAG_INEXACT);
            dotrap(DfpField.FLAG_INEXACT, ALIGN_TRAP, this, this);
        }

        return lostdigit;
    }

    /** Check if instance is less than x.
     * @param x number to check instance against
     * @return true if instance is less than x and neither are NaN, false otherwise
     */
    public boolean lessThan(final Dfp x) {

        // make sure we don't mix number with different precision
        if (field.getRadixDigits() != x.field.getRadixDigits()) {
            field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
            final Dfp result = newInstance(getZero());
            result.nans = QNAN;
            dotrap(DfpField.FLAG_INVALID, LESS_THAN_TRAP, x, result);
            return false;
        }

        /* if a nan is involved, signal invalid and return false */
        if (isNaN() || x.isNaN()) {
            field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
            dotrap(DfpField.FLAG_INVALID, LESS_THAN_TRAP, x, newInstance(getZero()));
            return false;
        }

        return compare(this, x) < 0;
    }

    /** Check if instance is greater than x.
     * @param x number to check instance against
     * @return true if instance is greater than x and neither are NaN, false otherwise
     */
    public boolean greaterThan(final Dfp x) {

        // make sure we don't mix number with different precision
        if (field.getRadixDigits() != x.field.getRadixDigits()) {
            field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
            final Dfp result = newInstance(getZero());
            result.nans = QNAN;
            dotrap(DfpField.FLAG_INVALID, GREATER_THAN_TRAP, x, result);
            return false;
        }

        /* if a nan is involved, signal invalid and return false */
        if (isNaN() || x.isNaN()) {
            field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
            dotrap(DfpField.FLAG_INVALID, GREATER_THAN_TRAP, x, newInstance(getZero()));
            return false;
        }

        return compare(this, x) > 0;
    }

    /** Check if instance is less than or equal to 0.
     * @return true if instance is not NaN and less than or equal to 0, false otherwise
     */
    public boolean negativeOrNull() {

        if (isNaN()) {
            field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
            dotrap(DfpField.FLAG_INVALID, LESS_THAN_TRAP, this, newInstance(getZero()));
            return false;
        }

        return (sign < 0) || ((mant[mant.length - 1] == 0) && !isInfinite());
    }

    /** Check if instance is strictly less than 0.
     * @return true if instance is not NaN and less than or equal to 0, false otherwise
     */
    public boolean strictlyNegative() {

        if (isNaN()) {
            field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
            dotrap(DfpField.FLAG_INVALID, LESS_THAN_TRAP, this, newInstance(getZero()));
            return false;
        }

        return (sign < 0) && ((mant[mant.length - 1] != 0) || isInfinite());
    }

    /** Check if instance is greater than or equal to 0.
     * @return true if instance is not NaN and greater than or equal to 0, false otherwise
     */
    public boolean positiveOrNull() {

        if (isNaN()) {
            field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
            dotrap(DfpField.FLAG_INVALID, LESS_THAN_TRAP, this, newInstance(getZero()));
            return false;
        }

        return (sign > 0) || ((mant[mant.length - 1] == 0) && !isInfinite());
    }

    /** Check if instance is strictly greater than 0.
     * @return true if instance is not NaN and greater than or equal to 0, false otherwise
     */
    public boolean strictlyPositive() {

        if (isNaN()) {
            field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
            dotrap(DfpField.FLAG_INVALID, LESS_THAN_TRAP, this, newInstance(getZero()));
            return false;
        }

        return (sign > 0) && ((mant[mant.length - 1] != 0) || isInfinite());
    }

    /** Get the absolute value of instance.
     * @return absolute value of instance
     * @since 3.2
     */
    @Override
    public Dfp abs() {
        Dfp result = newInstance(this);
        result.sign = 1;
        return result;
    }

    /** Check if instance is infinite.
     * @return true if instance is infinite
     */
    public boolean isInfinite() {
        return nans == INFINITE;
    }

    /** Check if instance is not a number.
     * @return true if instance is not a number
     */
    public boolean isNaN() {
        return (nans == QNAN) || (nans == SNAN);
    }

    /** Check if instance is equal to zero.
     * @return true if instance is equal to zero
     */
    public boolean isZero() {

        if (isNaN()) {
            field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
            dotrap(DfpField.FLAG_INVALID, LESS_THAN_TRAP, this, newInstance(getZero()));
            return false;
        }

        return (mant[mant.length - 1] == 0) && !isInfinite();
    }

    /** Check if instance is equal to x.
     * @param other object to check instance against
     * @return true if instance is equal to x and neither are NaN, false otherwise
     */
    @Override
    public boolean equals(final Object other) {

        if (other instanceof Dfp) {
            final Dfp x = (Dfp) other;
            if (isNaN() || x.isNaN() || field.getRadixDigits() != x.field.getRadixDigits()) {
                return false;
            }

            return compare(this, x) == 0;
        }

        return false;
    }

    /**
     * Gets a hashCode for the instance.
     * @return a hash code value for this object
     */
    @Override
    public int hashCode() {
        return 17 + (isZero() ? 0 : (sign << 8)) + (nans << 16) + exp + Arrays.hashCode(mant);
    }

    /** Check if instance is not equal to x.
     * @param x number to check instance against
     * @return true if instance is not equal to x and neither are NaN, false otherwise
     */
    public boolean unequal(final Dfp x) {
        if (isNaN() || x.isNaN() || field.getRadixDigits() != x.field.getRadixDigits()) {
            return false;
        }

        return greaterThan(x) || lessThan(x);
    }

    /** Compare two instances.
     * @param a first instance in comparison
     * @param b second instance in comparison
     * @return -1 if {@code a < b}, 1 if {@code a > b} and 0 if {@code a == b}
     *  Note this method does not properly handle NaNs or numbers with different precision.
     */
    private static int compare(final Dfp a, final Dfp b) {
        // Ignore the sign of zero
        if (a.mant[a.mant.length - 1] == 0 && b.mant[b.mant.length - 1] == 0 &&
            a.nans == FINITE && b.nans == FINITE) {
            return 0;
        }

        if (a.sign != b.sign) {
            if (a.sign == -1) {
                return -1;
            } else {
                return 1;
            }
        }

        // deal with the infinities
        if (a.nans == INFINITE && b.nans == FINITE) {
            return a.sign;
        }

        if (a.nans == FINITE && b.nans == INFINITE) {
            return -b.sign;
        }

        if (a.nans == INFINITE && b.nans == INFINITE) {
            return 0;
        }

        // Handle special case when a or b is zero, by ignoring the exponents
        if (b.mant[b.mant.length - 1] != 0 && a.mant[b.mant.length - 1] != 0) {
            if (a.exp < b.exp) {
                return -a.sign;
            }

            if (a.exp > b.exp) {
                return a.sign;
            }
        }

        // compare the mantissas
        for (int i = a.mant.length - 1; i >= 0; i--) {
            if (a.mant[i] > b.mant[i]) {
                return a.sign;
            }

            if (a.mant[i] < b.mant[i]) {
                return -a.sign;
            }
        }

        return 0;
    }

    /** Round to nearest integer using the round-half-even method.
     *  That is round to nearest integer unless both are equidistant.
     *  In which case round to the even one.
     *  @return rounded value
     * @since 3.2
     */
    @Override
    public Dfp rint() {
        return trunc(DfpField.RoundingMode.ROUND_HALF_EVEN);
    }

     /** Round to an integer using the round floor mode.
      * That is, round toward -Infinity
      *  @return rounded value
      * @since 3.2
      */
     @Override
     public Dfp floor() {
         return trunc(DfpField.RoundingMode.ROUND_FLOOR);
     }

     /** Round to an integer using the round ceil mode.
      * That is, round toward +Infinity
      *  @return rounded value
      * @since 3.2
      */
     @Override
     public Dfp ceil() {
         return trunc(DfpField.RoundingMode.ROUND_CEIL);
     }

     /** Returns the IEEE remainder.
      * @param d divisor
      * @return this less n &times; d, where n is the integer closest to this/d
      * @since 3.2
      */
     @Override
     public Dfp remainder(final Dfp d) {

         final Dfp result = this.subtract(this.divide(d).rint().multiply(d));

         // IEEE 854-1987 says that if the result is zero, then it carries the sign of this
         if (result.mant[mant.length - 1] == 0) {
             result.sign = sign;
         }

         return result;
     }

     /** Does the integer conversions with the specified rounding.
      * @param rmode rounding mode to use
      * @return truncated value
      */
     protected Dfp trunc(final DfpField.RoundingMode rmode) {
         boolean changed = false;

         if (isNaN()) {
             return newInstance(this);
         }

         if (nans == INFINITE) {
             return newInstance(this);
         }

         if (mant[mant.length - 1] == 0) {
             // a is zero
             return newInstance(this);
         }

         /* If the exponent is less than zero then we can certainly
          * return zero */
         if (exp < 0) {
             field.setIEEEFlagsBits(DfpField.FLAG_INEXACT);
             Dfp result = newInstance(getZero());
             result = dotrap(DfpField.FLAG_INEXACT, TRUNC_TRAP, this, result);
             return result;
         }

         /* If the exponent is greater than or equal to digits, then it
          * must already be an integer since there is no precision left
          * for any fractional part */

         if (exp >= mant.length) {
             return newInstance(this);
         }

         /* General case:  create another dfp, result, that contains the
          * a with the fractional part lopped off.  */

         Dfp result = newInstance(this);
         for (int i = 0; i < mant.length - result.exp; i++) {
             changed |= result.mant[i] != 0;
             result.mant[i] = 0;
         }

         if (changed) {
             switch (rmode) {
             case ROUND_FLOOR:
                 if (result.sign == -1) {
                     // then we must increment the mantissa by one
                     result = result.add(newInstance(-1));
                 }
                 break;

             case ROUND_CEIL:
                 if (result.sign == 1) {
                     // then we must increment the mantissa by one
                     result = result.add(getOne());
                 }
                 break;

             case ROUND_HALF_EVEN:
             default:
                 final Dfp half = newInstance("0.5");
                 Dfp a = subtract(result);  // difference between this and result
                 a.sign = 1;            // force positive (take abs)
                 if (a.greaterThan(half)) {
                     a = newInstance(getOne());
                     a.sign = sign;
                     result = result.add(a);
                 }

                 /* If exactly equal to 1/2 and odd then increment */
                 if (a.equals(half) && result.exp > 0 && (result.mant[mant.length - result.exp] & 1) != 0) {
                     a = newInstance(getOne());
                     a.sign = sign;
                     result = result.add(a);
                 }
                 break;
             }

             field.setIEEEFlagsBits(DfpField.FLAG_INEXACT);  // signal inexact
             result = dotrap(DfpField.FLAG_INEXACT, TRUNC_TRAP, this, result);
             return result;
         }

         return result;
     }

     /** Convert this to an integer.
      * If greater than 2147483647, it returns 2147483647. If less than -2147483648 it returns -2147483648.
      * @return converted number
      */
     public int intValue() {
         Dfp rounded;
         int result = 0;

         rounded = rint();

         if (rounded.greaterThan(newInstance(2147483647))) {
             return 2147483647;
         }

         if (rounded.lessThan(newInstance(-2147483648))) {
             return -2147483648;
         }

         for (int i = mant.length - 1; i >= mant.length - rounded.exp; i--) {
             result = result * RADIX + rounded.mant[i];
         }

         if (rounded.sign == -1) {
             result = -result;
         }

         return result;
     }

     /** Get the exponent of the greatest power of 10000 that is
      *  less than or equal to the absolute value of this.  I.E.  if
      *  this is 10<sup>6</sup> then log10K would return 1.
      *  @return integer base 10000 logarithm
      */
     public int log10K() {
         return exp - 1;
     }

     /** Get the specified  power of 10000.
      * @param e desired power
      * @return 10000<sup>e</sup>
      */
     public Dfp power10K(final int e) {
         Dfp d = newInstance(getOne());
         d.exp = e + 1;
         return d;
     }

     /** Get the exponent of the greatest power of 10 that is less than or equal to abs(this).
      *  @return integer base 10 logarithm
      * @since 3.2
      */
     public int intLog10()  {
         if (mant[mant.length - 1] > 1000) {
             return exp * 4 - 1;
         }
         if (mant[mant.length - 1] > 100) {
             return exp * 4 - 2;
         }
         if (mant[mant.length - 1] > 10) {
             return exp * 4 - 3;
         }
         return exp * 4 - 4;
     }

     /** Return the specified  power of 10.
      * @param e desired power
      * @return 10<sup>e</sup>
      */
     public Dfp power10(final int e) {
         Dfp d = newInstance(getOne());

         if (e >= 0) {
             d.exp = e / 4 + 1;
         } else {
             d.exp = (e + 1) / 4;
         }

         switch ((e % 4 + 4) % 4) {
         case 0:
             break;
         case 1:
             d = d.multiply(10);
             break;
         case 2:
             d = d.multiply(100);
             break;
         default:
             d = d.multiply(1000);
         }

         return d;
     }

     /** Negate the mantissa of this by computing the complement.
      *  Leaves the sign bit unchanged, used internally by add.
      *  Denormalized numbers are handled properly here.
      *  @param extra ???
      *  @return ???
      */
     protected int complement(int extra) {

         extra = RADIX - extra;
         for (int i = 0; i < mant.length; i++) {
             mant[i] = RADIX - mant[i] - 1;
         }

         int rh = extra / RADIX;
         extra -= rh * RADIX;
         for (int i = 0; i < mant.length; i++) {
             final int r = mant[i] + rh;
             rh = r / RADIX;
             mant[i] = r - rh * RADIX;
         }

         return extra;
     }

     /** Add x to this.
      * @param x number to add
      * @return sum of this and x
      */
     @Override
     public Dfp add(final Dfp x) {

         // make sure we don't mix number with different precision
         if (field.getRadixDigits() != x.field.getRadixDigits()) {
             field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
             final Dfp result = newInstance(getZero());
             result.nans = QNAN;
             return dotrap(DfpField.FLAG_INVALID, ADD_TRAP, x, result);
         }

         /* handle special cases */
         if (nans != FINITE || x.nans != FINITE) {
             if (isNaN()) {
                 return this;
             }

             if (x.isNaN()) {
                 return x;
             }

             if (nans == INFINITE && x.nans == FINITE) {
                 return this;
             }

             if (x.nans == INFINITE && nans == FINITE) {
                 return x;
             }

             if (x.nans == INFINITE && nans == INFINITE && sign == x.sign) {
                 return x;
             }

             if (x.nans == INFINITE && nans == INFINITE && sign != x.sign) {
                 field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
                 Dfp result = newInstance(getZero());
                 result.nans = QNAN;
                 result = dotrap(DfpField.FLAG_INVALID, ADD_TRAP, x, result);
                 return result;
             }
         }

         /* copy this and the arg */
         Dfp a = newInstance(this);
         Dfp b = newInstance(x);

         /* initialize the result object */
         Dfp result = newInstance(getZero());

         /* Make all numbers positive, but remember their sign */
         final byte asign = a.sign;
         final byte bsign = b.sign;

         a.sign = 1;
         b.sign = 1;

         /* The result will be signed like the arg with greatest magnitude */
         byte rsign = bsign;
         if (compare(a, b) > 0) {
             rsign = asign;
         }

         /*
          * Handle special case when a or b is zero, by setting the exponent of the zero
          * number equal to the other one. This avoids an alignment which would cause
          * catastropic loss of precision
          */
         if (b.mant[mant.length - 1] == 0) {
             b.exp = a.exp;
         }

         if (a.mant[mant.length - 1] == 0) {
             a.exp = b.exp;
         }

         /* align number with the smaller exponent */
         int aextradigit = 0;
         int bextradigit = 0;
         if (a.exp < b.exp) {
             aextradigit = a.align(b.exp);
         } else {
             bextradigit = b.align(a.exp);
         }

         /* complement the smaller of the two if the signs are different */
         if (asign != bsign) {
             if (asign == rsign) {
                 bextradigit = b.complement(bextradigit);
             } else {
                 aextradigit = a.complement(aextradigit);
             }
         }

         /* add the mantissas */
         int rh = 0; /* acts as a carry */
         for (int i = 0; i < mant.length; i++) {
             final int r = a.mant[i] + b.mant[i] + rh;
             rh = r / RADIX;
             result.mant[i] = r - rh * RADIX;
         }
         result.exp = a.exp;
         result.sign = rsign;

         /* handle overflow -- note, when asign!=bsign an overflow is
          * normal and should be ignored.  */

         if (rh != 0 && (asign == bsign)) {
             final int lostdigit = result.mant[0];
             result.shiftRight();
             result.mant[mant.length - 1] = rh;
             final int excp = result.round(lostdigit);
             if (excp != 0) {
                 result = dotrap(excp, ADD_TRAP, x, result);
             }
         }

         /* normalize the result */
         for (int i = 0; i < mant.length; i++) {
             if (result.mant[mant.length - 1] != 0) {
                 break;
             }
             result.shiftLeft();
             if (i == 0) {
                 result.mant[0] = aextradigit + bextradigit;
                 aextradigit = 0;
                 bextradigit = 0;
             }
         }

         /* result is zero if after normalization the most sig. digit is zero */
         if (result.mant[mant.length - 1] == 0) {
             result.exp = 0;

             if (asign != bsign) {
                 // Unless adding 2 negative zeros, sign is positive
                 result.sign = 1;  // Per IEEE 854-1987 Section 6.3
             }
         }

         /* Call round to test for over/under flows */
         final int excp = result.round(aextradigit + bextradigit);
         if (excp != 0) {
             result = dotrap(excp, ADD_TRAP, x, result);
         }

         return result;
     }

     /** Returns a number that is this number with the sign bit reversed.
      * @return the opposite of this
      */
     @Override
     public Dfp negate() {
         Dfp result = newInstance(this);
         result.sign = (byte) -result.sign;
         return result;
     }

     /** Subtract x from this.
      * @param x number to subtract
      * @return difference of this and a
      */
     @Override
     public Dfp subtract(final Dfp x) {
         return add(x.negate());
     }

     /** Round this given the next digit n using the current rounding mode.
      * @param n ???
      * @return the IEEE flag if an exception occurred
      */
     protected int round(int n) {
         boolean inc = false;
         switch (field.getRoundingMode()) {
         case ROUND_DOWN:
             inc = false;
             break;

         case ROUND_UP:
             inc = n != 0;       // round up if n!=0
             break;

         case ROUND_HALF_UP:
             inc = n >= 5000;  // round half up
             break;

         case ROUND_HALF_DOWN:
             inc = n > 5000;  // round half down
             break;

         case ROUND_HALF_EVEN:
             inc = n > 5000 || (n == 5000 && (mant[0] & 1) == 1);  // round half-even
             break;

         case ROUND_HALF_ODD:
             inc = n > 5000 || (n == 5000 && (mant[0] & 1) == 0);  // round half-odd
             break;

         case ROUND_CEIL:
             inc = sign == 1 && n != 0;  // round ceil
             break;

         case ROUND_FLOOR:
         default:
             inc = sign == -1 && n != 0;  // round floor
             break;
         }

         if (inc) {
             // increment if necessary
             int rh = 1;
             for (int i = 0; i < mant.length; i++) {
                 final int r = mant[i] + rh;
                 rh = r / RADIX;
                 mant[i] = r - rh * RADIX;
             }

             if (rh != 0) {
                 shiftRight();
                 mant[mant.length - 1] = rh;
             }
         }

         // check for exceptional cases and raise signals if necessary
         if (exp < MIN_EXP) {
             // Gradual Underflow
             field.setIEEEFlagsBits(DfpField.FLAG_UNDERFLOW);
             return DfpField.FLAG_UNDERFLOW;
         }

         if (exp > MAX_EXP) {
             // Overflow
             field.setIEEEFlagsBits(DfpField.FLAG_OVERFLOW);
             return DfpField.FLAG_OVERFLOW;
         }

         if (n != 0) {
             // Inexact
             field.setIEEEFlagsBits(DfpField.FLAG_INEXACT);
             return DfpField.FLAG_INEXACT;
         }

         return 0;
     }

     /** Multiply this by x.
      * @param x multiplicand
      * @return product of this and x
      */
     @Override
     public Dfp multiply(final Dfp x) {

         // make sure we don't mix number with different precision
         if (field.getRadixDigits() != x.field.getRadixDigits()) {
             field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
             final Dfp result = newInstance(getZero());
             result.nans = QNAN;
             return dotrap(DfpField.FLAG_INVALID, MULTIPLY_TRAP, x, result);
         }

         Dfp result = newInstance(getZero());

         /* handle special cases */
         if (nans != FINITE || x.nans != FINITE) {
             if (isNaN()) {
                 return this;
             }

             if (x.isNaN()) {
                 return x;
             }

             if (nans == INFINITE && x.nans == FINITE && x.mant[mant.length - 1] != 0) {
                 result = newInstance(this);
                 result.sign = (byte) (sign * x.sign);
                 return result;
             }

             if (x.nans == INFINITE && nans == FINITE && mant[mant.length - 1] != 0) {
                 result = newInstance(x);
                 result.sign = (byte) (sign * x.sign);
                 return result;
             }

             if (x.nans == INFINITE && nans == INFINITE) {
                 result = newInstance(this);
                 result.sign = (byte) (sign * x.sign);
                 return result;
             }

             if ((x.nans == INFINITE && nans == FINITE && mant[mant.length - 1] == 0) ||
                 (nans == INFINITE && x.nans == FINITE && x.mant[mant.length - 1] == 0)) {
                 field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
                 result = newInstance(getZero());
                 result.nans = QNAN;
                 result = dotrap(DfpField.FLAG_INVALID, MULTIPLY_TRAP, x, result);
                 return result;
             }
         }

         int[] product = new int[mant.length * 2];  // Big enough to hold even the largest result

         for (int i = 0; i < mant.length; i++) {
             int rh = 0; // acts as a carry
             for (int j = 0; j < mant.length; j++) {
                 int r = mant[i] * x.mant[j]; // multiply the 2 digits
                 r += product[i + j] + rh; // add to the product digit with carry in

                 rh = r / RADIX;
                 product[i + j] = r - rh * RADIX;
             }
             product[i + mant.length] = rh;
         }

         // Find the most sig digit
         int md = mant.length * 2 - 1;  // default, in case result is zero
         for (int i = mant.length * 2 - 1; i >= 0; i--) {
             if (product[i] != 0) {
                 md = i;
                 break;
             }
         }

         // Copy the digits into the result
         for (int i = 0; i < mant.length; i++) {
             result.mant[mant.length - i - 1] = product[md - i];
         }

         // Fixup the exponent.
         result.exp = exp + x.exp + md - 2 * mant.length + 1;
         result.sign = (byte) ((sign == x.sign) ? 1 : -1);

         if (result.mant[mant.length - 1] == 0) {
             // if result is zero, set exp to zero
             result.exp = 0;
         }

         final int excp;
         if (md > (mant.length - 1)) {
             excp = result.round(product[md - mant.length]);
         } else {
             excp = result.round(0); // has no effect except to check status
         }

         if (excp != 0) {
             result = dotrap(excp, MULTIPLY_TRAP, x, result);
         }

         return result;
     }

     /** Multiply this by a single digit x.
      * @param x multiplicand
      * @return product of this and x
      */
     @Override
     public Dfp multiply(final int x) {
         if (x >= 0 && x < RADIX) {
             return multiplyFast(x);
         } else {
             return multiply(newInstance(x));
         }
     }

     /** Multiply this by a single digit 0&lt;=x&lt;radix.
      * There are speed advantages in this special case.
      * @param x multiplicand
      * @return product of this and x
      */
     private Dfp multiplyFast(final int x) {
         Dfp result = newInstance(this);

         /* handle special cases */
         if (nans != FINITE) {
             if (isNaN()) {
                 return this;
             }

             if (nans == INFINITE && x != 0) {
                 result = newInstance(this);
                 return result;
             }

             if (nans == INFINITE && x == 0) {
                 field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
                 result = newInstance(getZero());
                 result.nans = QNAN;
                 result = dotrap(DfpField.FLAG_INVALID, MULTIPLY_TRAP, newInstance(getZero()), result);
                 return result;
             }
         }

         /* range check x */
         if (x < 0 || x >= RADIX) {
             field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
             result = newInstance(getZero());
             result.nans = QNAN;
             result = dotrap(DfpField.FLAG_INVALID, MULTIPLY_TRAP, result, result);
             return result;
         }

         int rh = 0;
         for (int i = 0; i < mant.length; i++) {
             final int r = mant[i] * x + rh;
             rh = r / RADIX;
             result.mant[i] = r - rh * RADIX;
         }

         int lostdigit = 0;
         if (rh != 0) {
             lostdigit = result.mant[0];
             result.shiftRight();
             result.mant[mant.length - 1] = rh;
         }

         if (result.mant[mant.length - 1] == 0) { // if result is zero, set exp to zero
             result.exp = 0;
         }

         final int excp = result.round(lostdigit);
         if (excp != 0) {
             result = dotrap(excp, MULTIPLY_TRAP, result, result);
         }

         return result;
     }

     /** Divide this by divisor.
      * @param divisor divisor
      * @return quotient of this by divisor
      */
     @Override
     public Dfp divide(Dfp divisor) {
         int[] dividend;  // current status of the dividend
         int[] quotient;  // quotient
         int[] remainder; // remainder
         int qd;          // current quotient digit we're working with
         int nsqd;        // number of significant quotient digits we have
         int trial = 0;   // trial quotient digit
         int minadj;      // minimum adjustment
         boolean trialgood; // Flag to indicate a good trail digit
         int md = 0;      // most sig digit in result
         int excp;        // exceptions

         // make sure we don't mix number with different precision
         if (field.getRadixDigits() != divisor.field.getRadixDigits()) {
             field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
             final Dfp result = newInstance(getZero());
             result.nans = QNAN;
             return dotrap(DfpField.FLAG_INVALID, DIVIDE_TRAP, divisor, result);
         }

         Dfp result = newInstance(getZero());

         /* handle special cases */
         if (nans != FINITE || divisor.nans != FINITE) {
             if (isNaN()) {
                 return this;
             }

             if (divisor.isNaN()) {
                 return divisor;
             }

             if (nans == INFINITE && divisor.nans == FINITE) {
                 result = newInstance(this);
                 result.sign = (byte) (sign * divisor.sign);
                 return result;
             }

             if (divisor.nans == INFINITE && nans == FINITE) {
                 result = newInstance(getZero());
                 result.sign = (byte) (sign * divisor.sign);
                 return result;
             }

             if (divisor.nans == INFINITE && nans == INFINITE) {
                 field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
                 result = newInstance(getZero());
                 result.nans = QNAN;
                 result = dotrap(DfpField.FLAG_INVALID, DIVIDE_TRAP, divisor, result);
                 return result;
             }
         }

         /* Test for divide by zero */
         if (divisor.mant[mant.length - 1] == 0) {
             field.setIEEEFlagsBits(DfpField.FLAG_DIV_ZERO);
             result = newInstance(getZero());
             result.sign = (byte) (sign * divisor.sign);
             result.nans = INFINITE;
             result = dotrap(DfpField.FLAG_DIV_ZERO, DIVIDE_TRAP, divisor, result);
             return result;
         }

         dividend = new int[mant.length + 1]; // one extra digit needed
         quotient = new int[mant.length + 2]; // two extra digits needed 1 for overflow, 1 for rounding
         remainder = new int[mant.length + 1]; // one extra digit needed

         /* Initialize our most significant digits to zero */

         dividend[mant.length] = 0;
         quotient[mant.length] = 0;
         quotient[mant.length + 1] = 0;
         remainder[mant.length] = 0;

         /* copy our mantissa into the dividend, initialize the quotient while we are at it */

         for (int i = 0; i < mant.length; i++) {
             dividend[i] = mant[i];
             quotient[i] = 0;
             remainder[i] = 0;
         }

         /* outer loop.  Once per quotient digit */
         nsqd = 0;
         for (qd = mant.length + 1; qd >= 0; qd--) {
             /* Determine outer limits of our quotient digit */

             // r =  most sig 2 digits of dividend
             final int divMsb = dividend[mant.length] * RADIX + dividend[mant.length - 1];
             int min = divMsb / (divisor.mant[mant.length - 1] + 1);
             int max = (divMsb + 1) / divisor.mant[mant.length - 1];

             trialgood = false;
             while (!trialgood) {
                 // try the mean
                 trial = (min + max) / 2;

                 /* Multiply by divisor and store as remainder */
                 int rh = 0;
                 for (int i = 0; i < mant.length + 1; i++) {
                     int dm = (i < mant.length) ? divisor.mant[i] : 0;
                     final int r = (dm * trial) + rh;
                     rh = r / RADIX;
                     remainder[i] = r - rh * RADIX;
                 }

                 /* subtract the remainder from the dividend */
                 rh = 1; // carry in to aid the subtraction
                 for (int i = 0; i < mant.length + 1; i++) {
                     final int r = ((RADIX - 1) - remainder[i]) + dividend[i] + rh;
                     rh = r / RADIX;
                     remainder[i] = r - rh * RADIX;
                 }

                 /* Lets analyze what we have here */
                 if (rh == 0) {
                     // trial is too big -- negative remainder
                     max = trial - 1;
                     continue;
                 }

                 /* find out how far off the remainder is telling us we are */
                 minadj = (remainder[mant.length] * RADIX) + remainder[mant.length - 1];
                 minadj /= divisor.mant[mant.length - 1] + 1;

                 if (minadj >= 2) {
                     min = trial + minadj; // update the minimum
                     continue;
                 }

                 /* May have a good one here, check more thoroughly. Basically its a good
                  * one if it is less than the divisor */
                 trialgood = false;  // assume false
                 for (int i = mant.length - 1; i >= 0; i--) {
                     if (divisor.mant[i] > remainder[i]) {
                         trialgood = true;
                         break;
                     }
                     if (divisor.mant[i] < remainder[i]) {
                         break;
                     }
                 }

                 if (remainder[mant.length] != 0) {
                     trialgood = false;
                 }

                 if (!trialgood) {
                     min = trial + 1;
                 }
             }

             /* Great we have a digit! */
             quotient[qd] = trial;
             if (trial != 0 || nsqd != 0) {
                 nsqd++;
             }

             if (field.getRoundingMode() == DfpField.RoundingMode.ROUND_DOWN && nsqd == mant.length) {
                 // We have enough for this mode
                 break;
             }

             if (nsqd > mant.length) {
                 // We have enough digits
                 break;
             }

             /* move the remainder into the dividend while left shifting */
             dividend[0] = 0;
             System.arraycopy(remainder, 0, dividend, 1, mant.length);
         }

         /* Find the most sig digit */
         md = mant.length;  // default
         for (int i = mant.length + 1; i >= 0; i--) {
             if (quotient[i] != 0) {
                 md = i;
                 break;
             }
         }

         /* Copy the digits into the result */
         for (int i = 0; i < mant.length; i++) {
             result.mant[mant.length - i - 1] = quotient[md - i];
         }

         /* Fixup the exponent. */
         result.exp = exp - divisor.exp + md - mant.length;
         result.sign = (byte) ((sign == divisor.sign) ? 1 : -1);

         if (result.mant[mant.length - 1] == 0) { // if result is zero, set exp to zero
             result.exp = 0;
         }

         if (md > (mant.length - 1)) {
             excp = result.round(quotient[md - mant.length]);
         } else {
             excp = result.round(0);
         }

         if (excp != 0) {
             result = dotrap(excp, DIVIDE_TRAP, divisor, result);
         }

         return result;
     }

     /** Divide by a single digit less than radix.
      *  Special case, so there are speed advantages. 0 &lt;= divisor &lt; radix
      * @param divisor divisor
      * @return quotient of this by divisor
      */
     public Dfp divide(int divisor) {

         // Handle special cases
         if (nans != FINITE) {
             if (isNaN()) {
                 return this;
             }

             if (nans == INFINITE) {
                 return newInstance(this);
             }
         }

         // Test for divide by zero
         if (divisor == 0) {
             field.setIEEEFlagsBits(DfpField.FLAG_DIV_ZERO);
             Dfp result = newInstance(getZero());
             result.sign = sign;
             result.nans = INFINITE;
             result = dotrap(DfpField.FLAG_DIV_ZERO, DIVIDE_TRAP, getZero(), result);
             return result;
         }

         // range check divisor
         if (divisor < 0 || divisor >= RADIX) {
             field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
             Dfp result = newInstance(getZero());
             result.nans = QNAN;
             result = dotrap(DfpField.FLAG_INVALID, DIVIDE_TRAP, result, result);
             return result;
         }

         Dfp result = newInstance(this);

         int rl = 0;
         for (int i = mant.length - 1; i >= 0; i--) {
             final int r = rl * RADIX + result.mant[i];
             final int rh = r / divisor;
             rl = r - rh * divisor;
             result.mant[i] = rh;
         }

         if (result.mant[mant.length - 1] == 0) {
             // normalize
             result.shiftLeft();
             final int r = rl * RADIX;        // compute the next digit and put it in
             final int rh = r / divisor;
             rl = r - rh * divisor;
             result.mant[0] = rh;
         }

         final int excp = result.round(rl * RADIX / divisor);  // do the rounding
         if (excp != 0) {
             result = dotrap(excp, DIVIDE_TRAP, result, result);
         }

         return result;
     }

     /** {@inheritDoc} */
     @Override
     public Dfp reciprocal() {
         return field.getOne().divide(this);
     }

     /** Compute the square root.
      * @return square root of the instance
      * @since 3.2
      */
     @Override
     public Dfp sqrt() {

         // check for unusual cases
         if (nans == FINITE && mant[mant.length - 1] == 0) {
             // if zero
             return newInstance(this);
         }

         if (nans != FINITE) {
             if (nans == INFINITE && sign == 1) {
                 // if positive infinity
                 return newInstance(this);
             }

             if (nans == QNAN) {
                 return newInstance(this);
             }

             if (nans == SNAN) {
                 Dfp result;

                 field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
                 result = newInstance(this);
                 result = dotrap(DfpField.FLAG_INVALID, SQRT_TRAP, null, result);
                 return result;
             }
         }

         if (sign == -1) {
             // if negative
             Dfp result;

             field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
             result = newInstance(this);
             result.nans = QNAN;
             result = dotrap(DfpField.FLAG_INVALID, SQRT_TRAP, null, result);
             return result;
         }

         Dfp x = newInstance(this);

         /* Lets make a reasonable guess as to the size of the square root */
         if (x.exp < -1 || x.exp > 1) {
             x.exp = this.exp / 2;
         }

         /* Coarsely estimate the mantissa */
         switch (x.mant[mant.length - 1] / 2000) {
         case 0:
             x.mant[mant.length - 1] = x.mant[mant.length - 1] / 2 + 1;
             break;
         case 2:
             x.mant[mant.length - 1] = 1500;
             break;
         case 3:
             x.mant[mant.length - 1] = 2200;
             break;
         default:
             x.mant[mant.length - 1] = 3000;
         }

         Dfp dx = newInstance(x);

         /* Now that we have the first pass estimate, compute the rest
        by the formula dx = (y - x*x) / (2x); */

         Dfp px  = getZero();
         Dfp ppx = getZero();
         while (x.unequal(px)) {
             dx = newInstance(x);
             dx.sign = -1;
             dx = dx.add(this.divide(x));
             dx = dx.divide(2);
             ppx = px;
             px = x;
             x = x.add(dx);

             if (x.equals(ppx)) {
                 // alternating between two values
                 break;
             }

             // if dx is zero, break.  Note testing the most sig digit
             // is a sufficient test since dx is normalized
             if (dx.mant[mant.length - 1] == 0) {
                 break;
             }
         }

         return x;
     }

     /** Get a string representation of the instance.
      * @return string representation of the instance
      */
     @Override
     public String toString() {
         if (nans != FINITE) {
             // if non-finite exceptional cases
             if (nans == INFINITE) {
                 return (sign < 0) ? NEG_INFINITY_STRING : POS_INFINITY_STRING;
             } else {
                 return NAN_STRING;
             }
         }

         if (exp > mant.length || exp < -1) {
             return dfp2sci();
         }

         return dfp2string();
     }

     /** Convert an instance to a string using scientific notation.
      * @return string representation of the instance in scientific notation
      */
     protected String dfp2sci() {
         char[] rawdigits    = new char[mant.length * 4];
         char[] outputbuffer = new char[mant.length * 4 + 20];
         int p;
         int q;
         int e;
         int ae;
         int shf;

         // Get all the digits
         p = 0;
         for (int i = mant.length - 1; i >= 0; i--) {
             rawdigits[p++] = (char) ((mant[i] / 1000) + '0');
             rawdigits[p++] = (char) (((mant[i] / 100) % 10) + '0');
             rawdigits[p++] = (char) (((mant[i] / 10) % 10) + '0');
             rawdigits[p++] = (char) (((mant[i]) % 10) + '0');
         }

         // Find the first non-zero one
         for (p = 0; p < rawdigits.length; p++) {
             if (rawdigits[p] != '0') {
                 break;
             }
         }
         shf = p;

         // Now do the conversion
         q = 0;
         if (sign == -1) {
             outputbuffer[q++] = '-';
         }

         if (p != rawdigits.length) {
             // there are non zero digits...
             outputbuffer[q++] = rawdigits[p++];
             outputbuffer[q++] = '.';

             while (p < rawdigits.length) {
                 outputbuffer[q++] = rawdigits[p++];
             }
         } else {
             outputbuffer[q++] = '0';
             outputbuffer[q++] = '.';
             outputbuffer[q++] = '0';
             outputbuffer[q++] = 'e';
             outputbuffer[q++] = '0';
             return String.valueOf(outputbuffer, 0, 5);
         }

         outputbuffer[q++] = 'e';

         // Find the msd of the exponent

         e = exp * 4 - shf - 1;
         ae = e;
         if (e < 0) {
             ae = -e;
         }

         // Find the largest p such that p < e
         p = 1000000000;
         while (p > ae) {
             p /= 10;
         }

         if (e < 0) {
             outputbuffer[q++] = '-';
         }

         while (p > 0) {
             outputbuffer[q++] = (char)(ae / p + '0');
             ae %= p;
             p /= 10;
         }

         return String.valueOf(outputbuffer, 0, q);
     }

     /** Convert an instance to a string using normal notation.
      * @return string representation of the instance in normal notation
      */
     protected String dfp2string() {
         char[] buffer = new char[mant.length * 4 + 20];
         int p = 1;
         int q;
         int e = exp;
         boolean pointInserted = false;

         buffer[0] = ' ';

         if (e <= 0) {
             buffer[p++] = '0';
             buffer[p++] = '.';
             pointInserted = true;
         }

         while (e < 0) {
             buffer[p++] = '0';
             buffer[p++] = '0';
             buffer[p++] = '0';
             buffer[p++] = '0';
             e++;
         }

         for (int i = mant.length - 1; i >= 0; i--) {
             buffer[p++] = (char) ((mant[i] / 1000) + '0');
             buffer[p++] = (char) (((mant[i] / 100) % 10) + '0');
             buffer[p++] = (char) (((mant[i] / 10) % 10) + '0');
             buffer[p++] = (char) (((mant[i]) % 10) + '0');
             if (--e == 0) {
                 buffer[p++] = '.';
                 pointInserted = true;
             }
         }

         while (e > 0) {
             buffer[p++] = '0';
             buffer[p++] = '0';
             buffer[p++] = '0';
             buffer[p++] = '0';
             e--;
         }

         if (!pointInserted) {
             // Ensure we have a radix point!
             buffer[p++] = '.';
         }

         // Suppress leading zeros
         q = 1;
         while (buffer[q] == '0') {
             q++;
         }
         if (buffer[q] == '.') {
             q--;
         }

         // Suppress trailing zeros
         while (buffer[p - 1] == '0') {
             p--;
         }

         // Insert sign
         if (sign < 0) {
             buffer[--q] = '-';
         }

         return String.valueOf(buffer, q, p - q);
     }

     /** Raises a trap.  This does not set the corresponding flag however.
      *  @param type the trap type
      *  @param what - name of routine trap occurred in
      *  @param oper - input operator to function
      *  @param result - the result computed prior to the trap
      *  @return The suggested return value from the trap handler
      */
     public Dfp dotrap(int type, String what, Dfp oper, Dfp result) {
         Dfp def = result;

         switch (type) {
         case DfpField.FLAG_INVALID:
             def = newInstance(getZero());
             def.sign = result.sign;
             def.nans = QNAN;
             break;

         case DfpField.FLAG_DIV_ZERO:
             if (nans == FINITE && mant[mant.length - 1] != 0) {
                 // normal case, we are finite, non-zero
                 def = newInstance(getZero());
                 def.sign = (byte) (sign * oper.sign);
                 def.nans = INFINITE;
             }

             if (nans == FINITE && mant[mant.length - 1] == 0) {
                 // 0/0
                 def = newInstance(getZero());
                 def.nans = QNAN;
             }

             if (nans == INFINITE || nans == QNAN) {
                 def = newInstance(getZero());
                 def.nans = QNAN;
             }

             if (nans == INFINITE || nans == SNAN) {
                 def = newInstance(getZero());
                 def.nans = QNAN;
             }
             break;

         case DfpField.FLAG_UNDERFLOW:
             if ((result.exp + mant.length) < MIN_EXP) {
                 def = newInstance(getZero());
                 def.sign = result.sign;
             } else {
                 def = newInstance(result);  // gradual underflow
             }
             result.exp += ERR_SCALE;
             break;

         case DfpField.FLAG_OVERFLOW:
             result.exp -= ERR_SCALE;
             def = newInstance(getZero());
             def.sign = result.sign;
             def.nans = INFINITE;
             break;

         default: def = result; break;
         }

         return trap(type, what, oper, def, result);
     }

     /** Trap handler.  Subclasses may override this to provide trap
      *  functionality per IEEE 854-1987.
      *
      *  @param type  The exception type - e.g. FLAG_OVERFLOW
      *  @param what  The name of the routine we were in e.g. divide()
      *  @param oper  An operand to this function if any
      *  @param def   The default return value if trap not enabled
      *  @param result    The result that is specified to be delivered per
      *                   IEEE 854, if any
      *  @return the value that should be return by the operation triggering the trap
      */
     protected Dfp trap(int type, String what, Dfp oper, Dfp def, Dfp result) {
         return def;
     }

     /** Returns the type - one of FINITE, INFINITE, SNAN, QNAN.
      * @return type of the number
      */
     public int classify() {
         return nans;
     }

     /** Creates an instance that is the same as x except that it has the sign of y.
      * abs(x) = dfp.copysign(x, dfp.one)
      * @param x number to get the value from
      * @param y number to get the sign from
      * @return a number with the value of x and the sign of y
      */
     public static Dfp copySign(final Dfp x, final Dfp y) {
         Dfp result = x.newInstance(x);
         result.sign = y.sign;
         return result;
     }

     /** Returns the next number greater than this one in the direction of x.
      * If this==x then simply returns this.
      * @param x direction where to look at
      * @return closest number next to instance in the direction of x
      */
     public Dfp nextAfter(final Dfp x) {

         // make sure we don't mix number with different precision
         if (field.getRadixDigits() != x.field.getRadixDigits()) {
             field.setIEEEFlagsBits(DfpField.FLAG_INVALID);
             final Dfp result = newInstance(getZero());
             result.nans = QNAN;
             return dotrap(DfpField.FLAG_INVALID, NEXT_AFTER_TRAP, x, result);
         }

         // if this is greater than x
         boolean up = this.lessThan(x);

         if (compare(this, x) == 0) {
             return newInstance(x);
         }

         if (lessThan(getZero())) {
             up = !up;
         }

         final Dfp inc;
         Dfp result;
         if (up) {
             inc = newInstance(getOne());
             inc.exp = this.exp - mant.length + 1;
             inc.sign = this.sign;

             if (this.equals(getZero())) {
                 inc.exp = MIN_EXP - mant.length;
             }

             result = add(inc);
         } else {
             inc = newInstance(getOne());
             inc.exp = this.exp;
             inc.sign = this.sign;

             if (this.equals(inc)) {
                 inc.exp = this.exp - mant.length;
             } else {
                 inc.exp = this.exp - mant.length + 1;
             }

             if (this.equals(getZero())) {
                 inc.exp = MIN_EXP - mant.length;
             }

             result = this.subtract(inc);
         }

         if (result.classify() == INFINITE && this.classify() != INFINITE) {
             field.setIEEEFlagsBits(DfpField.FLAG_INEXACT);
             result = dotrap(DfpField.FLAG_INEXACT, NEXT_AFTER_TRAP, x, result);
         }

         if (result.equals(getZero()) && !this.equals(getZero())) {
             field.setIEEEFlagsBits(DfpField.FLAG_INEXACT);
             result = dotrap(DfpField.FLAG_INEXACT, NEXT_AFTER_TRAP, x, result);
         }

         return result;
     }

     /** Convert the instance into a double.
      * @return a double approximating the instance
      * @see #toSplitDouble()
      */
     public double toDouble() {

         if (isInfinite()) {
             if (lessThan(getZero())) {
                 return Double.NEGATIVE_INFINITY;
             } else {
                 return Double.POSITIVE_INFINITY;
             }
         }

         if (isNaN()) {
             return Double.NaN;
         }

         Dfp y = this;
         boolean negate = false;
         int cmp0 = compare(this, getZero());
         if (cmp0 == 0) {
             return sign < 0 ? -0.0 : +0.0;
         } else if (cmp0 < 0) {
             y = negate();
             negate = true;
         }

         /* Find the exponent, first estimate by integer log10, then adjust.
          Should be faster than doing a natural logarithm.  */
         int exponent = (int)(y.intLog10() * 3.32);
         if (exponent < 0) {
             exponent--;
         }

         Dfp tempDfp = DfpMath.pow(getTwo(), exponent);
         while (tempDfp.lessThan(y) || tempDfp.equals(y)) {
             tempDfp = tempDfp.multiply(2);
             exponent++;
         }
         exponent--;

         /* We have the exponent, now work on the mantissa */

         y = y.divide(DfpMath.pow(getTwo(), exponent));
         if (exponent > -1023) {
             y = y.subtract(getOne());
         }

         if (exponent < -1074) {
             return 0;
         }

         if (exponent > 1023) {
             return negate ? Double.NEGATIVE_INFINITY : Double.POSITIVE_INFINITY;
         }


         y = y.multiply(newInstance(4503599627370496L)).rint();
         String str = y.toString();
         str = str.substring(0, str.length() - 1);
         long mantissa = Long.parseLong(str);

         if (mantissa == 4503599627370496L) {
             // Handle special case where we round up to next power of two
             mantissa = 0;
             exponent++;
         }

         /* Its going to be subnormal, so make adjustments */
         if (exponent <= -1023) {
             exponent--;
         }

         while (exponent < -1023) {
             exponent++;
             mantissa >>>= 1;
         }

         long bits = mantissa | ((exponent + 1023L) << 52);
         double x = Double.longBitsToDouble(bits);

         if (negate) {
             x = -x;
         }

         return x;
     }

     /** Convert the instance into a split double.
      * @return an array of two doubles which sum represent the instance
      * @see #toDouble()
      */
     public double[] toSplitDouble() {
         double[] split = new double[2];
         long mask = 0xffffffffc0000000L;

         split[0] = Double.longBitsToDouble(Double.doubleToLongBits(toDouble()) & mask);
         split[1] = subtract(newInstance(split[0])).toDouble();

         return split;
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public double getReal() {
         return toDouble();
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp add(final double a) {
         return add(newInstance(a));
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp subtract(final double a) {
         return subtract(newInstance(a));
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp multiply(final double a) {
         return multiply(newInstance(a));
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp divide(final double a) {
         return divide(newInstance(a));
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp remainder(final double a) {
         return remainder(newInstance(a));
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public long round() {
         return Math.round(toDouble());
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp signum() {
         if (isNaN() || isZero()) {
             return this;
         } else {
             return newInstance(sign > 0 ? +1 : -1);
         }
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp copySign(final Dfp s) {
         if ((sign >= 0 && s.sign >= 0) || (sign < 0 && s.sign < 0)) { // Sign is currently OK
             return this;
         }
         return negate(); // flip sign
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp copySign(final double s) {
         long sb = Double.doubleToLongBits(s);
         if ((sign >= 0 && sb >= 0) || (sign < 0 && sb < 0)) { // Sign is currently OK
             return this;
         }
         return negate(); // flip sign
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp scalb(final int n) {
         return multiply(DfpMath.pow(getTwo(), n));
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp hypot(final Dfp y) {
         return multiply(this).add(y.multiply(y)).sqrt();
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp cbrt() {
         return rootN(3);
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp rootN(final int n) {
         return (sign >= 0) ?
                DfpMath.pow(this, getOne().divide(n)) :
                DfpMath.pow(negate(), getOne().divide(n)).negate();
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp pow(final double p) {
         return DfpMath.pow(this, newInstance(p));
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp pow(final int n) {
         return DfpMath.pow(this, n);
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp pow(final Dfp e) {
         return DfpMath.pow(this, e);
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp exp() {
         return DfpMath.exp(this);
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp expm1() {
         return DfpMath.exp(this).subtract(getOne());
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp log() {
         return DfpMath.log(this);
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp log1p() {
         return DfpMath.log(this.add(getOne()));
     }

     /** {@inheritDoc}
      * @since 3.2, return type changed to {@link Dfp} in 4.0
      */
     @Override
     public Dfp log10() {
         return DfpMath.log(this).divide(DfpMath.log(newInstance(10)));
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp cos() {
         return DfpMath.cos(this);
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp sin() {
         return DfpMath.sin(this);
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp tan() {
         return DfpMath.tan(this);
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp acos() {
         return DfpMath.acos(this);
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp asin() {
         return DfpMath.asin(this);
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp atan() {
         return DfpMath.atan(this);
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp atan2(final Dfp x)
         throws DimensionMismatchException {

         // compute r = sqrt(x^2+y^2)
         final Dfp r = x.multiply(x).add(multiply(this)).sqrt();

         if (x.sign >= 0) {

             // compute atan2(y, x) = 2 atan(y / (r + x))
             return getTwo().multiply(divide(r.add(x)).atan());
         } else {

             // compute atan2(y, x) = +/- pi - 2 atan(y / (r - x))
             final Dfp tmp = getTwo().multiply(divide(r.subtract(x)).atan());
             final Dfp pmPi = newInstance((tmp.sign <= 0) ? -Math.PI : Math.PI);
             return pmPi.subtract(tmp);
         }
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp cosh() {
         return DfpMath.exp(this).add(DfpMath.exp(negate())).divide(2);
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp sinh() {
         return DfpMath.exp(this).subtract(DfpMath.exp(negate())).divide(2);
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp tanh() {
         final Dfp ePlus  = DfpMath.exp(this);
         final Dfp eMinus = DfpMath.exp(negate());
         return ePlus.subtract(eMinus).divide(ePlus.add(eMinus));
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp acosh() {
         return multiply(this).subtract(getOne()).sqrt().add(this).log();
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp asinh() {
         return multiply(this).add(getOne()).sqrt().add(this).log();
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp atanh() {
         return getOne().add(this).divide(getOne().subtract(this)).log().divide(2);
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp linearCombination(final Dfp[] a, final Dfp[] b)
         throws DimensionMismatchException {
         if (a.length != b.length) {
             throw new DimensionMismatchException(a.length, b.length);
         }
         Dfp r = getZero();
         for (int i = 0; i < a.length; ++i) {
             r = r.add(a[i].multiply(b[i]));
         }
         return r;
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp linearCombination(final double[] a, final Dfp[] b)
         throws DimensionMismatchException {
         if (a.length != b.length) {
             throw new DimensionMismatchException(a.length, b.length);
         }
         Dfp r = getZero();
         for (int i = 0; i < a.length; ++i) {
             r = r.add(b[i].multiply(a[i]));
         }
         return r;
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp linearCombination(final Dfp a1, final Dfp b1, final Dfp a2, final Dfp b2) {
         return a1.multiply(b1).add(a2.multiply(b2));
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp linearCombination(final double a1, final Dfp b1, final double a2, final Dfp b2) {
         return b1.multiply(a1).add(b2.multiply(a2));
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp linearCombination(final Dfp a1, final Dfp b1,
                                  final Dfp a2, final Dfp b2,
                                  final Dfp a3, final Dfp b3) {
         return a1.multiply(b1).add(a2.multiply(b2)).add(a3.multiply(b3));
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp linearCombination(final double a1, final Dfp b1,
                                  final double a2, final Dfp b2,
                                  final double a3, final Dfp b3) {
         return b1.multiply(a1).add(b2.multiply(a2)).add(b3.multiply(a3));
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp linearCombination(final Dfp a1, final Dfp b1, final Dfp a2, final Dfp b2,
                                  final Dfp a3, final Dfp b3, final Dfp a4, final Dfp b4) {
         return a1.multiply(b1).add(a2.multiply(b2)).add(a3.multiply(b3)).add(a4.multiply(b4));
     }

     /** {@inheritDoc}
      * @since 3.2
      */
     @Override
     public Dfp linearCombination(final double a1, final Dfp b1, final double a2, final Dfp b2,
                                  final double a3, final Dfp b3, final double a4, final Dfp b4) {
         return b1.multiply(a1).add(b2.multiply(a2)).add(b3.multiply(a3)).add(b4.multiply(a4));
     }
 }