    001    /*
    002     * Licensed to the Apache Software Foundation (ASF) under one or more
    003     * contributor license agreements.  See the NOTICE file distributed with
    004     * this work for additional information regarding copyright ownership.
    005     * The ASF licenses this file to You under the Apache License, Version 2.0
    006     * (the "License"); you may not use this file except in compliance with
    007     * the License.  You may obtain a copy of the License at
    008     *
    009     *      http://www.apache.org/licenses/LICENSE-2.0
    010     *
    011     * Unless required by applicable law or agreed to in writing, software
    012     * distributed under the License is distributed on an "AS IS" BASIS,
    013     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
    014     * See the License for the specific language governing permissions and
    015     * limitations under the License.
    016     */
    017    package org.apache.commons.lang3.math;
    018
    019    import java.math.BigInteger;
    020
    021    /**
    022     * <p><code>Fraction</code> is a <code>Number</code> implementation that
    023     * stores fractions accurately.</p>
    024     *
    025     * <p>This class is immutable, and interoperable with most methods that accept
    026     * a <code>Number</code>.</p>
    027     *
    028     * @author Apache Software Foundation
    029     * @author Travis Reeder
    030     * @author Tim O'Brien
    031     * @author Pete Gieser
    032     * @author C. Scott Ananian
    033     * @since 2.0
    034     * @version $Id: Fraction.java 889215 2009-12-10 11:56:38Z bayard $
    035     */
    036    public final class Fraction extends Number implements Comparable<Fraction> {
    037
    038        /**
    039         * Required for serialization support. Lang version 2.0.
    040         *
    041         * @see java.io.Serializable
    042         */
    043        private static final long serialVersionUID = 65382027393090L;
    044
    045        /**
    046         * <code>Fraction</code> representation of 0.
    047         */
    048        public static final Fraction ZERO = new Fraction(0, 1);
    049        /**
    050         * <code>Fraction</code> representation of 1.
    051         */
    052        public static final Fraction ONE = new Fraction(1, 1);
    053        /**
    054         * <code>Fraction</code> representation of 1/2.
    055         */
    056        public static final Fraction ONE_HALF = new Fraction(1, 2);
    057        /**
    058         * <code>Fraction</code> representation of 1/3.
    059         */
    060        public static final Fraction ONE_THIRD = new Fraction(1, 3);
    061        /**
    062         * <code>Fraction</code> representation of 2/3.
    063         */
    064        public static final Fraction TWO_THIRDS = new Fraction(2, 3);
    065        /**
    066         * <code>Fraction</code> representation of 1/4.
    067         */
    068        public static final Fraction ONE_QUARTER = new Fraction(1, 4);
    069        /**
    070         * <code>Fraction</code> representation of 2/4.
    071         */
    072        public static final Fraction TWO_QUARTERS = new Fraction(2, 4);
    073        /**
    074         * <code>Fraction</code> representation of 3/4.
    075         */
    076        public static final Fraction THREE_QUARTERS = new Fraction(3, 4);
    077        /**
    078         * <code>Fraction</code> representation of 1/5.
    079         */
    080        public static final Fraction ONE_FIFTH = new Fraction(1, 5);
    081        /**
    082         * <code>Fraction</code> representation of 2/5.
    083         */
    084        public static final Fraction TWO_FIFTHS = new Fraction(2, 5);
    085        /**
    086         * <code>Fraction</code> representation of 3/5.
    087         */
    088        public static final Fraction THREE_FIFTHS = new Fraction(3, 5);
    089        /**
    090         * <code>Fraction</code> representation of 4/5.
    091         */
    092        public static final Fraction FOUR_FIFTHS = new Fraction(4, 5);
    093
    094
    095        /**
    096         * The numerator number part of the fraction (the three in three sevenths).
    097         */
    098        private final int numerator;
    099        /**
    100         * The denominator number part of the fraction (the seven in three sevenths).
    101         */
    102        private final int denominator;
    103
    104        /**
    105         * Cached output hashCode (class is immutable).
    106         */
    107        private transient int hashCode = 0;
    108        /**
    109         * Cached output toString (class is immutable).
    110         */
    111        private transient String toString = null;
    112        /**
    113         * Cached output toProperString (class is immutable).
    114         */
    115        private transient String toProperString = null;
    116
    117        /**
    118         * <p>Constructs a <code>Fraction</code> instance with the 2 parts
    119         * of a fraction Y/Z.</p>
    120         *
    121         * @param numerator  the numerator, for example the three in 'three sevenths'
    122         * @param denominator  the denominator, for example the seven in 'three sevenths'
    123         */
    124        private Fraction(int numerator, int denominator) {
    125            super();
    126            this.numerator = numerator;
    127            this.denominator = denominator;
    128        }
    129
    130        /**
    131         * <p>Creates a <code>Fraction</code> instance with the 2 parts
    132         * of a fraction Y/Z.</p>
    133         *
    134         * <p>Any negative signs are resolved to be on the numerator.</p>
    135         *
    136         * @param numerator  the numerator, for example the three in 'three sevenths'
    137         * @param denominator  the denominator, for example the seven in 'three sevenths'
    138         * @return a new fraction instance
    139         * @throws ArithmeticException if the denomiator is <code>zero</code>
    140         */
    141        public static Fraction getFraction(int numerator, int denominator) {
    142            if (denominator == 0) {
    143                throw new ArithmeticException("The denominator must not be zero");
    144            }
    145            if (denominator < 0) {
    146                if (numerator==Integer.MIN_VALUE ||
    147                        denominator==Integer.MIN_VALUE) {
    148                    throw new ArithmeticException("overflow: can't negate");
    149                }
    150                numerator = -numerator;
    151                denominator = -denominator;
    152            }
    153            return new Fraction(numerator, denominator);
    154        }
    155
    156        /**
    157         * <p>Creates a <code>Fraction</code> instance with the 3 parts
    158         * of a fraction X Y/Z.</p>
    159         *
    160         * <p>The negative sign must be passed in on the whole number part.</p>
    161         *
    162         * @param whole  the whole number, for example the one in 'one and three sevenths'
    163         * @param numerator  the numerator, for example the three in 'one and three sevenths'
    164         * @param denominator  the denominator, for example the seven in 'one and three sevenths'
    165         * @return a new fraction instance
    166         * @throws ArithmeticException if the denomiator is <code>zero</code>
    167         * @throws ArithmeticException if the denominator is negative
    168         * @throws ArithmeticException if the numerator is negative
    169         * @throws ArithmeticException if the resulting numerator exceeds
    170         *  <code>Integer.MAX_VALUE</code>
    171         */
    172        public static Fraction getFraction(int whole, int numerator, int denominator) {
    173            if (denominator == 0) {
    174                throw new ArithmeticException("The denominator must not be zero");
    175            }
    176            if (denominator < 0) {
    177                throw new ArithmeticException("The denominator must not be negative");
    178            }
    179            if (numerator < 0) {
    180                throw new ArithmeticException("The numerator must not be negative");
    181            }
    182            long numeratorValue;
    183            if (whole < 0) {
    184                numeratorValue = whole * (long)denominator - numerator;
    185            } else {
    186                numeratorValue = whole * (long)denominator + numerator;
    187            }
    188            if (numeratorValue < Integer.MIN_VALUE ||
    189                    numeratorValue > Integer.MAX_VALUE)  {
    190                throw new ArithmeticException("Numerator too large to represent as an Integer.");
    191            }
    192            return new Fraction((int) numeratorValue, denominator);
    193        }
    194
    195        /**
    196         * <p>Creates a reduced <code>Fraction</code> instance with the 2 parts
    197         * of a fraction Y/Z.</p>
    198         *
    199         * <p>For example, if the input parameters represent 2/4, then the created
    200         * fraction will be 1/2.</p>
    201         *
    202         * <p>Any negative signs are resolved to be on the numerator.</p>
    203         *
    204         * @param numerator  the numerator, for example the three in 'three sevenths'
    205         * @param denominator  the denominator, for example the seven in 'three sevenths'
    206         * @return a new fraction instance, with the numerator and denominator reduced
    207         * @throws ArithmeticException if the denominator is <code>zero</code>
    208         */
    209        public static Fraction getReducedFraction(int numerator, int denominator) {
    210            if (denominator == 0) {
    211                throw new ArithmeticException("The denominator must not be zero");
    212            }
    213            if (numerator==0) {
    214                return ZERO; // normalize zero.
    215            }
    216            // allow 2^k/-2^31 as a valid fraction (where k>0)
    217            if (denominator==Integer.MIN_VALUE && (numerator&1)==0) {
    218                numerator/=2; denominator/=2;
    219            }
    220            if (denominator < 0) {
    221                if (numerator==Integer.MIN_VALUE ||
    222                        denominator==Integer.MIN_VALUE) {
    223                    throw new ArithmeticException("overflow: can't negate");
    224                }
    225                numerator = -numerator;
    226                denominator = -denominator;
    227            }
    228            // simplify fraction.
    229            int gcd = greatestCommonDivisor(numerator, denominator);
    230            numerator /= gcd;
    231            denominator /= gcd;
    232            return new Fraction(numerator, denominator);
    233        }
    234
    235        /**
    236         * <p>Creates a <code>Fraction</code> instance from a <code>double</code> value.</p>
    237         *
    238         * <p>This method uses the <a href="http://archives.math.utk.edu/articles/atuyl/confrac/">
    239         *  continued fraction algorithm</a>, computing a maximum of
    240         *  25 convergents and bounding the denominator by 10,000.</p>
    241         *
    242         * @param value  the double value to convert
    243         * @return a new fraction instance that is close to the value
    244         * @throws ArithmeticException if <code>|value| > Integer.MAX_VALUE</code>
    245         *  or <code>value = NaN</code>
    246         * @throws ArithmeticException if the calculated denominator is <code>zero</code>
    247         * @throws ArithmeticException if the the algorithm does not converge
    248         */
    249        public static Fraction getFraction(double value) {
    250            int sign = (value < 0 ? -1 : 1);
    251            value = Math.abs(value);
    252            if (value  > Integer.MAX_VALUE || Double.isNaN(value)) {
    253                throw new ArithmeticException
    254                    ("The value must not be greater than Integer.MAX_VALUE or NaN");
    255            }
    256            int wholeNumber = (int) value;
    257            value -= wholeNumber;
    258
    259            int numer0 = 0;  // the pre-previous
    260            int denom0 = 1;  // the pre-previous
    261            int numer1 = 1;  // the previous
    262            int denom1 = 0;  // the previous
    263            int numer2 = 0;  // the current, setup in calculation
    264            int denom2 = 0;  // the current, setup in calculation
    265            int a1 = (int) value;
    266            int a2 = 0;
    267            double x1 = 1;
    268            double x2 = 0;
    269            double y1 = value - a1;
    270            double y2 = 0;
    271            double delta1, delta2 = Double.MAX_VALUE;
    272            double fraction;
    273            int i = 1;
    274    //        System.out.println("---");
    275            do {
    276                delta1 = delta2;
    277                a2 = (int) (x1 / y1);
    278                x2 = y1;
    279                y2 = x1 - a2 * y1;
    280                numer2 = a1 * numer1 + numer0;
    281                denom2 = a1 * denom1 + denom0;
    282                fraction = (double) numer2 / (double) denom2;
    283                delta2 = Math.abs(value - fraction);
    284    //            System.out.println(numer2 + " " + denom2 + " " + fraction + " " + delta2 + " " + y1);
    285                a1 = a2;
    286                x1 = x2;
    287                y1 = y2;
    288                numer0 = numer1;
    289                denom0 = denom1;
    290                numer1 = numer2;
    291                denom1 = denom2;
    292                i++;
    293    //            System.out.println(">>" + delta1 +" "+ delta2+" "+(delta1 > delta2)+" "+i+" "+denom2);
    294            } while ((delta1 > delta2) && (denom2 <= 10000) && (denom2 > 0) && (i < 25));
    295            if (i == 25) {
    296                throw new ArithmeticException("Unable to convert double to fraction");
    297            }
    298            return getReducedFraction((numer0 + wholeNumber * denom0) * sign, denom0);
    299        }
    300
    301        /**
    302         * <p>Creates a Fraction from a <code>String</code>.</p>
    303         *
    304         * <p>The formats accepted are:</p>
    305         *
    306         * <ol>
    307         *  <li><code>double</code> String containing a dot</li>
    308         *  <li>'X Y/Z'</li>
    309         *  <li>'Y/Z'</li>
    310         *  <li>'X' (a simple whole number)</li>
    311         * </ol>
    312         * and a .</p>
    313         *
    314         * @param str  the string to parse, must not be <code>null</code>
    315         * @return the new <code>Fraction</code> instance
    316         * @throws IllegalArgumentException if the string is <code>null</code>
    317         * @throws NumberFormatException if the number format is invalid
    318         */
    319        public static Fraction getFraction(String str) {
    320            if (str == null) {
    321                throw new IllegalArgumentException("The string must not be null");
    322            }
    323            // parse double format
    324            int pos = str.indexOf('.');
    325            if (pos >= 0) {
    326                return getFraction(Double.parseDouble(str));
    327            }
    328
    329            // parse X Y/Z format
    330            pos = str.indexOf(' ');
    331            if (pos > 0) {
    332                int whole = Integer.parseInt(str.substring(0, pos));
    333                str = str.substring(pos + 1);
    334                pos = str.indexOf('/');
    335                if (pos < 0) {
    336                    throw new NumberFormatException("The fraction could not be parsed as the format X Y/Z");
    337                } else {
    338                    int numer = Integer.parseInt(str.substring(0, pos));
    339                    int denom = Integer.parseInt(str.substring(pos + 1));
    340                    return getFraction(whole, numer, denom);
    341                }
    342            }
    343
    344            // parse Y/Z format
    345            pos = str.indexOf('/');
    346            if (pos < 0) {
    347                // simple whole number
    348                return getFraction(Integer.parseInt(str), 1);
    349            } else {
    350                int numer = Integer.parseInt(str.substring(0, pos));
    351                int denom = Integer.parseInt(str.substring(pos + 1));
    352                return getFraction(numer, denom);
    353            }
    354        }
    355
    356        // Accessors
    357        //-------------------------------------------------------------------
    358
    359        /**
    360         * <p>Gets the numerator part of the fraction.</p>
    361         *
    362         * <p>This method may return a value greater than the denominator, an
    363         * improper fraction, such as the seven in 7/4.</p>
    364         *
    365         * @return the numerator fraction part
    366         */
    367        public int getNumerator() {
    368            return numerator;
    369        }
    370
    371        /**
    372         * <p>Gets the denominator part of the fraction.</p>
    373         *
    374         * @return the denominator fraction part
    375         */
    376        public int getDenominator() {
    377            return denominator;
    378        }
    379
    380        /**
    381         * <p>Gets the proper numerator, always positive.</p>
    382         *
    383         * <p>An improper fraction 7/4 can be resolved into a proper one, 1 3/4.
    384         * This method returns the 3 from the proper fraction.</p>
    385         *
    386         * <p>If the fraction is negative such as -7/4, it can be resolved into
    387         * -1 3/4, so this method returns the positive proper numerator, 3.</p>
    388         *
    389         * @return the numerator fraction part of a proper fraction, always positive
    390         */
    391        public int getProperNumerator() {
    392            return Math.abs(numerator % denominator);
    393        }
    394
    395        /**
    396         * <p>Gets the proper whole part of the fraction.</p>
    397         *
    398         * <p>An improper fraction 7/4 can be resolved into a proper one, 1 3/4.
    399         * This method returns the 1 from the proper fraction.</p>
    400         *
    401         * <p>If the fraction is negative such as -7/4, it can be resolved into
    402         * -1 3/4, so this method returns the positive whole part -1.</p>
    403         *
    404         * @return the whole fraction part of a proper fraction, that includes the sign
    405         */
    406        public int getProperWhole() {
    407            return numerator / denominator;
    408        }
    409
    410        // Number methods
    411        //-------------------------------------------------------------------
    412
    413        /**
    414         * <p>Gets the fraction as an <code>int</code>. This returns the whole number
    415         * part of the fraction.</p>
    416         *
    417         * @return the whole number fraction part
    418         */
    419        @Override
    420        public int intValue() {
    421            return numerator / denominator;
    422        }
    423
    424        /**
    425         * <p>Gets the fraction as a <code>long</code>. This returns the whole number
    426         * part of the fraction.</p>
    427         *
    428         * @return the whole number fraction part
    429         */
    430        @Override
    431        public long longValue() {
    432            return (long) numerator / denominator;
    433        }
    434
    435        /**
    436         * <p>Gets the fraction as a <code>float</code>. This calculates the fraction
    437         * as the numerator divided by denominator.</p>
    438         *
    439         * @return the fraction as a <code>float</code>
    440         */
    441        @Override
    442        public float floatValue() {
    443            return ((float) numerator) / ((float) denominator);
    444        }
    445
    446        /**
    447         * <p>Gets the fraction as a <code>double</code>. This calculates the fraction
    448         * as the numerator divided by denominator.</p>
    449         *
    450         * @return the fraction as a <code>double</code>
    451         */
    452        @Override
    453        public double doubleValue() {
    454            return ((double) numerator) / ((double) denominator);
    455        }
    456
    457        // Calculations
    458        //-------------------------------------------------------------------
    459
    460        /**
    461         * <p>Reduce the fraction to the smallest values for the numerator and
    462         * denominator, returning the result.</p>
    463         *
    464         * <p>For example, if this fraction represents 2/4, then the result
    465         * will be 1/2.</p>
    466         *
    467         * @return a new reduced fraction instance, or this if no simplification possible
    468         */
    469        public Fraction reduce() {
    470            if (numerator == 0) {
    471                return equals(ZERO) ? this : ZERO;
    472            }
    473            int gcd = greatestCommonDivisor(Math.abs(numerator), denominator);
    474            if (gcd == 1) {
    475                return this;
    476            }
    477            return Fraction.getFraction(numerator / gcd, denominator / gcd);
    478        }
    479
    480        /**
    481         * <p>Gets a fraction that is the inverse (1/fraction) of this one.</p>
    482         *
    483         * <p>The returned fraction is not reduced.</p>
    484         *
    485         * @return a new fraction instance with the numerator and denominator
    486         *         inverted.
    487         * @throws ArithmeticException if the fraction represents zero.
    488         */
    489        public Fraction invert() {
    490            if (numerator == 0) {
    491                throw new ArithmeticException("Unable to invert zero.");
    492            }
    493            if (numerator==Integer.MIN_VALUE) {
    494                throw new ArithmeticException("overflow: can't negate numerator");
    495            }
    496            if (numerator<0) {
    497                return new Fraction(-denominator, -numerator);
    498            } else {
    499                return new Fraction(denominator, numerator);
    500            }
    501        }
    502
    503        /**
    504         * <p>Gets a fraction that is the negative (-fraction) of this one.</p>
    505         *
    506         * <p>The returned fraction is not reduced.</p>
    507         *
    508         * @return a new fraction instance with the opposite signed numerator
    509         */
    510        public Fraction negate() {
    511            // the positive range is one smaller than the negative range of an int.
    512            if (numerator==Integer.MIN_VALUE) {
    513                throw new ArithmeticException("overflow: too large to negate");
    514            }
    515            return new Fraction(-numerator, denominator);
    516        }
    517
    518        /**
    519         * <p>Gets a fraction that is the positive equivalent of this one.</p>
    520         * <p>More precisely: <code>(fraction >= 0 ? this : -fraction)</code></p>
    521         *
    522         * <p>The returned fraction is not reduced.</p>
    523         *
    524         * @return <code>this</code> if it is positive, or a new positive fraction
    525         *  instance with the opposite signed numerator
    526         */
    527        public Fraction abs() {
    528            if (numerator >= 0) {
    529                return this;
    530            }
    531            return negate();
    532        }
    533
    534        /**
    535         * <p>Gets a fraction that is raised to the passed in power.</p>
    536         *
    537         * <p>The returned fraction is in reduced form.</p>
    538         *
    539         * @param power  the power to raise the fraction to
    540         * @return <code>this</code> if the power is one, <code>ONE</code> if the power
    541         * is zero (even if the fraction equals ZERO) or a new fraction instance
    542         * raised to the appropriate power
    543         * @throws ArithmeticException if the resulting numerator or denominator exceeds
    544         *  <code>Integer.MAX_VALUE</code>
    545         */
    546        public Fraction pow(int power) {
    547            if (power == 1) {
    548                return this;
    549            } else if (power == 0) {
    550                return ONE;
    551            } else if (power < 0) {
    552                if (power==Integer.MIN_VALUE) { // MIN_VALUE can't be negated.
    553                    return this.invert().pow(2).pow(-(power/2));
    554                }
    555                return this.invert().pow(-power);
    556            } else {
    557                Fraction f = this.multiplyBy(this);
    558                if ((power % 2) == 0) { // if even...
    559                    return f.pow(power/2);
    560                } else { // if odd...
    561                    return f.pow(power/2).multiplyBy(this);
    562                }
    563            }
    564        }
    565
    566        /**
    567         * <p>Gets the greatest common divisor of the absolute value of
    568         * two numbers, using the "binary gcd" method which avoids
    569         * division and modulo operations.  See Knuth 4.5.2 algorithm B.
    570         * This algorithm is due to Josef Stein (1961).</p>
    571         *
    572         * @param u  a non-zero number
    573         * @param v  a non-zero number
    574         * @return the greatest common divisor, never zero
    575         */
    576        private static int greatestCommonDivisor(int u, int v) {
    577            //if either op. is abs 0 or 1, return 1:
    578            if (Math.abs(u) <= 1 || Math.abs(v) <= 1) {
    579                return 1;
    580            }
    581            // keep u and v negative, as negative integers range down to
    582            // -2^31, while positive numbers can only be as large as 2^31-1
    583            // (i.e. we can't necessarily negate a negative number without
    584            // overflow)
    585            if (u>0) { u=-u; } // make u negative
    586            if (v>0) { v=-v; } // make v negative
    587            // B1. [Find power of 2]
    588            int k=0;
    589            while ((u&1)==0 && (v&1)==0 && k<31) { // while u and v are both even...
    590                u/=2; v/=2; k++; // cast out twos.
    591            }
    592            if (k==31) {
    593                throw new ArithmeticException("overflow: gcd is 2^31");
    594            }
    595            // B2. Initialize: u and v have been divided by 2^k and at least
    596            //     one is odd.
    597            int t = ((u&1)==1) ? v : -(u/2)/*B3*/;
    598            // t negative: u was odd, v may be even (t replaces v)
    599            // t positive: u was even, v is odd (t replaces u)
    600            do {
    601                /* assert u<0 && v<0; */
    602                // B4/B3: cast out twos from t.
    603                while ((t&1)==0) { // while t is even..
    604                    t/=2; // cast out twos
    605                }
    606                // B5 [reset max(u,v)]
    607                if (t>0) {
    608                    u = -t;
    609                } else {
    610                    v = t;
    611                }
    612                // B6/B3. at this point both u and v should be odd.
    613                t = (v - u)/2;
    614                // |u| larger: t positive (replace u)
    615                // |v| larger: t negative (replace v)
    616            } while (t!=0);
    617            return -u*(1<<k); // gcd is u*2^k
    618        }
    619
    620        // Arithmetic
    621        //-------------------------------------------------------------------
    622
    623        /**
    624         * Multiply two integers, checking for overflow.
    625         *
    626         * @param x a factor
    627         * @param y a factor
    628         * @return the product <code>x*y</code>
    629         * @throws ArithmeticException if the result can not be represented as
    630         *                             an int
    631         */
    632        private static int mulAndCheck(int x, int y) {
    633            long m = ((long)x)*((long)y);
    634            if (m < Integer.MIN_VALUE ||
    635                m > Integer.MAX_VALUE) {
    636                throw new ArithmeticException("overflow: mul");
    637            }
    638            return (int)m;
    639        }
    640
    641        /**
    642         *  Multiply two non-negative integers, checking for overflow.
    643         *
    644         * @param x a non-negative factor
    645         * @param y a non-negative factor
    646         * @return the product <code>x*y</code>
    647         * @throws ArithmeticException if the result can not be represented as
    648         * an int
    649         */
    650        private static int mulPosAndCheck(int x, int y) {
    651            /* assert x>=0 && y>=0; */
    652            long m = ((long)x)*((long)y);
    653            if (m > Integer.MAX_VALUE) {
    654                throw new ArithmeticException("overflow: mulPos");
    655            }
    656            return (int)m;
    657        }
    658
    659        /**
    660         * Add two integers, checking for overflow.
    661         *
    662         * @param x an addend
    663         * @param y an addend
    664         * @return the sum <code>x+y</code>
    665         * @throws ArithmeticException if the result can not be represented as
    666         * an int
    667         */
    668        private static int addAndCheck(int x, int y) {
    669            long s = (long)x+(long)y;
    670            if (s < Integer.MIN_VALUE ||
    671                s > Integer.MAX_VALUE) {
    672                throw new ArithmeticException("overflow: add");
    673            }
    674            return (int)s;
    675        }
    676
    677        /**
    678         * Subtract two integers, checking for overflow.
    679         *
    680         * @param x the minuend
    681         * @param y the subtrahend
    682         * @return the difference <code>x-y</code>
    683         * @throws ArithmeticException if the result can not be represented as
    684         * an int
    685         */
    686        private static int subAndCheck(int x, int y) {
    687            long s = (long)x-(long)y;
    688            if (s < Integer.MIN_VALUE ||
    689                s > Integer.MAX_VALUE) {
    690                throw new ArithmeticException("overflow: add");
    691            }
    692            return (int)s;
    693        }
    694
    695        /**
    696         * <p>Adds the value of this fraction to another, returning the result in reduced form.
    697         * The algorithm follows Knuth, 4.5.1.</p>
    698         *
    699         * @param fraction  the fraction to add, must not be <code>null</code>
    700         * @return a <code>Fraction</code> instance with the resulting values
    701         * @throws IllegalArgumentException if the fraction is <code>null</code>
    702         * @throws ArithmeticException if the resulting numerator or denominator exceeds
    703         *  <code>Integer.MAX_VALUE</code>
    704         */
    705        public Fraction add(Fraction fraction) {
    706            return addSub(fraction, true /* add */);
    707        }
    708
    709        /**
    710         * <p>Subtracts the value of another fraction from the value of this one,
    711         * returning the result in reduced form.</p>
    712         *
    713         * @param fraction  the fraction to subtract, must not be <code>null</code>
    714         * @return a <code>Fraction</code> instance with the resulting values
    715         * @throws IllegalArgumentException if the fraction is <code>null</code>
    716         * @throws ArithmeticException if the resulting numerator or denominator
    717         *   cannot be represented in an <code>int</code>.
    718         */
    719        public Fraction subtract(Fraction fraction) {
    720            return addSub(fraction, false /* subtract */);
    721        }
    722
    723        /**
    724         * Implement add and subtract using algorithm described in Knuth 4.5.1.
    725         *
    726         * @param fraction the fraction to subtract, must not be <code>null</code>
    727         * @param isAdd true to add, false to subtract
    728         * @return a <code>Fraction</code> instance with the resulting values
    729         * @throws IllegalArgumentException if the fraction is <code>null</code>
    730         * @throws ArithmeticException if the resulting numerator or denominator
    731         *   cannot be represented in an <code>int</code>.
    732         */
    733        private Fraction addSub(Fraction fraction, boolean isAdd) {
    734            if (fraction == null) {
    735                throw new IllegalArgumentException("The fraction must not be null");
    736            }
    737            // zero is identity for addition.
    738            if (numerator == 0) {
    739                return isAdd ? fraction : fraction.negate();
    740            }
    741            if (fraction.numerator == 0) {
    742                return this;
    743            }
    744            // if denominators are randomly distributed, d1 will be 1 about 61%
    745            // of the time.
    746            int d1 = greatestCommonDivisor(denominator, fraction.denominator);
    747            if (d1==1) {
    748                // result is ( (u*v' +/- u'v) / u'v')
    749                int uvp = mulAndCheck(numerator, fraction.denominator);
    750                int upv = mulAndCheck(fraction.numerator, denominator);
    751                return new Fraction
    752                    (isAdd ? addAndCheck(uvp, upv) : subAndCheck(uvp, upv),
    753                     mulPosAndCheck(denominator, fraction.denominator));
    754            }
    755            // the quantity 't' requires 65 bits of precision; see knuth 4.5.1
    756            // exercise 7.  we're going to use a BigInteger.
    757            // t = u(v'/d1) +/- v(u'/d1)
    758            BigInteger uvp = BigInteger.valueOf(numerator)
    759                .multiply(BigInteger.valueOf(fraction.denominator/d1));
    760            BigInteger upv = BigInteger.valueOf(fraction.numerator)
    761                .multiply(BigInteger.valueOf(denominator/d1));
    762            BigInteger t = isAdd ? uvp.add(upv) : uvp.subtract(upv);
    763            // but d2 doesn't need extra precision because
    764            // d2 = gcd(t,d1) = gcd(t mod d1, d1)
    765            int tmodd1 = t.mod(BigInteger.valueOf(d1)).intValue();
    766            int d2 = (tmodd1==0)?d1:greatestCommonDivisor(tmodd1, d1);
    767
    768            // result is (t/d2) / (u'/d1)(v'/d2)
    769            BigInteger w = t.divide(BigInteger.valueOf(d2));
    770            if (w.bitLength() > 31) {
    771                throw new ArithmeticException
    772                    ("overflow: numerator too large after multiply");
    773            }
    774            return new Fraction
    775                (w.intValue(),
    776                 mulPosAndCheck(denominator/d1, fraction.denominator/d2));
    777        }
    778
    779        /**
    780         * <p>Multiplies the value of this fraction by another, returning the
    781         * result in reduced form.</p>
    782         *
    783         * @param fraction  the fraction to multiply by, must not be <code>null</code>
    784         * @return a <code>Fraction</code> instance with the resulting values
    785         * @throws IllegalArgumentException if the fraction is <code>null</code>
    786         * @throws ArithmeticException if the resulting numerator or denominator exceeds
    787         *  <code>Integer.MAX_VALUE</code>
    788         */
    789        public Fraction multiplyBy(Fraction fraction) {
    790            if (fraction == null) {
    791                throw new IllegalArgumentException("The fraction must not be null");
    792            }
    793            if (numerator == 0 || fraction.numerator == 0) {
    794                return ZERO;
    795            }
    796            // knuth 4.5.1
    797            // make sure we don't overflow unless the result *must* overflow.
    798            int d1 = greatestCommonDivisor(numerator, fraction.denominator);
    799            int d2 = greatestCommonDivisor(fraction.numerator, denominator);
    800            return getReducedFraction
    801                (mulAndCheck(numerator/d1, fraction.numerator/d2),
    802                 mulPosAndCheck(denominator/d2, fraction.denominator/d1));
    803        }
    804
    805        /**
    806         * <p>Divide the value of this fraction by another.</p>
    807         *
    808         * @param fraction  the fraction to divide by, must not be <code>null</code>
    809         * @return a <code>Fraction</code> instance with the resulting values
    810         * @throws IllegalArgumentException if the fraction is <code>null</code>
    811         * @throws ArithmeticException if the fraction to divide by is zero
    812         * @throws ArithmeticException if the resulting numerator or denominator exceeds
    813         *  <code>Integer.MAX_VALUE</code>
    814         */
    815        public Fraction divideBy(Fraction fraction) {
    816            if (fraction == null) {
    817                throw new IllegalArgumentException("The fraction must not be null");
    818            }
    819            if (fraction.numerator == 0) {
    820                throw new ArithmeticException("The fraction to divide by must not be zero");
    821            }
    822            return multiplyBy(fraction.invert());
    823        }
    824
    825        // Basics
    826        //-------------------------------------------------------------------
    827
    828        /**
    829         * <p>Compares this fraction to another object to test if they are equal.</p>.
    830         *
    831         * <p>To be equal, both values must be equal. Thus 2/4 is not equal to 1/2.</p>
    832         *
    833         * @param obj the reference object with which to compare
    834         * @return <code>true</code> if this object is equal
    835         */
    836        @Override
    837        public boolean equals(Object obj) {
    838            if (obj == this) {
    839                return true;
    840            }
    841            if (obj instanceof Fraction == false) {
    842                return false;
    843            }
    844            Fraction other = (Fraction) obj;
    845            return (getNumerator() == other.getNumerator() &&
    846                    getDenominator() == other.getDenominator());
    847        }
    848
    849        /**
    850         * <p>Gets a hashCode for the fraction.</p>
    851         *
    852         * @return a hash code value for this object
    853         */
    854        @Override
    855        public int hashCode() {
    856            if (hashCode == 0) {
    857                // hashcode update should be atomic.
    858                hashCode = 37 * (37 * 17 + getNumerator()) + getDenominator();
    859            }
    860            return hashCode;
    861        }
    862
    863        /**
    864         * <p>Compares this object to another based on size.</p>
    865         *
    866         * <p>Note: this class has a natural ordering that is inconsistent
    867         * with equals, because, for example, equals treats 1/2 and 2/4 as
    868         * different, whereas compareTo treats them as equal.
    869         *
    870         * @param other  the object to compare to
    871         * @return -1 if this is less, 0 if equal, +1 if greater
    872         * @throws ClassCastException if the object is not a <code>Fraction</code>
    873         * @throws NullPointerException if the object is <code>null</code>
    874         */
    875        public int compareTo(Fraction other) {
    876            if (this==other) {
    877                return 0;
    878            }
    879            if (numerator == other.numerator && denominator == other.denominator) {
    880                return 0;
    881            }
    882
    883            // otherwise see which is less
    884            long first = (long) numerator * (long) other.denominator;
    885            long second = (long) other.numerator * (long) denominator;
    886            if (first == second) {
    887                return 0;
    888            } else if (first < second) {
    889                return -1;
    890            } else {
    891                return 1;
    892            }
    893        }
    894
    895        /**
    896         * <p>Gets the fraction as a <code>String</code>.</p>
    897         *
    898         * <p>The format used is '<i>numerator</i>/<i>denominator</i>' always.
    899         *
    900         * @return a <code>String</code> form of the fraction
    901         */
    902        @Override
    903        public String toString() {
    904            if (toString == null) {
    905                toString = new StringBuilder(32)
    906                    .append(getNumerator())
    907                    .append('/')
    908                    .append(getDenominator()).toString();
    909            }
    910            return toString;
    911        }
    912
    913        /**
    914         * <p>Gets the fraction as a proper <code>String</code> in the format X Y/Z.</p>
    915         *
    916         * <p>The format used in '<i>wholeNumber</i> <i>numerator</i>/<i>denominator</i>'.
    917         * If the whole number is zero it will be ommitted. If the numerator is zero,
    918         * only the whole number is returned.</p>
    919         *
    920         * @return a <code>String</code> form of the fraction
    921         */
    922        public String toProperString() {
    923            if (toProperString == null) {
    924                if (numerator == 0) {
    925                    toProperString = "0";
    926                } else if (numerator == denominator) {
    927                    toProperString = "1";
    928                } else if (numerator == -1 * denominator) {
    929                    toProperString = "-1";
    930                } else if ((numerator>0?-numerator:numerator) < -denominator) {
    931                    // note that we do the magnitude comparison test above with
    932                    // NEGATIVE (not positive) numbers, since negative numbers
    933                    // have a larger range.  otherwise numerator==Integer.MIN_VALUE
    934                    // is handled incorrectly.
    935                    int properNumerator = getProperNumerator();
    936                    if (properNumerator == 0) {
    937                        toProperString = Integer.toString(getProperWhole());
    938                    } else {
    939                        toProperString = new StringBuilder(32)
    940                            .append(getProperWhole()).append(' ')
    941                            .append(properNumerator).append('/')
    942                            .append(getDenominator()).toString();
    943                    }
    944                } else {
    945                    toProperString = new StringBuilder(32)
    946                        .append(getNumerator()).append('/')
    947                        .append(getDenominator()).toString();
    948                }
    949            }
    950            return toProperString;
    951        }
    952    }