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