Coverage Report

Coverage Report - org.apache.commons.lang3.math.Fraction

Classes in this File
Line Coverage
Branch Coverage
Complexity
Fraction
97%
259/265
91%
180/196
5.353
 /*
  * 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.lang3.math;
 
 import java.math.BigInteger;
 
 /**
  * <p><code>Fraction</code> is a <code>Number</code> implementation that
  * stores fractions accurately.</p>
  *
  * <p>This class is immutable, and interoperable with most methods that accept
  * a <code>Number</code>.</p>
  *
  * <p>Note that this class is intended for common use cases, it is <i>int</i>
  * based and thus suffers from various overflow issues. For a BigInteger based 
  * equivalent, please see the Commons Math BigFraction class. </p>
  *
  * @since 2.0
  * @version $Id: Fraction.java 1436770 2013-01-22 07:09:45Z ggregory $
  */
 public final class Fraction extends Number implements Comparable<Fraction> {
 
     /**
      * Required for serialization support. Lang version 2.0.
      * 
      * @see java.io.Serializable
      */
     private static final long serialVersionUID = 65382027393090L;
 
     /**
      * <code>Fraction</code> representation of 0.
      */
     public static final Fraction ZERO = new Fraction(0, 1);
     /**
      * <code>Fraction</code> representation of 1.
      */
     public static final Fraction ONE = new Fraction(1, 1);
     /**
      * <code>Fraction</code> representation of 1/2.
      */
     public static final Fraction ONE_HALF = new Fraction(1, 2);
     /**
      * <code>Fraction</code> representation of 1/3.
      */
     public static final Fraction ONE_THIRD = new Fraction(1, 3);
     /**
      * <code>Fraction</code> representation of 2/3.
      */
     public static final Fraction TWO_THIRDS = new Fraction(2, 3);
     /**
      * <code>Fraction</code> representation of 1/4.
      */
     public static final Fraction ONE_QUARTER = new Fraction(1, 4);
     /**
      * <code>Fraction</code> representation of 2/4.
      */
     public static final Fraction TWO_QUARTERS = new Fraction(2, 4);
     /**
      * <code>Fraction</code> representation of 3/4.
      */
     public static final Fraction THREE_QUARTERS = new Fraction(3, 4);
     /**
      * <code>Fraction</code> representation of 1/5.
      */
     public static final Fraction ONE_FIFTH = new Fraction(1, 5);
     /**
      * <code>Fraction</code> representation of 2/5.
      */
     public static final Fraction TWO_FIFTHS = new Fraction(2, 5);
     /**
      * <code>Fraction</code> representation of 3/5.
      */
     public static final Fraction THREE_FIFTHS = new Fraction(3, 5);
     /**
      * <code>Fraction</code> representation of 4/5.
      */
     public static final Fraction FOUR_FIFTHS = new Fraction(4, 5);
 
 
     /**
      * The numerator number part of the fraction (the three in three sevenths).
      */
     private final int numerator;
     /**
      * The denominator number part of the fraction (the seven in three sevenths).
      */
     private final int denominator;
 
     /**
      * Cached output hashCode (class is immutable).
      */
     private transient int hashCode = 0;
     /**
      * Cached output toString (class is immutable).
      */
     private transient String toString = null;
     /**
      * Cached output toProperString (class is immutable).
      */
     private transient String toProperString = null;
 
     /**
      * <p>Constructs a <code>Fraction</code> instance with the 2 parts
      * of a fraction Y/Z.</p>
      *
      * @param numerator  the numerator, for example the three in 'three sevenths'
      * @param denominator  the denominator, for example the seven in 'three sevenths'
      */
     private Fraction(final int numerator, final int denominator) {
         super();
         this.numerator = numerator;
         this.denominator = denominator;
     }
 
     /**
      * <p>Creates a <code>Fraction</code> instance with the 2 parts
      * of a fraction Y/Z.</p>
      *
      * <p>Any negative signs are resolved to be on the numerator.</p>
      *
      * @param numerator  the numerator, for example the three in 'three sevenths'
      * @param denominator  the denominator, for example the seven in 'three sevenths'
      * @return a new fraction instance
      * @throws ArithmeticException if the denominator is <code>zero</code>
      * or the denominator is {@code negative} and the numerator is {@code Integer#MIN_VALUE}
      */
     public static Fraction getFraction(int numerator, int denominator) {
         if (denominator == 0) {
             throw new ArithmeticException("The denominator must not be zero");
         }
         if (denominator < 0) {
             if (numerator==Integer.MIN_VALUE ||
                     denominator==Integer.MIN_VALUE) {
                 throw new ArithmeticException("overflow: can't negate");
             }
             numerator = -numerator;
             denominator = -denominator;
         }
         return new Fraction(numerator, denominator);
     }
 
     /**
      * <p>Creates a <code>Fraction</code> instance with the 3 parts
      * of a fraction X Y/Z.</p>
      *
      * <p>The negative sign must be passed in on the whole number part.</p>
      *
      * @param whole  the whole number, for example the one in 'one and three sevenths'
      * @param numerator  the numerator, for example the three in 'one and three sevenths'
      * @param denominator  the denominator, for example the seven in 'one and three sevenths'
      * @return a new fraction instance
      * @throws ArithmeticException if the denominator is <code>zero</code>
      * @throws ArithmeticException if the denominator is negative
      * @throws ArithmeticException if the numerator is negative
      * @throws ArithmeticException if the resulting numerator exceeds 
      *  <code>Integer.MAX_VALUE</code>
      */
     public static Fraction getFraction(final int whole, final int numerator, final int denominator) {
         if (denominator == 0) {
             throw new ArithmeticException("The denominator must not be zero");
         }
         if (denominator < 0) {
             throw new ArithmeticException("The denominator must not be negative");
         }
         if (numerator < 0) {
             throw new ArithmeticException("The numerator must not be negative");
         }
         long numeratorValue;
         if (whole < 0) {
             numeratorValue = whole * (long)denominator - numerator;
         } else {
             numeratorValue = whole * (long)denominator + numerator;
         }
         if (numeratorValue < Integer.MIN_VALUE ||
                 numeratorValue > Integer.MAX_VALUE)  {
             throw new ArithmeticException("Numerator too large to represent as an Integer.");
         }
         return new Fraction((int) numeratorValue, denominator);
     }
 
     /**
      * <p>Creates a reduced <code>Fraction</code> instance with the 2 parts
      * of a fraction Y/Z.</p>
      *
      * <p>For example, if the input parameters represent 2/4, then the created
      * fraction will be 1/2.</p>
      *
      * <p>Any negative signs are resolved to be on the numerator.</p>
      *
      * @param numerator  the numerator, for example the three in 'three sevenths'
      * @param denominator  the denominator, for example the seven in 'three sevenths'
      * @return a new fraction instance, with the numerator and denominator reduced
      * @throws ArithmeticException if the denominator is <code>zero</code>
      */
     public static Fraction getReducedFraction(int numerator, int denominator) {
         if (denominator == 0) {
             throw new ArithmeticException("The denominator must not be zero");
         }
         if (numerator==0) {
             return ZERO; // normalize zero.
         }
         // allow 2^k/-2^31 as a valid fraction (where k>0)
         if (denominator==Integer.MIN_VALUE && (numerator&1)==0) {
             numerator/=2; denominator/=2;
         }
         if (denominator < 0) {
             if (numerator==Integer.MIN_VALUE ||
                     denominator==Integer.MIN_VALUE) {
                 throw new ArithmeticException("overflow: can't negate");
             }
             numerator = -numerator;
             denominator = -denominator;
         }
         // simplify fraction.
         final int gcd = greatestCommonDivisor(numerator, denominator);
         numerator /= gcd;
         denominator /= gcd;
         return new Fraction(numerator, denominator);
     }
 
     /**
      * <p>Creates a <code>Fraction</code> instance from a <code>double</code> value.</p>
      *
      * <p>This method uses the <a href="http://archives.math.utk.edu/articles/atuyl/confrac/">
      *  continued fraction algorithm</a>, computing a maximum of
      *  25 convergents and bounding the denominator by 10,000.</p>
      *
      * @param value  the double value to convert
      * @return a new fraction instance that is close to the value
      * @throws ArithmeticException if <code>|value| > Integer.MAX_VALUE</code> 
      *  or <code>value = NaN</code>
      * @throws ArithmeticException if the calculated denominator is <code>zero</code>
      * @throws ArithmeticException if the the algorithm does not converge
      */
     public static Fraction getFraction(double value) {
         final int sign = value < 0 ? -1 : 1;
         value = Math.abs(value);
         if (value  > Integer.MAX_VALUE || Double.isNaN(value)) {
             throw new ArithmeticException
                 ("The value must not be greater than Integer.MAX_VALUE or NaN");
         }
         final int wholeNumber = (int) value;
         value -= wholeNumber;
         
         int numer0 = 0;  // the pre-previous
         int denom0 = 1;  // the pre-previous
         int numer1 = 1;  // the previous
         int denom1 = 0;  // the previous
         int numer2 = 0;  // the current, setup in calculation
         int denom2 = 0;  // the current, setup in calculation
         int a1 = (int) value;
         int a2 = 0;
         double x1 = 1;
         double x2 = 0;
         double y1 = value - a1;
         double y2 = 0;
         double delta1, delta2 = Double.MAX_VALUE;
         double fraction;
         int i = 1;
 //        System.out.println("---");
         do {
             delta1 = delta2;
             a2 = (int) (x1 / y1);
             x2 = y1;
             y2 = x1 - a2 * y1;
             numer2 = a1 * numer1 + numer0;
             denom2 = a1 * denom1 + denom0;
             fraction = (double) numer2 / (double) denom2;
             delta2 = Math.abs(value - fraction);
 //            System.out.println(numer2 + " " + denom2 + " " + fraction + " " + delta2 + " " + y1);
             a1 = a2;
             x1 = x2;
             y1 = y2;
             numer0 = numer1;
             denom0 = denom1;
             numer1 = numer2;
             denom1 = denom2;
             i++;
 //            System.out.println(">>" + delta1 +" "+ delta2+" "+(delta1 > delta2)+" "+i+" "+denom2);
         } while (delta1 > delta2 && denom2 <= 10000 && denom2 > 0 && i < 25);
         if (i == 25) {
             throw new ArithmeticException("Unable to convert double to fraction");
         }
         return getReducedFraction((numer0 + wholeNumber * denom0) * sign, denom0);
     }
 
     /**
      * <p>Creates a Fraction from a <code>String</code>.</p>
      *
      * <p>The formats accepted are:</p>
      *
      * <ol>
      *  <li><code>double</code> String containing a dot</li>
      *  <li>'X Y/Z'</li>
      *  <li>'Y/Z'</li>
      *  <li>'X' (a simple whole number)</li>
      * </ol>
      * and a .</p>
      *
      * @param str  the string to parse, must not be <code>null</code>
      * @return the new <code>Fraction</code> instance
      * @throws IllegalArgumentException if the string is <code>null</code>
      * @throws NumberFormatException if the number format is invalid
      */
     public static Fraction getFraction(String str) {
         if (str == null) {
             throw new IllegalArgumentException("The string must not be null");
         }
         // parse double format
         int pos = str.indexOf('.');
         if (pos >= 0) {
             return getFraction(Double.parseDouble(str));
         }
 
         // parse X Y/Z format
         pos = str.indexOf(' ');
         if (pos > 0) {
             final int whole = Integer.parseInt(str.substring(0, pos));
             str = str.substring(pos + 1);
             pos = str.indexOf('/');
             if (pos < 0) {
                 throw new NumberFormatException("The fraction could not be parsed as the format X Y/Z");
             } else {
                 final int numer = Integer.parseInt(str.substring(0, pos));
                 final int denom = Integer.parseInt(str.substring(pos + 1));
                 return getFraction(whole, numer, denom);
             }
         }
 
         // parse Y/Z format
         pos = str.indexOf('/');
         if (pos < 0) {
             // simple whole number
             return getFraction(Integer.parseInt(str), 1);
         } else {
             final int numer = Integer.parseInt(str.substring(0, pos));
             final int denom = Integer.parseInt(str.substring(pos + 1));
             return getFraction(numer, denom);
         }
     }
 
     // Accessors
     //-------------------------------------------------------------------
 
     /**
      * <p>Gets the numerator part of the fraction.</p>
      *
      * <p>This method may return a value greater than the denominator, an
      * improper fraction, such as the seven in 7/4.</p>
      *
      * @return the numerator fraction part
      */
     public int getNumerator() {
         return numerator;
     }
 
     /**
      * <p>Gets the denominator part of the fraction.</p>
      *
      * @return the denominator fraction part
      */
     public int getDenominator() {
         return denominator;
     }
 
     /**
      * <p>Gets the proper numerator, always positive.</p>
      *
      * <p>An improper fraction 7/4 can be resolved into a proper one, 1 3/4.
      * This method returns the 3 from the proper fraction.</p>
      *
      * <p>If the fraction is negative such as -7/4, it can be resolved into
      * -1 3/4, so this method returns the positive proper numerator, 3.</p>
      *
      * @return the numerator fraction part of a proper fraction, always positive
      */
     public int getProperNumerator() {
         return Math.abs(numerator % denominator);
     }
 
     /**
      * <p>Gets the proper whole part of the fraction.</p>
      *
      * <p>An improper fraction 7/4 can be resolved into a proper one, 1 3/4.
      * This method returns the 1 from the proper fraction.</p>
      *
      * <p>If the fraction is negative such as -7/4, it can be resolved into
      * -1 3/4, so this method returns the positive whole part -1.</p>
      *
      * @return the whole fraction part of a proper fraction, that includes the sign
      */
     public int getProperWhole() {
         return numerator / denominator;
     }
 
     // Number methods
     //-------------------------------------------------------------------
 
     /**
      * <p>Gets the fraction as an <code>int</code>. This returns the whole number
      * part of the fraction.</p>
      *
      * @return the whole number fraction part
      */
     @Override
     public int intValue() {
         return numerator / denominator;
     }
 
     /**
      * <p>Gets the fraction as a <code>long</code>. This returns the whole number
      * part of the fraction.</p>
      *
      * @return the whole number fraction part
      */
     @Override
     public long longValue() {
         return (long) numerator / denominator;
     }
 
     /**
      * <p>Gets the fraction as a <code>float</code>. This calculates the fraction
      * as the numerator divided by denominator.</p>
      *
      * @return the fraction as a <code>float</code>
      */
     @Override
     public float floatValue() {
         return (float) numerator / (float) denominator;
     }
 
     /**
      * <p>Gets the fraction as a <code>double</code>. This calculates the fraction
      * as the numerator divided by denominator.</p>
      *
      * @return the fraction as a <code>double</code>
      */
     @Override
     public double doubleValue() {
         return (double) numerator / (double) denominator;
     }
 
     // Calculations
     //-------------------------------------------------------------------
 
     /**
      * <p>Reduce the fraction to the smallest values for the numerator and
      * denominator, returning the result.</p>
      * 
      * <p>For example, if this fraction represents 2/4, then the result
      * will be 1/2.</p>
      *
      * @return a new reduced fraction instance, or this if no simplification possible
      */
     public Fraction reduce() {
         if (numerator == 0) {
             return equals(ZERO) ? this : ZERO;
         }
         final int gcd = greatestCommonDivisor(Math.abs(numerator), denominator);
         if (gcd == 1) {
             return this;
         }
         return Fraction.getFraction(numerator / gcd, denominator / gcd);
     }
 
     /**
      * <p>Gets a fraction that is the inverse (1/fraction) of this one.</p>
      * 
      * <p>The returned fraction is not reduced.</p>
      *
      * @return a new fraction instance with the numerator and denominator
      *         inverted.
      * @throws ArithmeticException if the fraction represents zero.
      */
     public Fraction invert() {
         if (numerator == 0) {
             throw new ArithmeticException("Unable to invert zero.");
         }
         if (numerator==Integer.MIN_VALUE) {
             throw new ArithmeticException("overflow: can't negate numerator");
         }
         if (numerator<0) {
             return new Fraction(-denominator, -numerator);
         } else {
             return new Fraction(denominator, numerator);
         }
     }
 
     /**
      * <p>Gets a fraction that is the negative (-fraction) of this one.</p>
      *
      * <p>The returned fraction is not reduced.</p>
      *
      * @return a new fraction instance with the opposite signed numerator
      */
     public Fraction negate() {
         // the positive range is one smaller than the negative range of an int.
         if (numerator==Integer.MIN_VALUE) {
             throw new ArithmeticException("overflow: too large to negate");
         }
         return new Fraction(-numerator, denominator);
     }
 
     /**
      * <p>Gets a fraction that is the positive equivalent of this one.</p>
      * <p>More precisely: <code>(fraction >= 0 ? this : -fraction)</code></p>
      *
      * <p>The returned fraction is not reduced.</p>
      *
      * @return <code>this</code> if it is positive, or a new positive fraction
      *  instance with the opposite signed numerator
      */
     public Fraction abs() {
         if (numerator >= 0) {
             return this;
         }
         return negate();
     }
 
     /**
      * <p>Gets a fraction that is raised to the passed in power.</p>
      *
      * <p>The returned fraction is in reduced form.</p>
      *
      * @param power  the power to raise the fraction to
      * @return <code>this</code> if the power is one, <code>ONE</code> if the power
      * is zero (even if the fraction equals ZERO) or a new fraction instance 
      * raised to the appropriate power
      * @throws ArithmeticException if the resulting numerator or denominator exceeds
      *  <code>Integer.MAX_VALUE</code>
      */
     public Fraction pow(final int power) {
         if (power == 1) {
             return this;
         } else if (power == 0) {
             return ONE;
         } else if (power < 0) {
             if (power==Integer.MIN_VALUE) { // MIN_VALUE can't be negated.
                 return this.invert().pow(2).pow(-(power/2));
             }
             return this.invert().pow(-power);
         } else {
             final Fraction f = this.multiplyBy(this);
             if (power % 2 == 0) { // if even...
                 return f.pow(power/2);
             } else { // if odd...
                 return f.pow(power/2).multiplyBy(this);
             }
         }
     }
 
     /**
      * <p>Gets the greatest common divisor of the absolute value of
      * two numbers, using the "binary gcd" method which avoids
      * division and modulo operations.  See Knuth 4.5.2 algorithm B.
      * This algorithm is due to Josef Stein (1961).</p>
      *
      * @param u  a non-zero number
      * @param v  a non-zero number
      * @return the greatest common divisor, never zero
      */
     private static int greatestCommonDivisor(int u, int v) {
         // From Commons Math:
         if (u == 0 || v == 0) {
             if (u == Integer.MIN_VALUE || v == Integer.MIN_VALUE) {
                 throw new ArithmeticException("overflow: gcd is 2^31");
             }
             return Math.abs(u) + Math.abs(v);
         }
         //if either operand is abs 1, return 1:
         if (Math.abs(u) == 1 || Math.abs(v) == 1) {
             return 1;
         }
         // keep u and v negative, as negative integers range down to
         // -2^31, while positive numbers can only be as large as 2^31-1
         // (i.e. we can't necessarily negate a negative number without
         // overflow)
         if (u>0) { u=-u; } // make u negative
         if (v>0) { v=-v; } // make v negative
         // B1. [Find power of 2]
         int k=0;
         while ((u&1)==0 && (v&1)==0 && k<31) { // while u and v are both even...
             u/=2; v/=2; k++; // cast out twos.
         }
         if (k==31) {
             throw new ArithmeticException("overflow: gcd is 2^31");
         }
         // B2. Initialize: u and v have been divided by 2^k and at least
         //     one is odd.
         int t = (u&1)==1 ? v : -(u/2)/*B3*/;
         // t negative: u was odd, v may be even (t replaces v)
         // t positive: u was even, v is odd (t replaces u)
         do {
             /* assert u<0 && v<0; */
             // B4/B3: cast out twos from t.
             while ((t&1)==0) { // while t is even..
                 t/=2; // cast out twos
             }
             // B5 [reset max(u,v)]
             if (t>0) {
                 u = -t;
             } else {
                 v = t;
             }
             // B6/B3. at this point both u and v should be odd.
             t = (v - u)/2;
             // |u| larger: t positive (replace u)
             // |v| larger: t negative (replace v)
         } while (t!=0);
         return -u*(1<<k); // gcd is u*2^k
     }
 
     // Arithmetic
     //-------------------------------------------------------------------
 
     /** 
      * Multiply two integers, checking for overflow.
      * 
      * @param x a factor
      * @param y a factor
      * @return the product <code>x*y</code>
      * @throws ArithmeticException if the result can not be represented as
      *                             an int
      */
     private static int mulAndCheck(final int x, final int y) {
         final long m = (long)x*(long)y;
         if (m < Integer.MIN_VALUE ||
             m > Integer.MAX_VALUE) {
             throw new ArithmeticException("overflow: mul");
         }
         return (int)m;
     }
     
     /**
      *  Multiply two non-negative integers, checking for overflow.
      * 
      * @param x a non-negative factor
      * @param y a non-negative factor
      * @return the product <code>x*y</code>
      * @throws ArithmeticException if the result can not be represented as
      * an int
      */
     private static int mulPosAndCheck(final int x, final int y) {
         /* assert x>=0 && y>=0; */
         final long m = (long)x*(long)y;
         if (m > Integer.MAX_VALUE) {
             throw new ArithmeticException("overflow: mulPos");
         }
         return (int)m;
     }
     
     /** 
      * Add two integers, checking for overflow.
      * 
      * @param x an addend
      * @param y an addend
      * @return the sum <code>x+y</code>
      * @throws ArithmeticException if the result can not be represented as
      * an int
      */
     private static int addAndCheck(final int x, final int y) {
         final long s = (long)x+(long)y;
         if (s < Integer.MIN_VALUE ||
             s > Integer.MAX_VALUE) {
             throw new ArithmeticException("overflow: add");
         }
         return (int)s;
     }
     
     /** 
      * Subtract two integers, checking for overflow.
      * 
      * @param x the minuend
      * @param y the subtrahend
      * @return the difference <code>x-y</code>
      * @throws ArithmeticException if the result can not be represented as
      * an int
      */
     private static int subAndCheck(final int x, final int y) {
         final long s = (long)x-(long)y;
         if (s < Integer.MIN_VALUE ||
             s > Integer.MAX_VALUE) {
             throw new ArithmeticException("overflow: add");
         }
         return (int)s;
     }
     
     /**
      * <p>Adds the value of this fraction to another, returning the result in reduced form.
      * The algorithm follows Knuth, 4.5.1.</p>
      *
      * @param fraction  the fraction to add, must not be <code>null</code>
      * @return a <code>Fraction</code> instance with the resulting values
      * @throws IllegalArgumentException if the fraction is <code>null</code>
      * @throws ArithmeticException if the resulting numerator or denominator exceeds
      *  <code>Integer.MAX_VALUE</code>
      */
     public Fraction add(final Fraction fraction) {
         return addSub(fraction, true /* add */);
     }
 
     /**
      * <p>Subtracts the value of another fraction from the value of this one, 
      * returning the result in reduced form.</p>
      *
      * @param fraction  the fraction to subtract, must not be <code>null</code>
      * @return a <code>Fraction</code> instance with the resulting values
      * @throws IllegalArgumentException if the fraction is <code>null</code>
      * @throws ArithmeticException if the resulting numerator or denominator
      *   cannot be represented in an <code>int</code>.
      */
     public Fraction subtract(final Fraction fraction) {
         return addSub(fraction, false /* subtract */);
     }
 
     /** 
      * Implement add and subtract using algorithm described in Knuth 4.5.1.
      * 
      * @param fraction the fraction to subtract, must not be <code>null</code>
      * @param isAdd true to add, false to subtract
      * @return a <code>Fraction</code> instance with the resulting values
      * @throws IllegalArgumentException if the fraction is <code>null</code>
      * @throws ArithmeticException if the resulting numerator or denominator
      *   cannot be represented in an <code>int</code>.
      */
     private Fraction addSub(final Fraction fraction, final boolean isAdd) {
         if (fraction == null) {
             throw new IllegalArgumentException("The fraction must not be null");
         }
         // zero is identity for addition.
         if (numerator == 0) {
             return isAdd ? fraction : fraction.negate();
         }
         if (fraction.numerator == 0) {
             return this;
         }     
         // if denominators are randomly distributed, d1 will be 1 about 61%
         // of the time.
         final int d1 = greatestCommonDivisor(denominator, fraction.denominator);
         if (d1==1) {
             // result is ( (u*v' +/- u'v) / u'v')
             final int uvp = mulAndCheck(numerator, fraction.denominator);
             final int upv = mulAndCheck(fraction.numerator, denominator);
             return new Fraction
                 (isAdd ? addAndCheck(uvp, upv) : subAndCheck(uvp, upv),
                  mulPosAndCheck(denominator, fraction.denominator));
         }
         // the quantity 't' requires 65 bits of precision; see knuth 4.5.1
         // exercise 7.  we're going to use a BigInteger.
         // t = u(v'/d1) +/- v(u'/d1)
         final BigInteger uvp = BigInteger.valueOf(numerator)
             .multiply(BigInteger.valueOf(fraction.denominator/d1));
         final BigInteger upv = BigInteger.valueOf(fraction.numerator)
             .multiply(BigInteger.valueOf(denominator/d1));
         final BigInteger t = isAdd ? uvp.add(upv) : uvp.subtract(upv);
         // but d2 doesn't need extra precision because
         // d2 = gcd(t,d1) = gcd(t mod d1, d1)
         final int tmodd1 = t.mod(BigInteger.valueOf(d1)).intValue();
         final int d2 = tmodd1==0?d1:greatestCommonDivisor(tmodd1, d1);
 
         // result is (t/d2) / (u'/d1)(v'/d2)
         final BigInteger w = t.divide(BigInteger.valueOf(d2));
         if (w.bitLength() > 31) {
             throw new ArithmeticException
                 ("overflow: numerator too large after multiply");
         }
         return new Fraction
             (w.intValue(),
              mulPosAndCheck(denominator/d1, fraction.denominator/d2));
     }
 
     /**
      * <p>Multiplies the value of this fraction by another, returning the 
      * result in reduced form.</p>
      *
      * @param fraction  the fraction to multiply by, must not be <code>null</code>
      * @return a <code>Fraction</code> instance with the resulting values
      * @throws IllegalArgumentException if the fraction is <code>null</code>
      * @throws ArithmeticException if the resulting numerator or denominator exceeds
      *  <code>Integer.MAX_VALUE</code>
      */
     public Fraction multiplyBy(final Fraction fraction) {
         if (fraction == null) {
             throw new IllegalArgumentException("The fraction must not be null");
         }
         if (numerator == 0 || fraction.numerator == 0) {
             return ZERO;
         }
         // knuth 4.5.1
         // make sure we don't overflow unless the result *must* overflow.
         final int d1 = greatestCommonDivisor(numerator, fraction.denominator);
         final int d2 = greatestCommonDivisor(fraction.numerator, denominator);
         return getReducedFraction
             (mulAndCheck(numerator/d1, fraction.numerator/d2),
              mulPosAndCheck(denominator/d2, fraction.denominator/d1));
     }
 
     /**
      * <p>Divide the value of this fraction by another.</p>
      *
      * @param fraction  the fraction to divide by, must not be <code>null</code>
      * @return a <code>Fraction</code> instance with the resulting values
      * @throws IllegalArgumentException if the fraction is <code>null</code>
      * @throws ArithmeticException if the fraction to divide by is zero
      * @throws ArithmeticException if the resulting numerator or denominator exceeds
      *  <code>Integer.MAX_VALUE</code>
      */
     public Fraction divideBy(final Fraction fraction) {
         if (fraction == null) {
             throw new IllegalArgumentException("The fraction must not be null");
         }
         if (fraction.numerator == 0) {
             throw new ArithmeticException("The fraction to divide by must not be zero");
         }
         return multiplyBy(fraction.invert());
     }
 
     // Basics
     //-------------------------------------------------------------------
 
     /**
      * <p>Compares this fraction to another object to test if they are equal.</p>.
      *
      * <p>To be equal, both values must be equal. Thus 2/4 is not equal to 1/2.</p>
      *
      * @param obj the reference object with which to compare
      * @return <code>true</code> if this object is equal
      */
     @Override
     public boolean equals(final Object obj) {
         if (obj == this) {
             return true;
         }
         if (obj instanceof Fraction == false) {
             return false;
         }
         final Fraction other = (Fraction) obj;
         return getNumerator() == other.getNumerator() &&
                 getDenominator() == other.getDenominator();
     }
 
     /**
      * <p>Gets a hashCode for the fraction.</p>
      *
      * @return a hash code value for this object
      */
     @Override
     public int hashCode() {
         if (hashCode == 0) {
             // hashcode update should be atomic.
             hashCode = 37 * (37 * 17 + getNumerator()) + getDenominator();
         }
         return hashCode;
     }
 
     /**
      * <p>Compares this object to another based on size.</p>
      *
      * <p>Note: this class has a natural ordering that is inconsistent
      * with equals, because, for example, equals treats 1/2 and 2/4 as
      * different, whereas compareTo treats them as equal.
      *
      * @param other  the object to compare to
      * @return -1 if this is less, 0 if equal, +1 if greater
      * @throws ClassCastException if the object is not a <code>Fraction</code>
      * @throws NullPointerException if the object is <code>null</code>
      */
     @Override
     public int compareTo(final Fraction other) {
         if (this==other) {
             return 0;
         }
         if (numerator == other.numerator && denominator == other.denominator) {
             return 0;
         }
 
         // otherwise see which is less
         final long first = (long) numerator * (long) other.denominator;
         final long second = (long) other.numerator * (long) denominator;
         if (first == second) {
             return 0;
         } else if (first < second) {
             return -1;
         } else {
             return 1;
         }
     }
 
     /**
      * <p>Gets the fraction as a <code>String</code>.</p>
      *
      * <p>The format used is '<i>numerator</i>/<i>denominator</i>' always.
      *
      * @return a <code>String</code> form of the fraction
      */
     @Override
     public String toString() {
         if (toString == null) {
             toString = new StringBuilder(32)
                 .append(getNumerator())
                 .append('/')
                 .append(getDenominator()).toString();
         }
         return toString;
     }
 
     /**
      * <p>Gets the fraction as a proper <code>String</code> in the format X Y/Z.</p>
      *
      * <p>The format used in '<i>wholeNumber</i> <i>numerator</i>/<i>denominator</i>'.
      * If the whole number is zero it will be omitted. If the numerator is zero,
      * only the whole number is returned.</p>
      *
      * @return a <code>String</code> form of the fraction
      */
     public String toProperString() {
         if (toProperString == null) {
             if (numerator == 0) {
                 toProperString = "0";
             } else if (numerator == denominator) {
                 toProperString = "1";
             } else if (numerator == -1 * denominator) {
                 toProperString = "-1";
             } else if ((numerator>0?-numerator:numerator) < -denominator) {
                 // note that we do the magnitude comparison test above with
                 // NEGATIVE (not positive) numbers, since negative numbers
                 // have a larger range.  otherwise numerator==Integer.MIN_VALUE
                 // is handled incorrectly.
                 final int properNumerator = getProperNumerator();
                 if (properNumerator == 0) {
                     toProperString = Integer.toString(getProperWhole());
                 } else {
                     toProperString = new StringBuilder(32)
                         .append(getProperWhole()).append(' ')
                         .append(properNumerator).append('/')
                         .append(getDenominator()).toString();
                 }
             } else {
                 toProperString = new StringBuilder(32)
                     .append(getNumerator()).append('/')
                     .append(getDenominator()).toString();
             }
         }
         return toProperString;
     }
 }