Puzzle 3 - Time for a Change

Consider the following word problem:

Tom goes to the auto parts store to buy a spark plug that costs $1.10, but all he has in his wallet are two-dollar bills. How much change should he get if he pays for the spark plug with a two-dollar bill?

Here is a program that attempts to solve the word problem. What does it print?


public class Change {

public static void main(String args[]) {

System.out.println(2.00 - 1.10);

}

}




Solution 2: Time for a Change
Naively, you might expect the program to print 0.90, but how could it know that you wanted two digits after the decimal point? If you know something about the rules for converting double values to strings, which are specified by the documentation for Double.toString [Java-API], you know that the program prints the shortest decimal fraction sufficient to distinguish the double value from its nearest neighbor, with at least one digit before and after the decimal point. It seems reasonable, then, that the program should print 0.9. Reasonable, perhaps, but not correct. If you ran the program, you found that it prints 0.8999999999999999.

The problem is that the number 1.1 can't be represented exactly as a double, so it is represented by the closest double value. The program subtracts this value from 2. Unfortunately, the result of this calculation is not the closest double value to 0.9. The shortest representation of the resulting double value is the hideous number that you see printed.

More generally, the problem is that not all decimals can be represented exactly using binary floating-point. If you are using release 5.0 or a later release, you might be tempted to fix the program by using the printf facility to set the precision of the output:


// Poor solution - still uses binary floating-point!

System.out.printf("%.2f%n", 2.00 - 1.10);




This prints the right answer but does not represent a general solution to the underlying problem: It still uses double arithmetic, which is binary floating-point. Floating-point arithmetic provides good approximations over a wide range of values but does not generally yield exact results. Binary floating-point is particularly ill-suited to monetary calculations, as it is impossible to represent 0.1—or any other negative power of 10—exactly as a finite-length binary fraction [EJ Item 31].

One way to solve the problem is to use an integral type, such as int or long, and to perform the computation in cents. If you go this route, make sure the integral type is large enough to represent all the values you will use in your program. For this puzzle, int is ample. Here is how the println looks if we rewrite it using int values to represent monetary values in cents. This version prints 90 cents, which is the right answer:

System.out.println((200 - 110) + " cents");


Another way to solve the problem is to use BigDecimal, which performs exact decimal arithmetic. It also interoperates with the SQL DECIMAL type via JDBC. There is one caveat: Always use the BigDecimal(String) constructor, never BigDecimal(double). The latter constructor creates an instance with the exact value of its argument: new BigDecimal(.1) returns a BigDecimal representing 0.1000000000000000055511151231257827021181583404541015625. Using BigDecimal correctly, the program prints the expected result of 0.90:


import java.math.BigDecimal;

public class Change {

public static void main(String args[]) {

System.out.println(new BigDecimal("2.00").

subtract(new BigDecimal("1.10")));

}

}




This version is not terribly pretty, as Java provides no linguistic support for BigDecimal. Calculations with BigDecimal are also likely to be slower than those with any primitive type, which might be an issue for some programs that make heavy use of decimal calculations. It is of no consequence for most programs.

In summary, avoid float and double where exact answers are required; for monetary calculations, use int, long, or BigDecimal. For language designers, consider providing linguistic support for decimal arithmetic. One approach is to offer limited support for operator overloading, so that arithmetic operators can be made to work with numerical reference types, such as BigDecimal. Another approach is to provide a primitive decimal type, as did COBOL and PL/I.