8.1.3Problems Remain in Java

There is one area where the designers of Java chose not to detect errors automatically: numerical computations. In Java, a value of type int is represented as a 32-bit binary number. With 32 bits, it’s possible to represent a little over four billion di erent values. The values of type int range from -2147483648 to 2147483647. What happens when the result of a computation lies outside this range? For example, what is 2147483647 + 1? And what is 2000000000 * 2? The mathematically correct result in each case cannot be represented as a value of type int. These are examples of integer overflow . In most cases, integer overflow should be considered an error. However, Java does not automatically detect such errors. For example, it will compute the value of 2147483647 + 1 to be the negative number, -2147483648. (What happens is that any extra bits beyond the 32-nd bit in the correct answer are discarded. Values greater than 2147483647 will “wrap around” to negative values. Mathematically speaking, the result is always “correct modulo 232”.)

For example, consider the 3N+1 program, which was discussed in Subsection 3.2.2. Starting from a positive integer N, the program computes a certain sequence of integers:

System.out.println(N);

}

But there is a problem here: If N is too large, then the value of 3*N+1 will not be mathematically correct because of integer overflow. The problem arises whenever 3*N+1 > 2147483647, that is when N > 2147483646/3. For a completely correct program, we should check for this possibility before computing 3*N+1:

if (N > 2147483646/3) {

System.out.println("Sorry, but the value of N has become"); System.out.println("too large for your computer!");

break;

}

N = 3 * N + 1;

}

System.out.println(N);

}

The problem here is not that the original algorithm for computing 3N+1 sequences was wrong. The problem is that it just can’t be correctly implemented using 32-bit integers. Many programs ignore this type of problem. But integer overflow errors have been responsible for their share of serious computer failures, and a completely robust program should take the possibility of integer overflow into account. (The infamous “Y2K” bug was, in fact, just this sort of error.)

For numbers of type double, there are even more problems. There are still overflow errors, which occur when the result of a computation is outside the range of values that can be represented as a value of type double. This range extends up to about 1.7 times 10 to the