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10.7: Rounding Errors

  • Page ID
    17656
  • Most floating-point numbers are only approximately correct. Some numbers, like reasonably-sized integers, can be represented exactly. But repeating fractions, like 1/3, and irrational numbers, like \( \pi \), cannot. To represent these numbers, computers have to round off to the nearest floating-point number.

    The difference between the number we want and the floating-point number we get is called rounding error. For example, the following two statements should be equivalent:

    System.out.println(0.1 * 10);
    System.out.println(0.1 + 0.1 + 0.1 + 0.1 + 0.1
                     + 0.1 + 0.1 + 0.1 + 0.1 + 0.1);
    

    But on many machines, the output is:

    1.0
    0.9999999999999999
    

    For many applications, like computer graphics, encryption, statistical analysis, and multimedia rendering, floating-point arithmetic has benefits that outweigh the costs. But if you need absolute precision, use integers instead. For example, consider a bank account with a balance of $123.45:

    double balance = 123.45;  // potential rounding error
    

    The problem is that 0.1, which is a terminating fraction in decimal, is a repeating fraction in binary. So its floating-point representation is only approximate. When we add up the approximations, the rounding errors accumulate.

    In this example, balances will become inaccurate over time as the variable is used in arithmetic operations like deposits and withdrawals. The result would be angry customers and potential lawsuits. You can avoid the problem by representing the balance as an integer:

    int balance = 12345;      // total number of cents
    

    This solution works as long as the number of cents doesn’t exceed the largest integer, which is about 2 billion.