Overview

Consider the two numbers

and

The first number, x, is a 16-digit approximation to ?, while the second number agrees with ? to only 12 digits. Their difference is

This difference is in the normalized range of the IEEE single format. However, if we compute the difference z = x ? y in a C program, using the single format to store the variables x and y before doing the subtraction, and display the result to single precision, we find that the result displayed for z is

The reason is simple enough. The input numbers x and y are first converted from decimal to the single binary format; they are not exact floating point numbers, so the decimal to binary conversion requires some rounding. Because they agree to 12 digits, both x and y round to exactly the same single format number. Thus, all bits in their binary representation cancel when the subtraction is done; we say that we have a complete loss of accuracy in the computation z = x ? y.

If we use the double format to store x and y and their difference z, and if we display the result to double precision, we find that z has the value

This agrees with the exact answer (11.1) to about four digits, but what is the meaning of the other digits? The answer is that the result displayed...