*by Jerry D. Cavin, Senior Software Engineer
*

How many times have you coded an equation and the results produced are not what you expected? Then you rearrange the equation and get completely different results! If the computer is calculating correctly, why does it produce incorrect answers? Could it be the ‘Ghost in the Machine?’

Gilbert Ryle’s notion of the ‘Ghost in the Machine’ was introduced in his book, *The Concept of Mind*, a critique of Rene Descartes’ discussion of the relationship between the mind and the body (the mind-body dualism). The expression has been widely used in the context of a computer’s tendency to make unexplainable numerical errors. Are the errors the fault of the human, or is it the fault of the ‘Ghost in the Machine?’ To gain insight into these mathematical errors made by computers let us examine how the ‘mind’ of the computer views our concept of numbers.

INTEGER NUMBERS

The ‘mind’ of the computer perceives finite number sets. The size of the number set depends on the size of the space where the number is stored. If the number is stored in 8 bits, the number set starts with 0 and ends at 255. One of the bits can be reserved to represent the sign of the value (i.e. + or -). In this case, the numbers that exist are from -128 to 127. Numbers can also be stored in a larger space, but that does not change the fact the ‘mind’ of the computer still perceives finite number sets. So what happens when 1 is added to the maximum value in a finite integer set? It wraps around to the smallest negative value in the set. In our previous example for 8-bit values, when the computer adds 1 to 127 it would give the answer of -128. This overflow, or wrap around error, can occur if you are adding, subtracting, multiplying or dividing. Integer overflow errors are dangerous because they cannot be detected after it has happened. The ISO C99 standard states an integer overflow causes ‘Undefined Behavior.’ This allows compiler manufacturers, conforming to the standard, to implement anything from completely ignoring the overflow to causing the program to abort. Most compilers totally ignore the overflow, resulting in erroneous results being calculated. Depending upon how the integer variable is used, the overflow can cause a multitude of serious problems. If an overflow occurs with a loop index variable, it may cause an infinite loop. If an overflow occurs with a table index variable, the buffer overflow may cause data corruption. Another strange condition can arise with integers that we do not see in the human world. Under some circumstances an equation may give an answer of +0 or it may give an answer of -0. These two values of 0 are the same, but it may cause some confusion. This is a problem to many older compilers; some compilers detect this condition and simply change the answer to +0. Newer compilers use two’s complement mathematical operations and never encounter this issue.

What does your computer do?