Computer arithmetic can behave in apparently strange ways, even though
each elementary floating point operation (+,-,/,*) is performed to high
accuracy. We describe various examples of how rounding errors can have
surprising or even calamitous consequences, including puzzling differences
between results from similar Cray computers, and the Patriot missile
software problem in the Gulf war. On the positive side, we describe IEEE
standard arithmetic and its advantages for the programmer. As an example
of how the effects of rounding errors can be understood we explain how to
obtain a sharp bound for the error in summing floating point numbers.