"how numbers are stored in computers"
TODO
The results of this section can be summarized by saying that a guard digit guarantees accuracy when nearby precisely known quantities are subtracted (benign cancellation). Sometimes a formula that gives inaccurate results can be rewritten to have much higher numerical accuracy by using benign cancellation; however, the procedure only works if subtraction is performed using a guard digit. The price of a guard digit is not high, because it merely requires making the adder one bit wider. For a 54 bit double precision adder, the additional cost is less than 2%. For this price, you gain the ability to run many algorithms such as formula (6) for computing the area of a triangle and the expression ln(1 + x). Although most modern computers have a guard digit, there are a few (such as Cray systems) that do not.
When floating-point operations are done with a guard digit, they are not as accurate as if they were computed exactly and then rounded to the nearest floating-point number. Operations performed in this manner are called exactly rounded.
The example immediately preceding Theorem 2 shows that a single guard digit will not always give exactly rounded results. The previous section gave several examples of algorithms that require a guard digit in order to work properly. This section gives examples of algorithms that require exact rounding.
A single guard digit is enough to prevent catastrophic cancellation, where all digits are wrong, but it doesn’t fully eliminate error. It ensures that the relative error in the result remains small—typically close to but possibly slightly greater than machine epsilon.