Unformatted text preview: Suppose x is a real number which is in the range of the ﬂoating point system (doesn’t underﬂow or overﬂow). Then ﬂ( x ) = x (1 + ± ) , where | ± | ≤ μ This is a statement about relative error, and can also be written as | x-ﬂ( x ) | | x | ≤ μ. The set of ﬂoats is not closed under our arithmetic operations. For example, when we add two ﬂoats, the result is not necessarily a ﬂoat, and it will need to be approximated by another ﬂoat. Computers today almost always satisfy the following rule: The Fundamental Axiom of Floating Point Arithmetic. Let x op y be some arithmetic operation. That is, op is one of +,-, × or ÷ . Suppose x and y are ﬂoats and that x op y doesn’t underﬂow or overﬂow. Then ﬂ( x op y ) = ( x op y )(1 + ± ) , where | ± | ≤ μ Notice that this is a statement about ﬂoats. Real numbers need to be represented by ﬂoats before we can do the arithmetic!...
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This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.
- Fall '10