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2.5.4 Algebraic Properties

Because of these inexact results, none of the three varieties of addition are associative, as shown below:

math6359

math6379

math6399

Similarly for multiplication: none of the three varieties are associative. Addition and multiplication do have the identities tex2html_wrap_inline32445 and tex2html_wrap_inline32409 , respectively. All varieties of addition and multiplication are commutative over the floats, but the lack of associativity causes any non-trivial symbolic manipulation of an expression to affect the expression's value. Negation is a total inverse for all three addition operators. Since the base is a fixed integer none of the three multiplication operators have total inverses. None of the tex2html_wrap_inline32449 possible distributive laws are obeyed. Algebraic properties of rounded computations are discussed in [37, 39, 38].


Common Practice

The floating point number system does not obey many nice formal rules [40]. Extensions and generalizations of IEEE 754 floating-point have been put forward [13, 6]. For many applications the use of floating point does not adversely affect the output, which has been envisioned as coming from computations using real numbers. With long streams of computations there is a worry that the floating point computation stream will radically diverge from the underlying real computation stream. In these cases, formal arguments involving particular implementations of the operators and particular sequences of computations must be made.


next up previous notation contents
Next: 2.6 Extended Real Numbers Up: 2.5 Floating Point Previous: 2.5.3 Rounding
Jeff TupperMarch 1996