Because of these inexact results, none of the three varieties of addition are associative, as shown below:
and
, 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 
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.
| Jeff Tupper | March 1996 |