TITLE: Algebraic structure for the floating-point arithmetic.
QUESTION [12 upvotes]: Is it possible to consider floating-point arithmetic as some common algebraic structure? 
For example, consider something like simplified IEEE754 single precision binary floating-point subset of numbers that consist of 1 sign bit, 23-bit mantissa and 8-bit exponent
Let $\mathcal{F}$ be a floating-point set which consists of:

finite floating-point numbers, such that each number is described by three integers: $(-1)^{s}\cdot M\cdot 2^{q} $. $s$ is a sign($0$ or $1$), $ \ M$ and $q$ must be integers in the ranges $0$ through $2^{23} -1 $ and $-128$ to $127$ respectively. 
two infinities: $+\infty$ and $-\infty$
NaN (not a number)

Let's consider four operations defined on the $\mathcal{F}$ : summation, subtraction, multiplication and division(as multiplication by a reciprocal).
The result is NaN in each of the following cases:
 - On of the operands is NaN
 - The divisions 0/0 and ±∞/±∞
 - The multiplications 0×±∞ and ±∞×0
 - The additions ∞ + (−∞), (−∞) + ∞ and equivalent subtractions
If sum or product of finite numbers cannot be represented as finite floating-point number, the result is infinity.
Finally, let $\mathfrak F$ be an algebraic structure with the set and operations we have defined.
So, is it possible to characterize this structure somehow?
Excuse me if my question is too absurd :)

REPLY [10 votes]: Look into D.E. Knuth's The Art of Computer Programming, Vol. 2, Section 4.2.2 A, p. 214-223, he gives an axiomatic approach of floating point arithmetic which seems to be what you are looking for.