"floating point arithmetic" is a terminology that refer to the arithmetic perform over (finite) representation of real number. See the wikipedia [article](https://en.wikipedia.org/wiki/Floating-point_arithmetic) for more details. 


In the [formal specification](https://en.wikipedia.org/wiki/IEEE_754)  of floating point arithmetic (that should be used by all the major programming languages), it is specified that a "Not A number" (NaN) value should be a number.


If we abstract the fact that floating point arithmetic care only of finitely many values and abstract a bit, we get a sort of arithmetic arithmetic over $\mathbb{R}\cup\{\textrm{NaN}\}$ with $\textrm{NaN}$ being a zero for any arithmetic operation. Formally, for all $x$,  

$$x\times \textrm{NaN} = x + \textrm{NaN} = \frac{x}{0} = \frac{0}{0}= \cdots =\textrm{NaN}$$


Is there any algebraic structures $(E,+,\times)$ having axioms allowing this or is it just to arbitrary to have been introduced even in weird part of algebra? 



Terminology: algebraic structure for "floating point" arithmetic
The abstract picture here is that of adjoining an undefined or bottom element to a (partial) algebra. Doing this to a total algebra is boring, but it is a useful trick to turn partial algebras into total ones.


One just picks an element $\bot$ not in the carrier set, and extends the operations as follows: Any operation applied to a tuple outside its original domain yields $\bot$, including any tuple containing a $\bot$ somewhere.


The structure you are describing is then just the $1$-point totalirization of the partial algebra $(\mathbb{R},+,\times,/)$. The algebras arising in such a manner will hardly ever have very familiar properties. In particular, I don't think people name them directly, and would rather speak about the underlying partial algebras.


