[Interval arithmetic](https://en.wikipedia.org/wiki/Interval_arithmetic) normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]+[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as $$\{a\}\_\delta=[a(1-\delta);a(1+\delta)]$$ or $$[a]\_\epsilon := \left[ae^{-\epsilon};ae^\epsilon\right]$$ which base on some kind of relative error around a number $a$. For example for the later definition one has the rule $$[a]\_\epsilon\cdot[b]\_\delta=[ab]\_{\epsilon+\delta}$$


**My Question:** Can you point me to textbooks/papers where interval arithmetic with different interval definitions such as the two above is discussed? I am looking for those interval definitions which base on some kind of relative error so that multiplication with those intervals is easy.


**Note:** It may help, that $$\begin{align}
[a]\_\epsilon &= \left\{ y \in \mathbb R^{+}: \left\|\frac{y}{a}\right\|\_{\log} = |\log(y)-\log(a)| < \epsilon \right\} \\
&= \left[ae^{-\epsilon};ae^{\epsilon}\right] \end{align}$$ whereby $\left\|\frac{y}{a}\right\|\_{\log}$ is the so called [log ratio distance](https://math.stackexchange.com/questions/1411899/name-for-some-kind-of-logarithmic-norm-error).



Interval arithmetic with different definitions of intervals
I know of a variant of interval arithmetic called ball arithmetic. It seems as though it may be like what you're looking for. Ball arithmetic is currently implemented in the C library, Arb.


Here is a link to a paper about this sort of arithmetic. Also, it discusses the differences between ball arithmetic and interval arithmetic.


<http://www.texmacs.org/joris/ball/ball-abs.html>


Also, you may be able to find what you're looking for at <http://www.cs.utep.edu/interval-comp/books.html>. Which contains a list of books on interval arithmetic.


Also, take a look at <http://fredrikj.net/blog/2012/04/high-precision-ball-arithmetic/>. This is extremely similar to what you wrote.


I can't find any other explicit definitions, but it shouldn't be hard to formulate ball arithmetic over general normed vector spaces.


