Rule-based explanations are a popular method to understand the rationale behind the answers of complex machine learning (ML) classifiers. Recent approaches, such as Anchors, focus on local explanations based on if-then rules that are applicable in the vicinity of a target instance. This has proved effective at producing faithful explanations, yet anchor-based explanations are not free of limitations. These include long overly specific rules as well as explanations of low fidelity. This work presents two simple methods that can mitigate such issues on tabular and textual data. The first approach proposes a careful selection of the discretization method for numerical attributes in tabular datasets. The second one applies the notion of pertinent negatives to explanations on textual data. Our experimental evaluation shows the positive impact of our contributions on the quality of anchor-based explanations.