Symbolic regression via genetic programming is a flexible approach to machine learning that does not require up-front specification of model structure. However, traditional approaches to symbolic regression require the use of protected operators, which can lead to perverse model characteristics and poor generalisation. In this paper, we revisit interval arithmetic as one possible solution to allow genetic programming to perform regression using unprotected operators. Using standard benchmarks, we show that using interval arithmetic within model evaluation does not prevent invalid solutions from entering the population, meaning that search performance remains compromised. We extend the basic interval arithmetic concept with `safe' search operators that integrate interval information into their process, thereby greatly reducing the number of invalid solutions produced during search. The resulting algorithms are able to more effectively identify good models that generalise well to unseen data. We conclude with an analysis of the sensitivity of interval arithmetic-based operators with respect to the accuracy of the supplied input feature intervals.