We present a method for automatic inference of conditions on the initial states of a program that guarantee that the safety assertions in the program are not violated. Constrained Horn clauses (CHCs) are used to model the program and assertions in a uniform way, and we use standard abstract interpretations to derive an over-approximation of the set of unsafe initial states. The precondition then is the constraint satisfied by the complement of that set under-approximating the set of safe initial states. This approach has been used before but demonstrated only on small transition systems. In this paper, we develop an iterative refinement algorithm for non-linear CHCs, and show that it produces much more precise, and in some cases optimal approximations of the safety conditions, and can scale to larger programs. The refinement algorithm uses partial evaluation and a form of counterexample-guided abstraction refinement techniques to focus on the relevant program states. Disjunctive constraints, which are essential to achieve good precision, are generated in a controlled way. The algorithm is implemented and tested on a benchmark suite of programs from the literature in precondition inference and software verification competitions.