The optimization of arithmetic circuits has always been essentially a manual task: arithmetic experts study the best architectures for arithmetic components and write libraries of generators, and designers instantiate library components and rely on logic synthesizers to obtain good implementations. In this paper we look at the capabilities of commercial synthesizers when it comes to arithmetic circuits, and observe that they are essentially unable to switch from one arithmetic architecture to another (e.g., from a ripple-carry to a carry-lookahead adder). Therefore, users relying on logic synthesis miss most optimization potentials. We therefore investigate algorithms for factorization which can prepare structured VHDL or Verilog for synthesizers to implement, and show first steps into pruning the search space from many irrelevant or equivalent solutions. Our results are still very limited in complexity but we show that our techniques successfully concentrate on the automatic exploration of very different solutions, and discover architectures known and unknown to expert designers, such as different types of adders, the carry-save representation, or improved multipliers. This is a first step toward a class of arithmetic optimizers which sit on top of classic logic synthesizers.