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In the past few years, new very successful bidirectional heuristic search algorithms have been proposed. Their key novelty is a lower bound on the cost of a solution that includes information from the g values in both directions. Kaindl and Kainz (1997) proposed measuring how inaccurate a heuristic is while expanding nodes in the opposite direction, and using this information to raise the f value of the evaluated nodes. However, this comes with a set of disadvantages and remains yet to be exploited to its full potential. Additionally, Sadhukhan (2013) presented BAE∗, a bidirectional best-first search algorithm based on the accumulated heuristic inaccuracy along a path. However, no complete comparison in regards to other bidirectional algorithms has yet been done, neither theoretical nor empirical. In this paper we define individual bounds within the lower-bound framework and show how both Kaindl and Kainz's and Sadhukhan's methods can be generalized thus creating new bounds. This overcomes previous shortcomings and allows newer algorithms to benefit from these techniques as well. Experimental results show a substantial improvement, up to an order of magnitude in the number of necessarily-expanded nodes compared to state-of-the-art near-optimal algorithms in common benchmarks.