It is now widely admitted that in order to provide state-of-the-art accuracy, a conditional branch predictor must combine several predictions. Recent research has shown that an adder tree is a very effective approach for the prediction combination function. In this paper, we present a more cost effective solution for this prediction combination function for predictors relying on several predictor components indexed with different history lengths. Using geometric history length as the O-GEHL predictor, the TAGE predictor uses (partially) tagged components as the PPM-like predictor. TAGE relies on (partial) hit-miss detection as the prediction computation function. TAGE provides state-of-the-art prediction accuracy on conditional branches. In particular, at equivalent storage budgets, the TAGE predictor significantly outperforms all the predictors that were presented at the Championship Branch Prediction in december 2004. The accuracy of the prediction of the targets of indirect branches is a major issue on some applications. We show that the principles of the TAGE predictor can be directly applied to the prediction of indirect branches. The ITTAGE predictor (Indirect Target TAgged GEometric history length) significantly outperforms previous state-of-the-art indirect target branch predictors. Both TAGE and ITTAGE predictors feature tagged predictor components indexed with distinct history lengths forming a geometric series. They can be associated in a single cost-effective predictor, sharing tables and predictor logic, the COTTAGE predictor (COnditional and indirect Target TAgged GEometric history length).