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Interval constraint solvers use local consistencies---among which one worth mentioning is box consistency---for computing verified solutions of real constraint systems. Though among the most efficient ones, the algorithm for enforcing box consistency suffers from the use of time-consuming operators. This paper first introduces box varphi consistency, a weakening of box consistency; this new notion then allows us to devise an adaptive algorithm that computes box consistency by enforcing box varphi consistency, decreasing the varphi parameter as variables' domains get tightened, then achieving eventually box0 consistency, which is equivalent to box consistency. A new propagation algorithm is also given, that intensifies the use of the most contracting pruning functions based on box varphi consistency. The resulting algorithm is finally shown to outperform the original scheme for enforcing box consistency on a set of standard benchmarks.