Recent analysis of classical algorithms resulted in their axiomatization as transition systems satisfying some simple postulates, and in the formulation of the Abstract State Machine Theorem, which assures us that any classical algorithm can be emulated step-by-step by a most general model of computation, called an ``abstract state machine''. We refine that analysis to take details of intra-step behavior into account, and show that there is in fact an abstract state machine that not only has the same state transitions as does a given algorithm but also performs the exact same tests on states when determining how to proceed to the next state. This enhancement allows the inclusion -- within the abstract-state-machine framework -- of algorithms whose states only have partially-defined equality, or employ other native partial functions, as is the case, for instance, with inversion of a matrix of computable reals.