What voltage do you measure while implementing the charge algorithm?
The state transition from pre-charge to constant current (fast) charge takes place when the battery voltage first reaches a preset voltage with the pre-charge current flowing. 

The state transition from fast charge to constant voltage takes place at when the battery voltage first reaches the termination voltage, with the fast charge current flowing. 

The constant voltage charge terminates when the charge current drops to the taper current threshold. Obviously, the termination voltage must be applied for the current to flow. 

(Of course there are usually time, temperature and other ways to terminate early if the battery is acting unhealthy)

You might find looking at the data sheets for charging chips useful, but there is no substitute for the detailed recommendations from your battery manufacturer. If you can't get that, try to find a similar battery from a reputable manufacturer.