Accurately
Vine's Expository Dictionary of NT Words [1]
is correctly translated in the RV of Luke 1:3 , "having traced the course of all things accurately" (AV, "having had perfect understanding"). It is used in Matthew 2:8 , of Herod's command to the wise men as to searching for the young Child (RV, "carefully;" AV, "diligently"); in Acts 18:25 , of Apollos' teaching of "the things concerning Jesus" (RV, "carefully;" AV, "diligently"); in Ephesians 5:15 , of the way in which believers are to walk (RV, "carefully;" AV, "circumspectly"); in 1—Thessalonians 5:2 , of the knowledge gained by the saints through the Apostle's teaching concerning the Day of the Lord (RV and AV, "perfectly"). The word expresses that "accuracy" which is the outcome of carefulness. It is connected with akros, "pointed."
CarefulDiligentExactlyPerfect.
Webster's Dictionary [2]
(adv.) In an accurate manner; exactly; precisely; without error or defect.