Reverend John Hale is a learned man, who has spent numerous years studying witchcraft and desires to be the best at diagnosing and eradicating witchcraft from the world. His mission in Salem is to help the people 'Fight the devil' and work out whether witchcraft exists in their community and take appropriate action to deal with it. He starts out with good intentions, but by the end of the play has questioned his own beliefs and values and the validity of them and discovered corruption in the trials at Salem.
The 'Devil's work is precise' basically means that Hale believed the Devil had made himself known and that it would be easy to recognise the work of the Devil. Hale believed that it would be clear to him and to other learned men, what true witchcraft was, and that there would be no doubt in the end as to who was telling the truth and who was not.
The worst thing a Puritan community could discover was sin in its midst, and the worst kind of sin would have been witchcraft because it was the mark of the devil. In Arthur Miller's The Crucible, when Reverend Parris has a sense that there might be something happening in his town, Salem, regarding wtchcraft, he sends for Reverend Hale. (It is interesting to note that while Reverend Parris is acrually afraid there is some kind of witchery afoot because he saw the girls in the forest last night, he is equally afraid that Hale will find witchcraft. This is Parris's worst nightmare--he either has rebellious children in his house who are going to tarnish his reputation or the devil has not only invaded his town and his church but his own home. He is one frightened man.)
Reverend Hale is a serious man with a reputation for taking the things of the devil seriously.
Mr. Hale is nearing forty, a tight-skinned, eager-eyed intellectual. This is a beloved errand for him; on being called here to ascertain witchcraft he felt the pride of the specialist whose unique knowledge has at last been publicly called for.
He has studied and knows that the “[w]e cannot look to superstition in this. The Devil is precise.” That means that he has studied the devil's ways and knows that there are certain signs and signals which illuminate his presence and working. He does want anyone to make rash assumptions about things and blame the devil for things for which he is not to blame. Ironically, of course, that is exactly what Hale himself is fooled into doing for most of the play.
A good example of this is his conversation with Tituba. He asks pointed questions which he thinks will tell him whether or not the devil is working here in Salem.
When the Devil comes to you does he ever come - with another person?... Perhaps another person in the village? Someone you know.
Unfortunately, his questions are leading, and the frightened slave says what she must to keep from being punished. The tragedy surrounding Reverend Hale is that he comes to Salem to seek truth and to help identify and rid the town of the devil's work; however, he is unable to draw accurate conclusions because people are not honest with him and so often do not tell the truth about anything.
Sadly, what Hale comes to realize is that the work of the devil takes many forms, including the lies of a bunch of young girls who are at first too scared of their parents to tell the truth and are then too scared of Abigail Williams to tell the truth. We do not know this for certain, but it is likely that Reverend Hale's view of witchcraft, though perhaps not his view of the devil, changed significantly after his experience in Salem. Nothing in his books or studies could have prepared him for what the devil did there.