There are two measures applied to the vertical and to the horizontal directions. Calculations are identical, and if the results are different, one should take the largest value that is the least favorable one for defining precision. The measures are expressed in the units of the original map, be they degrees or meters.
The general expression is: (bound Max - bound Min) / 2 billion
Example 1: an UTM map bounded in the X direction to 0 and 2,000,000 meters will have an internal X precision of 2 million / 2 billion meters, for example, 1 millimeter. If its Y bounds were 3,000,000 to 5,000,000 meters, the Y precision will also be 1 millimeter.
Example 2: a Long\Lat "unbounded" as a definition of 1 millionth of a degree; that gives 11.11 cm in the Y direction and 11.11*cos(lat) in the X.
Example 3: a Long\Lat bounded to 20 and 40 degrees North and -90 and -70 in Longitude has a precision of 1/100 million of a degree, 100 times superior to the unbounded one.