If you make a definition such as f[x_]:=value, the Wolfram Language will use the value you give for any f function it encounters. In some cases, however, you may want to define a value that is to be used specifically when you ask for numerical values.
define a numerical value to be used when n‐digit precision and any accuracy is requested
Defining numerical values that depend on numerical precision.
This defines a numerical value for the symbol const, using 4n+5 terms in the product for n‐digit precision:
Here is the value of const, computed to 30‐digit precision using the value you specified:
The Wolfram Language treats numerical values essentially like upvalues. When you define a numerical value for f, the Wolfram Language effectively enters your definition as an upvalue for f with respect to the numerical evaluation operation N.