SetPrecision within Block

Question

I am reading Mathematica Cookbook, chapter 1.

Author gives two examples, with the following explanation

You can control precision within a complex calculation (without using N[] on every intermediate result) by changing these values; however, you should only do so within a Block (a local context). For example, compare the difference between a calculation with automatic precision for intermediate results to the same calculation with fixed precision (obtained by making $MinPrecision == $MaxPrecision ).

In[1]:= SetPrecision[(1 + Exp[Sqrt[2] + Sqrt[3]])/2^25, 32]

Out[1]= 7.226780742612584668840452114476*10^-7

In[2]:= 
   Block[{$MinPrecision = 32, $MaxPrecision = 32}, 
     SetPrecision[(1 + Exp[Sqrt[2] + Sqrt[3]])/2^25, 32]]

Out[2]= 7.2267807426125846688404521144759*10^-7

Why two results are different, despite the fact that precision was set to 32 digits in both expression ? And what is "automatic precision" ?
Edit: user "rcollyer" points out that

running Precision on the two results gives 31.5935 and 32..

Answer 1

I don't think that SetPrecision[x,p] simply returns a version of x with precision p. Rather, it sets all numeric types in x to precison p and then evaluates x. This is important, if we want SetPrecision to work on symbolic arguments. For example,

SetPrecision[2 x, 4]

2.000 x

Thus, the following computation need not yield 4:

Precision[SetPrecision[Sqrt[2], 4]]

4.30103

In fact, this is equivalent to

Precision[Sqrt[SetPrecision[2, 4]]]

4.30103

Of course, Sqrt[SetPrecision[2, 4]] is subject to precision tracking using significance arithmetic. Thus, we don't expect it's precision to be 4.

On the other hand, setting $MaxPrecision==$MinPrecision forces all results to have this common value.