Global precision setting

How do I tell mathematica that all numbers e.g. 1.5 are actually 20 Digits precision? SetPrecision on all numbers or add the `20 everywhere?

You could force this with $PreRead. This naive definition is likely inefficient and probably breaks a number of corner cases I have not considered, but here is a rough demonstration:

$PreRead = (# /. 
     s_String /; 
       StringMatchQ[s, NumberString] && 
        Precision@ToExpression@s == MachinePrecision :> s <> "`20." &);

3/1.5 + Pi/7

Precision[%]

2.4487989505128276055

20.0879

As Alexey notes this breaks if the machine number string already has a "NumberMark" after it e.g. 1.23`. One could use a more precise string replacement to avoid this.

A different approach is to process at the expression rather than box level, though this simple first attempt probably fails in some cases as well:

$Pre = Function[Null, 
  Unevaluated[#] /. r_Real?MachineNumberQ :> RuleCondition@SetPrecision[r, 25], 
  HoldAllComplete]

Now:

MachineNumberQ[2.2]
ToString[3.14]

False

"3.140000000000000124344979"

There is a quick-n-dirty solution. Set

$MinPrecision = 100

And then enter numbers something like

x = 1.01`2;

You will be getting warnings as

Precision::precsm: Requested precision 2.` is smaller than $MinPrecision.
    Using $MinPrecision instead.

but in this way you if you want to change precision you just change $MinPrecision value.

In[21]:= x

Out[21]= 1.\
0100000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000