Order of operations, precedence in Mathematica [duplicate]

Question

I get confused in the order of operations in Mathematica. For example,

f /@ 10^{1, 2, 3}
{10, 100, 1000}

In my head, that should be

{f[10], f[100], f[1000]}

After using FullForm, I see that my command should be regarded as : (f /@ 10)^{1, 2, 3}. In fact, () is not often used in Mathematica code, and I'm always not sure about the order of the expression evaluation. And I can't find the information about this order.

For example, in C: http://en.wikipedia.org/wiki/Order_of_operations

On MathWorld, it is just the basic : http://mathworld.wolfram.com/Precedence.html

1. Parenthesization,

2. Factorial,

3. Exponentiation,

4. Multiplication and division,

5. Addition and subtraction.

Another example: let look at the order of this:

x + x /. x -> y
2 y

x + (x /. x -> y)
x + y

And it takes time to understand this:

x + y /. x -> 1 + y + x /. x -> 5
6 + 2 y

Without the order of operations, one can get confused by interpret this expression, and of course, we can use ().

((x + y) /. x -> 1) +(( y + x) /. x -> 5)  
x + y /. x -> (1 + y + x) /. x -> 5 

Another example, one can get confused:

Cases[{1, 2, 3}, _?#1 < 2 &]

or

Cases[{1, 2, 3}, _?(#1 < 2) &]

or

Cases[{1, 2, 3}, _?(#1 < 2 &)]

Another examples, which one & or /. is more privileged ?

x /. x -> y + #1 &

x + #1 & /. x -> y

I really appreciate any rule of thumbs or a guide of the operation ordered: /@, @@, _?, /., ->,& ...

Answer 1

You can check the precedence using Precedence. So for you first example it works that way because Precedence[Power] < Precedence[Map]. You have furthermore that Precedence[Function] < Precedence[ReplaceAll] for one of your later queries.

When you get confused and need to tell what the precedence is the fastest way though is to use FullForm and Hold. There are variations of this, I forget which one is considered best practice, but it works like this:

FullForm[Hold[f /@ 10^{1, 2, 3}]]
(* Out: Hold[Power[Map[f,10],List[1,2,3]]] *)