Question

So Mathematica is different from other dialects of lisp because it blurs the lines between functions and macros. In Mathematica if a user wanted to write a mathematical function they would likely use pattern matching like f[x_]:= x*x instead of f=Function[{x},x*x] though both would return the same result when called with f[x]. My understanding is that the first approach is something equivalent to a lisp macro and in my experience is favored because of the more concise syntax.

So I have two questions, is there a performance difference between executing functions versus the pattern matching/macro approach? Though part of me wouldn’t be surprised if functions were actually transformed into some version of macros to allow features like Listable to be implemented.

The reason I care about this question is because of the recent set of questions (1) (2) about trying to catch Mathematica errors in large programs. If most of the computations were defined in terms of Functions, it seems to me that keeping track of the order of evaluation and where the error originated would be easier than trying to catch the error after the input has been rewritten by the successive application of macros/patterns.

Answer 1

The way I understand Mathematica is that it is one giant search replace engine. All functions, variables, and other assignments are essentially stored as rules and during evaluation Mathematica goes through this global rule base and applies them until the resulting expression stops changing.

It follows that the fewer times you have to go through the list of rules the faster the evaluation. Looking at what happens using Trace (using gdelfino‘s function g and h)

In[1]:= Trace@(#*#)&@x
Out[1]= {x x,x^2}
In[2]:= Trace@g@x
Out[2]= {g[x],x x,x^2}
In[3]:= Trace@h@x
Out[3]= {{h,Function[{x},x x]},Function[{x},x x][x],x x,x^2}

it becomes clear why anonymous functions are fastest and why using Function introduces additional overhead over a simple SetDelayed. I recommend looking at the introduction of Leonid Shifrin’s excellent book, where these concepts are explained in some detail.

I have on occasion constructed a Dispatch table of all the functions I need and manually applied it to my starting expression. This provides a significant speed increase over normal evaluation as none of Mathematica’s inbuilt functions need to be matched against my expression.