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Making Mathematica Functions Evaluate Rapidly?

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  • Subject: [mg2206] Making Mathematica Functions Evaluate Rapidly?
  • From: siegman at (A. E. Siegman)
  • Date: Mon, 16 Oct 1995 11:53:04 -0400
  • Organization: Stanford University

I frequently want to evaluate (plot, numerically integrate, calculate
numerical moments of) functions which start out complex, e.g., something

      f[x,y,z] = (1/ f1[x,y,z] ) Exp[ f2[x,y,z] ]

where f1[x,y,z] and f2[x,y,z] may contain various purely real
coefficients, call 'em a,b,c,... which have fixed (predefined) values, as
well as the variables x,y,z , and some explicit I's (the imaginary unit
I), and some standard functions, e.g., Cos[], Sin[], etc.. In other words,
all input values are purely real, and all I's are explicit.

Then, what I really want to do is to generate purely real outputs, either 

      g1[x,y,z] = Abs[f[x,y,z]] 


      g2[x,y,z] = Abs[f[x,y,z]]^2 

in a form that will evaluate as rapidly as possible (on a PowerMac).

What's the best way to do this?  Should I define the initial functions
using = or := ?  At what stage should I compile (and how)?  Can I compile
a function that contains other functions that have already been compiled?

I sometimes seem to get very small complex values coming out of the
compiled versions of g1 or g2 (i.e., the Abs[] functions), even though
I've used _Real on x,y,z in the compilation, which makes me think they may
being evaluated with complex values even though everything is supposedly
real.  Do I need to use ComplexExpand somehow on f[x,y,z] to separate it
into Re and Im parts, then square them independently to get Abs[]^2 ?

And finally, what if Pi or Sqrt[2] or ... are also contained in the
functions f1, f2 ?  Do I need to do something to force those into
numerical form also?

(My experience is that depending on just how I set up a calculation like
the above I get wildly different speeds -- but there seems to be no
systematic way to know how to get the fastest evaluation.)

Email replies to siegman at appreciated  -- thanks.

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