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Expression timing; a black art?

  • To: mathgroup at
  • Subject: [mg63234] Expression timing; a black art?
  • From: AES <siegman at>
  • Date: Tue, 20 Dec 2005 04:19:30 -0500 (EST)
  • Organization: Stanford University
  • Sender: owner-wri-mathgroup at

OK, to follow up on my own recent post, I did some timing tests for the 
function I asked about earlier, as a function or as a Module[] with a 
repeated square root pulled out.  To say the results are puzzling (to 
me, anyway) is putting it mildly.

Approach: Create a notebook with three sections, each of the form:

   x; Remove["Global`*"];
   fn :=  (function as below)
   a=0.12; xmax=3.5; dx=0.1;
   tn=Timing[Table[{x, fn[a,x]//N}, {x,0,xmax,dx}]]; 

where the three functions "fn" are:

f1[a_,x_] := Sum[Exp[-(Pi a n)^2-
   ((x-n Sqrt[1-(Pi a^2)^2])/a)^2],

f2[a_,x_] := Module[{b},      
   b=Sqrt[1-(Pi a^2)^2];      
   Sum[Exp[-(Pi a n)^2-((x-n b)/a)^2],

f3[a_,x_] := Module[{b,c1,c2},      
   b=Sqrt[1-(Pi a^2)^2];      
   c1=(Pi a)^2;      
   Sum[Exp[-c1 n^2-c2(x-n b)^2],

Summary of results to date:

1)  Open Mathematica, run notebook first time with functions in f1, f2, 
f3 order.  Timings are (in round numbers, +/-10%) t1 = 30 sec, t2 = 5 
sec, t3 = 20 sec.

2)  Re-run same notebook from top:  Timings are now 5 sec, 5 sec, 5 sec.  
Clearly Mathematica is remembering *something* from the first run, despite the 
Remove[Global] in each section . . . ?

3)  Quit Mathematica, re-Open, reorder sections in f2, f1, f3 order.  Timings on 
first run are now t2 = 30 sec, t1 = 5 sec, t3 = 20 sec; timings on 
second run are again 5, 5, 5 sec.  

4)  Quit Mathematica, re-Open, reorder sections in f3, f2, f1 order.  Timings on 
first run are 30, 20, 5 sec.

Conclusion #1:  Running *either* f1 or f2 once leaves something (?) in 
the kernel that greatly speeds up the f2 or f1, and gives a little help 
to f3.  Running f3 first gives a little help to f2 (30 down to 20), and 
probably also to f1 (didn't try), but doesn't push it all the way down 
to 5.

Conclusion #2:  Using modular form with Sqrt[] pulled out doesn't help 
at all.

Conclusion #3:  If a naive user like me had only done the very first 
test above, I'd have been left believing that pulling the Sqrt[] out 
*did* help.

Conclusion #4:  Trying to understand Mathematica timing is a very black art.

Hypothesis:  Running any of these functions on a *random* set of values 
the first time, then another random set the second time, will *not* 
speed up the second run for either the same fn or any of the other ones.  
Anyone want to predict if this is so?

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