Re: Beware of Simplify[] bottlenecks in v 3.0

*To*: mathgroup at smc.vnet.net*Subject*: [mg8987] Re: [mg8925] Beware of Simplify[] bottlenecks in v 3.0*From*: David Withoff <withoff>*Date*: Tue, 7 Oct 1997 03:35:33 -0400*Sender*: owner-wri-mathgroup at wolfram.com

> After installing v 3.0 on my Max 8500/120 I have done some side > by side timing tests with old Notebooks comparing the speed of > Mma 2.2 vs 3.0. Generally v. 3.0 is 10% to 60% slower in most cases, > but ocassionally I found it to be an order of magnitude slower. > I traced the difference to the frequent use of the Simplify function. > This used to be a bottleneck in 2.2, and can be even more so in 3.0. > > The following code fragment, extracted from an old Notebook, > illustrates the dramatic speed difference: > > fp=(a*(2*a + a*Cos[phiQ])*(2*a*Cos[phiQ]*(1 - Cos[theta]) + > a*(1 - Cos[phiQ]*Cos[Pi/8]*Cos[theta] - Sin[phiQ]*Sin[Pi/8])))/ > (2*2^(1/2)*(5*a^2 - (2*a + a*Cos[phiQ])*(2*a + a*Cos[Pi/8])*Cos[theta] + > a*(2*a*(Cos[phiQ] + Cos[Pi/8]) - a*Sin[phiQ]*Sin[Pi/8]))^(3/2)); > Print[Timing[fp=Simplify[fp]]]; > > Mma 2.2: 15.75 Seconds > Mma 3.0: 140.517 Seconds - same result > >I tried to limit the operation to 15 seconds to see if 3.0 can return the > same answer quicker: > > Print[Timing[fp=Simplify[fp,TimeConstraint->15]]]; > > but the option is ignored: > > Mma 3.0: 142.417 Seconds > > The purpose of this post is to warn users converting to 3.0 about > such bottlenecks, and that they are uncontrollable until the > TimeConstraint option works. The TimeConstraint option limits the amount of time for individual transformations within Simplify, not the total time. On my computer I was able to use the TimeConstraint option to cut the simplification time in half. Here is the timing that I found using Mathematica Version 3.0.1 for Windows on my 100 MHz Pentium computer (which is apparently quite a bit slower than your computer): In[2]:= Timing[simp = Simplify[fp]] 3 Out[2]= {319.28 Second, -(a (2 + Cos[phiQ]) Pi (-1 + Cos[phiQ] (-2 + (2 + Cos[--]) Cos[theta]) + 8 Pi Sin[phiQ] Sin[--])) / 8 2 Pi (2 Sqrt[2] Power[a (5 + 2 (Cos[phiQ] + Cos[--]) - 8 Pi (2 + Cos[phiQ]) (2 + Cos[--]) Cos[theta] - 8 Pi Sin[phiQ] Sin[--]), 3/2])} 8 This result is significantly simpler (as measured by LeafCount) that the original expression: In[3]:= LeafCount[fp] Out[3]= 116 In[4]:= LeafCount[simp] Out[4]= 90 I got a similar result in about half the time by including the TimeConstraint option: In[9]:=Timing[LeafCount[Simplify[fp, TimeConstraint -> 1]]] Simplify::time: Time spent on a transformation exceeded 1 seconds, and the transformation was aborted. Increasing the value of TimeConstraint option may improve the result of simplification. Out[9]= {151.7 Second, 92} This can be compared with the result using Mathematica Version 2.2.3 on the same computer. A result is returned about four times faster in that version, but there is almost no simplification. (I did not observe the factor of nine speed difference that you reported. Perhaps you were using a different version of Mathematica.) In[2]:= Timing[simp = Simplify[fp]][[1]] Out[2]= 80.728 Second In[3]:= LeafCount[simp] Out[3]= 113 The Simplify function in Version 3.0 will be slower for some examples because it tries more transformations. In this example, Version 3.0 finds some simplifications that Version 2.2 didn't find. You can use the TimeConstraint option in Version 3.0 to exclude the transformations that use the most time. If you are concerned about speed, you can get considerable speed improvement (and the same result) by turning off the trigonometric transformations. Here is a timing in Version 3.0.1: In[10]:= LeafCount[Simplify[fp, Trig -> False]] //Timing Out[10]= {11.15 Second, 90} The Simplify function applies various transformations to the expression and to every subexpression, and returns the simplest result that it finds. If you are even more concerned about speed, you may want to be more selective in choosing transformations. You can get the simplifications in this example in a fraction of a second with careful application of Collect and FactorSquareFree. Dave Withoff Wolfram Research

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