Re: ComplexityFunction affects set of transformations tried by Simplify

• To: mathgroup at smc.vnet.net
• Subject: [mg71530] Re: ComplexityFunction affects set of transformations tried by Simplify
• From: "dimitris" <dimmechan at yahoo.com>
• Date: Wed, 22 Nov 2006 05:22:12 -0500 (EST)
• References: <200611202311.SAA18396@smc.vnet.net><ejuqdp\$htk\$1@smc.vnet.net>

```\$VersionNumber
5.2

I got also the same results;

Regards
Dimitris

Andrzej Kozlowski wrote:
> On 21 Nov 2006, at 08:11, Schochet wrote:
>
> > Oops! I originally sent this as a reply to an unrelated thread.
> >
> > As part of a large computation, I wanted Mathematica to
> > simplify expressions of the form   Sqrt[c + d/k](c k + d)2,
> > where k is positive, to   (c k + d)^(5/2)/Sqrt[k]
> >
> > Note that Mathematica knows that the two expressions
> > are equivalent:
> >
> > In[1]:=expr1 = Sqrt[c + d/k](c k + d)2;
> > expr2 = (c k + d)^(5/2)/Sqrt[k];
> > Simplify[expr1 == expr2, k > 0]
> >
> > Out[1]=True
>
> Rather unlikely, I would say. Consider that my Mathematica 5.1 gives:
>
> expr1 = Sqrt[c + d/k]*(d + c*k)*2;
> expr2 = (d + c*k)^(5/2)/Sqrt[k];
>
>
> Simplify[expr1 == expr2, k > 0]
>
>
> (d + c*k - 2)*Sqrt[d + c*k] == 0
>
> furthermore
>
> expr1 /. {d -> 1, c -> 0, k -> 1}
>
>
> 2
>
>
> expr2 /. {d -> 1, c -> 0, k -> 1}
>
>
> 1
>
> It kind of makes the rest of the message less interesting but...
>
> >
> > However Simplify with LeafCount as the ComplexityFunction
> > leaves expr1 unchanged, while Simplify with Depth as the
> > ComplexityFunction changes expr1 to expr2:
> >
> > In[3]:=Simplify[expr1, k > 0,
> > ComplexityFunction -> LeafCount] // InputForm
> >
> > Out[3]//InputForm=Sqrt[c + d/k]*(d + c*k)2
> >
> > In[4]:=Simplify[expr1, k > 0,
> > ComplexityFunction -> Depth] // InputForm
> >
> > Out[4]//InputForm=(d + c*k)^(5/2)/Sqrt[k]
> >
> > What explains this difference in behavior? Changing the order
> > of calculations 3 and 4 has no effect.  Since  calculation 2
> > shows that the form expr2 would be preferred if it were tried,
> > while calculation 4 shows that the form expr2 is tried when
> > the ComplexityFunction is Depth, it seems that the choice of
> > ComplexityFunction affects the set of transformations tried
> > by Simplify.
>
> Here I get:
>
>
> In[4]:=
> Simplify[expr1, k > 0, ComplexityFunction -> LeafCount]
>
> Out[4]=
> (2*(d + c*k)^(3/2))/Sqrt[k]
>
> In[6]:=
> Simplify[expr1, k > 0, ComplexityFunction -> Depth]
>
> Out[6]=
> (2*(d + c*k)^(3/2))/Sqrt[k]
>
>
> So, the way I see it there are two possibilities. One is, that there
> was something funny with the state of your Mathematica when you got
> those results. If not, then I am  pleased that I have not upgraded
> from version 5.1 to 5.2 ;-)
>
> Andrzej Kozlowski
> Tokyo, Japan

```

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