FullSimplify results

*To*: mathgroup at smc.vnet.net*Subject*: [mg12800] FullSimplify results*From*: Ersek_Ted%PAX1A at mr.nawcad.navy.mil*Date*: Fri, 12 Jun 1998 04:05:32 -0400*Sender*: owner-wri-mathgroup at wolfram.com

Consider the following: In[1]:= FullSimplify[ -E^(-x) ]//InputForm Out[1]//InputForm= -Cosh[x] + Sinh[x] In[2]:= { LeafCount[-Cosh[x]+Sinh[x]], LeafCount[ -E^(-x) ] } Out[2]= {7, 7} Note: The output of FullSimplify has the same LeafCount as the expression it started with. How does FullSimplify decide that it likes (-Cosh[x]+Sinh[x]) better? The online help for ComplexityFunction indicates: "With the default setting ComplexityFunction->Automatic, forms are ranked primarily according to their LeafCount, with corrections to treat integers with more digits as more complex". There has to be more to it than that because that doesn't explain the result above. I came up with a ComplexityFunction (see in In[3] below) that will often give the same result as the default ComplexityFunction, but it sometimes gives a result that I like a little better. My ComplexityFunction adds a little bit of complexity for each symbol in the expression. NonNumeric symbols are considered a bit more complex than Numeric symbols. In[3]:= SetOptions[FullSimplify, ComplexityFunction->(LeafCount[#]+ 0.0001*Count[#,var_Symbol/;Not[NumericQ[var]],Infinity]+ 0.00006*Count[#,var_Symbol/;NumericQ[var],Infinity]&)]; In[4]:= FullSimplify[-Cosh[x]+Sinh[x]]//InputForm Out[4]//InputForm= -E^(-x) In[5]:= FullSimplify[-E^(-x)]//InputForm Out[5]//InputForm= -E^(-x) In the lines above we see FullSimplify now prefers (-E^(-x)) over (-Cosh[x]+Sinh[x] ). The online help ComplexityFunction has corrections to treat integers with more digits as more complex. I haven't come up with a case where that would be relevant. If anyone can come up with an example please let me know. Can anyone demonstrate that the default ComplexityFunction is (in some sense) better than the one I give above? Does anyone want to propose their own ComplexityFunction? Ted Ersek

**Follow-Ups**:**Re: FullSimplify results***From:*Daniel Lichtblau <danl>