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simplifying expressions - How to find the most compact form of an equation


I would like to know whether there is any general approach to obtain the most compact form of an equation with Mathematica.


I used to believe that the FullSimplify command would do that for me, but I recently found out that it will not necessarily give the most compact form. I found out with the following equation, which has been FullSimplified:


$$\left[\sec (\alpha ) \left(\coth \left(L \sqrt{\text{Bo} \cos (\alpha )}\right)+\text{csch}\left(L \sqrt{\text{Bo} \cos (\alpha )}\right) (\text{Bo} L R \sin (\alpha )-1)-R \sin (\alpha ) \sqrt{\text{Bo} \sec (\alpha )}\right)\right]\left(R \sqrt{\text{Bo} \sec (\alpha )}\right)^{-1}$$


Or, in inputform for your convenience:



eq1 = (Sec[α] (Coth[L Sqrt[Bo Cos[α]]] - R Sqrt[Bo Sec[α]] Sin[α] + 
      Csch[L Sqrt[Bo Cos[α]]] (-1 + Bo L R Sin[α])))/(R Sqrt[Bo Sec[α]])

However, if I now run the following:


Collect[eq1,Bo] //FullSimplify

I get a much more compact form, namely: $$ -\tan (\alpha )+L \sin (\alpha ) \sqrt{\text{Bo} \sec (\alpha )} \text{csch}\left(L \sqrt{\text{Bo} \cos (\alpha )}\right)+\frac{\tanh \left(\frac{1}{2} L \sqrt{\text{Bo} \cos (\alpha )}\right)}{R \sqrt{\text{Bo} \cos (\alpha )}}$$


Can someone explain why I get a more compact solution when I first collect the equation for a certain parameter and then run FullSimplify and whether there is a general 'recipe' to get the most compact form of an equation?


P.S. sorry for the long equation, but it seems that most of the smaller equations do not have this behaviour which makes sort of intuitive sense



Answer




There are at least two aspects to this. The first is the fact that FullSimplify will not try all possible transformations, even those it is aware of. See: Why does Simplify ignore an assumption? The second is that Mathematica does not see "compact" and "simple" as the same thing; if you give it a different ComplexityFunction it will do much better in this case. Compare:


eq1 = (Sec[α] (Coth[L Sqrt[Bo Cos[α]]] - R Sqrt[Bo Sec[α]] Sin[α] + 
      Csch[L Sqrt[Bo Cos[α]]] (-1 + Bo L R Sin[α])))/(R Sqrt[Bo Sec[α]])

short1 = Collect[eq1, Bo] // FullSimplify


-Tan[α] + (
 Sec[α] (Bo L R Csch[L Sqrt[Bo Cos[α]]] Sin[α] + 
    Tanh[1/2 L Sqrt[Bo Cos[α]]]))/(R Sqrt[Bo Sec[α]])


short2 = FullSimplify[eq1, ComplexityFunction -> Composition[StringLength, ToString]]


(Csch[
   L Sqrt[Bo Cos[α]]] Sec[α] (-1 + Cosh[L Sqrt[Bo Cos[α]]] + 
    Bo L R Sin[α]))/(R Sqrt[Bo Sec[α]]) - Tan[α]

The three expressions measured by both LeafCount and my ComplexityFunction:


LeafCount /@ {eq1, short1, short2}

Composition[StringLength, ToString] /@ {eq1, short1, short2}


{59, 51, 49}

{275, 374, 212}

Note that though short1 and short2 appear to be roughly the same length my ComplexityFunction sees short1 as being much longer. That is because it is not a very good metric, and it is using the OutputForm of the expression rendering "2D" text like this:


ToString[short1]



                                                               L Sqrt[Bo Cos[α]]
          Sec[α] (Bo L R Csch[L Sqrt[Bo Cos[α]]] Sin[α] + Tanh[-----------------])
                                                                       2
-Tan[α] + ------------------------------------------------------------------------
                                     R Sqrt[Bo Sec[α]]

The point of all of this is that you need to be quite specific with ComplexityFunction if you want the expression simplified in the way you see as simple or compact.


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