I have an application where I strongly prefer $1+\frac{3}{2+3x}$ over $\frac{5+3x}{2+3x}$, and $\sin(2x)$ over $2\sin(x)\cos(x)$. In general, I prefer to minimize the number of occurrences of the variables.
Is there a way to set the "cost" function for Simplify to achieve this?
For the example in OP FullSimplify gives the desired result:
FullSimplify [(5 + 3 x)/(2 + 3 x)]
1 + 3/(2 + 3 x)
To take a different example for which neither Simplify nor FullSimplify makes the desired simplification without additional work:
expr = (5 + 2 x)/(2 + 3 x);
Simplify @ expr
(5 + 2 x)/(2 + 3 x)
FullSimplify @ expr
(5 + 2 x)/(2 + 3 x)
"to minimize the number of occurrences of the variables" you can use a custom ComplexityFunction that penalizes multiple occurences of symbols. For example,
cF = Simplify`SimplifyCount[#] + 100 Count[#, _Symbol, All] &;
Using it with FullSimplify
FullSimplify[expr, ComplexityFunction -> cF]
2/3 + 11/(6 + 9 x)
With Simplify we need to do more:
Simplify[expr, ComplexityFunction -> cF]
(5 + 2 x)/(2 + 3 x)
Adding Apart to the Automatic TransformationFunctions used by Simplify gives the desired result:
Simplify[expr, ComplexityFunction -> cF, TransformationFunctions -> {Automatic, Apart}]
2/3 + 11/(6 + 9 x)
Apart enough? It gives 2/3 + 11/(3 (2+3 x)). One can then also expand denominator with ExpandAll to get 2/3 + 11/(6+9 x).
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