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Re: Re: How to simplify ArcCos[x/Sqrt[x^2+y^2]] to Pi/2-ArcTan[x/Abs[y]]?

On Thu, Sep 11, 2008 at 8:03 PM, Andrzej Kozlowski <akoz at> wrote:
> On 12 Sep 2008, at 09:46, Peng Yu wrote:
>> My question was raised because I had a very complex expression, which
>> is too long to understand. The case in the title is just one little
>> sub expression in that I could pinpoint in that long expression. I
>> don't know the simplest form of the long expression. But it is of
>> course not the simplest, because I pinpoint the sub expression that
>> can be made simpler.
>> I could do this by hand, but then why I need Mathematica?
> You can never hope to get "the simplest" form, firstly because there is no
> unique defintion of "simplest" (that is why the functions Simplify and
> FullSimplify have the ComplexityFunction option, which allows you to decide
> which of two equivalent forms of an expression is "simpler".). Secondly,
> there are only a certain finite number of transformations that Simplify and
> FullSimplify will use (FullSimplify uses more). Some very natural functions
> ar enot included because they have too high complexity and applying them all
> the time would essentially stall Mathematica. You can add your own functions
> using the option TransformationFucntions in Simplify and FullSimplify. In
> general, only you can know what kind of simplifications will be useful in
> your particular case. Computers cannot think and make choices, so if one
> adds too many transformation functions as default they will be applied all
> the time even in situations when they have no hope of being useful. Human
> beings are (hopefully) intelligent and often can see easily that a certain
> transformation will be of no use in a certain situation. Computer programs
> usually can't.
> In any case, all you can expect from Simplify is to obtain an expression
> that will have the lowest value of ComplexityFunction under transformations
> that are built up of the ones Simplify uses automatically plus any that you
> have added yourself. FullSimplify uses more transformation functions than
> Simplify and also uses them recursively, so it will often take a very long
> time to return an answer. That's all there really is to it, and short of
> developing software than can reason like human mathematicians there is
> little, I think, that can in principle be changed, although of course more
> efficient algorithms are being developed all the time (and implemented in
> Mathematica).

There are only a few examples in the help of ComplexityFunction. I'm
wondering what kind of complexity functions I should use for the two
cases that I raised.


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