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Re: Replacement Rule with Sqrt in denominator

• To: mathgroup at smc.vnet.net
• Subject: [mg114426] Re: Replacement Rule with Sqrt in denominator
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Sat, 4 Dec 2010 06:12:43 -0500 (EST)

Richard Fateman wrote:
> [...]
> Each time someone defends Mathematica via "you have to look
> at FullForm" he/she is essentially saying, "Mathematica is now too
> stupid to do what you want,

A more accurate translation would be "Mathematica is to stupid to read 
your mind and figure out what you really want."


> and the programmers of the system
> are too resistant to making it smarter.

I cannot speak for all of the programmers here. For myself, I'll say 
I've been doing this stuff a long time, and I recognize a fool's errand 
in what you propose here.


> Thus the burden falls on
> you to look at the data representation underlying the abstraction
> to make your program work."

I'd change "abstraction" to "representation". At which point I would 
agree, regarding the burden.


> It's not so hard to make the system smarter.

I'll indicate below why I am not convinced what you want equates to 
smarter software.


> But you pay for what
> you get.

Here I agree.

Actually one of the things I really appreciate about Mathematica, and 
for which, alas, I can take no credit, is that rule replacement works so 
reliably well. When I get frustrated with it failing to do what I intend 
(about once every month or two), there is the inevitable session with 
MatchQ to sort things out.


> RJF

I think any nonsyntactic rule replacement will need very clear 
semantics. The notion "replace it if it shows up visually" is not 
terribly clear. It relies, among other things, on specifics of the 
format settings, vagaries of e.g. StandardForm, and other details.

To give an idea of what is meant by the formatting vagaries, consider 
that we could represent 1/Sqrt[x+1] explicitly as (x+1)^(-1/2) (mentally 
format that so it looks pretty, with the exponent raised, no caret, 
etc.) So the substitution semantics being proposed would depend on 
formatting?! Or should it apply to any power of (x+1) when that power 
visually has a 2 in the denominator? (Does it then apply to 
(x+1)^(a/2+b/4)? Or only to the equivalent (x+1)^(1/2*(a+b/2))?)

These are the questions that would need to be answered before even 
embarking on this journey.

I will add that about the last thing anyone wants is that StandardForm 
formatting be influenced by the desire to avoid any possibility of 
showing "misleading" expression structure. I don't think you are 
proposing otherwise, but it would be a natural outcome to any effort to 
try to curtail this misunderstanding.

I have not raised the I->3*I issue. I do wonder what should be the 
outcome of Sin[x] /. I->3*I . Maybe the trig equivalent of
(Exp[3*I*x] - Exp[-3*I*x]) / (6*I), which is Sin[3*x]/3? (Maybe we 
should make it case insensitive, and return S3in[x]?)

Daniel Lichtblau
Wolfram Research