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Replacement rule limitations

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  • Subject: [mg110550] Replacement rule limitations
  • From: "S. B. Gray" <stevebg at ROADRUNNER.COM>
  • Date: Fri, 25 Jun 2010 07:26:47 -0400 (EDT)
  • Reply-to: stevebg at ROADRUNNER.COM

Several responders have told me that replacement rules are the way to 
simplify complex expressions and to reduce redundant computations (?).

So let's try an arbitrary expression as a very simple example of 
something that could be much more complicated:

exp = 1/Sqrt[
    x^2+y^2+z^2] - (x^2+y^2+z^2) + (x^2+y^2+z^2)^(
    1/3) /. x^2+y^2+z^2->dd               This gives

1/Sqrt[dd] + dd^(1/3)-x^2-y^2-z^2        which is not that useful.

But if I introduce a superfluous multiplier "s":

exp = 1/Sqrt[x^2+y^2+z^2] -
    s(x^2+y^2+z^2) + (x^2+y^2+z^2)^(
    1/3) /. {x^2+y^2+z^2->dd, s->1}     I get

1/Sqrt[dd] + dd^(1/3) - dd             which is better. Asking for

exp^2                                  gives, as desired,

(1/Sqrt[dd] + dd^(1/3) - dd)^2 .       But trying to proceed as if this 
were regular algebra where cascaded substitutions are routine, I try:

exp/.{x->a^2, y->3b, z->Sqrt[d + e]} , I get the useless result

1/Sqrt[dd] + dd^(1/3) - dd .

Unless I am missing something important (it wouldn't be the first 
time!), replacement rules are not a good substitute for real 
intermediate variables. This does not even address a feature I'd like to 
see in Mathematica in which it would figure out what subexpressions appear 
repeatedly and make up its own simplifying intermediate variables. This 
could be incorporated into FullSimplify.

Comments will be greatly appreciated.

Steve Gray

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