Simplifications by symbolic algebra programs

[Gerenuk]

DH comment:
This thread was split from [thread=413916]this thread[/thread][/color]

Hint:
http://www.wolframalpha.com/input/?i=tan(pi/4+x)

 

[D H]

Hint:
http://www.wolframalpha.com/input/?i=tan(pi/4+x)

From that site:

Alternate form:
$$\frac{\cos(x)}{\sqrt 2\left(\frac {\cos(x)}{\sqrt 2} - \frac {\sin(x)}{\sqrt 2}\right)} +<br /> \frac{\sin(x)}{\sqrt 2\left(\frac {\cos(x)}{\sqrt 2} - \frac {\sin(x)}{\sqrt 2}\right)}$$​

It's amazing how amazingly stupid those symbolic tools such as Mathematica can be, even though 40+ years have transpired since the development of Macsyma and Schoonschip in the late 1960s.

 

[Gerenuk]

Actually that was a motivation for me to start programming a program where you can do transformations manually.

Because everyone knows that mathematica might get you the most amazing integral, but it ***** at the most basic simplifications.

 

[Gerenuk]

Could be interesting to hear if anyone knows of an algebra program that let's you do simplifications your way (by hand), so that you get useful results. I was often asked whether such a program doesn't exist yet, and I replied "no". I haven't heard of any.

Unfortunately the development of my program has come to a halt, since I had to move for a job and don't have much stuff here yet :)

 

[D H]

So, what is it that makes symbolic algebra so hard? Why can't Mathematica make the obvious simplification of the expression in post #2? I'm not dinging Mathematica specifically here. Maple, for example, is just as bad as Mathematica when it comes to basic simplifications.

 

[Hurkyl]

It's amazing how amazingly stupid those symbolic tools such as Mathematica can be, even though 40+ years have transpired since the development of Macsyma and Schoonschip in the late 1960s.

Computers do what you tell them. I suspect wolframalpha simply did something like

(Sin[:pi:/4] Cos[x] + Cos[:pi:/4] Sin[x])/(Cos[:pi:/4] Cos[x] - Sin[pi:/4] Sin[x])​

and didn't ask Mathematica to simplify. Mathematica simply parsed the expression, decided it was worth evaluating Sin[:pi:/4] -> 1 / Sqrt[2] immediately (same for cosine), then returned the parse tree in expression form.


Sure, one might imagine "that's stupid. Mathematica should always simplify before it outputs a result" -- but such a person has never been in a situation where he wanted to look at the unsimplified expression, or had a different "simplified form" in mind than what Mathematica wanted. Wolframalpha may have even been specifically designed not to ask Mathematica to simplify, because it is likely to change it into an unintended form. (e.g. I would be unsurprised if it converted right back to Tan[x + :pi:/4])

 

[Hurkyl]

an algebra program that let's you do simplifications your way (by hand)

Mathematica gives you a good degree of flexibility in that regard. You can apply any symbolic transformation rule you like -- e.g. don't like tangent? Apply the rule Tan[x_] -> Sin[x] / Cos[x]. It also gives you functions that allow you to manipulate the expression tree directly.

 

[Office_Shredder]

Well hold on now. Alternate form doesn't necessarily imply simplified form