Hello,

I'm kind of thinking about doing this ^, making a complete CAS for the Ti-83/84+(se) family (written in Axe). However I'm currently working on something else, but this is still something fun/interesting to talk about, for the future and whatnot.

A list of lots of math functions can be found at http://functions.wolfram.com/ , and from there we can narrow it down to probably about 50-100 (the site has something like 300,000+ functions submitted).

Anyways, just something interesting to think about / expect for the future. What do you guys think? Any must-have features or anything like that? Any tricks about doing math stuff in Axe?

First of all, you're almost certainly going to have to at least be familiar with ASM to write a good CAS. Sorry, it just works out that way.

Secondly, I would highly recommend a numerical core to any math product. The symbolic manipulations available in a CAS are useful. However, they're far more powerful when combined with a numerical package. The majority of those functions you see on the Wolfram site (~99% of which are never used) are numerical algorithms, not symbolic functions. To get a basic CAS/numerical math package up, you pretty much need the following capabilities:

Symbolic differentiation
Symbolic integration (which is a formally undecidable problem, so don't worry about restricting the range of functions allowed as input)
Symbolic simplification
Array generation (generate arrays of numbers according to some rule)
Map (maps a function accepting 1 argument to each element of an array)
Symbolic functions
Eval (Numerically evaluates a user defined symbolic function at a point)
Plot <function>
Sin/Cos/Tan/ArcSin/ArcCos/ArcTan/CoSecant/Secant/Cotangent
Prime(n) (returns the n^th prime number)
Numerical differentiation
Numerical integeration
Elementary operations (+,-,*,/,^,Log_n,Abs, etc... These are generally expected to work on all numerical data types without complaint)
Some method of writing programs (either with a built-in scripting language or with a Well-documented API. Yes, this is entirely necessary.)
Rand (generates a random decimal number)
RandInt (generates a random integer)


Keep in mind that this is about the minimum to have a useful CAS.

If I have a CAS I'm considering, there are really only three things I look at:
-Flexibility of the integrator
-Flexibility of a symbolic equation solver (i.e. could it solve ((x+1)*x+1)*x+1=0? tan(4x)+tan(6x)=0? tan(4x)+tan(6x)=tan(x)?)
-Numerical integrator supports a wide variety of functions, can do arbitrary (or at least very high) precision

It's generally tricky to get the first to a point where it's better than any Joe who's passed their AP Calc. A solver can generally try many things at once, and is reliant on a good simplification/manipulation repertoire, which I think is usually hard-coded in largely. (eg. If you have (x-y), try multiplying by (x+y). If you have 1/x+1/y, try to combine the fractions. If you have a*b/c, try replacing with a/(c/b).)
Numerical integration, finally, is something that humans can't really do by themselves, so having something with high precision and the ability for a lot of inputs is good.

Hmm, you know, I'm wondering now how good of arbitrary precision libraries there for the z80? I mean I know there's arbitrary multiplication/addition, what about, say, sin(x) or e^x or log(x)? I've made a fast implementation of e^x before for my own entertainment (in Java, though)... I would be pretty impressed if some just implemented TI-OS's functions with arbitrary precision.

Note: Extraordinary high expectations for CAS above (not so much the arbitrary precision, I think that part's realistic) - I've been spoiled with free Mathematica since age 11.

Sqrt(time), you're kind of implicitly assuming that CASes solve equations the same way humans do. As it turns out, the algorithms (normal) humans typically use are either remarkably inefficient (e.g. the partial fractions method of Rational Integration) or almost impossible to implement (e.g. algebraic solving by heuristics).

Oh.. I did think that, yes.. I admit I don't know that much about what algorithms a CAS uses for that kind of thing, sorry. The only one I've really heard of is the Risch algorithm, which is kind of a monstrosity and largely above my head. If you know more about how they work, could you maybe provide some info/links?

...thanks! I'll check it out.