Arithmetic Expression Compiler

[Smaug]

Basically, you're trying to write a computer algebera system, am I right? That is a very interesting task. Do you plan to develop it to some usable version or just write it to learn?

I myself made a simple parser for analytic expressions a year ago or so. I wrote it in C# since I had no intent to make it optimal or something (I also don't know low-level languages nearly good enaugh to get into optimization). The goal was to learn from practice some of the the principles of how systems like Maple and Matlab operate. The parcer could parse algebraic expressions into trees, differentiate in respect to a variable or to time and prepare Lagrange's DE's.

Wait, how did you make it output the derivative? It's certainly harder than making it output assembly, or even doing the basic algebraic operations.

Unlike finding an indefinite integral, finding a derivative is a fully algorithmic procedure so it isn't decisively harder than doing basic algebraic operations. First you have to expand the initial expression to get rid of all the brackets. Then you have to go through all the terms and apply the chain rule which is somewhat similar to dealing with nested brackets. This step may take several iterations. While doing that it is handy to use 'delayed evaluation'. I. e. you just mark functions with a derivative sign and do not substitute anything from the table of derivatives. After doing so you obtain a tree which has derivatives of elementary functions as its leafs. The final step is to evaluate the leafs by substituting from the table.

Differentiating functions of time-dependant coortinates in respect to time is somewhat harder but it's basically the same thing.