A Mathematica interpreter in Typescript
I’ve been using Mathematica to help with learning math (and using math to help learn Mathematica). Too often my Mathematica code has surprising behavior, which means I don’t really understand the language or how it’s evaluated. So I thought I’d do two things to get better.
First, write a bunch of Mathematica rules to implement a toy symbolic differentiation operator. Second, write a toy Mathematica interpreter in Typescript that’s good enough to correctly run my custom differentiation code.
Writing a toy differentiator turns out to be shockingly easy. It’s a near verbatim transcription of differentiation rules from any calculus textbook:
My first implementation had three more rules (one for each of dxk/dx,(c⋅f)′(x)dx^k/dx, (cf)’(x) d x k / d x , ( c ⋅ f ) ′ ( x ) , and (1/f)′(x)(1/f)’(x) ( 1/ f ) ′ ( x ) ), which I later realized you don’t need. These are shortcuts for human differentiators, but they’re automagically covered by the multiplication and exponentiation rules above.
Some problems I encountered while writing these rules: infinite recursion, rules not matching, and rules matching but evaluating to a surprising result. The hobby edition of Mathematica has some tools for debugging these problems, but not much. Ultimately I fixed bad rules by staring at the code and thinking really hard. I found Trace impossible to read, and TracePrint (which is supposed to be better) not much better. Also MatchQ is good, but somehow not as useful for debugging as I would have liked.
