Quantifying simplicity, in the case of trigonometric and exponential functions

Question

The pair of identities the sine and cosine of a sum of two terms as functions of the sines and cosines of the terms separately is not as simple as the identity that expresses the exponential of a sum as a product. But in some senses the former says the same thing as the latter.

It seems there are also some identities that are simpler when stated in terms of sines and cosines than as equivalent identities involving exponential functions. E.g. if $a+b+c=\pi$ then $\sin(2a)+\sin(2b)+\sin(2c)=4\sin a\sin b\sin c$.

Is there some sensible objective way of quantifying simplicity and stating things like this precisely, and proving results that say specified kinds of identities are simpler in one form than in the other?

Answer 1

LeafCount quantifies simplicity in Mathematica.

For instance, LeafCount[ Sin[2x] + Sin[2y] + Sin[2z] ] = 13, where the leaves are one Plus, three Sin's, three Times, three 2's, and x, y, and z.

Mathematica uses LeafCount in determining how to Simplify an expression. Similarly, you'd quantify the simplicity of your identity as the LeafCount of the difference of sides.

Answer 2

A way to quantify the simplicity of any identity is to regard it as an object consisting of two expressions. Then, we can quantify their simplicity with the concepts from the literature on symbolic algebra tools. One of them is the notion of the “expression entropy”:

“the number of binary bits inherent in an expression (…) a very simple practical means of measuring such entropy. Namely: run an expression as a text file, through an established entropy compressor.”
Borwein, Jonathan M., O-Yeat Chan, and Richard E. Crandall. "Higher-dimensional box integrals." Experimental Mathematics 19.4 (2010): 431-446. (see Section 6) Link

The authors suggest applying text files as with different symbolic languages we can get widely varying text-character counts.

The different – more subjective - measure is described by Billing and Wehmeier (see p. 40).