Horner's rule for two-variable polynomial

Question

Horner\'s rule is used to simplify the process of evaluating a polynomial at specific variable values. https://rosettacode.org/wiki/Horner%27s_rule_for_polynomial_evaluation

Answer 1

Your second approach seems to be on the right track. If you have already defined horner, what you need to do is to apply horner to the result of mapping horner applied to inner list x over the outer list, something like:

fun evalXY coeffLists x y = horner (map (fn coeffList => horner coeffList x) coeffLists) y

You could replace the two calls to horner by the corresponding folds, but it would be much less readable.

Note that if you reverse the order of the two parameters in horner then you can shorted evalXY:

fun horner x coeffList = foldr (fn (a, b) => a + b * x) (0.0) coeffList
fun evalXY x y coeffLists = horner y (map (horner x) coeffLists)

The point being that the way currying works, if you use this second order then horner x is already a function of coeffList so you no longer need the anonymous function fn coeffList => horner coeffList x. The moral of the story is that when defining a curried function, you should think carefully about the order of the parameters since it will make some partial applications easier to create than others.

By the way, SML is fussy. In your discussion of horner you said that you would call it like horner list 2. It would need to be horner list 2.0. Similarly, in your second attempt, using 0 rather than 0.0 is problematic.