[Haskell-cafe] Are there arithmetic composition of functions?

[wren ng thornton]

On 3/19/12 12:57 PM, sdiyazg at sjtu.edu.cn wrote:
> By arithmetic I mean the everyday arithmetic operations used in engineering.
> In signal processing for example, we write a lot of expressions like f(t)=g(t)+h(t)+g'(t) or f(t)=g(t)*h(t).
> I feel it would be very natural to have in haskell something like

You should take a look at Ralf Hinze's _The Lifting Lemma_:

     http://www.cs.ox.ac.uk/ralf.hinze/WG2.8/26/slides/ralf.pdf

The fact that you can lift arithmetic to work on functions comes from 
the fact that for every type T, the type (T->) is a monad and therefore 
is an applicative functor. The output type of the function doesn't 
matter, except inasmuch as the arithmetic operations themselves care.


This pattern has been observed repeatedly, even long before Haskell was 
around. But one reason it's not too common in production Haskell code is 
that it's all too easy to make a mistake when programming (e.g., you 
don't mean to be adding functions but you accidentally forget some 
argument), and if you're using this trick implicitly by providing a Num 
instance, then you can get arcane/unexpected/unhelpful error messages 
during type checking.

But then you do get some fun line noise :)

     twiceTheSumOf  = (+) + (+)
     squareTheSumOf = (+) * (+)
     cubeTheSumOf   = (+) * (+) * (+)

     -- N.B., the names only make sense if all arguments
     -- are numeric literals. Don't look at the types.
     addThreeThings = (+) . (+)
     addFourThings  = (+) . (+) . (+)
     addFiveThings  = (+) . (+) . (+) . (+)

-- 
Live well,
~wren