[sordina]

One of my favourite Haskell "one-liners" is combining the AD package with Number.Symbolic:

    {-# LANGUAGE ImportQualifiedPost #-}
    
    module Module_1663406024_9206 where
    
    import Numeric.AD qualified as Ad
    import Data.Number.Symbolic qualified as Sym
    
    -- >>> f x = x^2 + 3 * x
    -- >>> Ad.diff f 1
    -- >>> Ad.diff f (Sym.var "a")
    -- 5
    -- 3+a+a

    -- >>> Ad.diff sin pi
    -- >>> Ad.diff sin (Sym.var "a")
    -- -1.0
    -- cos a

The package authors did not need to coordinate to make this possible which is pretty wild.

Saw it first here: https://twitter.com/GabriellaG439/status/647601518871359489 and https://www.reddit.com/r/haskell/comments/3r75hq/comment/cwm...

[cryptonector]

> The package authors did not need to coordinate to make this possible which is pretty wild.

It works because `f` is polymorphic. The type of its `x` argument is not constrained in `f`'s definition, so you can plug in any `x` of any type you want provided that `x`'s type implements the methods used in `f`'s definition. With the `Dual` scheme you get to use as `x` a "dual" of `y` (`f x`, for some `f`) and `y'`, and then you get an `f` applied to that `x` where the actual `f` is parameterized by the actual `x`'s type, and so the methods called by `f` are those that apply to `x`'s type. So instead of the traditional numeric addition and multiplication, you'd get the "dual" addition and multiplication, and then everything "chains" through and you end up with `diff f x` being the `y'` in the dual of `y` and `y'` (you don't care about the `y`, just the `y'` because you want the `diff` -- the differential or derivative).

It's brilliant.

[magicalhippo]

I still don't get how sin ends up as cos, without any coordination.