Merge pull request #573 from JuliaDiff/ox/tandoc

Improve documentation of tangent types

docs/src/rule_author/tangents.md

@@ -8,12 +8,23 @@ or it might be one of the [`AbstractTangent`](@ref ChainRulesCore.AbstractTangen
 Tangents support a number of operations.
 Most importantly: `+` and `*`, which let them act as mathematical objects.
 
-The most important `AbstractTangent`s when getting started are the ones about avoiding work:
+To be more formal they support operations which let them act as a vector space.
 
- - [`Thunk`](@ref): this is a deferred computation. A thunk is a [word for a zero argument closure](https://en.wikipedia.org/wiki/Thunk). A computation wrapped in a `@thunk` doesn't get evaluated until [`unthunk`](@ref) is called on the thunk. `unthunk` is a no-op on non-thunked inputs.
- - [`ZeroTangent`](@ref): It is a special representation of `0`. It does great things around avoiding expanding `Thunks` in addition.
+## Operations on a tangent type
+Any tangent type must support:
+ - `zero` which returns an additive identity for that type (though it can just return `ZeroTangent()` (see below))
+ - `+` for addition between two tangents of this primal, returning another tangent of this primal. This allows gradient accumulation.
+ - `*` for multiplication (scaling) by a scalar.
+- `+` between a tangent and its primal type returning another tangent of the primal type -- differential geometers sometimes call this exponential map.
 
-### Other `AbstractTangent`s:
- - [`Tangent{P}`](@ref Tangent): this is the tangent for tuples and  structs. Use it like a `Tuple` or `NamedTuple`. The type parameter `P` is for the primal type.
+Further they often support other linear operators for convenience in writing rules. 
+
+## The subtypes of AbstractTangent
+
+Not all tangents need to subtype the AbstractTangent type -- infact most don't: most are just numbers or arrays -- but ChainRulesCore does provide a number of special tangent types that can be very useful. 
+
+ - [`ZeroTangent`](@ref): It is a special representation of `0`. It does great things around avoiding expanding `Thunks` in addition.
  - [`NoTangent`](@ref): Zero-like, represents that the operation on this input is not differentiable. Its primal type is normally `Integer` or `Bool`.
- - [`InplaceableThunk`](@ref): it is like a `Thunk` but it can do in-place `add!`.
+ - [`Tangent{P}`](@ref Tangent): this is the tangent for tuples and  structs. Use it like a `Tuple` or `NamedTuple`. The type parameter `P` is for the primal type.
+ - [`Thunk`](@ref): this is a deferred computation. A thunk is a [word for a zero argument closure](https://en.wikipedia.org/wiki/Thunk). A computation wrapped in a `@thunk` doesn't get evaluated until [`unthunk`](@ref) is called on the thunk. `unthunk` is a no-op on non-thunked inputs.
+ - [`InplaceableThunk`](@ref): it is like a `Thunk` but it can do in-place `add!` which allows for avoiding allocation during gradient accumulation.