# Tangent types

The values that come back from pullbacks or pushforwards are not always the same type as the input/outputs of the primal function. They are tangents, which correspond roughly to something able to represent the difference between two values of the primal types. A tangent might be such a regular type, like a `Number`

, or a `Matrix`

, matching to the original type; or it might be one of the `AbstractTangent`

subtypes.

Tangents support a number of operations. Most importantly: `+`

and `*`

, which let them act as mathematical objects.

To be more formal they support operations which let them act as a vector space.

## 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.

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 – in fact 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`

: It is a special representation of`0`

. It does great things around avoiding expanding`Thunks`

in addition.`NoTangent`

: Zero-like, represents that the operation on this input is not differentiable. Its primal type is normally`Integer`

or`Bool`

.`Tangent{P}`

: 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`

: this is a deferred computation. A thunk is a word for a zero argument closure. A computation wrapped in a`@thunk`

doesn't get evaluated until`unthunk`

is called on the thunk.`unthunk`

is a no-op on non-thunked inputs.`InplaceableThunk`

: it is like a`Thunk`

but it can do in-place`add!`

which allows for avoiding allocation during gradient accumulation.