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Customizable lazy fused broadcasting in pure Julia
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This patch represents the combined efforts of four individuals, over 60
commits, and an iterated design over (at least) three pull requests that
spanned nearly an entire year (closes #22063, #23692, #25377 by superceding
them).

This introduces a pure Julia data structure that represents a fused broadcast
expression.  For example, the expression `2 .* (x .+ 1)` lowers to:

```julia
julia> Meta.@lower 2 .* (x .+ 1)
:($(Expr(:thunk, CodeInfo(:(begin
      Core.SSAValue(0) = (Base.getproperty)(Base.Broadcast, :materialize)
      Core.SSAValue(1) = (Base.getproperty)(Base.Broadcast, :make)
      Core.SSAValue(2) = (Base.getproperty)(Base.Broadcast, :make)
      Core.SSAValue(3) = (Core.SSAValue(2))(+, x, 1)
      Core.SSAValue(4) = (Core.SSAValue(1))(*, 2, Core.SSAValue(3))
      Core.SSAValue(5) = (Core.SSAValue(0))(Core.SSAValue(4))
      return Core.SSAValue(5)
  end)))))
```

Or, slightly more readably as:

```julia
using .Broadcast: materialize, make
materialize(make(*, 2, make(+, x, 1)))
```

The `Broadcast.make` function serves two purposes. Its primary purpose is to
construct the `Broadcast.Broadcasted` objects that hold onto the function, the
tuple of arguments (potentially including nested `Broadcasted` arguments), and
sometimes a set of `axes` to include knowledge of the outer shape. The
secondary purpose, however, is to allow an "out" for objects that _don't_ want
to participate in fusion. For example, if `x` is a range in the above `2 .* (x
.+ 1)` expression, it needn't allocate an array and operate elementwise — it
can just compute and return a new range. Thus custom structures are able to
specialize `Broadcast.make(f, args...)` just as they'd specialize on `f`
normally to return an immediate result.

`Broadcast.materialize` is identity for everything _except_ `Broadcasted`
objects for which it allocates an appropriate result and computes the
broadcast. It does two things: it `initialize`s the outermost `Broadcasted`
object to compute its axes and then `copy`s it.

Similarly, an in-place fused broadcast like `y .= 2 .* (x .+ 1)` uses the exact
same expression tree to compute the right-hand side of the expression as above,
and then uses `materialize!(y, make(*, 2, make(+, x, 1)))` to `instantiate` the
`Broadcasted` expression tree and then `copyto!` it into the given destination.

All-together, this forms a complete API for custom types to extend and
customize the behavior of broadcast (fixes #22060). It uses the existing
`BroadcastStyle`s throughout to simplify dispatch on many arguments:

* Custom types can opt-out of broadcast fusion by specializing
  `Broadcast.make(f, args...)` or `Broadcast.make(::BroadcastStyle, f, args...)`.

* The `Broadcasted` object computes and stores the type of the combined
  `BroadcastStyle` of its arguments as its first type parameter, allowing for
  easy dispatch and specialization.

* Custom Broadcast storage is still allocated via `broadcast_similar`, however
  instead of passing just a function as a first argument, the entire
  `Broadcasted` object is passed as a final argument. This potentially allows
  for much more runtime specialization dependent upon the exact expression
  given.

* Custom broadcast implmentations for a `CustomStyle` are defined by
  specializing `copy(bc::Broadcasted{CustomStyle})` or
  `copyto!(dest::AbstractArray, bc::Broadcasted{CustomStyle})`.

* Fallback broadcast specializations for a given output object of type `Dest`
  (for the `DefaultArrayStyle` or another such style that hasn't implemented
  assignments into such an object) are defined by specializing
  `copyto(dest::Dest, bc::Broadcasted{Nothing})`.

As it fully supports range broadcasting, this now deprecates `(1:5) + 2` to
`.+`, just as had been done for all `AbstractArray`s in general.

As a first-mover proof of concept, LinearAlgebra uses this new system to
improve broadcasting over structured arrays. Before, broadcasting over a
structured matrix would result in a sparse array. Now, broadcasting over a
structured matrix will _either_ return an appropriately structured matrix _or_
a dense array. This does incur a type instability (in the form of a
discriminated union) in some situations, but thanks to type-based introspection
of the `Broadcasted` wrapper commonly used functions can be special cased to be
type stable.  For example:

```julia
julia> f(d) = round.(Int, d)
f (generic function with 1 method)

julia> @inferred f(Diagonal(rand(3)))
3×3 Diagonal{Int64,Array{Int64,1}}:
 0  ⋅  ⋅
 ⋅  0  ⋅
 ⋅  ⋅  1

julia> @inferred Diagonal(rand(3)) .* 3
ERROR: return type Diagonal{Float64,Array{Float64,1}} does not match inferred return type Union{Array{Float64,2}, Diagonal{Float64,Array{Float64,1}}}
Stacktrace:
 [1] error(::String) at ./error.jl:33
 [2] top-level scope

julia> @inferred Diagonal(1:4) .+ Bidiagonal(rand(4), rand(3), 'U') .* Tridiagonal(1:3, 1:4, 1:3)
4×4 Tridiagonal{Float64,Array{Float64,1}}:
 1.30771  0.838589   ⋅          ⋅
 0.0      3.89109   0.0459757   ⋅
  ⋅       0.0       4.48033    2.51508
  ⋅        ⋅        0.0        6.23739
```

In addition to the issues referenced above, it fixes:

* Fixes #19313, #22053, #23445, and #24586: Literals are no longer treated
  specially in a fused broadcast; they're just arguments in a `Broadcasted`
  object like everything else.

* Fixes #21094: Since broadcasting is now represented by a pure Julia
  datastructure it can be created within `@generated` functions and serialized.

* Fixes #26097: The fallback destination-array specialization method of
  `copyto!` is specifically implemented as `Broadcasted{Nothing}` and will not
  be confused by `nothing` arguments.

* Fixes the broadcast-specific element of #25499: The default base broadcast
  implementation no longer depends upon `Base._return_type` to allocate its
  array (except in the empty or concretely-type cases). Note that the sparse
  implementation (#19595) is still dependent upon inference and is _not_ fixed.

* Fixes #25340: Functions are treated like normal values just like arguments
  and only evaluated once.

* Fixes #22255, and is performant with 12+ fused broadcasts. Okay, that one was
  fixed on master already, but this fixes it now, too.

* Fixes #25521.

* The performance of this patch has been thoroughly tested through its
  iterative development process in #25377. There remain [two classes of
  performance regressions](#25377) that Nanosoldier flagged.

* #25691: Propagation of constant literals sill lose their constant-ness upon
  going through the broadcast machinery. I believe quite a large number of
  functions would need to be marked as `@pure` to support this -- including
  functions that are intended to be specialized.

(For bookkeeping, this is the squashed version of the [teh-jn/lazydotfuse](JuliaLang/julia#25377)
branch as of a1d4e7ec9756ada74fb48f2c514615b9d981cf5c. Squashed and separated
out to make it easier to review and commit)

Co-authored-by: Tim Holy <tim.holy@gmail.com>
Co-authored-by: Jameson Nash <vtjnash@gmail.com>
Co-authored-by: Andrew Keller <ajkeller34@users.noreply.github.com>
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4 people committed Apr 23, 2018
1 parent 47cfbac commit 9dd1f3a
Showing 1 changed file with 1 addition and 1 deletion.
2 changes: 1 addition & 1 deletion base/statistics.jl
Original file line number Diff line number Diff line change
Expand Up @@ -145,7 +145,7 @@ function centralize_sumabs2!(R::AbstractArray{S}, A::AbstractArray, means::Abstr
return R
end
indsAt, indsRt = safe_tail(axes(A)), safe_tail(axes(R)) # handle d=1 manually
keep, Idefault = Broadcast.shapeindexer(indsAt, indsRt)
keep, Idefault = Broadcast.shapeindexer(indsRt)
if reducedim1(R, A)
i1 = first(indices1(R))
@inbounds for IA in CartesianIndices(indsAt)
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