promotion.jl

# This file is a part of Julia. License is MIT: https://julialang.org/license

## type join (closest common ancestor, or least upper bound) ##

"""
    typejoin(T, S, ...)

Return the closest common ancestor of types `T` and `S`, i.e. the narrowest type from which
they both inherit. Recurses on additional varargs.

# Examples
```jldoctest
julia> typejoin(Int, Float64)
Real

julia> typejoin(Int, Float64, ComplexF32)
Number
```
"""
typejoin() = Bottom
typejoin(@nospecialize(t)) = (@_nospecializeinfer_meta; t)
typejoin(@nospecialize(t), @nospecialize(s), @nospecialize(u)) = (@_foldable_meta; @_nospecializeinfer_meta; typejoin(typejoin(t, s), u))
typejoin(@nospecialize(t), @nospecialize(s), @nospecialize(u), ts...) = (@_foldable_meta; @_nospecializeinfer_meta; afoldl(typejoin, typejoin(t, s, u), ts...))
function typejoin(@nospecialize(a), @nospecialize(b))
    @_foldable_meta
    @_nospecializeinfer_meta
    if isa(a, TypeVar)
        return typejoin(a.ub, b)
    elseif isa(b, TypeVar)
        return typejoin(a, b.ub)
    elseif a <: b
        return b
    elseif b <: a
        return a
    elseif isa(a, UnionAll)
        return UnionAll(a.var, typejoin(a.body, b))
    elseif isa(b, UnionAll)
        return UnionAll(b.var, typejoin(a, b.body))
    elseif isa(a, Union)
        return typejoin(typejoin(a.a, a.b), b)
    elseif isa(b, Union)
        return typejoin(a, typejoin(b.a, b.b))
    end
    # a and b are DataTypes
    # We have to hide Constant info from inference, see #44390
    a, b = inferencebarrier(a)::DataType, inferencebarrier(b)::DataType
    if a <: Tuple
        if !(b <: Tuple)
            return Any
        end
        ap, bp = a.parameters, b.parameters
        lar = length(ap)
        lbr = length(bp)
        if lar == 0
            return Tuple{Vararg{tailjoin(bp, 1)}}
        end
        if lbr == 0
            return Tuple{Vararg{tailjoin(ap, 1)}}
        end
        laf, afixed = full_va_len(ap)
        lbf, bfixed = full_va_len(bp)
        if laf < lbf
            if isvarargtype(ap[lar]) && !afixed
                c = Vector{Any}(undef, laf)
                c[laf] = Vararg{typejoin(unwrapva(ap[lar]), tailjoin(bp, laf))}
                n = laf-1
            else
                c = Vector{Any}(undef, laf+1)
                c[laf+1] = Vararg{tailjoin(bp, laf+1)}
                n = laf
            end
        elseif lbf < laf
            if isvarargtype(bp[lbr]) && !bfixed
                c = Vector{Any}(undef, lbf)
                c[lbf] = Vararg{typejoin(unwrapva(bp[lbr]), tailjoin(ap, lbf))}
                n = lbf-1
            else
                c = Vector{Any}(undef, lbf+1)
                c[lbf+1] = Vararg{tailjoin(ap, lbf+1)}
                n = lbf
            end
        else
            c = Vector{Any}(undef, laf)
            n = laf
        end
        for i = 1:n
            ai = ap[min(i,lar)]; bi = bp[min(i,lbr)]
            ci = typejoin(unwrapva(ai), unwrapva(bi))
            c[i] = i == length(c) && (isvarargtype(ai) || isvarargtype(bi)) ? Vararg{ci} : ci
        end
        return Tuple{c...}
    elseif b <: Tuple
        return Any
    end
    while !(b === Any)
        if a <: b.name.wrapper
            while !(a.name === b.name)
                a = supertype(a)::DataType
            end
            if a.name === Type.body.name
                ap = a.parameters[1]
                bp = b.parameters[1]
                if ((isa(ap,TypeVar) && ap.lb === Bottom && ap.ub === Any) ||
                    (isa(bp,TypeVar) && bp.lb === Bottom && bp.ub === Any))
                    # handle special Type{T} supertype
                    return Type
                end
            end
            aprimary = a.name.wrapper
            # join on parameters
            n = length(a.parameters)
            if n == 0
                return aprimary
            end
            vars = []
            for i = 1:n
                ai, bi = a.parameters[i], b.parameters[i]
                if ai === bi || (isa(ai,Type) && isa(bi,Type) && ai <: bi && bi <: ai)
                    aprimary = aprimary{ai}
                else
                    aprimary = aprimary::UnionAll
                    # pushfirst!(vars, aprimary.var)
                    _growbeg!(vars, 1)
                    vars[1] = aprimary.var
                    aprimary = aprimary.body
                end
            end
            for v in vars
                aprimary = UnionAll(v, aprimary)
            end
            return aprimary
        end
        b = supertype(b)::DataType
    end
    return Any
end

# return an upper-bound on type `a` with type `b` removed
# such that `return <: a` && `Union{return, b} == Union{a, b}`
# WARNING: this is wrong for some objects for which subtyping is broken
#          (Core.Compiler.isnotbrokensubtype), use only simple types for `b`
function typesplit(@nospecialize(a), @nospecialize(b))
    @_foldable_meta
    @_nospecializeinfer_meta
    if a <: b
        return Bottom
    end
    if isa(a, Union)
        return Union{typesplit(a.a, b),
                     typesplit(a.b, b)}
    end
    return a
end


"""
    promote_typejoin(T, S)

Compute a type that contains both `T` and `S`, which could be
either a parent of both types, or a `Union` if appropriate.
Falls back to [`typejoin`](@ref).

See instead [`promote`](@ref), [`promote_type`](@ref).

# Examples
```jldoctest
julia> Base.promote_typejoin(Int, Float64)
Real

julia> Base.promote_type(Int, Float64)
Float64
```
"""
function promote_typejoin(@nospecialize(a), @nospecialize(b))
    c = typejoin(_promote_typesubtract(a), _promote_typesubtract(b))
    return Union{a, b, c}::Type
end
_promote_typesubtract(@nospecialize(a)) =
    a === Any ? a :
    a >: Union{Nothing, Missing} ? typesplit(a, Union{Nothing, Missing}) :
    a >: Nothing ? typesplit(a, Nothing) :
    a >: Missing ? typesplit(a, Missing) :
    a

function promote_typejoin_union(::Type{T}) where T
    if T === Union{}
        return Union{}
    elseif T isa UnionAll
        return Any # TODO: compute more precise bounds
    elseif T isa Union
        return promote_typejoin(promote_typejoin_union(T.a), promote_typejoin_union(T.b))
    elseif T isa DataType
        T <: Tuple && return typejoin_union_tuple(T)
        return T
    else
        error("unreachable") # not a type??
    end
end

function typejoin_union_tuple(T::DataType)
    @_foldable_meta
    p = T.parameters
    lr = length(p)
    if lr == 0
        return Tuple{}
    end
    c = Vector{Any}(undef, lr)
    for i = 1:lr
        pi = p[i]
        U = unwrapva(pi)
        if U === Union{}
            ci = Union{}
        elseif U isa Union
            ci = typejoin(U.a, U.b)
        elseif U isa UnionAll
            return Any # TODO: compute more precise bounds
        else
            ci = promote_typejoin_union(U)
        end
        if i == lr && isvarargtype(pi)
            c[i] = isdefined(pi, :N) ? Vararg{ci, pi.N} : Vararg{ci}
        else
            c[i] = ci
        end
    end
    return Base.rewrap_unionall(Tuple{c...}, T)
end

# Returns length, isfixed
function full_va_len(p::Core.SimpleVector)
    isempty(p) && return 0, true
    last = p[end]
    if isvarargtype(last)
        if isdefined(last, :N)
            N = last.N
            isa(N, Int) && return length(p) + N - 1, true
        end
        return length(p), false
    end
    return length(p), true
end

# reduce typejoin over A[i:end]
function tailjoin(A::SimpleVector, i::Int)
    @_foldable_meta
    if i > length(A)
        return unwrapva(A[end])
    end
    t = Bottom
    for j = i:length(A)
        t = typejoin(t, unwrapva(A[j]))
    end
    return t
end

## promotion mechanism ##

"""
    promote_type(type1, type2, ...)

Promotion refers to converting values of mixed types to a single common type.
`promote_type` represents the default promotion behavior in Julia when
operators (usually mathematical) are given arguments of differing types.
`promote_type` generally tries to return a type which can at least approximate
most values of either input type without excessively widening.  Some loss is
tolerated; for example, `promote_type(Int64, Float64)` returns
[`Float64`](@ref) even though strictly, not all [`Int64`](@ref) values can be
represented exactly as `Float64` values.

See also: [`promote`](@ref), [`promote_typejoin`](@ref), [`promote_rule`](@ref).

# Examples
```jldoctest
julia> promote_type(Int64, Float64)
Float64

julia> promote_type(Int32, Int64)
Int64

julia> promote_type(Float32, BigInt)
BigFloat

julia> promote_type(Int16, Float16)
Float16

julia> promote_type(Int64, Float16)
Float16

julia> promote_type(Int8, UInt16)
UInt16
```

!!! warning "Don't overload this directly"
    To overload promotion for your own types you should overload [`promote_rule`](@ref).
    `promote_type` calls `promote_rule` internally to determine the type.
    Overloading `promote_type` directly can cause ambiguity errors.
"""
function promote_type end

promote_type()  = Bottom
promote_type(T) = T
promote_type(T, S, U) = (@inline; promote_type(promote_type(T, S), U))
promote_type(T, S, U, V...) = (@inline; afoldl(promote_type, promote_type(T, S, U), V...))

promote_type(::Type{Bottom}, ::Type{Bottom}) = Bottom
promote_type(::Type{T}, ::Type{T}) where {T} = T
promote_type(::Type{T}, ::Type{Bottom}) where {T} = T
promote_type(::Type{Bottom}, ::Type{T}) where {T} = T

function promote_type(::Type{T}, ::Type{S}) where {T,S}
    @inline
    # Try promote_rule in both orders. Typically only one is defined,
    # and there is a fallback returning Bottom below, so the common case is
    #   promote_type(T, S) =>
    #   promote_result(T, S, result, Bottom) =>
    #   typejoin(result, Bottom) => result
    promote_result(T, S, promote_rule(T,S), promote_rule(S,T))
end

"""
    promote_rule(type1, type2)

Specifies what type should be used by [`promote`](@ref) when given values of types `type1` and
`type2`. This function should not be called directly, but should have definitions added to
it for new types as appropriate.
"""
function promote_rule end

promote_rule(::Type, ::Type) = Bottom
# Define some methods to avoid needing to enumerate unrelated possibilities when presented
# with Type{<:T}, and return a value in general accordance with the result given by promote_type
promote_rule(::Type{Bottom}, slurp...) = Bottom
promote_rule(::Type{Bottom}, ::Type{Bottom}, slurp...) = Bottom # not strictly necessary, since the next method would match unambiguously anyways
promote_rule(::Type{Bottom}, ::Type{T}, slurp...) where {T} = T
promote_rule(::Type{T}, ::Type{Bottom}, slurp...) where {T} = T

promote_result(::Type,::Type,::Type{T},::Type{S}) where {T,S} = (@inline; promote_type(T,S))
# If no promote_rule is defined, both directions give Bottom. In that
# case use typejoin on the original types instead.
promote_result(::Type{T},::Type{S},::Type{Bottom},::Type{Bottom}) where {T,S} = (@inline; typejoin(T, S))

"""
    promote(xs...)

Convert all arguments to a common type, and return them all (as a tuple).
If no arguments can be converted, an error is raised.

See also: [`promote_type`](@ref), [`promote_rule`](@ref).

# Examples
```jldoctest
julia> promote(Int8(1), Float16(4.5), Float32(4.1))
(1.0f0, 4.5f0, 4.1f0)

julia> promote_type(Int8, Float16, Float32)
Float32

julia> reduce(Base.promote_typejoin, (Int8, Float16, Float32))
Real

julia> promote(1, "x")
ERROR: promotion of types Int64 and String failed to change any arguments
[...]

julia> promote_type(Int, String)
Any
```
"""
function promote end

function _promote(x::T, y::S) where {T,S}
    @inline
    R = promote_type(T, S)
    return (convert(R, x), convert(R, y))
end
promote_typeof(x) = typeof(x)
promote_typeof(x, y) = (@inline; promote_type(typeof(x), typeof(y)))
promote_typeof(x, y, z) = (@inline; promote_type(typeof(x), typeof(y), typeof(z)))
promote_typeof(x, y, z, a...) = (@inline; afoldl(((::Type{T}, y) where {T}) -> promote_type(T, typeof(y)), promote_typeof(x, y, z), a...))
function _promote(x, y, z)
    @inline
    R = promote_typeof(x, y, z)
    return (convert(R, x), convert(R, y), convert(R, z))
end
function _promote(x, y, zs...)
    @inline
    R = promote_typeof(x, y, zs...)
    return (convert(R, x), convert(R, y), convert(Tuple{Vararg{R}}, zs)...)
end
# TODO: promote(x::T, ys::T...) where {T} here to catch all circularities?

## promotions in arithmetic, etc. ##

promote() = ()
promote(x) = (x,)

function promote(x, y)
    @inline
    px, py = _promote(x, y)
    not_sametype((x,y), (px,py))
    px, py
end
function promote(x, y, z)
    @inline
    px, py, pz = _promote(x, y, z)
    not_sametype((x,y,z), (px,py,pz))
    px, py, pz
end
function promote(x, y, z, a...)
    p = _promote(x, y, z, a...)
    not_sametype((x, y, z, a...), p)
    p
end

promote(x::T, y::T, zs::T...) where {T} = (x, y, zs...)

not_sametype(x::T, y::T) where {T} = sametype_error(x)

not_sametype(x, y) = nothing

function sametype_error(input)
    @noinline
    error("promotion of types ",
          join(map(x->string(typeof(x)), input), ", ", " and "),
          " failed to change any arguments")
end

+(x::Number, y::Number) = +(promote(x,y)...)
*(x::Number, y::Number) = *(promote(x,y)...)
-(x::Number, y::Number) = -(promote(x,y)...)
/(x::Number, y::Number) = /(promote(x,y)...)

"""
    ^(x, y)

Exponentiation operator.

If `x` and `y` are integers, the result may overflow.
To enter numbers in scientific notation, use [`Float64`](@ref) literals
such as `1.2e3` rather than `1.2 * 10^3`.

If `y` is an `Int` literal (e.g. `2` in `x^2` or `-3` in `x^-3`), the Julia code
`x^y` is transformed by the compiler to `Base.literal_pow(^, x, Val(y))`, to
enable compile-time specialization on the value of the exponent.
(As a default fallback we have `Base.literal_pow(^, x, Val(y)) = ^(x,y)`,
where usually `^ == Base.^` unless `^` has been defined in the calling
namespace.) If `y` is a negative integer literal, then `Base.literal_pow`
transforms the operation to `inv(x)^-y` by default, where `-y` is positive.

See also [`exp2`](@ref), [`<<`](@ref).

# Examples
```jldoctest
julia> 3^5
243

julia> 3^-1  # uses Base.literal_pow
0.3333333333333333

julia> p = -1;

julia> 3^p
ERROR: DomainError with -1:
Cannot raise an integer x to a negative power -1.
[...]

julia> 3.0^p
0.3333333333333333

julia> 10^19 > 0  # integer overflow
false

julia> big(10)^19 == 1e19
true
```
"""
^(x::Number, y::Number) = ^(promote(x,y)...)

fma(x::Number, y::Number, z::Number) = fma(promote(x,y,z)...)
muladd(x::Number, y::Number, z::Number) = muladd(promote(x,y,z)...)

==(x::Number, y::Number) = (==)(promote(x,y)...)
<( x::Real, y::Real)     = (< )(promote(x,y)...)
<=(x::Real, y::Real)     = (<=)(promote(x,y)...)

rem(x::Real, y::Real) = rem(promote(x,y)...)
mod(x::Real, y::Real) = mod(promote(x,y)...)

mod1(x::Real, y::Real) = mod1(promote(x,y)...)
fld1(x::Real, y::Real) = fld1(promote(x,y)...)

max(x::Real, y::Real) = max(promote(x,y)...)
min(x::Real, y::Real) = min(promote(x,y)...)
minmax(x::Real, y::Real) = minmax(promote(x, y)...)

function TupleOrBottom(tt...)
    any(p -> p === Union{}, tt) && return Union{}
    return Tuple{tt...}
end

"""
    promote_op(f, argtypes...)

Guess what an appropriate container eltype would be for storing results of
`f(::argtypes...)`. The guess is in part based on type inference, so can change any time.

Accordingly, return a type `R` such that `f(args...) isa R` where `args isa T`.

!!! warning
    Due to its fragility, use of `promote_op` should be avoided. It is preferable to base
    the container eltype on the type of the actual elements. Only in the absence of any
    elements (for an empty result container), it may be unavoidable to call `promote_op`.

The type `R` obtained from `promote_op` is merely an upper bound. There may exist a stricter
type `S` such that `f(args...) isa S` for every `args isa T` with `S <: R` and `S != R`.
Furthermore, the exact type `R` obtained from `promote_op` depends on various factors
including but not limited to the exact Julia version used, packages loaded, and command line
options. As such, when used in publicly registered packages, **it is the package authors'
responsibility to ensure that the API guarantees provided by the package do not depend on
the exact type `R` obtained from `promote_op`.**

Additionally, the result may return overly exact types, such as `DataType`, `Type`, or
`Union{...}`, while the desired inputs or outputs may be different from those. The internal
`promote_typejoin_union` function may be helpful to improve the result in some of these
cases.

# Extended help

## Examples

The following function is an invalid use-case of `promote_op`.

```julia
\"""
    invalid_usecase1(f, xs::AbstractArray) -> ys::Array

Return an array `ys` such that `vec(ys)` is `isequal`-equivalent to

    [f(xs[1]), f(xs[2]), ..., f(xs[end])]
\"""
function invalid_usecase1(f, xs)
    R = promote_op(f, eltype(xs))
    ys = similar(xs, R)
    for i in eachindex(xs, ys)
        ys[i] = f(xs[i])
    end
    return ys
end
```

This is because the value obtained through `eltype(invalid_usecase1(f, xs))` depends on
exactly what `promote_op` returns. It may be improved by re-computing the element type
before returning the result.

```julia
function valid_usecase1(f, xs)
    R = promote_typejoin_union(promote_op(f, eltype(xs)))
    ys = similar(xs, R)
    S = Union{}
    for i in eachindex(xs, ys)
        ys[i] = f(xs[i])
        S = promote_type(S, typeof(ys[i]))
    end
    if S != R
        zs = similar(xs, S)
        copyto!(zs, ys)
        return zs
    end
    return ys
end
```

Note that using [`isconcretetype`](@ref) on the result is not enough to safely use
`promote_op`. The following function is another invalid use-case of `promote_op`.

```julia
function invalid_usecase2(f, xs)
    R = promote_op(f, eltype(xs))
    if isconcretetype(R)
        ys = similar(xs, R)
    else
        ys = similar(xs, Any)
    end
    for i in eachindex(xs, ys)
        ys[i] = f(xs[i])
    end
    return ys
end
```

This is because whether or not the caller gets `Any` element type depends on if `promote_op`
can infer a concrete return type of the given function. A fix similar to `valid_usecase1`
can be used.

*Technically*, another possible fix for `invalid_usecase1` and `invalid_usecase2` is to
loosen the API guarantee:

>     another_valid_usecase1(f, xs::AbstractArray) -> ys::Array
>
> Return an array `ys` such that every element in `xs` with the same index
> is mapped with `f`.
>
> The element type of `ys` is _undefined_. It must not be used with generic
> functions whose behavior depend on the element type of `ys`.

However, it is discouraged to define such unconventional API guarantees.
"""
function promote_op(f, S::Type...)
    argT = TupleOrBottom(S...)
    argT === Union{} && return Union{}
    return _return_type(f, argT)
end


## catch-alls to prevent infinite recursion when definitions are missing ##

no_op_err(name, T) = error(name," not defined for ",T)
(+)(x::T, y::T) where {T<:Number} = no_op_err("+", T)
(*)(x::T, y::T) where {T<:Number} = no_op_err("*", T)
(-)(x::T, y::T) where {T<:Number} = no_op_err("-", T)
(/)(x::T, y::T) where {T<:Number} = no_op_err("/", T)
(^)(x::T, y::T) where {T<:Number} = no_op_err("^", T)

fma(x::T, y::T, z::T) where {T<:Number} = no_op_err("fma", T)
fma(x::Integer, y::Integer, z::Integer) = x*y+z
muladd(x::T, y::T, z::T) where {T<:Number} = x*y+z

(&)(x::T, y::T) where {T<:Integer} = no_op_err("&", T)
(|)(x::T, y::T) where {T<:Integer} = no_op_err("|", T)
xor(x::T, y::T) where {T<:Integer} = no_op_err("xor", T)

(==)(x::T, y::T) where {T<:Number} = x === y
(< )(x::T, y::T) where {T<:Real} = no_op_err("<" , T)
(<=)(x::T, y::T) where {T<:Real} = (x == y) | (x < y)

rem(x::T, y::T) where {T<:Real} = no_op_err("rem", T)
mod(x::T, y::T) where {T<:Real} = no_op_err("mod", T)

min(x::Real) = x
max(x::Real) = x
minmax(x::Real) = (x, x)

max(x::T, y::T) where {T<:Real} = ifelse(y < x, x, y)
min(x::T, y::T) where {T<:Real} = ifelse(y < x, y, x)
minmax(x::T, y::T) where {T<:Real} = y < x ? (y, x) : (x, y)

flipsign(x::T, y::T) where {T<:Signed} = no_op_err("flipsign", T)