int.jl

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

## integer arithmetic ##

# The tuples and types that do not include 128 bit sizes are necessary to handle
# certain issues on 32-bit machines, and also to simplify promotion rules, as
# they are also used elsewhere where Int128/UInt128 support is separated out,
# such as in hashing2.jl

const BitSigned32_types      = (Int8, Int16, Int32)
const BitUnsigned32_types    = (UInt8, UInt16, UInt32)
const BitInteger32_types     = (BitSigned32_types..., BitUnsigned32_types...)

const BitSigned64_types      = (BitSigned32_types..., Int64)
const BitUnsigned64_types    = (BitUnsigned32_types..., UInt64)
const BitInteger64_types     = (BitSigned64_types..., BitUnsigned64_types...)

const BitSigned_types        = (BitSigned64_types..., Int128)
const BitUnsigned_types      = (BitUnsigned64_types..., UInt128)
const BitInteger_types       = (BitSigned_types..., BitUnsigned_types...)

const BitSignedSmall_types   = Int === Int64 ? ( Int8,  Int16,  Int32) : ( Int8,  Int16)
const BitUnsignedSmall_types = Int === Int64 ? (UInt8, UInt16, UInt32) : (UInt8, UInt16)
const BitIntegerSmall_types  = (BitSignedSmall_types..., BitUnsignedSmall_types...)

const BitSigned32      = Union{BitSigned32_types...}
const BitUnsigned32    = Union{BitUnsigned32_types...}
const BitInteger32     = Union{BitInteger32_types...}

const BitSigned64      = Union{BitSigned64_types...}
const BitUnsigned64    = Union{BitUnsigned64_types...}
const BitInteger64     = Union{BitInteger64_types...}

const BitSigned        = Union{BitSigned_types...}
const BitUnsigned      = Union{BitUnsigned_types...}
const BitInteger       = Union{BitInteger_types...}

const BitSignedSmall   = Union{BitSignedSmall_types...}
const BitUnsignedSmall = Union{BitUnsignedSmall_types...}
const BitIntegerSmall  = Union{BitIntegerSmall_types...}

const BitSigned64T     = Union{Type{Int8}, Type{Int16}, Type{Int32}, Type{Int64}}
const BitUnsigned64T   = Union{Type{UInt8}, Type{UInt16}, Type{UInt32}, Type{UInt64}}

const BitIntegerType = Union{map(T->Type{T}, BitInteger_types)...}

## integer comparisons ##

(<)(x::T, y::T) where {T<:BitSigned}  = slt_int(x, y)

(-)(x::BitInteger)                    = neg_int(x)
(-)(x::T, y::T) where {T<:BitInteger} = sub_int(x, y)
(+)(x::T, y::T) where {T<:BitInteger} = add_int(x, y)
(*)(x::T, y::T) where {T<:BitInteger} = mul_int(x, y)

inv(x::Integer) = float(one(x)) / float(x)
(/)(x::T, y::T) where {T<:Integer} = float(x) / float(y)
# skip promotion for system integer types
(/)(x::BitInteger, y::BitInteger) = float(x) / float(y)

"""
    isodd(x::Integer) -> Bool

Return `true` if `x` is odd (that is, not divisible by 2), and `false` otherwise.

```jldoctest
julia> isodd(9)
true

julia> isodd(10)
false
```
"""
isodd(n::Integer) = rem(n, 2) != 0

"""
    iseven(x::Integer) -> Bool

Return `true` is `x` is even (that is, divisible by 2), and `false` otherwise.

```jldoctest
julia> iseven(9)
false

julia> iseven(10)
true
```
"""
iseven(n::Integer) = !isodd(n)

signbit(x::Integer) = x < 0
signbit(x::Unsigned) = false

flipsign(x::T, y::T) where {T<:BitSigned} = flipsign_int(x, y)
flipsign(x::BitSigned, y::BitSigned) = flipsign_int(promote(x, y)...) % typeof(x)

flipsign(x::Signed, y::Float16) = flipsign(x, bitcast(Int16, y))
flipsign(x::Signed, y::Float32) = flipsign(x, bitcast(Int32, y))
flipsign(x::Signed, y::Float64) = flipsign(x, bitcast(Int64, y))
flipsign(x::Signed, y::Real)    = flipsign(x, -oftype(x, signbit(y)))

copysign(x::Signed, y::Signed)  = flipsign(x, x ⊻ y)
copysign(x::Signed, y::Float16) = copysign(x, bitcast(Int16, y))
copysign(x::Signed, y::Float32) = copysign(x, bitcast(Int32, y))
copysign(x::Signed, y::Float64) = copysign(x, bitcast(Int64, y))
copysign(x::Signed, y::Real)    = copysign(x, -oftype(x, signbit(y)))

"""
    abs(x)

The absolute value of `x`.

When `abs` is applied to signed integers, overflow may occur,
resulting in the return of a negative value. This overflow occurs only
when `abs` is applied to the minimum representable value of a signed
integer. That is, when `x == typemin(typeof(x))`, `abs(x) == x < 0`,
not `-x` as might be expected.

```jldoctest
julia> abs(-3)
3

julia> abs(1 + im)
1.4142135623730951

julia> abs(typemin(Int64))
-9223372036854775808
```
"""
function abs end

abs(x::Unsigned) = x
abs(x::Signed) = flipsign(x,x)

~(n::Integer) = -n-1

unsigned(x::BitSigned) = reinterpret(typeof(convert(Unsigned, zero(x))), x)
unsigned(x::Bool) = convert(Unsigned, x)

"""
    unsigned(x) -> Unsigned

Convert a number to an unsigned integer. If the argument is signed, it is reinterpreted as
unsigned without checking for negative values.

# Examples
```jldoctest
julia> unsigned(-2)
0xfffffffffffffffe

julia> unsigned(2)
0x0000000000000002

julia> signed(unsigned(-2))
-2
```
"""
unsigned(x) = convert(Unsigned, x)
signed(x::Unsigned) = reinterpret(typeof(convert(Signed, zero(x))), x)

"""
    signed(x)

Convert a number to a signed integer. If the argument is unsigned, it is reinterpreted as
signed without checking for overflow.
"""
signed(x) = convert(Signed, x)

div(x::BitSigned, y::Unsigned) = flipsign(signed(div(unsigned(abs(x)), y)), x)
div(x::Unsigned, y::BitSigned) = unsigned(flipsign(signed(div(x, unsigned(abs(y)))), y))

rem(x::BitSigned, y::Unsigned) = flipsign(signed(rem(unsigned(abs(x)), y)), x)
rem(x::Unsigned, y::BitSigned) = rem(x, unsigned(abs(y)))

fld(x::Signed, y::Unsigned) = div(x, y) - (signbit(x) & (rem(x, y) != 0))
fld(x::Unsigned, y::Signed) = div(x, y) - (signbit(y) & (rem(x, y) != 0))


"""
    mod(x, y)
    rem(x, y, RoundDown)

The reduction of `x` modulo `y`, or equivalently, the remainder of `x` after floored
division by `y`, i.e.
```julia
x - y*fld(x,y)
```
if computed without intermediate rounding.

The result will have the same sign as `y`, and magnitude less than `abs(y)` (with some
exceptions, see note below).

!!! note

    When used with floating point values, the exact result may not be representable by the
    type, and so rounding error may occur. In particular, if the exact result is very
    close to `y`, then it may be rounded to `y`.

```jldoctest
julia> mod(8, 3)
2

julia> mod(9, 3)
0

julia> mod(8.9, 3)
2.9000000000000004

julia> mod(eps(), 3)
2.220446049250313e-16

julia> mod(-eps(), 3)
3.0
```
"""
function mod(x::T, y::T) where T<:Integer
    y == -1 && return T(0)   # avoid potential overflow in fld
    return x - fld(x, y) * y
end
mod(x::BitSigned, y::Unsigned) = rem(y + unsigned(rem(x, y)), y)
mod(x::Unsigned, y::Signed) = rem(y + signed(rem(x, y)), y)
mod(x::T, y::T) where {T<:Unsigned} = rem(x, y)

cld(x::Signed, y::Unsigned) = div(x, y) + (!signbit(x) & (rem(x, y) != 0))
cld(x::Unsigned, y::Signed) = div(x, y) + (!signbit(y) & (rem(x, y) != 0))

# Don't promote integers for div/rem/mod since there is no danger of overflow,
# while there is a substantial performance penalty to 64-bit promotion.
div(x::T, y::T) where {T<:BitSigned64} = checked_sdiv_int(x, y)
rem(x::T, y::T) where {T<:BitSigned64} = checked_srem_int(x, y)
div(x::T, y::T) where {T<:BitUnsigned64} = checked_udiv_int(x, y)
rem(x::T, y::T) where {T<:BitUnsigned64} = checked_urem_int(x, y)


# fld(x,y) == div(x,y) - ((x>=0) != (y>=0) && rem(x,y) != 0 ? 1 : 0)
fld(x::T, y::T) where {T<:Unsigned} = div(x,y)
function fld(x::T, y::T) where T<:Integer
    d = div(x, y)
    return d - (signbit(x ⊻ y) & (d * y != x))
end

# cld(x,y) = div(x,y) + ((x>0) == (y>0) && rem(x,y) != 0 ? 1 : 0)
function cld(x::T, y::T) where T<:Unsigned
    d = div(x, y)
    return d + (d * y != x)
end
function cld(x::T, y::T) where T<:Integer
    d = div(x, y)
    return d + (((x > 0) == (y > 0)) & (d * y != x))
end

## integer bitwise operations ##

"""
    ~(x)

Bitwise not.

# Examples
```jldoctest
julia> ~4
-5

julia> ~10
-11

julia> ~true
false
```
"""
(~)(x::BitInteger)             = not_int(x)

"""
    &(x, y)

Bitwise and.

# Examples
```jldoctest
julia> 4 & 10
0

julia> 4 & 12
4
```
"""
(&)(x::T, y::T) where {T<:BitInteger} = and_int(x, y)

"""
    |(x, y)

Bitwise or.

# Examples
```jldoctest
julia> 4 | 10
14

julia> 4 | 1
5
```
"""
(|)(x::T, y::T) where {T<:BitInteger} = or_int(x, y)
xor(x::T, y::T) where {T<:BitInteger} = xor_int(x, y)

"""
    bswap(n)

Byte-swap an integer. Flip the bits of its binary representation.

# Examples
```jldoctest
julia> a = bswap(4)
288230376151711744

julia> bswap(a)
4

julia> bin(1)
"1"

julia> bin(bswap(1))
"100000000000000000000000000000000000000000000000000000000"
```
"""
bswap(x::Union{Int8, UInt8}) = x
bswap(x::Union{Int16, UInt16, Int32, UInt32, Int64, UInt64, Int128, UInt128}) =
    bswap_int(x)

"""
    count_ones(x::Integer) -> Integer

Number of ones in the binary representation of `x`.

```jldoctest
julia> count_ones(7)
3
```
"""
count_ones(x::BitInteger) = Int(ctpop_int(x))

"""
    leading_zeros(x::Integer) -> Integer

Number of zeros leading the binary representation of `x`.

```jldoctest
julia> leading_zeros(Int32(1))
31
```
"""
leading_zeros(x::BitInteger) = Int(ctlz_int(x))

"""
    trailing_zeros(x::Integer) -> Integer

Number of zeros trailing the binary representation of `x`.

```jldoctest
julia> trailing_zeros(2)
1
```
"""
trailing_zeros(x::BitInteger) = Int(cttz_int(x))

"""
    count_zeros(x::Integer) -> Integer

Number of zeros in the binary representation of `x`.

```jldoctest
julia> count_zeros(Int32(2 ^ 16 - 1))
16
```
"""
count_zeros(x::Integer) = count_ones(~x)

"""
    leading_ones(x::Integer) -> Integer

Number of ones leading the binary representation of `x`.

```jldoctest
julia> leading_ones(UInt32(2 ^ 32 - 2))
31
```
"""
leading_ones(x::Integer) = leading_zeros(~x)

"""
    trailing_ones(x::Integer) -> Integer

Number of ones trailing the binary representation of `x`.

```jldoctest
julia> trailing_ones(3)
2
```
"""
trailing_ones(x::Integer) = trailing_zeros(~x)

## integer comparisons ##

(< )(x::T, y::T) where {T<:BitUnsigned} = ult_int(x, y)
(<=)(x::T, y::T) where {T<:BitSigned}   = sle_int(x, y)
(<=)(x::T, y::T) where {T<:BitUnsigned} = ule_int(x, y)

==(x::BitSigned,   y::BitUnsigned) = (x >= 0) & (unsigned(x) == y)
==(x::BitUnsigned, y::BitSigned  ) = (y >= 0) & (x == unsigned(y))
<( x::BitSigned,   y::BitUnsigned) = (x <  0) | (unsigned(x) <  y)
<( x::BitUnsigned, y::BitSigned  ) = (y >= 0) & (x <  unsigned(y))
<=(x::BitSigned,   y::BitUnsigned) = (x <  0) | (unsigned(x) <= y)
<=(x::BitUnsigned, y::BitSigned  ) = (y >= 0) & (x <= unsigned(y))

## integer shifts ##

# unsigned shift counts always shift in the same direction
>>(x::BitSigned,   y::BitUnsigned) = ashr_int(x, y)
>>(x::BitUnsigned, y::BitUnsigned) = lshr_int(x, y)
<<(x::BitInteger,  y::BitUnsigned) = shl_int(x, y)
>>>(x::BitInteger, y::BitUnsigned) = lshr_int(x, y)
# signed shift counts can shift in either direction
# note: this early during bootstrap, `>=` is not yet available
# note: we only define Int shift counts here; the generic case is handled later
>>(x::BitInteger, y::Int) =
    select_value(0 <= y, x >> unsigned(y), x << unsigned(-y))
<<(x::BitInteger, y::Int) =
    select_value(0 <= y, x << unsigned(y), x >> unsigned(-y))
>>>(x::BitInteger, y::Int) =
    select_value(0 <= y, x >>> unsigned(y), x << unsigned(-y))

for to in BitInteger_types, from in (BitInteger_types..., Bool)
    if !(to === from)
        if to.size < from.size
            @eval rem(x::($from), ::Type{$to}) = trunc_int($to, x)
        elseif from === Bool
            @eval rem(x::($from), ::Type{$to}) = convert($to, x)
        elseif from.size < to.size
            if from <: Signed
                @eval rem(x::($from), ::Type{$to}) = sext_int($to, x)
            else
                @eval rem(x::($from), ::Type{$to}) = convert($to, x)
            end
        else
            @eval rem(x::($from), ::Type{$to}) = bitcast($to, x)
        end
    end
end

# @doc isn't available when running in Core at this point.
# Tuple syntax for documention two function signatures at the same time
# doesn't work either at this point.
if nameof(@__MODULE__) === :Base
    for fname in (:mod, :rem)
        @eval @doc ("""
            rem(x::Integer, T::Type{<:Integer}) -> T
            mod(x::Integer, T::Type{<:Integer}) -> T
            %(x::Integer, T::Type{<:Integer}) -> T

        Find `y::T` such that `x` ≡ `y` (mod n), where n is the number of integers representable
        in `T`, and `y` is an integer in `[typemin(T),typemax(T)]`.
        If `T` can represent any integer (e.g. `T == BigInt`), then this operation corresponds to
        a conversion to `T`.

        ```jldoctest
        julia> 129 % Int8
        -127
        ```
        """ -> $fname(x::Integer, T::Type{<:Integer}))
    end
end

rem(x::T, ::Type{T}) where {T<:Integer} = x
rem(x::Integer, T::Type{<:Integer}) = convert(T, x)  # `x % T` falls back to `convert`
rem(x::Integer, ::Type{Bool}) = ((x & 1) != 0)
mod(x::Integer, ::Type{T}) where {T<:Integer} = rem(x, T)

unsafe_trunc(::Type{T}, x::Integer) where {T<:Integer} = rem(x, T)

"""
    trunc([T,] x, [digits, [base]])

`trunc(x)` returns the nearest integral value of the same type as `x` whose absolute value
is less than or equal to `x`.

`trunc(T, x)` converts the result to type `T`, throwing an `InexactError` if the value is
not representable.

`digits` and `base` work as for [`round`](@ref).
"""
function trunc end

"""
    floor([T,] x, [digits, [base]])

`floor(x)` returns the nearest integral value of the same type as `x` that is less than or
equal to `x`.

`floor(T, x)` converts the result to type `T`, throwing an `InexactError` if the value is
not representable.

`digits` and `base` work as for [`round`](@ref).
"""
function floor end

"""
    ceil([T,] x, [digits, [base]])

`ceil(x)` returns the nearest integral value of the same type as `x` that is greater than or
equal to `x`.

`ceil(T, x)` converts the result to type `T`, throwing an `InexactError` if the value is not
representable.

`digits` and `base` work as for [`round`](@ref).
"""
function ceil end

round(x::Integer) = x
trunc(x::Integer) = x
floor(x::Integer) = x
 ceil(x::Integer) = x

round(::Type{T}, x::Integer) where {T<:Integer} = convert(T, x)
trunc(::Type{T}, x::Integer) where {T<:Integer} = convert(T, x)
floor(::Type{T}, x::Integer) where {T<:Integer} = convert(T, x)
 ceil(::Type{T}, x::Integer) where {T<:Integer} = convert(T, x)

## integer construction ##

macro int128_str(s)
    return parse(Int128, s)
end

macro uint128_str(s)
    return parse(UInt128, s)
end

macro big_str(s)
    if '_' in s
        # remove _ in s[2:end-1]
        bf = IOBuffer(lastindex(s))
        print(bf, s[1])
        for c in SubString(s, 2, lastindex(s)-1)
            c != '_' && print(bf, c)
        end
        print(bf, s[end])
        seekstart(bf)
        n = tryparse(BigInt, String(take!(bf)))
        n === nothing || return n
    else
        n = tryparse(BigInt, s)
        n === nothing || return n
        n = tryparse(BigFloat, s)
        n === nothing || return n
    end
    message = "invalid number format $s for BigInt or BigFloat"
    return :(throw(ArgumentError($message)))
end

## integer promotions ##

# with different sizes, promote to larger type
promote_rule(::Type{Int16}, ::Union{Type{Int8}, Type{UInt8}}) = Int16
promote_rule(::Type{Int32}, ::Union{Type{Int16}, Type{Int8}, Type{UInt16}, Type{UInt8}}) = Int32
promote_rule(::Type{Int64}, ::Union{Type{Int16}, Type{Int32}, Type{Int8}, Type{UInt16}, Type{UInt32}, Type{UInt8}}) = Int64
promote_rule(::Type{Int128}, ::Union{Type{Int16}, Type{Int32}, Type{Int64}, Type{Int8}, Type{UInt16}, Type{UInt32}, Type{UInt64}, Type{UInt8}}) = Int128
promote_rule(::Type{UInt16}, ::Union{Type{Int8}, Type{UInt8}}) = UInt16
promote_rule(::Type{UInt32}, ::Union{Type{Int16}, Type{Int8}, Type{UInt16}, Type{UInt8}}) = UInt32
promote_rule(::Type{UInt64}, ::Union{Type{Int16}, Type{Int32}, Type{Int8}, Type{UInt16}, Type{UInt32}, Type{UInt8}}) = UInt64
promote_rule(::Type{UInt128}, ::Union{Type{Int16}, Type{Int32}, Type{Int64}, Type{Int8}, Type{UInt16}, Type{UInt32}, Type{UInt64}, Type{UInt8}}) = UInt128
# with mixed signedness and same size, Unsigned wins
promote_rule(::Type{UInt8},   ::Type{Int8}  ) = UInt8
promote_rule(::Type{UInt16},  ::Type{Int16} ) = UInt16
promote_rule(::Type{UInt32},  ::Type{Int32} ) = UInt32
promote_rule(::Type{UInt64},  ::Type{Int64} ) = UInt64
promote_rule(::Type{UInt128}, ::Type{Int128}) = UInt128

_default_type(::Type{Unsigned}) = UInt
_default_type(::Union{Type{Integer},Type{Signed}}) = Int

## traits ##

"""
    typemin(T)

The lowest value representable by the given (real) numeric DataType `T`.

# Examples
```jldoctest
julia> typemin(Float16)
-Inf16

julia> typemin(Float32)
-Inf32
```
"""
function typemin end

"""
    typemax(T)

The highest value representable by the given (real) numeric `DataType`.
"""
function typemax end

typemin(::Type{Int8  }) = Int8(-128)
typemax(::Type{Int8  }) = Int8(127)
typemin(::Type{UInt8 }) = UInt8(0)
typemax(::Type{UInt8 }) = UInt8(255)
typemin(::Type{Int16 }) = Int16(-32768)
typemax(::Type{Int16 }) = Int16(32767)
typemin(::Type{UInt16}) = UInt16(0)
typemax(::Type{UInt16}) = UInt16(65535)
typemin(::Type{Int32 }) = Int32(-2147483648)
typemax(::Type{Int32 }) = Int32(2147483647)
typemin(::Type{UInt32}) = UInt32(0)
typemax(::Type{UInt32}) = UInt32(4294967295)
typemin(::Type{Int64 }) = -9223372036854775808
typemax(::Type{Int64 }) = 9223372036854775807
typemin(::Type{UInt64}) = UInt64(0)
typemax(::Type{UInt64}) = 0xffffffffffffffff
@eval typemin(::Type{UInt128}) = $(convert(UInt128, 0))
@eval typemax(::Type{UInt128}) = $(bitcast(UInt128, convert(Int128, -1)))
@eval typemin(::Type{Int128} ) = $(convert(Int128, 1) << 127)
@eval typemax(::Type{Int128} ) = $(bitcast(Int128, typemax(UInt128) >> 1))

widen(::Type{<:Union{Int8, Int16}}) = Int32
widen(::Type{Int32}) = Int64
widen(::Type{Int64}) = Int128
widen(::Type{<:Union{UInt8, UInt16}}) = UInt32
widen(::Type{UInt32}) = UInt64
widen(::Type{UInt64}) = UInt128

# a few special cases,
# Int64*UInt64 => Int128
# |x|<=2^(k-1), |y|<=2^k-1   =>   |x*y|<=2^(2k-1)-1
widemul(x::Signed,y::Unsigned) = widen(x) * signed(widen(y))
widemul(x::Unsigned,y::Signed) = signed(widen(x)) * widen(y)
# multplication by Bool doesn't require widening
widemul(x::Bool,y::Bool) = x * y
widemul(x::Bool,y::Number) = x * y
widemul(x::Number,y::Bool) = x * y


## wide multiplication, Int128 multiply and divide ##

if Core.sizeof(Int) == 4
    function widemul(u::Int64, v::Int64)
        local u0::UInt64, v0::UInt64, w0::UInt64
        local u1::Int64, v1::Int64, w1::UInt64, w2::Int64, t::UInt64

        u0 = u & 0xffffffff; u1 = u >> 32
        v0 = v & 0xffffffff; v1 = v >> 32
        w0 = u0 * v0
        t = reinterpret(UInt64, u1) * v0 + (w0 >>> 32)
        w2 = reinterpret(Int64, t) >> 32
        w1 = u0 * reinterpret(UInt64, v1) + (t & 0xffffffff)
        hi = u1 * v1 + w2 + (reinterpret(Int64, w1) >> 32)
        lo = w0 & 0xffffffff + (w1 << 32)
        return Int128(hi) << 64 + Int128(lo)
    end

    function widemul(u::UInt64, v::UInt64)
        local u0::UInt64, v0::UInt64, w0::UInt64
        local u1::UInt64, v1::UInt64, w1::UInt64, w2::UInt64, t::UInt64

        u0 = u & 0xffffffff; u1 = u >>> 32
        v0 = v & 0xffffffff; v1 = v >>> 32
        w0 = u0 * v0
        t = u1 * v0 + (w0 >>> 32)
        w2 = t >>> 32
        w1 = u0 * v1 + (t & 0xffffffff)
        hi = u1 * v1 + w2 + (w1 >>> 32)
        lo = w0 & 0xffffffff + (w1 << 32)
        return UInt128(hi) << 64 + UInt128(lo)
    end

    function *(u::Int128, v::Int128)
        u0 = u % UInt64; u1 = Int64(u >> 64)
        v0 = v % UInt64; v1 = Int64(v >> 64)
        lolo = widemul(u0, v0)
        lohi = widemul(reinterpret(Int64, u0), v1)
        hilo = widemul(u1, reinterpret(Int64, v0))
        t = reinterpret(UInt128, hilo) + (lolo >>> 64)
        w1 = reinterpret(UInt128, lohi) + (t & 0xffffffffffffffff)
        return Int128(lolo & 0xffffffffffffffff) + reinterpret(Int128, w1) << 64
    end

    function *(u::UInt128, v::UInt128)
        u0 = u % UInt64; u1 = UInt64(u>>>64)
        v0 = v % UInt64; v1 = UInt64(v>>>64)
        lolo = widemul(u0, v0)
        lohi = widemul(u0, v1)
        hilo = widemul(u1, v0)
        t = hilo + (lolo >>> 64)
        w1 = lohi + (t & 0xffffffffffffffff)
        return (lolo & 0xffffffffffffffff) + UInt128(w1) << 64
    end

    function div(x::Int128, y::Int128)
        (x == typemin(Int128)) & (y == -1) && throw(DivideError())
        return Int128(div(BigInt(x), BigInt(y)))
    end
    function div(x::UInt128, y::UInt128)
        return UInt128(div(BigInt(x), BigInt(y)))
    end

    function rem(x::Int128, y::Int128)
        return Int128(rem(BigInt(x), BigInt(y)))
    end
    function rem(x::UInt128, y::UInt128)
        return UInt128(rem(BigInt(x), BigInt(y)))
    end

    function mod(x::Int128, y::Int128)
        return Int128(mod(BigInt(x), BigInt(y)))
    end
else
    *(x::T, y::T) where {T<:Union{Int128,UInt128}}  = mul_int(x, y)

    div(x::Int128,  y::Int128)  = checked_sdiv_int(x, y)
    div(x::UInt128, y::UInt128) = checked_udiv_int(x, y)

    rem(x::Int128,  y::Int128)  = checked_srem_int(x, y)
    rem(x::UInt128, y::UInt128) = checked_urem_int(x, y)
end

# issue #15489: since integer ops are unchecked, they shouldn't check promotion
for op in (:+, :-, :*, :&, :|, :xor)
    @eval function $op(a::Integer, b::Integer)
        T = promote_typeof(a, b)
        return $op(a % T, b % T)
    end
end