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 BitSigned64_types      = (Int8, Int16, Int32, Int64)
const BitUnsigned64_types    = (UInt8, UInt16, UInt32, 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 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}}

## 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

Returns `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

Returns `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::Signed)  = convert(typeof(x), flipsign(promote_noncircular(x, y)...))
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::Signed) = 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::Signed, y::Unsigned) = flipsign(signed(div(unsigned(abs(x)), y)), x)
div(x::Unsigned, y::Signed) = unsigned(flipsign(signed(div(x, unsigned(abs(y)))), y))

rem(x::Signed, y::Unsigned) = flipsign(signed(rem(unsigned(abs(x)), y)), x)
rem(x::Unsigned, y::Signed) = 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::Signed, 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(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::Signed,   y::Unsigned) = (x >= 0) & (unsigned(x) == y)
==(x::Unsigned, y::Signed  ) = (y >= 0) & (x == unsigned(y))
<( x::Signed,   y::Unsigned) = (x <  0) | (unsigned(x) <  y)
<( x::Unsigned, y::Signed  ) = (y >= 0) & (x <  unsigned(y))
<=(x::Signed,   y::Unsigned) = (x <  0) | (unsigned(x) <= y)
<=(x::Unsigned, y::Signed  ) = (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))

function is_top_bit_set(x::BitInteger)
    @_inline_meta
    lshr_int(x, (sizeof(x) << 0x03) - 1) == rem(0x01, typeof(x))
end
function check_top_bit(x::BitInteger)
    @_inline_meta
    is_top_bit_set(x) && throw(InexactError(:check_top_bit, Integer, x))
    x
end

## integer conversions ##

function checked_trunc_sint{To,From}(::Type{To}, x::From)
    @_inline_meta
    y = trunc_int(To, x)
    back = sext_int(From, y)
    x == back || throw(InexactError(:trunc, To, x))
    y
end

function checked_trunc_uint{To,From}(::Type{To}, x::From)
    @_inline_meta
    y = trunc_int(To, x)
    back = zext_int(From, y)
    x == back || throw(InexactError(:trunc, To, x))
    y
end

for to in BitInteger_types, from in (BitInteger_types..., Bool)
    if !(to === from)
        if to.size < from.size
            if issubtype(to, Signed)
                if issubtype(from, Unsigned)
                    @eval convert(::Type{$to}, x::($from)) =
                        checked_trunc_sint($to, check_top_bit(x))
                else
                    @eval convert(::Type{$to}, x::($from)) =
                        checked_trunc_sint($to, x)
                end
            else
                @eval convert(::Type{$to}, x::($from)) =
                    checked_trunc_uint($to, x)
            end
            @eval rem(x::($from), ::Type{$to}) = trunc_int($to, x)
        elseif from === Bool
            # Bools use i8 storage and may have garbage in their 7 high bits
            @eval convert(::Type{$to}, x::($from)) = zext_int($to, x) & $to(1)
            @eval rem(x::($from), ::Type{$to}) = convert($to, x)
        elseif from.size < to.size
            if issubtype(from, Signed)
                if issubtype(to, Unsigned)
                    @eval convert(::Type{$to}, x::($from)) =
                        sext_int($to, check_top_bit(x))
                else
                    @eval convert(::Type{$to}, x::($from)) =
                        sext_int($to, x)
                end
                @eval rem(x::($from), ::Type{$to}) = sext_int($to, x)
            else
                @eval convert(::Type{$to}, x::($from)) = zext_int($to, x)
                @eval rem(x::($from), ::Type{$to}) = convert($to, x)
            end
        else
            if !(issubtype(from, Signed) === issubtype(to, Signed))
                # raise InexactError if x's top bit is set
                @eval convert(::Type{$to}, x::($from)) = bitcast($to, check_top_bit(x))
            else
                @eval convert(::Type{$to}, x::($from)) = bitcast($to, x)
            end
            @eval rem(x::($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 module_name(@__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, ::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)
for (Ts, Tu) in ((Int8, UInt8), (Int16, UInt16), (Int32, UInt32), (Int64, UInt64), (Int128, UInt128))
    @eval convert(::Type{Signed}, x::$Tu) = convert($Ts, x)
    @eval convert(::Type{Unsigned}, x::$Ts) = convert($Tu, x)
end

convert(::Type{Signed}, x::Union{Float32, Float64, Bool}) = convert(Int, x)
convert(::Type{Unsigned}, x::Union{Float32, Float64, Bool}) = convert(UInt, x)

convert(::Type{Integer}, x::Integer) = x
convert(::Type{Integer}, x::Real) = convert(Signed, x)

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)
    n = tryparse(BigInt, s)
    !isnull(n) && return get(n)
    n = tryparse(BigFloat, s)
    !isnull(n) && return get(n)
    message = "invalid number format $s for BigInt or BigFloat"
    return :(throw(ArgumentError($message)))
end

## integer promotions ##

promote_rule(::Type{Int8}, ::Type{Int16})   = Int16
promote_rule(::Type{UInt8}, ::Type{UInt16}) = UInt16
promote_rule(::Type{Int32}, ::Type{<:Union{Int8,Int16}})    = Int32
promote_rule(::Type{UInt32}, ::Type{<:Union{UInt8,UInt16}}) = UInt32
promote_rule(::Type{Int64}, ::Type{<:Union{Int8,Int16,Int32}})     = Int64
promote_rule(::Type{UInt64}, ::Type{<:Union{UInt8,UInt16,UInt32}}) = UInt64
promote_rule(::Type{Int128}, ::Type{<:BitSigned64})    = Int128
promote_rule(::Type{UInt128}, ::Type{<:BitUnsigned64}) = UInt128
for T in BitSigned_types
    @eval promote_rule(::Type{<:Union{UInt8,UInt16}}, ::Type{$T}) =
        $(sizeof(T) < sizeof(Int) ? Int : T)
end
@eval promote_rule(::Type{UInt32}, ::Type{<:Union{Int8,Int16,Int32}}) =
    $(Core.sizeof(Int) == 8 ? Int : UInt)
promote_rule(::Type{UInt32}, ::Type{Int64}) = Int64
promote_rule(::Type{UInt64}, ::Type{<:BitSigned64}) = UInt64
promote_rule(::Type{<:Union{UInt32, UInt64}}, ::Type{Int128}) = Int128
promote_rule(::Type{UInt128}, ::Type{<:BitSigned}) = UInt128

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

## traits ##

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