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# This file is a part of Julia. License is MIT: https://julialang.org/license
"""
Rational{T<:Integer} <: Real
Rational number type, with numerator and denominator of type `T`.
"""
struct Rational{T<:Integer} <: Real
num::T
den::T
function Rational{T}(num::Integer, den::Integer) where T<:Integer
num == den == zero(T) && throw(ArgumentError("invalid rational: zero($T)//zero($T)"))
g = den < 0 ? -gcd(den, num) : gcd(den, num)
new(div(num, g), div(den, g))
end
end
Rational(n::T, d::T) where {T<:Integer} = Rational{T}(n,d)
Rational(n::Integer, d::Integer) = Rational(promote(n,d)...)
Rational(n::Integer) = Rational(n,one(n))
function divgcd(x::Integer,y::Integer)
g = gcd(x,y)
div(x,g), div(y,g)
end
"""
//(num, den)
Divide two integers or rational numbers, giving a [`Rational`](@ref) result.
```jldoctest
julia> 3 // 5
3//5
julia> (3 // 5) // (2 // 1)
3//10
```
"""
//(n::Integer, d::Integer) = Rational(n,d)
function //(x::Rational, y::Integer)
xn,yn = divgcd(x.num,y)
xn//checked_mul(x.den,yn)
end
function //(x::Integer, y::Rational)
xn,yn = divgcd(x,y.num)
checked_mul(xn,y.den)//yn
end
function //(x::Rational, y::Rational)
xn,yn = divgcd(x.num,y.num)
xd,yd = divgcd(x.den,y.den)
checked_mul(xn,yd)//checked_mul(xd,yn)
end
//(x::Complex, y::Real) = complex(real(x)//y,imag(x)//y)
//(x::Number, y::Complex) = x*y'//abs2(y)
//(X::AbstractArray, y::Number) = X .// y
function show(io::IO, x::Rational)
show(io, numerator(x))
print(io, "//")
show(io, denominator(x))
end
function read(s::IO, ::Type{Rational{T}}) where T<:Integer
r = read(s,T)
i = read(s,T)
r//i
end
function write(s::IO, z::Rational)
write(s,numerator(z),denominator(z))
end
convert(::Type{Rational{T}}, x::Rational) where {T<:Integer} = Rational{T}(convert(T,x.num),convert(T,x.den))
convert(::Type{Rational{T}}, x::Integer) where {T<:Integer} = Rational{T}(convert(T,x), convert(T,1))
convert(::Type{Rational}, x::Rational) = x
convert(::Type{Rational}, x::Integer) = convert(Rational{typeof(x)},x)
convert(::Type{Bool}, x::Rational) = x==0 ? false : x==1 ? true :
throw(InexactError(:convert, Bool, x)) # to resolve ambiguity
convert(::Type{Integer}, x::Rational) = (isinteger(x) ? convert(Integer, x.num) :
throw(InexactError(:convert, Integer, x)))
convert(::Type{T}, x::Rational) where {T<:Integer} = (isinteger(x) ? convert(T, x.num) :
throw(InexactError(:convert, T, x)))
convert(::Type{AbstractFloat}, x::Rational) = float(x.num)/float(x.den)
function convert(::Type{T}, x::Rational{S}) where T<:AbstractFloat where S
P = promote_type(T,S)
convert(T, convert(P,x.num)/convert(P,x.den))
end
function convert(::Type{Rational{T}}, x::AbstractFloat) where T<:Integer
r = rationalize(T, x, tol=0)
x == convert(typeof(x), r) || throw(InexactError(:convert, Rational{T}, x))
r
end
convert(::Type{Rational}, x::Float64) = convert(Rational{Int64}, x)
convert(::Type{Rational}, x::Float32) = convert(Rational{Int}, x)
big(z::Complex{<:Rational{<:Integer}}) = Complex{Rational{BigInt}}(z)
promote_rule(::Type{Rational{T}}, ::Type{S}) where {T<:Integer,S<:Integer} = Rational{promote_type(T,S)}
promote_rule(::Type{Rational{T}}, ::Type{Rational{S}}) where {T<:Integer,S<:Integer} = Rational{promote_type(T,S)}
promote_rule(::Type{Rational{T}}, ::Type{S}) where {T<:Integer,S<:AbstractFloat} = promote_type(T,S)
widen(::Type{Rational{T}}) where {T} = Rational{widen(T)}
"""
rationalize([T<:Integer=Int,] x; tol::Real=eps(x))
Approximate floating point number `x` as a [`Rational`](@ref) number with components
of the given integer type. The result will differ from `x` by no more than `tol`.
If `T` is not provided, it defaults to `Int`.
```jldoctest
julia> rationalize(5.6)
28//5
julia> a = rationalize(BigInt, 10.3)
103//10
julia> typeof(numerator(a))
BigInt
```
"""
function rationalize(::Type{T}, x::AbstractFloat, tol::Real) where T<:Integer
if tol < 0
throw(ArgumentError("negative tolerance $tol"))
end
isnan(x) && return zero(T)//zero(T)
isinf(x) && return (x < 0 ? -one(T) : one(T))//zero(T)
p, q = (x < 0 ? -one(T) : one(T)), zero(T)
pp, qq = zero(T), one(T)
x = abs(x)
a = trunc(x)
r = x-a
y = one(x)
tolx = oftype(x, tol)
nt, t, tt = tolx, zero(tolx), tolx
ia = np = nq = zero(T)
# compute the successive convergents of the continued fraction
# np // nq = (p*a + pp) // (q*a + qq)
while r > nt
try
ia = convert(T,a)
np = checked_add(checked_mul(ia,p),pp)
nq = checked_add(checked_mul(ia,q),qq)
p, pp = np, p
q, qq = nq, q
catch e
isa(e,InexactError) || isa(e,OverflowError) || rethrow(e)
return p // q
end
# naive approach of using
# x = 1/r; a = trunc(x); r = x - a
# is inexact, so we store x as x/y
x, y = y, r
a, r = divrem(x,y)
# maintain
# x0 = (p + (-1)^i * r) / q
t, tt = nt, t
nt = a*t+tt
end
# find optimal semiconvergent
# smallest a such that x-a*y < a*t+tt
a = cld(x-tt,y+t)
try
ia = convert(T,a)
np = checked_add(checked_mul(ia,p),pp)
nq = checked_add(checked_mul(ia,q),qq)
return np // nq
catch e
isa(e,InexactError) || isa(e,OverflowError) || rethrow(e)
return p // q
end
end
rationalize(::Type{T}, x::AbstractFloat; tol::Real = eps(x)) where {T<:Integer} = rationalize(T, x, tol)::Rational{T}
rationalize(x::AbstractFloat; kvs...) = rationalize(Int, x; kvs...)
"""
numerator(x)
Numerator of the rational representation of `x`.
```jldoctest
julia> numerator(2//3)
2
julia> numerator(4)
4
```
"""
numerator(x::Integer) = x
numerator(x::Rational) = x.num
"""
denominator(x)
Denominator of the rational representation of `x`.
```jldoctest
julia> denominator(2//3)
3
julia> denominator(4)
1
```
"""
denominator(x::Integer) = one(x)
denominator(x::Rational) = x.den
sign(x::Rational) = oftype(x, sign(x.num))
signbit(x::Rational) = signbit(x.num)
copysign(x::Rational, y::Real) = copysign(x.num,y) // x.den
copysign(x::Rational, y::Rational) = copysign(x.num,y.num) // x.den
typemin(::Type{Rational{T}}) where {T<:Integer} = -one(T)//zero(T)
typemax(::Type{Rational{T}}) where {T<:Integer} = one(T)//zero(T)
isinteger(x::Rational) = x.den == 1
-(x::Rational) = (-x.num) // x.den
function -(x::Rational{T}) where T<:Signed
x.num == typemin(T) && throw(OverflowError("rational numerator is typemin(T)"))
(-x.num) // x.den
end
function -(x::Rational{T}) where T<:Unsigned
x.num != zero(T) && throw(OverflowError("cannot negate unsigned number"))
x
end
for (op,chop) in ((:+,:checked_add), (:-,:checked_sub),
(:rem,:rem), (:mod,:mod))
@eval begin
function ($op)(x::Rational, y::Rational)
xd, yd = divgcd(x.den, y.den)
Rational(($chop)(checked_mul(x.num,yd), checked_mul(y.num,xd)), checked_mul(x.den,yd))
end
end
end
function *(x::Rational, y::Rational)
xn,yd = divgcd(x.num,y.den)
xd,yn = divgcd(x.den,y.num)
checked_mul(xn,yn) // checked_mul(xd,yd)
end
/(x::Rational, y::Rational) = x//y
/(x::Rational, y::Complex{<:Union{Integer,Rational}}) = x//y
inv(x::Rational) = Rational(x.den, x.num)
fma(x::Rational, y::Rational, z::Rational) = x*y+z
==(x::Rational, y::Rational) = (x.den == y.den) & (x.num == y.num)
<( x::Rational, y::Rational) = x.den == y.den ? x.num < y.num :
widemul(x.num,y.den) < widemul(x.den,y.num)
<=(x::Rational, y::Rational) = x.den == y.den ? x.num <= y.num :
widemul(x.num,y.den) <= widemul(x.den,y.num)
==(x::Rational, y::Integer ) = (x.den == 1) & (x.num == y)
==(x::Integer , y::Rational) = y == x
<( x::Rational, y::Integer ) = x.num < widemul(x.den,y)
<( x::Integer , y::Rational) = widemul(x,y.den) < y.num
<=(x::Rational, y::Integer ) = x.num <= widemul(x.den,y)
<=(x::Integer , y::Rational) = widemul(x,y.den) <= y.num
function ==(x::AbstractFloat, q::Rational)
if isfinite(x)
(count_ones(q.den) == 1) & (x*q.den == q.num)
else
x == q.num/q.den
end
end
==(q::Rational, x::AbstractFloat) = x == q
for rel in (:<,:<=,:cmp)
for (Tx,Ty) in ((Rational,AbstractFloat), (AbstractFloat,Rational))
@eval function ($rel)(x::$Tx, y::$Ty)
if isnan(x) || isnan(y)
$(rel == :cmp ? :(throw(DomainError((x,y), "Inputs cannot be NaN."))) :
:(return false))
end
xn, xp, xd = decompose(x)
yn, yp, yd = decompose(y)
if xd < 0
xn = -xn
xd = -xd
end
if yd < 0
yn = -yn
yd = -yd
end
xc, yc = widemul(xn,yd), widemul(yn,xd)
xs, ys = sign(xc), sign(yc)
if xs != ys
return ($rel)(xs,ys)
elseif xs == 0
# both are zero or ±Inf
return ($rel)(xn,yn)
end
xb, yb = ndigits0z(xc,2) + xp, ndigits0z(yc,2) + yp
if xb == yb
xc, yc = promote(xc,yc)
if xp > yp
xc = (xc<<(xp-yp))
else
yc = (yc<<(yp-xp))
end
return ($rel)(xc,yc)
else
return xc > 0 ? ($rel)(xb,yb) : ($rel)(yb,xb)
end
end
end
end
# needed to avoid ambiguity between ==(x::Real, z::Complex) and ==(x::Rational, y::Number)
==(z::Complex , x::Rational) = isreal(z) & (real(z) == x)
==(x::Rational, z::Complex ) = isreal(z) & (real(z) == x)
for op in (:div, :fld, :cld)
@eval begin
function ($op)(x::Rational, y::Integer )
xn,yn = divgcd(x.num,y)
($op)(xn, checked_mul(x.den,yn))
end
function ($op)(x::Integer, y::Rational)
xn,yn = divgcd(x,y.num)
($op)(checked_mul(xn,y.den), yn)
end
function ($op)(x::Rational, y::Rational)
xn,yn = divgcd(x.num,y.num)
xd,yd = divgcd(x.den,y.den)
($op)(checked_mul(xn,yd), checked_mul(xd,yn))
end
end
end
trunc(::Type{T}, x::Rational) where {T} = convert(T,div(x.num,x.den))
floor(::Type{T}, x::Rational) where {T} = convert(T,fld(x.num,x.den))
ceil(::Type{T}, x::Rational) where {T} = convert(T,cld(x.num,x.den))
function round(::Type{T}, x::Rational{Tr}, ::RoundingMode{:Nearest}) where {T,Tr}
if denominator(x) == zero(Tr) && T <: Integer
throw(DivideError())
elseif denominator(x) == zero(Tr)
return convert(T, copysign(one(Tr)//zero(Tr), numerator(x)))
end
q,r = divrem(numerator(x), denominator(x))
s = q
if abs(r) >= abs((denominator(x)-copysign(Tr(4), numerator(x))+one(Tr)+iseven(q))>>1 + copysign(Tr(2), numerator(x)))
s += copysign(one(Tr),numerator(x))
end
convert(T, s)
end
round(::Type{T}, x::Rational) where {T} = round(T, x, RoundNearest)
function round(::Type{T}, x::Rational{Tr}, ::RoundingMode{:NearestTiesAway}) where {T,Tr}
if denominator(x) == zero(Tr) && T <: Integer
throw(DivideError())
elseif denominator(x) == zero(Tr)
return convert(T, copysign(one(Tr)//zero(Tr), numerator(x)))
end
q,r = divrem(numerator(x), denominator(x))
s = q
if abs(r) >= abs((denominator(x)-copysign(Tr(4), numerator(x))+one(Tr))>>1 + copysign(Tr(2), numerator(x)))
s += copysign(one(Tr),numerator(x))
end
convert(T, s)
end
function round(::Type{T}, x::Rational{Tr}, ::RoundingMode{:NearestTiesUp}) where {T,Tr}
if denominator(x) == zero(Tr) && T <: Integer
throw(DivideError())
elseif denominator(x) == zero(Tr)
return convert(T, copysign(one(Tr)//zero(Tr), numerator(x)))
end
q,r = divrem(numerator(x), denominator(x))
s = q
if abs(r) >= abs((denominator(x)-copysign(Tr(4), numerator(x))+one(Tr)+(numerator(x)<0))>>1 + copysign(Tr(2), numerator(x)))
s += copysign(one(Tr),numerator(x))
end
convert(T, s)
end
function round(::Type{T}, x::Rational{Bool}) where T
if denominator(x) == false && (T <: Union{Integer, Bool})
throw(DivideError())
end
convert(T, x)
end
round(::Type{T}, x::Rational{Bool}, ::RoundingMode{:Nearest}) where {T} = round(T, x)
round(::Type{T}, x::Rational{Bool}, ::RoundingMode{:NearestTiesAway}) where {T} = round(T, x)
round(::Type{T}, x::Rational{Bool}, ::RoundingMode{:NearestTiesUp}) where {T} = round(T, x)
round(::Type{T}, x::Rational{Bool}, ::RoundingMode) where {T} = round(T, x)
trunc(x::Rational{T}) where {T} = Rational(trunc(T,x))
floor(x::Rational{T}) where {T} = Rational(floor(T,x))
ceil(x::Rational{T}) where {T} = Rational(ceil(T,x))
round(x::Rational{T}) where {T} = Rational(round(T,x))
function ^(x::Rational, n::Integer)
n >= 0 ? power_by_squaring(x,n) : power_by_squaring(inv(x),-n)
end
^(x::Number, y::Rational) = x^(y.num/y.den)
^(x::T, y::Rational) where {T<:AbstractFloat} = x^convert(T,y)
^(x::Complex{T}, y::Rational) where {T<:AbstractFloat} = x^convert(T,y)
^(z::Complex{<:Rational}, n::Bool) = n ? z : one(z) # to resolve ambiguity
function ^(z::Complex{<:Rational}, n::Integer)
n >= 0 ? power_by_squaring(z,n) : power_by_squaring(inv(z),-n)
end
iszero(x::Rational) = iszero(numerator(x))
isone(x::Rational) = isone(numerator(x)) & isone(denominator(x))
function lerpi(j::Integer, d::Integer, a::Rational, b::Rational)
((d-j)*a)/d + (j*b)/d
end
float(::Type{Rational{T}}) where {T<:Integer} = float(T)