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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_rational_argerror(T) num2, den2 = (sign(den) < 0) ? divgcd(-num, -den) : divgcd(num, den) new(num2, den2) end end @noinline __throw_rational_argerror(T) = throw(ArgumentError("invalid rational: zero(\$T)//zero(\$T)")) 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. # Examples ```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*conj(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 Rational{T}(x::Rational) where {T<:Integer} = Rational{T}(convert(T,x.num), convert(T,x.den)) Rational{T}(x::Integer) where {T<:Integer} = Rational{T}(convert(T,x), convert(T,1)) Rational(x::Rational) = x Bool(x::Rational) = x==0 ? false : x==1 ? true : throw(InexactError(:Bool, Bool, x)) # to resolve ambiguity (::Type{T})(x::Rational) where {T<:Integer} = (isinteger(x) ? convert(T, x.num) : throw(InexactError(Symbol(string(T)), T, x))) AbstractFloat(x::Rational) = float(x.num)/float(x.den) function (::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 Rational{T}(x::AbstractFloat) where T<:Integer r = rationalize(T, x, tol=0) x == convert(typeof(x), r) || throw(InexactError(:Rational, Rational{T}, x)) r end Rational(x::Float64) = Rational{Int64}(x) Rational(x::Float32) = 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`. # Examples ```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 T(x)//one(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() 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() 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`. # Examples ```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`. # Examples ```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 abs(x::Rational) = Rational(abs(x.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<:BitSigned 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) \$(rel == :cmp ? :(return isnan(y) ? 0 : 1) : :(return false)) end if isnan(y) \$(rel == :cmp ? :(return -1) : :(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) ^(z::Complex{T}, p::Rational) where {T<:Real} = z^convert(typeof(one(T)^p), p) ^(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)