# publicbenyang/julia forked from JuliaLang/julia

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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 `type Rational{T<:Integer} <: Real    num::T    den::T    function Rational(num::T, den::T)        if num == 0 && den == 0            error("invalid rational: 0//0")        end        g = gcd(den, num)        new(div(num, g), div(den, g))    endendRational{T<:Integer}(n::T, d::T) = Rational{T}(n,d)Rational(n::Integer, d::Integer) = Rational(promote(n,d)...)Rational(n::Integer) = Rational(n,one(n))//(n::Integer, d::Integer ) = Rational(n,d)//(x::Rational, y::Integer ) = x.num//(x.den*y)//(x::Integer, y::Rational) = (x*y.den)//y.num//(x::Rational, y::Rational) = (x.num*y.den)//(x.den*y.num)//(x::Complex, y::Real ) = complex(real(x)//y,imag(x)//y)//(x::Real, y::Complex ) = x*y'//real(y*y')function //(x::Complex, y::Complex)    xy = x*y'    yy = real(y*y')    complex(real(xy)//yy, imag(xy)//yy)endfunction show(io, x::Rational)    if isinf(x)        print(io, x.num > 0 ? "Inf" : "-Inf")    else        show(io, num(x)); print(io, "//"); show(io, den(x))    endendconvert{T<:Integer}(::Type{Rational{T}}, x::Rational) = Rational(convert(T,x.num),convert(T,x.den))convert{T<:Integer}(::Type{Rational{T}}, x::Integer) = Rational(convert(T,x), convert(T,1))function convert{T<:Integer}(::Type{Rational{T}}, x::Float, tol::Real)    if isnan(x); return zero(T)//zero(T); end    if x < typemin(T); return -one(T)//zero(T); end    if typemax(T) < x; return one(T)//zero(T); end    y = x    a = d = one(T)    b = c = zero(T)    while true        f = convert(T,trunc(y)); y -= f        a, b, c, d = f*a+c, f*b+d, a, b        if y == 0 || abs(a/b-x) <= tol            return a//b        end        y = 1/y    endendconvert{T<:Integer}(rt::Type{Rational{T}}, x::Float) = convert(rt,x,0)convert(::Type{Bool}, x::Rational) = (x!=0) # to resolve ambiguityconvert{T<:Rational}(::Type{T}, x::Rational) = xconvert{T<:Real}(::Type{T}, x::Rational) = convert(T, x.num/x.den)promote_rule{T<:Integer}(::Type{Rational{T}}, ::Type{T}) = Rational{T}promote_rule{T<:Integer,S<:Integer}(::Type{Rational{T}}, ::Type{S}) = Rational{promote_type(T,S)}promote_rule{T<:Integer,S<:Integer}(::Type{Rational{T}}, ::Type{Rational{S}}) = Rational{promote_type(T,S)}promote_rule{T<:Integer,S<:Float}(::Type{Rational{T}}, ::Type{S}) = promote_type(T,S)num(x::Integer) = xden(x::Integer) = one(x)num(x::Rational) = x.numden(x::Rational) = x.densign(x::Rational) = sign(x.num)signbit(x::Rational) = signbit(x.num)copysign(x::Rational, y::Real) = copysign(x.num,y) // x.dencopysign(x::Rational, y::Rational) = copysign(x.num,y.num) // x.denisnan(x::Rational) = falseisinf(x::Rational) = x.den == 0isfinite(x::Rational) = x.den != 0typemin{T<:Integer}(::Type{Rational{T}}) = -one(T)//zero(T)typemax{T<:Integer}(::Type{Rational{T}}) = one(T)//zero(T)integer_valued(x::Rational) = x.den == 1float64_valued(x::Rational) = abs(x.num) <= x.den*maxintfloat(Float64)hash(x::Rational) = integer_valued(x) ? hash(x.num) :                    float64_valued(x) ? hash(float64(x)) :                    bitmix(hash(x.num),hash(x.den))-(x::Rational) = (-x.num) // x.den+(x::Rational, y::Rational) = (x.num*y.den + x.den*y.num) // (x.den*y.den)-(x::Rational, y::Rational) = (x.num*y.den - x.den*y.num) // (x.den*y.den)*(x::Rational, y::Rational) = (x.num*y.num) // (x.den*y.den)/(x::Rational, y::Rational) = (x.num*y.den) // (x.den*y.num)/(x::Rational, z::ComplexPair) = inv(z/x)==(x::Rational, y::Rational) = x.den == y.den && x.num == y.num==(x::Rational, y::Integer ) = x.den == 1 && x.num == y==(x::Integer , y::Rational) = y == x# needed to avoid ambiguity between ==(x::Real, z::Complex) and ==(x::Rational, y::Number)==(z::Complex , x::Rational) = real_valued(z) && real(z) == x==(x::Rational, z::Complex ) = real_valued(z) && real(z) == x==(x::Rational, y::Number ) = x.num == x.den*y==(x::Number , y::Rational) = y == x==(x::Rational, y::Float ) = x.den==0 ? oftype(y,x)==y : x.num == x.den*y< (x::Rational, y::Rational) = x.den == y.den ? x.num < y.num : x.num*y.den < x.den*y.num< (x::Rational, y::Real ) = x.num < x.den*y< (x::Real , y::Rational) = x*y.den < y.num<=(x::Rational, y::Rational) = x.den == y.den ? x.num <= y.num : x.num*y.den <= x.den*y.num<=(x::Rational, y::Real ) = x.num <= x.den*y<=(x::Real , y::Rational) = x*y.den <= y.numdiv(x::Rational, y::Rational) = div(x.num*y.den, x.den*y.num)div(x::Rational, y::Real ) = div(x.num, x.den*y)div(x::Real , y::Rational) = div(x*y.den, y.num)fld(x::Rational, y::Rational) = fld(x.num*y.den, x.den*y.num)fld(x::Rational, y::Real ) = fld(x.num, x.den*y)fld(x::Real , y::Rational) = fld(x*y.den, y.num)itrunc(x::Rational) = div(x.num,x.den)ifloor(x::Rational) = fld(x.num,x.den)iceil (x::Rational) = -fld(-x.num,x.den)iround(x::Rational) = div(x.num*2 + copysign(x.den,x.num), x.den*2)trunc(x::Rational) = Rational(itrunc(x))floor(x::Rational) = Rational(ifloor(x))ceil (x::Rational) = Rational(iceil(x))round(x::Rational) = Rational(iround(x))rational(x::Real) = rational(x, 0)rational(x::Rational, tol::Real) = xrational(x::Integer) = x // one(x)rational(x::Integer, tol::Real) = x // one(x)rational(x::Float32, tol::Real) = convert(Rational{Int32}, x, tol)rational(x::Float64, tol::Real) = convert(Rational{Int64}, x, tol)rational(z::Complex) = complex(rational(real(z)), rational(imag(z)))rational(z::Complex, tol::Real) =    (tol /= sqrt(2); complex(rational(real(z), tol), rational(imag(z), tol)))## rational to int coercion ##for f in (:int8, :int16, :int32, :int64, :int128,          :uint8, :uint16, :uint32, :uint64, :uint128,          :signed, :integer, :unsigned, :int, :uint)    @eval (\$f)(x::Rational) = (\$f)(iround(x))end`
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