Switch to using HalfIntegers (#6)

* swich to using HalfIntegers, add project.toml, bump version, update CI

* add Random and add seed to avoid unlikely test failure

.travis.yml

@@ -4,9 +4,9 @@ os:
   - linux
   - osx
 julia:
-  - 0.7
   - 1.0
   - 1.1
+  - 1.2
   - nightly
 notifications:
   email: false

Project.toml

@@ -0,0 +1,19 @@
+name = "WignerSymbols"
+uuid = "9f57e263-0b3d-5e2e-b1be-24f2bb48858b"
+authors = ["Jutho Haegeman"]
+version = "1.0.0"
+
+[deps]
+HalfIntegers = "f0d1745a-41c9-11e9-1dd9-e5d34d218721"
+Primes = "27ebfcd6-29c5-5fa9-bf4b-fb8fc14df3ae"
+
+[compat]
+julia = "1"
+
+[extras]
+Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
+LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
+Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
+
+[targets]
+test = ["Test", "LinearAlgebra", "Random"]

REQUIRE

@@ -1,2 +0,0 @@
-julia 0.7
-Primes

appveyor.yml

@@ -1,7 +1,8 @@
 environment:
   matrix:
-  - julia_version: 0.7
   - julia_version: 1
+  - julia_version: 1.1
+  - julia_version: 1.2
   - julia_version: nightly
 
 platform:

src/WignerSymbols.jl

@@ -3,8 +3,8 @@ module WignerSymbols
 export δ, Δ, clebschgordan, wigner3j, wigner6j, racahV, racahW, HalfInteger
 
 using Base.GMP.MPZ
+using HalfIntegers
 
-include("halfinteger.jl")
 include("primefactorization.jl")
 
 const Wigner3j = Dict{Tuple{UInt,UInt,UInt,Int,Int},Tuple{Rational{BigInt},Rational{BigInt}}}()
@@ -29,13 +29,7 @@ end
 
 Checks the triangle conditions `j₃ <= j₁ + j₂`, `j₁ <= j₂ + j₃` and `j₂ <= j₃ + j₁`.
 """
-function δ(j₁, j₂, j₃)
+δ(j₁, j₂, j₃) = (j₃ <= j₁ + j₂) && (j₁ <= j₂ + j₃) && (j₂ <= j₃ + j₁) && isinteger(j₁+j₂+j₃)
-    j₃ <= j₁ + j₂ || return false
-    j₁ <= j₂ + j₃ || return false
-    j₂ <= j₃ + j₁ || return false
-    isinteger(j₁+j₂+j₃) || return false
-    return true
-end
 
 # triangle coefficient
 """
@@ -51,7 +45,7 @@ throws a `DomainError` if the `jᵢ`s are not (half)integer
 Δ(j₁, j₂, j₃) = Δ(Float64, j₁, j₂, j₃)
 function Δ(T::Type{<:AbstractFloat}, j₁, j₂, j₃)
     for jᵢ in (j₁, j₂, j₃)
-        (ishalfinteger(jᵢ) && jᵢ >= 0) || throw(DomainError("invalid jᵢ", jᵢ))
+        (ishalfinteger(jᵢ) && jᵢ >= zero(jᵢ)) || throw(DomainError("invalid jᵢ", jᵢ))
     end
     if !δ(j₁, j₂, j₃)
         return zero(T)
@@ -109,7 +103,8 @@ function wigner3j(T::Type{<:AbstractFloat}, j₁, j₂, j₃, m₁, m₂, m₃ =
         Wigner3j[(β₁, β₂, β₃, α₁, α₂)] = (r,s)
     end
 
-    return sgn*sqrt(convert(T, r.num)/convert(T, r.den))*(convert(T, s.num)/convert(T, s.den))
+    sn, sd, rn, rd = convert.(T, (s.num, s.den, r.num, r.den))
+    return sgn*(sn/sd)*sqrt(rn/rd)
 end
 
 """
@@ -121,7 +116,8 @@ as a type `T` floating point number. By default, `T = Float64` and `m₃ = m₁+
 Returns `zero(T)` if the triangle condition `δ(j₁, j₂, j₃)` is not satisfied, but
 throws a `DomainError` if the `jᵢ`s and `mᵢ`s are not (half)integer or `abs(mᵢ) > jᵢ`.
 """
-clebschgordan(j₁, m₁, j₂, m₂, j₃, m₃ = m₁+m₂) = clebschgordan(Float64, j₁, m₁, j₂, m₂, j₃, m₃)
+clebschgordan(j₁, m₁, j₂, m₂, j₃, m₃ = m₁+m₂) =
+    clebschgordan(Float64, j₁, m₁, j₂, m₂, j₃, m₃)
 function clebschgordan(T::Type{<:AbstractFloat}, j₁, m₁, j₂, m₂, j₃, m₃ = m₁+m₂)
     s = wigner3j(T, j₁, j₂, j₃, m₁, m₂, -m₃)
     iszero(s) && return s
@@ -162,13 +158,13 @@ wigner6j(j₁, j₂, j₃, j₄, j₅, j₆) = wigner6j(Float64, j₁, j₂, j
 function wigner6j(T::Type{<:AbstractFloat}, j₁, j₂, j₃, j₄, j₅, j₆)
     # check validity of `jᵢ`s
     for jᵢ in (j₁, j₂, j₃, j₄, j₅, j₆)
-        (ishalfinteger(jᵢ) && jᵢ >= 0) || throw(DomainError("invalid jᵢ", jᵢ))
+        (ishalfinteger(jᵢ) && jᵢ >= zero(jᵢ)) || throw(DomainError("invalid jᵢ", jᵢ))
     end
 
-    α̂₁ = map(converthalfinteger, (j₁, j₂, j₃))
+    α̂₁ = (j₁, j₂, j₃)
-    α̂₂ = map(converthalfinteger, (j₁, j₆, j₅))
+    α̂₂ = (j₁, j₆, j₅)
-    α̂₃ = map(converthalfinteger, (j₂, j₄, j₆))
+    α̂₃ = (j₂, j₄, j₆)
-    α̂₄ = map(converthalfinteger, (j₃, j₄, j₅))
+    α̂₄ = (j₃, j₄, j₅)
 
     # check triangle conditions
     if !(δ(α̂₁...) && δ(α̂₂...) && δ(α̂₃...) && δ(α̂₄...))
@@ -206,7 +202,8 @@ function wigner6j(T::Type{<:AbstractFloat}, j₁, j₂, j₃, j₄, j₅, j₆)
         Wigner6j[(β₁, β₂, β₃, α₁, α₂, α₃)] = (r, s)
     end
 
-    return sqrt(convert(T, r.num)/convert(T, r.den))*(convert(T, s.num)/convert(T, s.den))
+    sn, sd, rn, rd = convert.(T, (s.num, s.den, r.num, r.den))
+    return (sn/sd)*sqrt(rn/rd)
 end
 
 """

src/halfinteger.jl

@@ -1,190 +0,0 @@
-# HalfInteger
-
-"""
-    struct HalfInteger <: Real
-
-Represents half-integer values.
-
----
-
-HalfInteger(numerator::Integer, denominator::Integer)
-
-Constructs a `HalfInteger` object as a rational number from the given integer numerator
-and denominator values.
-
-# Examples
-
-```jldoctest
-julia> HalfInteger(1, 2)
-1/2
-
-julia> HalfInteger(-2, 1)
--2
-```
-"""
-struct HalfInteger <: Real
-    numerator::Int # with an implicit denominator of 2
-
-    function HalfInteger(num::Integer, den::Integer)
-        (den == 2) && return new(num)
-        (den == 1) && return new(2*num)
-        (den == 0) && throw(ArgumentError("Denominator can not be zero."))
-        # If non-trivial, we'll see if we can reduce it down to a half-integer
-        numerator, r = divrem(2*num, den)
-        if r == 0
-            return new(numerator)
-        else
-            throw(ArgumentError("$num // $den is not a half-integer value."))
-        end
-    end
-end
-
-"""
-    HalfInteger(x::Real)
-
-Attempts to create a `HalfInteger` out of the real number `x`. Throws an `InexactError` if
-`x` can not be represented as a half-integer value.
-
-# Examples
-
-```jldoctest
-julia> HalfInteger(3)
-3
-
-julia> HalfInteger(1.5)
-3/2
-```
-"""
-HalfInteger(x::Real) = convert(HalfInteger, x)
-
-Base.promote_rule(::Type{HalfInteger}, ::Type{<:Integer}) = HalfInteger
-Base.promote_rule(::Type{HalfInteger}, T::Type{<:Rational}) = T
-Base.promote_rule(::Type{HalfInteger}, T::Type{<:Real}) = T
-
-Base.convert(::Type{HalfInteger}, n::Integer) = HalfInteger(2*n, 2)
-function Base.convert(::Type{HalfInteger}, r::Rational)
-    if r.den == 1
-        return HalfInteger(2*r.num, 2)
-    elseif r.den == 2
-        return HalfInteger(r.num, 2)
-    else
-        throw(InexactError(:HalfInteger, HalfInteger, r))
-    end
-end
-function Base.convert(::Type{HalfInteger}, r::Real)
-    num = 2*r
-    if isinteger(num)
-        return HalfInteger(convert(Int, num), 2)
-    else
-        throw(InexactError(:HalfInteger, HalfInteger, r))
-    end
-end
-Base.convert(T::Type{<:Integer}, s::HalfInteger) = iseven(s.numerator) ? convert(T, s.numerator>>1) : throw(InexactError(Symbol(T), T, s))
-Base.convert(T::Type{<:Rational}, s::HalfInteger) = convert(T, s.numerator//2)
-Base.convert(T::Type{<:AbstractFloat}, s::HalfInteger) = convert(T, s.numerator) / T(2)
-Base.convert(::Type{HalfInteger}, s::HalfInteger) = s
-
-# Arithmetic
-
-Base.:+(a::HalfInteger, b::HalfInteger) = HalfInteger(a.numerator+b.numerator, 2)
-Base.:-(a::HalfInteger, b::HalfInteger) = HalfInteger(a.numerator-b.numerator, 2)
-Base.:-(a::HalfInteger) = HalfInteger(-a.numerator, 2)
-Base.:*(a::Integer, b::HalfInteger) = HalfInteger(a * b.numerator, 2)
-Base.:*(a::HalfInteger, b::Integer) = b * a
-Base.:<=(a::HalfInteger, b::HalfInteger) = a.numerator <= b.numerator
-Base.:<(a::HalfInteger, b::HalfInteger) = a.numerator < b.numerator
-Base.one(::Type{HalfInteger}) = HalfInteger(2, 2)
-Base.zero(::Type{HalfInteger}) = HalfInteger(0, 2)
-
-Base.floor(x::HalfInteger) = isinteger(x) ? x : x - HalfInteger(1, 2)
-Base.floor(::Type{T}, x::HalfInteger) where T <: Integer = convert(T, floor(x))
-
-Base.ceil(x::HalfInteger) = isinteger(x) ? x : x + HalfInteger(1, 2)
-Base.ceil(::Type{T}, x::HalfInteger) where T <: Integer = convert(T, ceil(x))
-
-# Hashing
-
-function Base.hash(a::HalfInteger, h::UInt)
-    iseven(a.numerator) && return hash(a.numerator>>1, h)
-    num, den = a.numerator, 2
-    den = 1
-    pow = -1
-    if abs(num) < 9007199254740992
-        return hash(ldexp(Float64(num),pow), h)
-    end
-    h = Base.hash_integer(den, h)
-    h = Base.hash_integer(pow, h)
-    h = Base.hash_integer(num, h)
-    return h
-end
-
-# Parsing and printing
-
-"""
-    parse(HalfInteger, s)
-
-Parses the string `s` into the corresponding `HalfInteger`-value. String can either be a
-number or a fraction of the form `n/2`.
-"""
-function Base.parse(::Type{HalfInteger}, s::AbstractString)
-    if in('/', s)
-        num, den = split(s, '/'; limit=2)
-        parse(Int, den) == 2 ||
-            throw(ArgumentError("Denominator not 2 in HalfInteger string '$s'."))
-        HalfInteger(parse(Int, num), 2)
-    elseif !isempty(strip(s))
-        HalfInteger(parse(Int, s))
-    else
-        throw(ArgumentError("input string is empty or only contains whitespace"))
-    end
-end
-
-Base.show(io::IO, x::HalfInteger) =
-    print(io, iseven(x.numerator) ? "$(div(x.numerator, 2))" : "$(x.numerator)/2")
-
-# Other methods
-
-Base.isinteger(a::HalfInteger) = iseven(a.numerator)
-ishalfinteger(a::HalfInteger) = true
-ishalfinteger(a::Integer) = true
-ishalfinteger(a::Rational) = a.den == 1 || a.den == 2
-ishalfinteger(a::Real) = isinteger(2*a)
-
-converthalfinteger(a::Number) = convert(HalfInteger, a)
-
-Base.numerator(a::HalfInteger) = iseven(a.numerator) ? div(a.numerator, 2) : a.numerator
-Base.denominator(a::HalfInteger) = iseven(a.numerator) ? 1 : 2
-
-# Range of HalfIntegers
-
-"""
-    struct HalfIntegerRange <: AbstractVector{HalfInteger}
-
-A range of `HalfInteger` values from `start` to `stop`, spaced by `1`. The `a:b` syntax
-where both `a` and `b` are `HalfInteger`s can also be use to construct this range.
-"""
-struct HalfIntegerRange <: AbstractVector{HalfInteger}
-    start :: HalfInteger
-    stop :: HalfInteger
-
-    function HalfIntegerRange(start::HalfInteger, stop::HalfInteger)
-        (start <= stop) ||
-            throw(ArgumentError("Second argument must be greater or equal to the first."))
-        return new(start, stop)
-    end
-end
-Base.iterate(it::HalfIntegerRange) = (it.start, it.start + 1)
-Base.iterate(it::HalfIntegerRange, s) = (s <= it.stop) ? (s, s+1) : nothing
-Base.length(it::HalfIntegerRange) = floor(Int, it.stop - it.start) + 1
-Base.size(it::HalfIntegerRange) = (length(it),)
-function Base.getindex(it::HalfIntegerRange, i::Integer)
-    1 <= i <= length(it) || throw(BoundsError(it, i))
-    it.start + i - 1
-end
-
-"""
-    (:)(i::HalfInteger, j::HalfInteger)
-
-Constructs a `HalfIntegerRange` out of two `HalfInteger` values.
-"""
-Base.:(:)(i::HalfInteger, j::HalfInteger) = HalfIntegerRange(i, j)

test/halfinteger.jl

@@ -1,194 +0,0 @@
-using Test
-using WignerSymbols: HalfInteger, ishalfinteger, HalfIntegerRange
-
-@testset "HalfInteger" begin
-    @testset "HalfInteger type" begin
-        # HalfInteger constructors
-        @test HalfInteger(1, 2).numerator == 1
-        @test HalfInteger(1, 1).numerator == 2
-        @test HalfInteger(0, 1).numerator == 0
-        @test HalfInteger(0, 2).numerator == 0
-        @test HalfInteger(0, 5).numerator == 0
-        @test HalfInteger(10, 5).numerator == 4
-        @test HalfInteger(21, 14).numerator == 3
-        @test HalfInteger(-3, 2).numerator == -3
-        @test HalfInteger(3, -2).numerator == -3
-        @test HalfInteger(-3, -2).numerator == 3
-        @test_throws ArgumentError HalfInteger(1, 0)
-        @test_throws ArgumentError HalfInteger(1, 3)
-        @test_throws ArgumentError HalfInteger(1, -3)
-        @test_throws ArgumentError HalfInteger(-5, 3)
-        @test_throws ArgumentError HalfInteger(-1000, -999)
-
-        # convert methods
-        @test convert(HalfInteger, 2) === HalfInteger(2, 1)
-        @test convert(HalfInteger, 1//2) === HalfInteger(1, 2)
-        @test convert(HalfInteger, 1.5) === HalfInteger(3, 2)
-        @test_throws InexactError convert(HalfInteger, 1//3)
-        @test_throws InexactError convert(HalfInteger, 0.6)
-        @test convert(HalfInteger, 2) === HalfInteger(2, 1)
-        @test convert(HalfInteger, 1//2) === HalfInteger(1, 2)
-        @test convert(HalfInteger, 1.5) === HalfInteger(3, 2)
-
-        @test convert(Integer, HalfInteger(2, 1)) === 2
-        @test_throws InexactError convert(Integer, HalfInteger(1, 2))
-        @test convert(Float64, HalfInteger(3, 2)) isa Float64
-        @test convert(Float32, HalfInteger(3, 2)) isa Float32
-        @test convert(Float64, HalfInteger(3, 2)) == 1.5
-        @test convert(Real, HalfInteger(3, 2)) === HalfInteger(3, 2)
-
-        # single-argument constructor
-        @test HalfInteger(0) == HalfInteger(0, 2)
-        @test HalfInteger(1) == HalfInteger(1, 1)
-        @test HalfInteger(2) == HalfInteger(2, 1)
-        @test HalfInteger(-30) == HalfInteger(-60, 2)
-        @test HalfInteger(0//2) == HalfInteger(0, 1)
-        @test HalfInteger(1//2) == HalfInteger(1, 2)
-        @test HalfInteger(-5//2) == HalfInteger(-5, 2)
-    end
-
-    a = HalfInteger(2)
-    b = HalfInteger(3, 2)
-
-    @testset "HalfInteger arithmetic" begin
-        @test a + b == 2 + 3//2
-        @test a - b == 2 - 3//2
-        @test zero(a) == 0
-        @test one(a) == 1
-        @test a > b
-        @test b < a
-        @test b <= a
-        @test a >= b
-        @test a == a
-        @test a != b
-        @test 2 * HalfInteger(0) == HalfInteger(0)
-        @test 2 * HalfInteger(1, 2) == HalfInteger(1)
-        @test HalfInteger(1) * 2 == HalfInteger(2)
-        @test 2 * a == HalfInteger(4)
-        @test (-1) * b == HalfInteger(-3//2)
-
-        @test floor(HalfInteger(0)) === HalfInteger(0)
-        @test floor(HalfInteger(-1)) === HalfInteger(-1)
-        @test floor(HalfInteger(1, 2)) === HalfInteger(0)
-        @test floor(HalfInteger(-1, 2)) === HalfInteger(-1)
-        @test floor(Int, HalfInteger(0)) === 0
-        @test floor(Int, HalfInteger(1, 2)) === 0
-        @test floor(Int32, HalfInteger(-5, 2)) === Int32(-3)
-        @test floor(Int32, HalfInteger(5)) === Int32(5)
-
-        @test ceil(HalfInteger(0)) === HalfInteger(0)
-        @test ceil(HalfInteger(-1)) === HalfInteger(-1)
-        @test ceil(HalfInteger(1, 2)) === HalfInteger(1)
-        @test ceil(HalfInteger(-1, 2)) === HalfInteger(0)
-        @test ceil(Int, HalfInteger(0)) === 0
-        @test ceil(Int, HalfInteger(1, 2)) === 1
-        @test ceil(Int32, HalfInteger(-5, 2)) === Int32(-2)
-        @test ceil(Int32, HalfInteger(5)) === Int32(5)
-
-        for n in -98:7:98
-            halfint, rat = HalfInteger(n, 2), n // 2
-            @test halfint == rat
-            @test halfint == HalfInteger(n / 2)
-            iseven(n) && @test halfint == HalfInteger(div(n, 2))
-            @test ceil(halfint) == ceil(rat)
-            @test floor(halfint) == floor(rat)
-        end
-    end
-
-    @testset "Parsing and printing" begin
-        @test string(HalfInteger(0)) == "0"
-        @test string(HalfInteger(1)) == "1"
-        @test string(HalfInteger(-1)) == "-1"
-        @test string(HalfInteger(1, 2)) == "1/2"
-        @test string(HalfInteger(-3, 2)) == "-3/2"
-
-        @test parse(HalfInteger, "0") == HalfInteger(0)
-        @test parse(HalfInteger, "1") == HalfInteger(1)
-        @test parse(HalfInteger, "210938") == HalfInteger(210938)
-        @test parse(HalfInteger, "-15") == HalfInteger(-15)
-        @test parse(HalfInteger, "1/2") == HalfInteger(1//2)
-        @test parse(HalfInteger, "-3/2") == HalfInteger(-3//2)
-        @test_throws ArgumentError parse(HalfInteger, "")
-        @test_throws ArgumentError parse(HalfInteger, "-50/100")
-        @test_throws ArgumentError parse(HalfInteger, "1/3")
-    end
-
-    @testset "HalfInteger hashing" begin
-        @test hash(a) == hash(2)
-        @test hash(b) == hash(1.5)
-    end
-
-    @testset "Other HalfInteger methods" begin
-        @test isinteger(HalfInteger(0))
-        @test isinteger(HalfInteger(1))
-        @test !isinteger(HalfInteger(1, 2))
-
-        @test ishalfinteger(1)
-        @test ishalfinteger(1.0)
-        @test ishalfinteger(-0.5)
-        @test ishalfinteger(HalfInteger(0))
-        @test ishalfinteger(HalfInteger(1, 2))
-        @test ishalfinteger(1//1)
-        @test ishalfinteger(1//2)
-        @test !ishalfinteger(0.3)
-        @test !ishalfinteger(-5//7)
-
-        @test numerator(HalfInteger(0)) == 0
-        @test numerator(HalfInteger(1, 2)) == 1
-        @test numerator(HalfInteger(1)) == 1
-        @test numerator(HalfInteger(-3, 2)) == -3
-
-        @test denominator(HalfInteger(0)) == 1
-        @test denominator(HalfInteger(1, 2)) == 2
-        @test denominator(HalfInteger(1)) == 1
-        @test denominator(HalfInteger(-3, 2)) == 2
-    end
-
-    @testset "HalfIntegerRange" begin
-        hi(x) = HalfInteger(x)
-
-        @test length(HalfIntegerRange(hi(0), hi(0))) == 1
-        @test length(HalfIntegerRange(hi(0), hi(2))) == 3
-        let hirange = HalfIntegerRange(hi(-1//2), hi(1//2))
-            @test length(hirange) == 2
-            @test size(hirange) == (2,)
-            @test collect(hirange) == [hi(-1//2), hi(1//2)]
-        end
-        let hirange = HalfIntegerRange(hi(0), hi(1//2))
-            @test length(hirange) == 1
-            @test size(hirange) == (1,)
-            @test collect(hirange) == [hi(0)]
-        end
-        let hirange = HalfIntegerRange(hi(1//2), hi(3))
-            @test length(hirange) == 3
-            @test size(hirange) == (3,)
-            @test collect(hirange) == [hi(1//2), hi(3//2), hi(5//2)]
-        end
-
-        @test hi(5):hi(7) == HalfIntegerRange(hi(5), hi(7))
-        @test hi(-1//2):hi(1//2) == HalfIntegerRange(hi(-1//2), hi(1//2))
-
-        @test collect(hi(0) : hi(2)) == [hi(0), hi(1), hi(2)]
-        @test collect(hi(-3//2) : hi(1//2)) == [hi(-3//2), hi(-1//2), hi(1//2)]
-
-        let hirange = hi(-3//2):hi(0)
-            @test length(hirange) == 2
-            @test size(hirange) == (2,)
-            @test collect(hirange) == [hi(-3//2), hi(-1//2)]
-        end
-
-        @test hi(1//2) ∈ hi(-1//2) : hi(1//2)
-        @test 1 ∈ hi(0) : hi(2)
-        @test 1//2 ∈ hi(-1//2) : hi(7//2)
-        @test !(hi(1//2) ∈ hi(0) : hi(1))
-        @test !(1//2 ∈ hi(-1) : hi(7))
-
-        r = hi(-3//2) : hi(3//2)
-        @test r[1] == hi(-3//2)
-        @test r[2] == hi(-1//2)
-        @test r[3] == hi(1//2)
-        @test r[4] == hi(3//2)
-        @test_throws BoundsError r[0]
-        @test_throws BoundsError r[5]
-    end
-end

test/runtests.jl

@@ -1,8 +1,9 @@
 using Test
 using WignerSymbols
 using LinearAlgebra
+using Random
 
-include("halfinteger.jl")
+Random.seed!(1234)
 
 smalljlist = 0:1//2:10
 largejlist = 0:1//2:1000
@@ -99,8 +100,9 @@ end
             Y = (2*j+1)*( j*(j+1)*( -j*(j+1) + j2*(j2+1) + j3*(j3+1) - 2*l1*(l1+1)) +
                             l2*(l2+1)*( j*(j+1) + j2*(j2+1) - j3*(j3+1) ) +
                             l3*(l3+1)*( j*(j+1) - j2*(j2+1) + j3*(j3+1) ) )
-            Z = (j+1)*sqrt((j^2-(j2-j3)^2)*((j2+j3+1)^2-j^2)*(j^2-(l2-l3)^2)*((l2+l3+1)^2 - j^2))
+            Z = (j+1) * sqrt( (j^2-(j2-j3)^2) * ((j2+j3+1)^2-j^2) *
-            tol = 10*max(abs(X),abs(Y),abs(Z))*eps(BigFloat)
+                                (j^2-(l2-l3)^2) * ((l2+l3+1)^2-j^2) )
+            tol = 10 * max(abs(X), abs(Y), abs(Z)) * eps(BigFloat)
             @test (X*wigner6j(BigFloat,j+1,j2,j3,l1,l2,l3) + Z*wigner6j(BigFloat,j-1,j2,j3,l1,l2,l3))≈(-Y*wigner6j(BigFloat,j,j2,j3,l1,l2,l3)) atol=tol
         end
     end
@@ -121,15 +123,15 @@ end
                 M = rand(-J:J) # only test for one instance of M in -J:J, should be independent of M anyway
                 fill!(V1,0)
                 fill!(V2,0)
-                for (k1,m1) in enumerate(m1range)
+                for (k1,m1) in enumerate(m1range), (k2,m2) in enumerate(m2range)
-                    for (k2,m2) in enumerate(m2range)
                     abs(m1+m2)<=J12 || continue
                     for (k3,m3) in enumerate(m3range)
                         abs(m2+m3)<=J23 || continue
                         m1+m2+m3==M || continue
-                            V1[k1,k2,k3] = clebschgordan(j1,m1,j2,m2,J12)*clebschgordan(J12,m1+m2,j3,m3,J)
+                        V1[k1,k2,k3] = clebschgordan(j1,m1,j2,m2,J12) *
-                            V2[k1,k2,k3] = clebschgordan(j2,m2,j3,m3,J23)*clebschgordan(j1,m1,J23,m2+m3,J)
+                                        clebschgordan(J12,m1+m2,j3,m3,J)
-                        end
+                        V2[k1,k2,k3] = clebschgordan(j2,m2,j3,m3,J23) *
+                                        clebschgordan(j1,m1,J23,m2+m3,J)
                     end
                 end
                 @test racahW(j1,j2,J,j3,J12,J23) ≈ dot(V2,V1)/sqrt((2*J12+1)*(2*J23+1)) atol=10*eps(Float64)