Add exact rules for irrational numbers

src/IrrationalConstants.jl

@@ -29,5 +29,6 @@ export
     log4π       # log(4π)
 
 include("stats.jl")
+include("rules.jl")
 
 end # module

src/rules.jl

@@ -0,0 +1,64 @@
+## Square
+SQUARE_PAIRS = (
+    (sqrt2, 2.0),
+    (sqrt3, 3.0),
+    (sqrtπ, π),
+    (sqrt2π, twoπ),
+    (sqrt4π, fourπ),
+    (sqrthalfπ, halfπ),
+    (invsqrt2, 0.5),
+    (invsqrtπ, invπ),
+    (invsqrt2π, inv2π),
+)
+for (a,b) in SQUARE_PAIRS
+    Base.:(*)(::typeof(a), ::typeof(a)) = b
+    Base.literal_pow(::typeof(^), ::typeof(a), ::Val{2}) = b
+end
+
+## Inverse
+INVERSE_PAIRS = (
+    (π, invπ),
+    (twoπ, inv2π),
+    (twoinvπ, halfπ),
+    (quartπ, fourinvπ),
+    (fourπ, inv4π),
+    (sqrt2π, invsqrt2π),
+    (sqrt2, invsqrt2),
+    (sqrtπ, invsqrtπ),
+)
+for (a,b) in INVERSE_PAIRS
+    if a !== π  # Avoid type piracy
+        Base.inv(::typeof(a)) = b
+        Base.literal_pow(::typeof(^), ::typeof(a), ::Val{-1}) = b
+    end
+    Base.inv(::typeof(b)) = a
+    Base.literal_pow(::typeof(^), ::typeof(b), ::Val{-1}) = a
+    Base.:(*)(::typeof(a), ::typeof(b)) = one(Irrational)
+end
+
+## Triangular
+Base.sin(::Irrational{:twoπ}) = 0.0
+Base.cos(::Irrational{:twoπ}) = 1.0
+Base.sincos(::Irrational{:twoπ}) = (0.0, 1.0)
+Base.sin(::Irrational{:fourπ}) = 0.0
+Base.cos(::Irrational{:fourπ}) = 1.0
+Base.sincos(::Irrational{:fourπ}) = (0.0, 1.0)
+Base.sin(::Irrational{:quartπ}) = invsqrt2
+Base.cos(::Irrational{:quartπ}) = invsqrt2
+Base.sincos(::Irrational{:quartπ}) = (invsqrt2, invsqrt2)
+Base.sin(::Irrational{:halfπ}) = 1.0
+Base.cos(::Irrational{:halfπ}) = 0.0
+Base.sincos(::Irrational{:halfπ}) = (1.0, 0.0)
+
+## Exponential
+Base.exp(::Irrational{:loghalf}) = 0.5
+Base.exp(::Irrational{:logtwo}) = 2.0
+Base.exp(::Irrational{:logten}) = 10.0
+Base.exp(::Irrational{:logπ}) = π
+Base.exp(::Irrational{:log2π}) = twoπ
+Base.exp(::Irrational{:log4π}) = fourπ
+
+## Logarithm
+# Base.log(::Irrational{:π}) = logπ  # Avoid type piracy
+Base.log(::Irrational{:twoπ}) = log2π
+Base.log(::Irrational{:fourπ}) = log4π

test/runtests.jl

@@ -1,42 +1,114 @@
 using IrrationalConstants
 using Test
 
-@testset "k*pi" begin
-  @test isapprox(2*pi, twoπ)
-  @test isapprox(4*pi, fourπ)
-  @test isapprox(pi/2, halfπ)
-  @test isapprox(pi/4, quartπ)
-end
+const IRRATIONALS = (
+    twoπ,
+    fourπ,
+    halfπ,
+    quartπ,
+    invπ,
+    twoinvπ,
+    fourinvπ,
+    inv2π,
+    inv4π,
+    sqrt2,
+    sqrt3,
+    sqrtπ,
+    sqrt2π,
+    sqrt4π,
+    sqrthalfπ,
+    invsqrt2,
+    invsqrtπ,
+    invsqrt2π,
+    loghalf,
+    logtwo,
+    logten,
+    logπ,
+    log2π,
+    log4π,
+)
 
-@testset "k/pi" begin
-  @test isapprox(1/pi, invπ)
-  @test isapprox(2/pi, twoinvπ)
-  @test isapprox(4/pi, fourinvπ)
-end
+function test_with_function(f, a::Irrational)
+    b = f(a)
+    @test b ≈ f(float(a)) atol=1e-14
 
-@testset "1/(k*pi)" begin  
-  @test isapprox(1/(2pi), inv2π)
-  @test isapprox(1/(4pi), inv4π)
-end
+    # If f(a) is approximately equal to a value in IRRATIONALS, f(a) should be Irrational.
+    @test (b .≈ IRRATIONALS) == (b .=== IRRATIONALS)
 
-@testset "sqrt" begin
-  @test isapprox(sqrt(2), sqrt2)
-  @test isapprox(sqrt(3), sqrt3)
-  @test isapprox(sqrt(pi), sqrtπ)
-  @test isapprox(sqrt(2pi), sqrt2π)
-  @test isapprox(sqrt(4pi), sqrt4π)
-  @test isapprox(sqrt(pi/2), sqrthalfπ)
-  @test isapprox(sqrt(1/2), invsqrt2)
-  @test isapprox(sqrt(1/(pi)), invsqrtπ)
-  @test isapprox(sqrt(1/(2pi)), invsqrt2π)
+    # If f(a) is close to integer, it should be a integer.
+    if abs(b - round(b)) < 1e-14
+        @test isinteger(b)
+    end
 end
 
-@testset "log" begin
-  @test isapprox(log(1/2), loghalf)
-  @test isapprox(log(2), logtwo)
-  @test isapprox(log(10), logten)
-  @test isapprox(log(pi), logπ)
-  @test isapprox(log(2pi), log2π)
-  @test isapprox(log(4pi), log4π)
+@testset "approximately equal" begin
+    @testset "k*pi" begin
+        @test isapprox(2*pi, twoπ)
+        @test isapprox(4*pi, fourπ)
+        @test isapprox(pi/2, halfπ)
+        @test isapprox(pi/4, quartπ)
+    end
+
+    @testset "k/pi" begin
+        @test isapprox(1/pi, invπ)
+        @test isapprox(2/pi, twoinvπ)
+        @test isapprox(4/pi, fourinvπ)
+    end
+
+    @testset "1/(k*pi)" begin
+        @test isapprox(1/(2pi), inv2π)
+        @test isapprox(1/(4pi), inv4π)
+    end
+
+    @testset "sqrt" begin
+        @test isapprox(sqrt(2), sqrt2)
+        @test isapprox(sqrt(3), sqrt3)
+        @test isapprox(sqrt(pi), sqrtπ)
+        @test isapprox(sqrt(2pi), sqrt2π)
+        @test isapprox(sqrt(4pi), sqrt4π)
+        @test isapprox(sqrt(pi/2), sqrthalfπ)
+        @test isapprox(sqrt(1/2), invsqrt2)
+        @test isapprox(sqrt(1/(pi)), invsqrtπ)
+        @test isapprox(sqrt(1/(2pi)), invsqrt2π)
+    end
+
+    @testset "log" begin
+        @test isapprox(log(1/2), loghalf)
+        @test isapprox(log(2), logtwo)
+        @test isapprox(log(10), logten)
+        @test isapprox(log(pi), logπ)
+        @test isapprox(log(2pi), log2π)
+        @test isapprox(log(4pi), log4π)
+    end
 end
 
+@testset "rules for $(a)" for a in IRRATIONALS
+    @testset "log" begin
+        if a > 0
+            test_with_function(log, a)
+        else
+            @test_throws DomainError log(a)
+        end
+    end
+
+    @testset "inv" begin
+        test_with_function(inv, a)
+        test_with_function(t->t^-1, a)
+    end
+
+    @testset "exp" begin
+        test_with_function(exp, a)
+    end
+
+    @testset "sin" begin
+        test_with_function(sin, a)
+        test_with_function(cos, a)
+        @test sincos(a) == (sin(a),cos(a))
+    end
+
+    @testset "square" begin
+        test_with_function(t->t*t, a)
+        test_with_function(abs2, a)
+        test_with_function(t->t^2, a)
+    end
+end