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complex.jl
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complex.jl
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# This file is a part of Julia. License is MIT: https://julialang.org/license
using LinearAlgebra
@test reim(2 + 3im) == (2, 3)
for T in (Int64, Float64)
@test real(T) == T
@test real(Complex{T}) == T
@test complex(T) == Complex{T}
@test complex(Complex{T}) == Complex{T}
end
#show
@test sprint(show, complex(1, 0), context=:compact => true) == "1+0im"
@test sprint(show, complex(true, true)) == "Complex(true,true)"
@testset "arithmetic" begin
@testset for T in (Float16, Float32, Float64, BigFloat)
t = true
f = false
@testset "add and subtract" begin
@test isequal(T(+0.0) + im, Complex(T(+0.0), T(+1.0)))
@test isequal(T(-0.0) + im, Complex(T(-0.0), T(+1.0)))
@test isequal(T(+0.0) - im, Complex(T(+0.0), T(-1.0)))
@test isequal(T(-0.0) - im, Complex(T(-0.0), T(-1.0)))
@test isequal(T(+1.0) + im, Complex(T(+1.0), T(+1.0)))
@test isequal(T(-1.0) + im, Complex(T(-1.0), T(+1.0)))
@test isequal(T(+1.0) - im, Complex(T(+1.0), T(-1.0)))
@test isequal(T(-1.0) - im, Complex(T(-1.0), T(-1.0)))
@test isequal(im + T(+0.0), Complex(T(+0.0), T(+1.0)))
@test isequal(im + T(-0.0), Complex(T(-0.0), T(+1.0)))
@test isequal(im - T(+0.0), Complex(T(+0.0), T(+1.0)))
@test isequal(im - T(-0.0), Complex(T(+0.0), T(+1.0)))
@test isequal(im + T(+1.0), Complex(T(+1.0), T(+1.0)))
@test isequal(im + T(-1.0), Complex(T(-1.0), T(+1.0)))
@test isequal(im - T(+1.0), Complex(T(-1.0), T(+1.0)))
@test isequal(im - T(-1.0), Complex(T(+1.0), T(+1.0)))
@test isequal(T(f) + im, Complex(T(+0.0), T(+1.0)))
@test isequal(T(t) + im, Complex(T(+1.0), T(+1.0)))
@test isequal(T(f) - im, Complex(T(+0.0), T(-1.0)))
@test isequal(T(t) - im, Complex(T(+1.0), T(-1.0)))
@test isequal(im + T(f), Complex(T(+0.0), T(+1.0)))
@test isequal(im + T(t), Complex(T(+1.0), T(+1.0)))
@test isequal(im - T(f), Complex(T(+0.0), T(+1.0)))
@test isequal(im - T(t), Complex(T(-1.0), T(+1.0)))
end
@testset "multiply" begin
@test isequal(T(+0.0) * im, Complex(T(+0.0), T(+0.0)))
@test isequal(T(-0.0) * im, Complex(T(-0.0), T(-0.0)))
@test isequal(T(+1.0) * im, Complex(T(+0.0), T(+1.0)))
@test isequal(T(-1.0) * im, Complex(T(-0.0), T(-1.0)))
@test isequal(im * T(+0.0), Complex(T(+0.0), T(+0.0)))
@test isequal(im * T(-0.0), Complex(T(-0.0), T(-0.0)))
@test isequal(im * T(+1.0), Complex(T(+0.0), T(+1.0)))
@test isequal(im * T(-1.0), Complex(T(-0.0), T(-1.0)))
end
@testset "divide" begin
@test isequal(T(+0.0) / im, Complex(T(+0.0), T(-0.0)))
@test isequal(T(-0.0) / im, Complex(T(-0.0), T(+0.0)))
@test isequal(T(+1.0) / im, Complex(T(+0.0), T(-1.0)))
@test isequal(T(-1.0) / im, Complex(T(-0.0), T(+1.0)))
end
end
@test isequal(true + complex(true,false), complex(true,false) + complex(true,false))
@test isequal(complex(true,false) + true, complex(true,false) + complex(true,false))
@test isequal(true - complex(true,false), complex(true,false) - complex(true,false))
@test isequal(complex(true,false) - true, complex(true,false) - complex(true,false))
@test isequal(true * complex(true,false), complex(true,false) * complex(true,false))
@test isequal(complex(true,false) * true, complex(true,false) * complex(true,false))
end
@testset "basic math functions" begin
# We compare to BigFloat instead of hard-coding
# values, assuming that BigFloat has an independent and independently
# tested implementation.
@testset for T in (Float32, Float64)
x = Complex{T}(1//3 + 1//4*im)
y = Complex{T}(1//2 + 1//5*im)
yi = 4
@testset "Random values" begin
@test x^y ≈ big(x)^big(y)
@test x^yi ≈ big(x)^yi
@test x^true ≈ big(x)^true
@test x^false ≈ big(x)^false
@test x^1 ≈ big(x)^1
@test abs(x) ≈ abs(big(x))
@test abs2(x) ≈ abs2(big(x))
@test acos(x) ≈ acos(big(x))
@test acosh(1+x) ≈ acosh(1+big(x))
@test angle(x) ≈ angle(big(x))
@test asin(x) ≈ asin(big(x))
@test asinh(x) ≈ asinh(big(x))
@test atan(x) ≈ atan(big(x))
@test atanh(x) ≈ atanh(big(x))
@test cis(real(x)) ≈ cis(real(big(x)))
@test cis(x) ≈ cis(big(x))
@test cos(x) ≈ cos(big(x))
@test cosh(x) ≈ cosh(big(x))
@test exp(x) ≈ exp(big(x))
@test exp10(x) ≈ exp10(big(x))
@test exp2(x) ≈ exp2(big(x))
@test expm1(x) ≈ expm1(big(x)) atol=eps(T)
@test log(x) ≈ log(big(x))
@test log10(x) ≈ log10(big(x))
@test log1p(x) ≈ log1p(big(x))
@test log2(x) ≈ log2(big(x))
@test sin(x) ≈ sin(big(x))
@test sinh(x) ≈ sinh(big(x))
@test sqrt(x) ≈ sqrt(big(x))
@test tan(x) ≈ tan(big(x))
@test tanh(x) ≈ tanh(big(x))
end
@testset "Inverses" begin
@test acos(cos(x)) ≈ x
@test acosh(cosh(x)) ≈ x
@test asin(sin(x)) ≈ x
@test asinh(sinh(x)) ≈ x
@test atan(tan(x)) ≈ x
@test atanh(tanh(x)) ≈ x
@test cos(acos(x)) ≈ x
@test cosh(acosh(1+x)) ≈ 1+x
@test exp(log(x)) ≈ x
@test exp10(log10(x)) ≈ x
@test exp2(log2(x)) ≈ x
@test expm1(log1p(x)) ≈ x
@test log(exp(x)) ≈ x
@test log10(exp10(x)) ≈ x
@test log1p(expm1(x)) ≈ x
@test log2(exp2(x)) ≈ x
@test sin(asin(x)) ≈ x
@test sinh(asinh(x)) ≈ x
@test sqrt(x)^2 ≈ x
@test sqrt(x^2) ≈ x
@test tan(atan(x)) ≈ x
@test tanh(atanh(x)) ≈ x
end
@testset "Relations between functions" begin
@test cosh(x) ≈ (exp(x)+exp(-x))/2
@test cosh(x)^2-sinh(x)^2 ≈ 1
@test sin(x)^2+cos(x)^2 ≈ 1
@test sinh(x) ≈ (exp(x)-exp(-x))/2
@test tan(x) ≈ sin(x)/cos(x)
@test tanh(x) ≈ sinh(x)/cosh(x)
end
end
end
@testset "isinf" begin
@test iszero(real(complex(0.0,1.0))) # isimag deprecated
@test !iszero(real(complex(1.0,1.0))) # isimag deprecated
@test isinf(complex(Inf,0))
@test isinf(complex(-Inf,0))
@test isinf(complex(0,Inf))
@test isinf(complex(0,-Inf))
@test !isinf(complex(0,0))
end
@testset "flipsign" begin
@test isequal(complex( 0.0, 0.0 ), flipsign(complex( 0.0, 0.0 ), 1))
@test isequal(complex( -0.0, -0.0 ), flipsign(complex( 0.0, 0.0 ), -1))
@test isequal(complex( Inf, 0.0 ), flipsign(complex( Inf, 0.0 ), 1))
@test isequal(complex( -Inf, -0.0 ), flipsign(complex( Inf, 0.0 ), -1))
@test isequal(complex( 0.0, NaN ), flipsign(complex( 0.0, NaN ), 1.0))
@test isequal(complex( -0.0, NaN ), flipsign(complex( 0.0, NaN ), -1.0))
@test isequal(complex( 5.0, 4.0 ), flipsign(complex(-5.0, -4.0), -1))
@test isequal(complex( 0.5, -0.5 ), flipsign(complex(-0.5, 0.5), -2))
end
@testset "sqrt" begin
# tests special values from csqrt man page
# as well as conj(sqrt(z)) = sqrt(conj(z))
@test isequal(sqrt(complex( 0.0, 0.0)), complex( 0.0, 0.0))
@test isequal(sqrt(complex( 0.0,-0.0)), complex( 0.0,-0.0))
@test isequal(sqrt(complex( 0.0, Inf)), complex( Inf, Inf))
@test isequal(sqrt(complex( 0.0,-Inf)), complex( Inf,-Inf))
@test isequal(sqrt(complex( 0.0, NaN)), complex( NaN, NaN))
@test isequal(sqrt(complex(-0.0, 0.0)), complex( 0.0, 0.0))
@test isequal(sqrt(complex(-0.0,-0.0)), complex( 0.0,-0.0))
@test isequal(sqrt(complex( 5.0, 0.0)), complex(sqrt(5.0), 0.0))
@test isequal(sqrt(complex( 5.0,-0.0)), complex(sqrt(5.0),-0.0))
@test isequal(sqrt(complex(-5.0, 0.0)), complex( 0.0, sqrt(5.0)))
@test isequal(sqrt(complex(-5.0,-0.0)), complex( 0.0,-sqrt(5.0)))
@test isequal(sqrt(complex( Inf, 0.0)), complex( Inf, 0.0))
@test isequal(sqrt(complex( Inf,-0.0)), complex( Inf,-0.0))
@test isequal(sqrt(complex( Inf, 5.0)), complex( Inf, 0.0))
@test isequal(sqrt(complex( Inf,-5.0)), complex( Inf,-0.0))
@test isequal(sqrt(complex( Inf, Inf)), complex( Inf, Inf))
@test isequal(sqrt(complex( Inf,-Inf)), complex( Inf,-Inf))
@test isequal(sqrt(complex( Inf, NaN)), complex( Inf, NaN))
@test isequal(sqrt(complex(-Inf, 0.0)), complex( 0.0, Inf))
@test isequal(sqrt(complex(-Inf,-0.0)), complex( 0.0,-Inf))
@test isequal(sqrt(complex(-Inf, 5.0)), complex( 0.0, Inf))
@test isequal(sqrt(complex(-Inf,-5.0)), complex( 0.0,-Inf))
@test isequal(sqrt(complex(-Inf, Inf)), complex( Inf, Inf))
@test isequal(sqrt(complex(-Inf,-Inf)), complex( Inf,-Inf))
@test isequal(sqrt(complex(-Inf, NaN)), complex( NaN, Inf))
@test isequal(sqrt(complex( NaN, 0.0)), complex( NaN, NaN))
@test isequal(sqrt(complex( NaN, 0.0)), complex( NaN, NaN))
@test isequal(sqrt(complex( NaN, Inf)), complex( Inf, Inf))
@test isequal(sqrt(complex( NaN,-Inf)), complex( Inf,-Inf))
end
@testset "log(conj(z)) == conj(log(z))" begin
@test isequal(log(complex( 0.0, 0.0)), complex(-Inf, 0.0))
@test isequal(log(complex( 0.0,-0.0)), complex(-Inf,-0.0))
@test isequal(log(complex( 0.0, 1.0)), complex( 0.0, pi/2))
@test isequal(log(complex( 0.0,-1.0)), complex( 0.0,-pi/2))
@test isequal(log(complex( 0.0, Inf)), complex( Inf, pi/2))
@test isequal(log(complex( 0.0,-Inf)), complex( Inf,-pi/2))
@test isequal(log(complex( 0.0, NaN)), complex( NaN, NaN))
@test isequal(log(complex(-0.0, 0.0)), complex(-Inf, pi))
@test isequal(log(complex(-0.0,-0.0)), complex(-Inf,-pi))
@test isequal(log(complex( 5.0, 0.0)),complex(log(5.0), 0.0))
@test isequal(log(complex( 5.0,-0.0)),complex(log(5.0),-0.0))
@test isequal(log(complex( Inf, 5.0)), complex( Inf, 0.0))
@test isequal(log(complex( Inf,-5.0)), complex( Inf,-0.0))
@test isequal(log(complex( Inf, Inf)), complex( Inf, pi/4))
@test isequal(log(complex( Inf,-Inf)), complex( Inf,-pi/4))
@test isequal(log(complex( Inf, NaN)), complex( Inf, NaN))
@test isequal(log(complex(-Inf, 5.0)), complex( Inf, pi))
@test isequal(log(complex(-Inf,-5.0)), complex( Inf,-pi))
@test isequal(log(complex(-Inf, Inf)), complex( Inf, 3*pi/4))
@test isequal(log(complex(-Inf,-Inf)), complex( Inf,-3*pi/4))
@test isequal(log(complex(-Inf, NaN)), complex( Inf, NaN))
@test isequal(log(complex( NaN, 0.0)), complex( NaN, NaN))
@test isequal(log(complex( NaN, Inf)), complex( Inf, NaN))
@test isequal(log(complex( NaN,-Inf)), complex( Inf, NaN))
@test isequal(log(complex( NaN, NaN)), complex( NaN, NaN))
end
@testset "exp(conj(z)) == conj(exp(z))" begin
@test isequal(exp(complex( 0.0, 0.0)), complex(1.0, 0.0))
@test isequal(exp(complex( 0.0,-0.0)), complex(1.0,-0.0))
@test isequal(exp(complex( 0.0, Inf)), complex(NaN, NaN))
@test isequal(exp(complex( 0.0,-Inf)), complex(NaN, NaN))
@test isequal(exp(complex( 0.0, NaN)), complex(NaN, NaN))
@test isequal(exp(complex(-0.0, 0.0)), complex(1.0, 0.0))
@test isequal(exp(complex(-0.0,-0.0)), complex(1.0,-0.0))
@test isequal(exp(complex( 5.0, Inf)), complex(NaN, NaN))
@test isequal(exp(complex( Inf, 0.0)), complex(Inf, 0.0))
@test isequal(exp(complex( Inf,-0.0)), complex(Inf,-0.0))
@test isequal(exp(complex( Inf, 5.0)), complex(cos(5.0)*Inf,sin(5.0)* Inf))
@test isequal(exp(complex( Inf,-5.0)), complex(cos(5.0)*Inf,sin(5.0)*-Inf))
@test isequal(exp(complex( Inf, NaN)), complex(-Inf, NaN))
@test isequal(exp(complex( Inf, Inf)), complex(-Inf, NaN))
@test isequal(exp(complex( Inf,-Inf)), complex(-Inf, NaN))
@test isequal(exp(complex(-Inf, 0.0)), complex(0.0, 0.0))
@test isequal(exp(complex(-Inf,-0.0)), complex(0.0,-0.0))
@test isequal(exp(complex(-Inf, 5.0)), complex(cos(5.0)*0.0,sin(5.0)* 0.0))
@test isequal(exp(complex(-Inf,-5.0)), complex(cos(5.0)*0.0,sin(5.0)*-0.0))
@test isequal(exp(complex(-Inf, Inf)), complex(-0.0, 0.0))
@test isequal(exp(complex(-Inf,-Inf)), complex(-0.0,-0.0))
@test isequal(exp(complex(-Inf, NaN)), complex(-0.0, 0.0))
@test isequal(exp(complex( NaN, 0.0)), complex( NaN, 0.0))
@test isequal(exp(complex( NaN,-0.0)), complex( NaN,-0.0))
@test isequal(exp(complex( NaN, 5.0)), complex( NaN, NaN))
@test isequal(exp(complex( NaN, NaN)), complex( NaN, NaN))
end
@testset "expm1(conj(z)) == conj(expm1(z))" begin
@test isequal(expm1(complex( 0.0, 0.0)), complex(0.0, 0.0))
@test isequal(expm1(complex( 0.0,-0.0)), complex(0.0,-0.0))
@test isequal(expm1(complex( 0.0, Inf)), complex(NaN, NaN))
@test isequal(expm1(complex( 0.0,-Inf)), complex(NaN, NaN))
@test isequal(expm1(complex( 0.0, NaN)), complex(NaN, NaN))
@test isequal(expm1(complex(-0.0, 0.0)), complex(-0.0, 0.0))
@test isequal(expm1(complex(-0.0,-0.0)), complex(-0.0,-0.0))
@test isequal(expm1(complex( 5.0, Inf)), complex(NaN, NaN))
@test isequal(expm1(complex( Inf, 0.0)), complex(Inf, 0.0))
@test isequal(expm1(complex( Inf,-0.0)), complex(Inf,-0.0))
@test isequal(expm1(complex( Inf, 5.0)), complex(cos(5.0)*Inf,sin(5.0)* Inf))
@test isequal(expm1(complex( Inf,-5.0)), complex(cos(5.0)*Inf,sin(5.0)*-Inf))
@test isequal(expm1(complex( Inf, NaN)), complex(-Inf, NaN))
@test isequal(expm1(complex( Inf, Inf)), complex(-Inf, NaN))
@test isequal(expm1(complex( Inf,-Inf)), complex(-Inf, NaN))
@test isequal(expm1(complex(-Inf, 0.0)), complex(-1.0, 0.0))
@test isequal(expm1(complex(-Inf,-0.0)), complex(-1.0,-0.0))
@test isequal(expm1(complex(-Inf, 5.0)), complex(-1.0,sin(5.0)* 0.0))
@test isequal(expm1(complex(-Inf,-5.0)), complex(-1.0,sin(5.0)*-0.0))
@test isequal(expm1(complex(-Inf, Inf)), complex(-1.0, 0.0))
@test isequal(expm1(complex(-Inf,-Inf)), complex(-1.0,-0.0))
@test isequal(expm1(complex(-Inf, NaN)), complex(-1.0, 0.0))
@test isequal(expm1(complex( NaN, 0.0)), complex( NaN, 0.0))
@test isequal(expm1(complex( NaN,-0.0)), complex( NaN,-0.0))
@test isequal(expm1(complex( NaN, 5.0)), complex( NaN, NaN))
@test isequal(expm1(complex( NaN, NaN)), complex( NaN, NaN))
@test isequal(expm1(complex( 1e-20, 0.0)), complex(expm1( 1e-20), 0.0))
@test isequal(expm1(complex(-1e-20, 0.0)), complex(expm1(-1e-20), 0.0))
@test expm1(complex( 1e-20, 1e-10)) ≈ complex( 5e-21, 1e-10)
@test expm1(complex( 1e-20,-1e-10)) ≈ complex( 5e-21,-1e-10)
@test expm1(complex(-1e-20, 1e-10)) ≈ complex(-1.5e-20, 1e-10)
@test expm1(complex(-1e-20,-1e-10)) ≈ complex(-1.5e-20,-1e-10)
@test expm1(complex( 10.0, 10.0)) ≈ exp(complex( 10.0, 10.0))-1
@test expm1(complex( 10.0,-10.0)) ≈ exp(complex( 10.0,-10.0))-1
@test expm1(complex(-10.0, 10.0)) ≈ exp(complex(-10.0, 10.0))-1
@test expm1(complex(-10.0,-10.0)) ≈ exp(complex(-10.0,-10.0))-1
end
import Base.Math.@horner
@testset "log1p" begin
@test isequal(log1p(complex(Inf, 3)), complex(Inf, +0.))
@test isequal(log1p(complex(Inf, -3)), complex(Inf, -0.))
@test isequal(log1p(complex(-Inf, 3)), complex(Inf, +pi))
@test isequal(log1p(complex(-Inf, -3)), complex(Inf, -pi))
@test isequal(log1p(complex(Inf, NaN)), complex(Inf, NaN))
@test isequal(log1p(complex(NaN, 0)), complex(NaN, NaN))
@test isequal(log1p(complex(0, NaN)), complex(NaN, NaN))
@test isequal(log1p(complex(-1, +0.)), complex(-Inf, +0.))
@test isequal(log1p(complex(-1, -0.)), complex(-Inf, -0.))
@test isequal(log1p(complex(-2, 1e-10)), log(1 + complex(-2, 1e-10)))
@test isequal(log1p(complex(1, Inf)), complex(Inf, pi/2))
@test isequal(log1p(complex(1, -Inf)), complex(Inf, -pi/2))
for z in (1e-10+1e-9im, 1e-10-1e-9im, -1e-10+1e-9im, -1e-10-1e-9im)
@test log1p(z) ≈ @horner(z, 0, 1, -0.5, 1/3, -0.25, 0.2)
end
for z in (15+4im, 0.2+3im, 0.08-0.9im)
@test log1p(z) ≈ log(1+z)
end
end
@testset "^ (cpow)" begin
# equivalent to exp(y*log(x))
# except for 0^0?
# conj(x)^conj(y) = conj(x^y)
@test isequal(complex( 0.0, 0.0)^complex( 0.0, 0.0), complex(1.0, 0.0))
@test isequal(complex( 0.0, 0.0)^complex( 0.0,-0.0), complex(1.0, 0.0))
@test isequal(complex( 0.0, 0.0)^complex(-0.0, 0.0), complex(1.0,-0.0))
@test isequal(complex( 0.0, 0.0)^complex(-0.0,-0.0), complex(1.0,-0.0))
@test isequal(complex( 0.0,-0.0)^complex( 0.0, 0.0), complex(1.0,-0.0))
@test isequal(complex( 0.0,-0.0)^complex( 0.0,-0.0), complex(1.0,-0.0))
@test isequal(complex( 0.0,-0.0)^complex(-0.0, 0.0), complex(1.0, 0.0))
@test isequal(complex( 0.0,-0.0)^complex(-0.0,-0.0), complex(1.0, 0.0))
@test isequal(complex(-0.0, 0.0)^complex( 0.0, 0.0), complex(1.0, 0.0))
@test isequal(complex(-0.0, 0.0)^complex( 0.0,-0.0), complex(1.0, 0.0))
@test isequal(complex(-0.0, 0.0)^complex(-0.0, 0.0), complex(1.0,-0.0))
@test isequal(complex(-0.0, 0.0)^complex(-0.0,-0.0), complex(1.0,-0.0))
@test isequal(complex(-0.0,-0.0)^complex( 0.0, 0.0), complex(1.0,-0.0))
@test isequal(complex(-0.0,-0.0)^complex( 0.0,-0.0), complex(1.0,-0.0))
@test isequal(complex(-0.0,-0.0)^complex(-0.0, 0.0), complex(1.0, 0.0))
@test isequal(complex(-0.0,-0.0)^complex(-0.0,-0.0), complex(1.0, 0.0))
@test complex(0.0,1.0)^complex(2.0,0) ≈ complex(-1.0, 0.0)
@test complex(1.0,2.0)^complex(3.0,0) ≈ complex(-11.0, -2.0)
@test isequal(complex(0.0,0.0)^false, complex(1.0,0.0))
@test isequal(complex(0.0,0.0)^0, complex(1.0,0.0))
end
@testset "sinh and sin" begin
# sinh: has properties
# sinh(conj(z)) = conj(sinh(z))
# sinh(-z) = -sinh(z)
# sin: defined in terms of sinh
# sin(z) = -i*sinh(i*z)
# i.e. if sinh(a+ib) = x+iy
# then sin(b-ia) = y-ix
# sin(conj(z)) = conj(sin(z))
# sin(-z) = -sin(z)
# @test isequal(sin(complex( 0, 10000)),complex( 0.0, Inf))
# @test isequal(sin(complex( 0,-10000)),complex( 0.0,-Inf))
for (x,y) in [(complex( 0.0, 0.0), complex( 0.0, 0.0)),
(complex( 0.0, Inf), complex( 0.0, NaN)),
(complex( 0.0, NaN), complex( 0.0, NaN)),
(complex( 7.2, Inf), complex( NaN, NaN)),
(complex( 7.2, NaN), complex( NaN, NaN)),
(complex( 7.2, 0.0), complex( sinh(7.2), 0.0)),
(complex( 1e5, 0.0), complex( sinh(1e5), 0.0)),
(complex( Inf, 0.0), complex( Inf, 0.0)),
(complex( Inf, 7.2), Inf*cis(7.2)),
(complex( Inf, Inf), complex( Inf, NaN)),
(complex( Inf, NaN), complex( Inf, NaN)),
(complex( NaN, 0.0), complex( NaN, 0.0)),
(complex( NaN, 7.2), complex( NaN, NaN)),
(complex( NaN, NaN), complex( NaN, NaN)),
]
@test isequal(sinh(x), y)
@test isequal(sinh(conj(x)), conj(y))
@test isequal(sinh(-x), -y)
@test isequal(sinh(-conj(x)), -conj(y))
xx = complex(imag(x),-real(x))
yy = complex(imag(y),-real(y))
@test isequal(sin(xx),yy)
@test isequal(sin(conj(xx)), conj(yy))
@test isequal(sin(-xx), -yy)
@test isequal(sin(-conj(xx)), -conj(yy))
yyy = sin(pi*xx)
@test isequal(sinpi(xx), yyy)
@test isequal(sinpi(conj(xx)),conj(yyy))
@test isequal(sinpi(-xx),-yyy)
@test isequal(sinpi(-conj(xx)),-conj(yyy))
end
end
@testset "cosh and cos" begin
# cosh: has properties
# cosh(conj(z)) = conj(cosh(z))
# coshh(-z) = cosh(z)
# cos
# cos(z) = cosh(iz)
# i.e cos(b-ia) = cosh(a+ib)
# and cos(b+ia) = cosh(a-ib)
# cos(conj(z)) = conj(cos(z))
# cos(-z) = cos(z)
for (x,y) in [(complex( 0.0, 0.0), complex( 1.0, 0.0)),
(complex( 0.0, Inf), complex( NaN, 0.0)),
(complex( 0.0, NaN), complex( NaN, 0.0)),
(complex( 7.2, Inf), complex( NaN, NaN)),
(complex( 7.2, NaN), complex( NaN, NaN)),
(complex( 7.2, 0.0), complex( cosh(7.2), 0.0)),
(complex( 1e5, 0.0), complex( Inf, 0.0)),
(complex( Inf, 0.0), complex( Inf, 0.0)),
(complex( Inf, 7.2), Inf*cis(7.2)),
(complex( Inf, Inf), complex( Inf, NaN)),
(complex( Inf, NaN), complex( Inf, NaN)),
(complex( NaN, 0.0), complex( NaN, 0.0)),
(complex( NaN, 7.2), complex( NaN, NaN)),
(complex( NaN, NaN), complex( NaN, NaN)),
]
undef_sign = isequal(x,complex( NaN, 0.0)) || isequal(x,complex( 0.0, NaN))
@test isequal(cosh(x), y)
if !undef_sign
@test isequal(cosh(conj(x)), conj(y))
@test isequal(cosh(-x), y)
@test isequal(cosh(-conj(x)), conj(y))
end
xx = complex(imag(x),-real(x))
yy = y
@test isequal(cos(xx),yy)
if !undef_sign
@test isequal(cos(conj(xx)), conj(yy))
@test isequal(cos(-xx), yy)
@test isequal(cos(-conj(xx)), conj(yy))
end
yyy = cos(pi*xx)
@test isequal(cospi(xx), yyy)
if !undef_sign
@test isequal(cospi(conj(xx)), conj(yyy))
@test isequal(cospi(-xx), yyy)
@test isequal(cospi(-conj(xx)), conj(yyy))
end
end
end
@testset "tanh(op(z)) == op(tanh(z)) for op in (conj, -)" begin
@test isequal(tanh(complex( 0, 0)),complex(0.0,0.0)) #integer fallback
@test isequal(tanh(complex( 0.0, 0.0)),complex(0.0,0.0))
@test isequal(tanh(complex( 0.0,-0.0)),complex(0.0,-0.0))
@test_throws DomainError tanh(complex( 0.0, Inf))
@test_throws DomainError tanh(complex( 0.0,-Inf))
@test isequal(tanh(complex( 0.0, NaN)),complex(NaN,NaN))
@test isequal(tanh(complex(-0.0, 0.0)),complex(-0.0,0.0))
@test isequal(tanh(complex(-0.0,-0.0)),complex(-0.0,-0.0))
@test_throws DomainError tanh(complex( 5.0, Inf))
@test isequal(tanh(complex( 5.0, NaN)),complex(NaN,NaN))
@test isequal(tanh(complex( Inf, 0.0)),complex(1.0, 0.0))
@test isequal(tanh(complex( Inf,-0.0)),complex(1.0,-0.0))
@test isequal(tanh(complex( Inf, 5.0)),complex(1.0,sin(2*5.0)* 0.0))
@test isequal(tanh(complex( Inf,-5.0)),complex(1.0,sin(2*5.0)*-0.0))
@test isequal(tanh(complex( Inf, Inf)),complex(1.0, 0.0))
@test isequal(tanh(complex( Inf,-Inf)),complex(1.0,-0.0))
@test isequal(tanh(complex( Inf, NaN)),complex(1.0, 0.0))
@test isequal(tanh(complex(-Inf, 0.0)),complex(-1.0, 0.0))
@test isequal(tanh(complex(-Inf,-0.0)),complex(-1.0,-0.0))
@test isequal(tanh(complex(-Inf, 5.0)),complex(-1.0,sin(2*5.0)* 0.0))
@test isequal(tanh(complex(-Inf,-5.0)),complex(-1.0,sin(2*5.0)*-0.0))
@test isequal(tanh(complex(-Inf, Inf)),complex(-1.0, 0.0))
@test isequal(tanh(complex(-Inf,-Inf)),complex(-1.0,-0.0))
@test isequal(tanh(complex(-Inf, NaN)),complex(-1.0, 0.0))
@test isequal(tanh(complex( NaN, 0.0)),complex(NaN, 0.0))
@test isequal(tanh(complex( NaN,-0.0)),complex(NaN,-0.0))
@test isequal(tanh(complex( NaN, 5.0)),complex(NaN, NaN))
@test isequal(tanh(complex( NaN,-5.0)),complex(NaN, NaN))
@test isequal(tanh(complex( NaN, NaN)),complex(NaN, NaN))
end
@testset "tan(z) == -i tanh(iz)" begin
@test isequal(tan(complex( 0.0, Inf)),complex( 0.0, 1.0))
@test isequal(tan(complex( 0.0,-Inf)),complex( 0.0,-1.0))
@test isequal(tan(complex( 0.0, NaN)),complex( 0.0, NaN))
@test isequal(tan(complex(-0.0,-Inf)),complex(-0.0,-1.0))
@test isequal(tan(complex(-0.0, Inf)),complex(-0.0, 1.0))
@test isequal(tan(complex(-0.0, NaN)),complex(-0.0, NaN))
@test isequal(tan(complex( 5.0, Inf)),complex(sin(2*5.0)* 0.0, 1.0))
@test isequal(tan(complex( 5.0,-Inf)),complex(sin(2*5.0)* 0.0,-1.0))
@test isequal(tan(complex( 5.0, NaN)),complex( NaN, NaN))
@test isequal(tan(complex(-5.0, Inf)),complex(sin(2*5.0)*-0.0, 1.0))
@test isequal(tan(complex(-5.0,-Inf)),complex(sin(2*5.0)*-0.0,-1.0))
@test isequal(tan(complex(-5.0, NaN)),complex( NaN, NaN))
@test_throws DomainError tan(complex( Inf, 5.0))
@test isequal(tan(complex( Inf, Inf)),complex( 0.0, 1.0))
@test isequal(tan(complex( Inf,-Inf)),complex( 0.0,-1.0))
@test isequal(tan(complex(-Inf, Inf)),complex(-0.0, 1.0))
@test isequal(tan(complex(-Inf,-Inf)),complex(-0.0,-1.0))
@test isequal(tan(complex( NaN, 5.0)),complex( NaN, NaN))
@test isequal(tan(complex( NaN, Inf)),complex( 0.0, 1.0))
@test isequal(tan(complex( NaN,-Inf)),complex( 0.0,-1.0))
@test isequal(tan(complex( NaN, NaN)),complex( NaN, NaN))
end
@testset "acosh(conj(z)) == conj(acosh(z))" begin
@test isequal(acosh(complex( 0.0, 0.0)), complex( 0.0, pi/2))
@test isequal(acosh(complex( 0.0,-0.0)), complex( 0.0,-pi/2))
@test isequal(acosh(complex( 0.0, Inf)), complex( Inf, pi/2))
@test isequal(acosh(complex( 0.0,-Inf)), complex( Inf,-pi/2))
@test isequal(acosh(complex(-0.0, 0.0)), complex( 0.0, pi/2))
@test isequal(acosh(complex(-0.0,-0.0)), complex( 0.0,-pi/2))
@test isequal(acosh(complex( 5.0, Inf)), complex( Inf, pi/2))
@test isequal(acosh(complex( 5.0,-Inf)), complex( Inf,-pi/2))
@test isequal(acosh(complex( 5.0, NaN)), complex( NaN, NaN))
@test isequal(acosh(complex( Inf, 0.0)), complex( Inf, 0.0))
@test isequal(acosh(complex( Inf,-0.0)), complex( Inf,-0.0))
@test isequal(acosh(complex( Inf, 5.0)), complex( Inf, 0.0))
@test isequal(acosh(complex( Inf,-5.0)), complex( Inf,-0.0))
@test isequal(acosh(complex( Inf, Inf)), complex( Inf, pi/4))
@test isequal(acosh(complex( Inf,-Inf)), complex( Inf,-pi/4))
@test isequal(acosh(complex( Inf, NaN)), complex( Inf, NaN))
@test isequal(acosh(complex(-Inf, 0.0)), complex( Inf, pi))
@test isequal(acosh(complex(-Inf,-0.0)), complex( Inf,-pi))
@test isequal(acosh(complex(-Inf, 5.0)), complex( Inf, pi))
@test isequal(acosh(complex(-Inf,-5.0)), complex( Inf,-pi))
@test isequal(acosh(complex(-Inf, Inf)), complex( Inf, 3*pi/4))
@test isequal(acosh(complex(-Inf,-Inf)), complex( Inf,-3*pi/4))
@test isequal(acosh(complex(-Inf, NaN)), complex( Inf, NaN))
@test isequal(acosh(complex( NaN, Inf)), complex( Inf, NaN))
@test isequal(acosh(complex( NaN,-Inf)), complex( Inf, NaN))
@test isequal(acosh(complex( NaN, NaN)), complex( NaN, NaN))
end
@testset "acos(conj(z)) == conj(acos(z))" begin
@test isequal(acos(complex( 0, 0)),complex(pi/2,-0.0)) #integer fallback
@test isequal(acos(complex( 0.0, 0.0)),complex(pi/2,-0.0))
@test isequal(acos(complex( 0.0,-0.0)),complex(pi/2, 0.0))
@test isequal(acos(complex( 0.0, Inf)),complex(pi/2,-Inf))
@test isequal(acos(complex( 0.0,-Inf)),complex(pi/2, Inf))
@test isequal(acos(complex( 0.0, NaN)),complex(pi/2, NaN))
@test isequal(acos(complex(-0.0, 0.0)),complex(pi/2,-0.0))
@test isequal(acos(complex(-0.0,-0.0)),complex(pi/2, 0.0))
@test isequal(acos(complex(-0.0, NaN)),complex(pi/2, NaN))
@test isequal(acos(complex( 5.0, Inf)),complex(pi/2,-Inf))
@test isequal(acos(complex( 5.0,-Inf)),complex(pi/2, Inf))
@test isequal(acos(complex( 5.0, NaN)),complex( NaN, NaN))
@test isequal(acos(complex( Inf, 0.0)),complex( 0.0,-Inf))
@test isequal(acos(complex( Inf,-0.0)),complex( 0.0, Inf))
@test isequal(acos(complex( Inf, 5.0)),complex( 0.0,-Inf))
@test isequal(acos(complex( Inf,-5.0)),complex( 0.0, Inf))
@test isequal(acos(complex( Inf, Inf)),complex(pi/4,-Inf))
@test isequal(acos(complex( Inf,-Inf)),complex(pi/4, Inf))
@test isequal(acos(complex( Inf, NaN)),complex( NaN, Inf))
@test isequal(acos(complex(-Inf, 0.0)),complex(pi,-Inf))
@test isequal(acos(complex(-Inf,-0.0)),complex(pi, Inf))
@test isequal(acos(complex(-Inf, 5.0)),complex(pi,-Inf))
@test isequal(acos(complex(-Inf,-5.0)),complex(pi, Inf))
@test isequal(acos(complex(-Inf, Inf)),complex(3*pi/4,-Inf))
@test isequal(acos(complex(-Inf,-Inf)),complex(3*pi/4, Inf))
@test isequal(acos(complex(-Inf, NaN)),complex( NaN, Inf))
@test isequal(acos(complex( NaN, 0.0)),complex( NaN, NaN))
@test isequal(acos(complex( NaN, 5.0)),complex( NaN, NaN))
@test isequal(acos(complex( NaN, Inf)),complex( NaN,-Inf))
@test isequal(acos(complex( NaN,-Inf)),complex( NaN, Inf))
@test isequal(acos(complex( NaN, NaN)),complex( NaN, NaN))
end
@testset "asinh(op(z)) == op(asinh(z)) for op in (conj, -)" begin
@test isequal(asinh(complex( 0.0, 0.0)),complex( 0.0, 0.0))
@test isequal(asinh(complex( 0.0,-0.0)),complex( 0.0,-0.0))
@test isequal(asinh(complex( 0.0, Inf)),complex( Inf, pi/2))
@test isequal(asinh(complex( 0.0,-Inf)),complex( Inf,-pi/2))
@test isequal(asinh(complex( 0.0, NaN)),complex( NaN, NaN))
@test isequal(asinh(complex(-0.0, 0.0)),complex(-0.0, 0.0))
@test isequal(asinh(complex(-0.0,-0.0)),complex(-0.0,-0.0))
@test isequal(asinh(complex(-0.0, Inf)),complex(-Inf, pi/2))
@test isequal(asinh(complex(-0.0,-Inf)),complex(-Inf,-pi/2))
@test isequal(asinh(complex( 5.0, Inf)),complex( Inf, pi/2))
@test isequal(asinh(complex( 5.0,-Inf)),complex( Inf,-pi/2))
@test isequal(asinh(complex( 5.0, NaN)),complex( NaN, NaN))
@test isequal(asinh(complex(-5.0, Inf)),complex(-Inf, pi/2))
@test isequal(asinh(complex(-5.0,-Inf)),complex(-Inf,-pi/2))
@test isequal(asinh(complex( Inf, Inf)),complex( Inf, pi/4))
@test isequal(asinh(complex( Inf,-Inf)),complex( Inf,-pi/4))
@test isequal(asinh(complex( Inf, NaN)),complex( Inf, NaN))
@test isequal(asinh(complex(-Inf, Inf)),complex(-Inf, pi/4))
@test isequal(asinh(complex(-Inf,-Inf)),complex(-Inf,-pi/4))
@test isequal(asinh(complex(-Inf, NaN)),complex(-Inf, NaN))
@test isequal(asinh(complex( NaN, 0.0)),complex( NaN, 0.0))
@test isequal(asinh(complex( NaN,-0.0)),complex( NaN,-0.0))
@test isequal(asinh(complex( NaN, 5.0)),complex( NaN, NaN))
@test isequal(asinh(complex( NaN, Inf)),complex( Inf, NaN))
@test isequal(asinh(complex( NaN,-Inf)),complex( Inf, NaN))
@test isequal(asinh(complex( NaN, NaN)),complex( NaN, NaN))
end
@testset "asin(z) == -i*asinh(iz)" begin
@test isequal(asin(complex( 0.0, 0.0)),complex( 0.0, 0.0))
@test isequal(asin(complex( 0.0,-0.0)),complex( 0.0,-0.0))
@test isequal(asin(complex(-0.0, 0.0)),complex(-0.0, 0.0))
@test isequal(asin(complex( 0.0, NaN)),complex( 0.0, NaN))
@test isequal(asin(complex(-0.0,-0.0)),complex(-0.0,-0.0))
@test isequal(asin(complex(-0.0, NaN)),complex(-0.0, NaN))
@test isequal(asin(complex( 5.0, NaN)),complex( NaN, NaN))
@test isequal(asin(complex( Inf, 0.0)),complex( pi/2, Inf))
@test isequal(asin(complex( Inf,-0.0)),complex( pi/2,-Inf))
@test isequal(asin(complex( Inf, 5.0)),complex( pi/2, Inf))
@test isequal(asin(complex( Inf,-5.0)),complex( pi/2,-Inf))
@test isequal(asin(complex( Inf, Inf)),complex( pi/4, Inf))
@test isequal(asin(complex( Inf,-Inf)),complex( pi/4,-Inf))
@test isequal(asin(complex( Inf, NaN)),complex( NaN, Inf))
@test isequal(asin(complex(-Inf, 0.0)),complex(-pi/2, Inf))
@test isequal(asin(complex(-Inf,-0.0)),complex(-pi/2,-Inf))
@test isequal(asin(complex(-Inf, 5.0)),complex(-pi/2, Inf))
@test isequal(asin(complex(-Inf,-5.0)),complex(-pi/2,-Inf))
@test isequal(asin(complex(-Inf, Inf)),complex(-pi/4, Inf))
@test isequal(asin(complex(-Inf,-Inf)),complex(-pi/4,-Inf))
@test isequal(asin(complex(-Inf, NaN)),complex( NaN, Inf))
@test isequal(asin(complex( NaN, 0.0)),complex( NaN, NaN))
@test isequal(asin(complex( NaN, 5.0)),complex( NaN, NaN))
@test isequal(asin(complex( NaN, Inf)),complex( NaN, Inf))
@test isequal(asin(complex( NaN,-Inf)),complex( NaN,-Inf))
@test isequal(asin(complex( NaN, NaN)),complex( NaN, NaN))
end
@testset "atanh(op(z)) == op(atanh(z)) for op in (conj, -)" begin
@test isequal(atanh(complex( 0, 0)),complex( 0.0, 0.0)) #integer fallback
@test isequal(atanh(complex( 0.0, 0.0)),complex( 0.0, 0.0))
@test isequal(atanh(complex( 0.0,-0.0)),complex( 0.0,-0.0))
@test isequal(atanh(complex( 0.0, NaN)),complex( 0.0, NaN))
@test isequal(atanh(complex( 0.0, Inf)),complex( 0.0, pi/2))
@test isequal(atanh(complex( 0.0,-Inf)),complex( 0.0,-pi/2))
@test isequal(atanh(complex(-0.0, NaN)),complex(-0.0, NaN))
@test isequal(atanh(complex(-0.0, 0.0)),complex(-0.0, 0.0))
@test isequal(atanh(complex(-0.0,-0.0)),complex(-0.0,-0.0))
@test isequal(atanh(complex(-0.0, Inf)),complex(-0.0, pi/2))
@test isequal(atanh(complex(-0.0,-Inf)),complex(-0.0,-pi/2))
@test isequal(atanh(complex( 1.0, 0.0)),complex( Inf, 0.0))
@test isequal(atanh(complex(-1.0, 0.0)),complex(-Inf, 0.0))
@test isequal(atanh(complex( 5.0, Inf)),complex( 0.0, pi/2))
@test isequal(atanh(complex( 5.0,-Inf)),complex( 0.0,-pi/2))
@test isequal(atanh(complex( 5.0, NaN)),complex( NaN, NaN))
@test isequal(atanh(complex(-5.0, Inf)),complex(-0.0, pi/2))
@test isequal(atanh(complex(-5.0,-Inf)),complex(-0.0,-pi/2))
@test isequal(atanh(complex(-5.0, NaN)),complex( NaN, NaN))
@test isequal(atanh(complex( Inf, 0.0)),complex(0.0, pi/2))
@test isequal(atanh(complex( Inf,-0.0)),complex(0.0,-pi/2))
@test isequal(atanh(complex( Inf, 5.0)),complex(0.0, pi/2))
@test isequal(atanh(complex( Inf,-5.0)),complex(0.0,-pi/2))
@test isequal(atanh(complex( Inf, Inf)),complex(0.0, pi/2))
@test isequal(atanh(complex( Inf,-Inf)),complex(0.0,-pi/2))
@test isequal(atanh(complex( Inf, NaN)),complex(0.0, NaN))
@test isequal(atanh(complex(-Inf, 0.0)),complex(-0.0, pi/2))
@test isequal(atanh(complex(-Inf,-0.0)),complex(-0.0,-pi/2))
@test isequal(atanh(complex(-Inf, 5.0)),complex(-0.0, pi/2))
@test isequal(atanh(complex(-Inf,-5.0)),complex(-0.0,-pi/2))
@test isequal(atanh(complex(-Inf, Inf)),complex(-0.0, pi/2))
@test isequal(atanh(complex(-Inf,-Inf)),complex(-0.0,-pi/2))
@test isequal(atanh(complex(-Inf, NaN)),complex(-0.0, NaN))
@test isequal(atanh(complex( NaN, 0.0)),complex( NaN, NaN))
@test isequal(atanh(complex( NaN,-0.0)),complex( NaN, NaN))
@test isequal(atanh(complex( NaN, 5.0)),complex( NaN, NaN))
@test isequal(atanh(complex( NaN,-5.0)),complex( NaN, NaN))
@test isequal(atanh(complex( NaN, Inf)),complex( 0.0, pi/2))
@test isequal(atanh(complex( NaN,-Inf)),complex( 0.0,-pi/2))
@test isequal(atanh(complex( NaN, NaN)),complex( NaN, NaN))
end
@testset "atan(z) == -i*atanh(iz)" begin
@test isequal(atan(complex( 0.0, 0.0)),complex( 0.0, 0.0))
@test isequal(atan(complex( 0.0,-0.0)),complex( 0.0,-0.0))
@test isequal(atan(complex( 0.0, 1.0)),complex( 0.0, Inf))
@test isequal(atan(complex( 0.0, Inf)),complex( pi/2, 0.0))
@test isequal(atan(complex( 0.0,-Inf)),complex( pi/2,-0.0))
@test isequal(atan(complex( 0.0, NaN)),complex( NaN, NaN))
@test isequal(atan(complex(-0.0, 0.0)),complex(-0.0, 0.0))
@test isequal(atan(complex(-0.0,-0.0)),complex(-0.0,-0.0))
@test isequal(atan(complex(-0.0, Inf)),complex(-pi/2, 0.0))
@test isequal(atan(complex(-0.0,-Inf)),complex(-pi/2,-0.0))
@test isequal(atan(complex(-0.0, NaN)),complex( NaN, NaN))
@test isequal(atan(complex( 5.0, Inf)),complex( pi/2, 0.0))
@test isequal(atan(complex( 5.0,-Inf)),complex( pi/2,-0.0))
@test isequal(atan(complex( 5.0, NaN)),complex( NaN, NaN))
@test isequal(atan(complex(-5.0, Inf)),complex(-pi/2, 0.0))
@test isequal(atan(complex(-5.0,-Inf)),complex(-pi/2,-0.0))
@test isequal(atan(complex(-5.0, NaN)),complex( NaN, NaN))
@test isequal(atan(complex( Inf, 0.0)),complex( pi/2, 0.0))
@test isequal(atan(complex( Inf,-0.0)),complex( pi/2,-0.0))
@test isequal(atan(complex( Inf, 5.0)),complex( pi/2, 0.0))
@test isequal(atan(complex( Inf,-5.0)),complex( pi/2,-0.0))
@test isequal(atan(complex( Inf, Inf)),complex( pi/2, 0.0))
@test isequal(atan(complex( Inf,-Inf)),complex( pi/2,-0.0))
@test isequal(atan(complex( Inf, NaN)),complex( pi/2, 0.0))
@test isequal(atan(complex(-Inf, 0.0)),complex(-pi/2, 0.0))
@test isequal(atan(complex(-Inf,-0.0)),complex(-pi/2,-0.0))
@test isequal(atan(complex(-Inf, 5.0)),complex(-pi/2, 0.0))
@test isequal(atan(complex(-Inf,-5.0)),complex(-pi/2,-0.0))
@test isequal(atan(complex(-Inf, Inf)),complex(-pi/2, 0.0))
@test isequal(atan(complex(-Inf,-Inf)),complex(-pi/2,-0.0))
@test isequal(atan(complex(-Inf, NaN)),complex(-pi/2, 0.0))
@test isequal(atan(complex( NaN, 0.0)),complex( NaN, 0.0))
@test isequal(atan(complex( NaN,-0.0)),complex( NaN,-0.0))
@test isequal(atan(complex( NaN, 5.0)),complex( NaN, NaN))
@test isequal(atan(complex( NaN,-5.0)),complex( NaN, NaN))
@test isequal(atan(complex( NaN, Inf)),complex( NaN, 0.0))
@test isequal(atan(complex( NaN,-Inf)),complex( NaN,-0.0))
@test isequal(atan(complex( NaN, NaN)),complex( NaN, NaN))
end
# misc.
@test complex(1//2,1//3)^2 === complex(5//36, 1//3)
@test complex(2,2)^2 === complex(0,8)
let p = -2
@test_throws DomainError complex(2,2)^p
end
@test complex(2,2)^(-2) === complex(2.0,2.0)^(-2) === complex(0.0, -0.125)
@test complex.(1.0, [1.0, 1.0]) == [complex(1.0, 1.0), complex(1.0, 1.0)]
@test complex.([1.0, 1.0], 1.0) == [complex(1.0, 1.0), complex(1.0, 1.0)]
# robust division of Float64
# hard complex divisions from Fig 6 of arxiv.1210.4539
z7 = Complex{Float64}(3.898125604559113300e289, 8.174961907852353577e295)
z9 = Complex{Float64}(0.001953125, -0.001953125)
z10 = Complex{Float64}( 1.02951151789360578e-84, 6.97145987515076231e-220)
harddivs = ((1.0+im*1.0, 1.0+im*2^1023.0, 2^-1023.0-im*2^-1023.0), #1
(1.0+im*1.0, 2^-1023.0+im*2^-1023.0, 2^1023.0+im*0.0), #2
(2^1023.0+im*2^-1023.0, 2^677.0+im*2^-677.0, 2^346.0-im*2^-1008.0), #3
(2^1023.0+im*2^1023.0, 1.0+im*1.0, 2^1023.0+im*0.0), #4
(2^1020.0+im*2^-844., 2^656.0+im*2^-780.0, 2^364.0-im*2^-1072.0), #5
(2^-71.0+im*2^1021., 2^1001.0+im*2^-323.0, 2^-1072.0+im*2^20.0), #6
(2^-347.0+im*2^-54., 2^-1037.0+im*2^-1058.0, z7), #7
(2^-1074.0+im*2^-1074., 2^-1073.0+im*2^-1074., 0.6+im*0.2), #8
(2^1015.0+im*2^-989., 2^1023.0+im*2^1023.0, z9), #9
(2^-622.0+im*2^-1071., 2^-343.0+im*2^-798.0, z10) #10
)
# calculate "accurate bits" in range 0:53 by algorithm given in arxiv.1210.4539
function sb_accuracy(x,expected)
min(logacc(real(x),real(expected)),
logacc(imag(x),imag(expected)))
end
relacc(x,expected) = abs(x-expected)/abs(expected)
function logacc(x::Float64,expected::Float64)
x == expected && (return 53)
expected == 0 && (return 0)
(x == Inf || x == -Inf) && (return 0)
isnan(x) && (return 0)
ra = relacc(BigFloat(x),BigFloat(expected))
max(floor(Int,-log2(ra)),0)
end
# the robust division algorithm should have 53 or 52
# bits accuracy for each of the hard divisions
@test 52 <= minimum([sb_accuracy(h[1]/h[2],h[3]) for h in harddivs])
# division of non-Float64
function cdiv_test(a,b)
c=convert(Complex{Float64},a)/convert(Complex{Float64},b)
50 <= sb_accuracy(c,convert(Complex{Float64},a/b))
end
@test cdiv_test(complex(1//2, 3//4), complex(17//13, 4//5))
@test cdiv_test(complex(1,2), complex(8997,2432))
@testset "inv" begin
@test inv(1e300+0im) == 1e-300 - 0.0im
@test inv(0+1e300im) == 0.0 - 1e-300im
end
@testset "issue #7904" begin
@test log10(10+0im) === 1.0 + 0.0im
@test log2(2+0im) === 1.0 + 0.0im
end
@testset "sign" begin
for T in (Float32, Float64)
z = Complex{T}(1)
@test typeof(sign(z)) == typeof(z)
z = Complex{T}(0)
@test typeof(sign(z)) == typeof(z)
end
for T in (Int32, Int64)
z = Complex{T}(1)
@test typeof(sign(z)) == typeof(float(z))
z = Complex{T}(0)
@test typeof(sign(z)) == typeof(float(z))
end
@test sign(0 + 0im) == 0
@test sign(2 + 0im) == 1
@test sign(-2 + 0im) == -1
@test sign(1 + im) ≈ (1 + im) / sqrt(2)
@test sign(1 - im) ≈ (1 - im) / sqrt(2)
for T in (Float16, Float32, Float64)
z = Complex(zero(T), zero(T))
@test sign(z) === z
@test sign(-z) === -z
@test sign(conj(z)) === conj(z)
@test sign(-conj(z)) === -conj(z)
end
end
@testset "cis" begin
@test cis(0.0+1.0im) ≈ 0.367879441171442321595523770161460867445811131031767834507836+0.0im
@test cis(1.0+0.0im) ≈ 0.54030230586813971740093660744297660373231042061+0.84147098480789650665250232163029899962256306079im
@test cis(pi) ≈ -1.0+0.0im
@test cis(pi/2) ≈ 0.0+1.0im
end
@testset "exp2" begin
@test exp2(0.0+0.0im) == 1.0+0.0im
@test exp2(1.0+0.0im) == 2.0+0.0im
#wolframalpha
@test exp2(1.0+3.0im) ≈ -0.9739888359315627962096198412+1.74681016354974281701922im
@test exp2(im) ≈ 0.7692389013639721 + 0.6389612763136348im
end
@testset "exp10" begin
@test exp10(0.0+0.0im) == 1.0+0.0im
@test exp10(1.0+0.0im) == 10.0+0.0im
#wolframalpha
@test exp10(1.0+2.0im) ≈ -1.0701348355877020772086517528518239460495529361-9.9425756941378968736161937190915602112878340717im
@test exp10(im) ≈ -0.6682015101903132 + 0.7439803369574931im
end
@testset "round and float, PR #8291" begin
@test round(Complex(1.125, 0.875), digits=2) == Complex(1.12, 0.88)
@test round(Complex(1.5, 0.5), RoundDown, RoundUp) == Complex(1.0, 1.0)
@test round.([1:5;] .+ im) == [1:5;] .+ im
@test round.([1:5;] .+ 0.5im) == [1.0:5.0;]
@test float(Complex(1, 2)) == Complex(1.0, 2.0)
@test round(float(Complex(π, ℯ)), digits=3) == Complex(3.142, 2.718)
end
@testset "ComplexF16 arithmetic, PR #10003" begin
@test Float16(1)+Float16(1)im === ComplexF16(1, 1)
@test Float16(1)-Float16(1)im === Float16(1)+Float16(-1)im === ComplexF16(1, -1)
@test Float16(1)*im === ComplexF16(im)
@test Float16(1)/im === ComplexF16(0,-1)
@test Float16(1)^im === ComplexF16(1) === Float16(1)+Float16(0)im
end
# issue/PR #10148
@test typeof(Int8(1) - im) == Complex{Int8}
# issue #10926
@test typeof(π - 1im) == Complex{Float64}
@testset "issue #15969" begin
# specialized muladd for complex types
for x in (3, 3+13im), y in (2, 2+7im), z in (5, 5+11im)
@test muladd(x,y,z) === x*y + z
end
end
@testset "issue #11839" begin
# type stability for Complex{Int64}
let x = 1+im
@inferred sin(x)
@inferred cos(x)
@inferred norm(x)
@inferred vecnorm(x)
end
end
@testset "issue #18785" begin
# type stability for exp, expm1 for Complex{Int64}
let x = 2*im
@inferred exp(x)
@inferred expm1(x)
end
end
# issue #19240
@test big(1)/(10+10im) ≈ (5-5im)/big(100) ≈ big"0.05" - big"0.05"*im
@testset "Complex Irrationals, issue #21204" begin
for x in (pi, ℯ, Base.MathConstants.catalan) # No need to test all of them
z = Complex(x, x)
@test typeof(z) == Complex{typeof(x)}
@test exp(z) ≈ exp(x) * cis(x)
@test log1p(z) ≈ log(1 + z)
@test exp2(z) ≈ exp(z * log(2))
@test exp10(z) ≈ exp(z * log(10))
end
end
@testset "expm1 type stability" begin
x = @inferred expm1(0.1im)
@test x isa ComplexF64
x = @inferred expm1(0.1f0im)
@test x isa ComplexF32
end
@testset "array printing with exponent format" begin
a = [1.0 + 1e-10im, 2.0e-15 - 2.0e-5im, 1.0e-15 + 2im, 1.0 + 2e-15im]
@test sprint((io, x) -> show(io, MIME("text/plain"), x), a) ==
join([
"4-element Array{Complex{Float64},1}:",
" 1.0 + 1.0e-10im",
" 2.0e-15 - 2.0e-5im ",
" 1.0e-15 + 2.0im ",
" 1.0 + 2.0e-15im"], "\n")
end
@testset "corner cases of division, issue #22983" begin
# These results abide by ISO/IEC 10967-3:2006(E) and
# mathematical definition of division of complex numbers.
for T in (Float32, Float64, BigFloat)
@test isequal(one(T) / zero(Complex{T}), one(Complex{T}) / zero(Complex{T}))
@test isequal(one(T) / zero(Complex{T}), Complex{T}(NaN, NaN))
@test isequal(one(Complex{T}) / zero(T), Complex{T}(Inf, NaN))
@test isequal(one(Complex{T}) / one(Complex{T}), one(Complex{T}))
@test isequal(one(T) / complex(one(T), zero(T)), Complex(one(T), -zero(T)))
@test isequal(one(T) / complex(one(T), -zero(T)), Complex(one(T), zero(T)))
end
end
@testset "complex^real, issue #14342" begin
for T in (Float32, Float64, BigFloat), p in (T(-21//10), -21//10)
z = T(2)+0im
@test real(z^p) ≈ 2^p
@test signbit(imag(z^p))
end
@test (2+0im)^(-21//10) === (2//1+0im)^(-21//10) === 2^-2.1 - 0.0im
end
@testset "more cpow" begin
# for testing signs of zeros, it is useful to convert ±0.0 to ±1e-15
zero2small(r::Real) = iszero(r) ? copysign(1e-15, r) : r