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math.jl
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# This file is a part of Julia. License is MIT: https://julialang.org/license
using Random
using LinearAlgebra
using Base.Experimental: @force_compile
function isnan_type(::Type{T}, x) where T
isa(x, T) && isnan(x)
end
# has_fma has no runtime support.
# So we need function wrappers to make this work.
has_fma_Int() = Core.Compiler.have_fma(Int)
has_fma_Float32() = Core.Compiler.have_fma(Float32)
has_fma_Float64() = Core.Compiler.have_fma(Float64)
has_fma = Dict(
Int => has_fma_Int(),
Rational{Int} => has_fma_Int(),
Float32 => has_fma_Float32(),
Float64 => has_fma_Float64(),
BigFloat => true,
)
@testset "clamp" begin
@test clamp(0, 1, 3) == 1
@test clamp(1, 1, 3) == 1
@test clamp(2, 1, 3) == 2
@test clamp(3, 1, 3) == 3
@test clamp(4, 1, 3) == 3
@test clamp(0.0, 1, 3) == 1.0
@test clamp(1.0, 1, 3) == 1.0
@test clamp(2.0, 1, 3) == 2.0
@test clamp(3.0, 1, 3) == 3.0
@test clamp(4.0, 1, 3) == 3.0
@test clamp.([0, 1, 2, 3, 4], 1.0, 3.0) == [1.0, 1.0, 2.0, 3.0, 3.0]
@test clamp.([0 1; 2 3], 1.0, 3.0) == [1.0 1.0; 2.0 3.0]
@test clamp(-200, Int8) === typemin(Int8)
@test clamp(100, Int8) === Int8(100)
@test clamp(200, Int8) === typemax(Int8)
begin
x = [0.0, 1.0, 2.0, 3.0, 4.0]
clamp!(x, 1, 3)
@test x == [1.0, 1.0, 2.0, 3.0, 3.0]
end
end
@testset "constants" begin
@test pi != ℯ
@test ℯ != 1//2
@test 1//2 <= ℯ
@test ℯ <= 15//3
@test big(1//2) < ℯ
@test ℯ < big(20//6)
@test ℯ^pi == exp(pi)
@test ℯ^2 == exp(2)
@test ℯ^2.4 == exp(2.4)
@test ℯ^(2//3) == exp(2//3)
@test Float16(3.0) < pi
@test pi < Float16(4.0)
@test widen(pi) === pi
@test occursin("3.14159", sprint(show, MIME"text/plain"(), π))
@test repr(Any[pi ℯ; ℯ pi]) == "Any[π ℯ; ℯ π]"
@test string(pi) == "π"
@test sin(π) == sind(180) === sinpi(1) === sinpi(1//1) == tan(π) == 0
@test tan(π) == tand(180) === tanpi(1) === tanpi(1//1) === -0.0
@test cos(π) == cosd(180) === cospi(1) === cospi(1//1) == sec(π) == -1
@test csc(π) == 1/0 && cot(π) == -1/0
@test sincos(π) === sincospi(1) == (0, -1)
end
@testset "frexp,ldexp,significand,exponent" begin
@testset "$T" for T in (Float16,Float32,Float64)
for z in (zero(T),-zero(T))
frexp(z) === (z,0)
significand(z) === z
@test_throws DomainError exponent(z)
end
for (a,b) in [(T(12.8),T(0.8)),
(prevfloat(floatmin(T)), prevfloat(one(T), 2)),
(prevfloat(floatmin(T)), prevfloat(one(T), 2)),
(prevfloat(floatmin(T)), nextfloat(one(T), -2)),
(nextfloat(zero(T), 3), T(0.75)),
(prevfloat(zero(T), -3), T(0.75)),
(nextfloat(zero(T)), T(0.5))]
n = Int(log2(a/b))
@test frexp(a) == (b,n)
@test ldexp(b,n) == a
@test ldexp(a,-n) == b
@test significand(a) == 2b
@test exponent(a) == n-1
@test frexp(-a) == (-b,n)
@test ldexp(-b,n) == -a
@test ldexp(-a,-n) == -b
@test significand(-a) == -2b
@test exponent(-a) == n-1
end
@test_throws DomainError exponent(convert(T,NaN))
@test isnan_type(T, significand(convert(T,NaN)))
x,y = frexp(convert(T,NaN))
@test isnan_type(T, x)
@test y == 0
@testset "ldexp function" begin
@test ldexp(T(0.0), 0) === T(0.0)
@test ldexp(T(-0.0), 0) === T(-0.0)
@test ldexp(T(Inf), 1) === T(Inf)
@test ldexp(T(Inf), 10000) === T(Inf)
@test ldexp(T(-Inf), 1) === T(-Inf)
@test isnan_type(T, ldexp(T(NaN), 10))
@test ldexp(T(1.0), 0) === T(1.0)
@test ldexp(T(0.8), 4) === T(12.8)
@test ldexp(T(-0.854375), 5) === T(-27.34)
@test ldexp(T(1.0), typemax(Int)) === T(Inf)
@test ldexp(T(1.0), typemin(Int)) === T(0.0)
@test ldexp(prevfloat(floatmin(T)), typemax(Int)) === T(Inf)
@test ldexp(prevfloat(floatmin(T)), typemin(Int)) === T(0.0)
@test ldexp(T(0.0), Int128(0)) === T(0.0)
@test ldexp(T(-0.0), Int128(0)) === T(-0.0)
@test ldexp(T(1.0), Int128(0)) === T(1.0)
@test ldexp(T(0.8), Int128(4)) === T(12.8)
@test ldexp(T(-0.854375), Int128(5)) === T(-27.34)
@test ldexp(T(1.0), typemax(Int128)) === T(Inf)
@test ldexp(T(1.0), typemin(Int128)) === T(0.0)
@test ldexp(prevfloat(floatmin(T)), typemax(Int128)) === T(Inf)
@test ldexp(prevfloat(floatmin(T)), typemin(Int128)) === T(0.0)
@test ldexp(T(0.0), BigInt(0)) === T(0.0)
@test ldexp(T(-0.0), BigInt(0)) === T(-0.0)
@test ldexp(T(1.0), BigInt(0)) === T(1.0)
@test ldexp(T(0.8), BigInt(4)) === T(12.8)
@test ldexp(T(-0.854375), BigInt(5)) === T(-27.34)
@test ldexp(T(1.0), BigInt(typemax(Int128))) === T(Inf)
@test ldexp(T(1.0), BigInt(typemin(Int128))) === T(0.0)
@test ldexp(prevfloat(floatmin(T)), BigInt(typemax(Int128))) === T(Inf)
@test ldexp(prevfloat(floatmin(T)), BigInt(typemin(Int128))) === T(0.0)
# Test also against BigFloat reference. Needs to be exactly rounded.
@test ldexp(floatmin(T), -1) == T(ldexp(big(floatmin(T)), -1))
@test ldexp(floatmin(T), -2) == T(ldexp(big(floatmin(T)), -2))
@test ldexp(floatmin(T)/2, 0) == T(ldexp(big(floatmin(T)/2), 0))
@test ldexp(floatmin(T)/3, 0) == T(ldexp(big(floatmin(T)/3), 0))
@test ldexp(floatmin(T)/3, -1) == T(ldexp(big(floatmin(T)/3), -1))
@test ldexp(floatmin(T)/3, 11) == T(ldexp(big(floatmin(T)/3), 11))
@test ldexp(floatmin(T)/11, -10) == T(ldexp(big(floatmin(T)/11), -10))
@test ldexp(-floatmin(T)/11, -10) == T(ldexp(big(-floatmin(T)/11), -10))
end
end
end
# We compare to BigFloat instead of hard-coding
# values, assuming that BigFloat has an independently tested implementation.
@testset "basic math functions" begin
@testset "$T" for T in (Float16, Float32, Float64)
x = T(1//3)
y = T(1//2)
yi = 4
@testset "Random values" begin
@test x^y === T(big(x)^big(y))
@test x^1 === x
@test x^yi === T(big(x)^yi)
@test (-x)^yi == x^yi
@test (-x)^(yi+1) == -(x^(yi+1))
@test acos(x) ≈ acos(big(x))
@test acosh(1+x) ≈ acosh(big(1+x))
@test asin(x) ≈ asin(big(x))
@test asinh(x) ≈ asinh(big(x))
@test atan(x) ≈ atan(big(x))
@test atan(x,y) ≈ atan(big(x),big(y))
@test atanh(x) ≈ atanh(big(x))
@test cbrt(x) ≈ cbrt(big(x))
@test fourthroot(x) ≈ fourthroot(big(x))
@test cos(x) ≈ cos(big(x))
@test cosh(x) ≈ cosh(big(x))
@test cospi(x) ≈ cospi(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))
@test expm1(T(-1.1)) ≈ expm1(big(T(-1.1)))
@test hypot(x,y) ≈ hypot(big(x),big(y))
@test hypot(x,x,y) ≈ hypot(hypot(big(x),big(x)),big(y))
@test hypot(x,x,y,y) ≈ hypot(hypot(big(x),big(x)),hypot(big(y),big(y)))
@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 sinpi(x) ≈ sinpi(big(x))
@test sqrt(x) ≈ sqrt(big(x))
@test tan(x) ≈ tan(big(x))
@test tanh(x) ≈ tanh(big(x))
@test tanpi(x) ≈ tanpi(big(x))
@test sec(x) ≈ sec(big(x))
@test csc(x) ≈ csc(big(x))
@test secd(x) ≈ secd(big(x))
@test cscd(x) ≈ cscd(big(x))
@test sech(x) ≈ sech(big(x))
@test csch(x) ≈ csch(big(x))
end
@testset "Special values" begin
@test isequal(T(1//4)^T(1//2), T(1//2))
@test isequal(T(1//4)^2, T(1//16))
@test isequal(acos(T(1)), T(0))
@test isequal(acosh(T(1)), T(0))
@test asin(T(1)) ≈ T(pi)/2 atol=eps(T)
@test atan(T(1)) ≈ T(pi)/4 atol=eps(T)
@test atan(T(1),T(1)) ≈ T(pi)/4 atol=eps(T)
@test isequal(cbrt(T(0)), T(0))
@test isequal(cbrt(T(1)), T(1))
@test isequal(cbrt(T(1000000000))^3, T(1000)^3)
@test isequal(fourthroot(T(0)), T(0))
@test isequal(fourthroot(T(1)), T(1))
@test isequal(fourthroot(T(100000000))^4, T(100)^4)
@test isequal(cos(T(0)), T(1))
@test cos(T(pi)/2) ≈ T(0) atol=eps(T)
@test isequal(cos(T(pi)), T(-1))
@test exp(T(1)) ≈ T(ℯ) atol=2*eps(T)
@test isequal(exp10(T(1)), T(10))
@test isequal(exp2(T(1)), T(2))
@test isequal(expm1(T(0)), T(0))
@test isequal(expm1(-floatmax(T)), -one(T))
@test isequal(expm1(floatmax(T)), T(Inf))
@test expm1(T(1)) ≈ T(ℯ)-1 atol=2*eps(T)
@test isequal(hypot(T(3),T(4)), T(5))
@test isequal(hypot(floatmax(T),T(1)),floatmax(T))
@test isequal(hypot(floatmin(T)*sqrt(eps(T)),T(0)),floatmin(T)*sqrt(eps(T)))
@test isequal(floatmin(T)*hypot(1.368423059742933,1.3510496552495361),hypot(floatmin(T)*1.368423059742933,floatmin(T)*1.3510496552495361))
@test isequal(log(T(1)), T(0))
@test isequal(log(ℯ,T(1)), T(0))
@test log(T(ℯ)) ≈ T(1) atol=eps(T)
@test isequal(log10(T(1)), T(0))
@test isequal(log10(T(10)), T(1))
@test isequal(log1p(T(0)), T(0))
@test log1p(T(ℯ)-1) ≈ T(1) atol=eps(T)
@test isequal(log2(T(1)), T(0))
@test isequal(log2(T(2)), T(1))
@test isequal(sin(T(0)), T(0))
@test isequal(sin(T(pi)/2), T(1))
@test sin(T(pi)) ≈ T(0) atol=eps(T)
@test isequal(sqrt(T(0)), T(0))
@test isequal(sqrt(T(1)), T(1))
@test isequal(sqrt(T(100000000))^2, T(10000)^2)
@test isequal(tan(T(0)), T(0))
@test tan(T(pi)/4) ≈ T(1) atol=eps(T)
@test isequal(sec(T(pi)), -one(T))
@test isequal(csc(T(pi)/2), one(T))
@test isequal(secd(T(180)), -one(T))
@test isequal(cscd(T(90)), one(T))
@test isequal(sech(log(one(T))), one(T))
@test isequal(csch(zero(T)), T(Inf))
@test zero(T)^y === zero(T)
@test zero(T)^zero(T) === one(T)
@test zero(T)^(-y) === T(Inf)
@test zero(T)^T(NaN) === T(NaN)
@test one(T)^y === one(T)
@test one(T)^zero(T) === one(T)
@test one(T)^T(NaN) === one(T)
@test isnan(T(NaN)^T(-.5))
end
@testset "Inverses" begin
@test acos(cos(x)) ≈ x
@test acosh(cosh(x)) ≈ x
@test asin(sin(x)) ≈ x
@test cbrt(x)^3 ≈ x
@test cbrt(x^3) ≈ x
@test fourthroot(x)^4 ≈ x
@test fourthroot(x^4) ≈ x
@test asinh(sinh(x)) ≈ x
@test atan(tan(x)) ≈ x
@test atan(x,y) ≈ atan(x/y)
@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 hypot(x,y) ≈ sqrt(x^2+y^2)
@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)
@test sec(x) ≈ inv(cos(x))
@test csc(x) ≈ inv(sin(x))
@test secd(x) ≈ inv(cosd(x))
@test cscd(x) ≈ inv(sind(x))
@test sech(x) ≈ inv(cosh(x))
@test csch(x) ≈ inv(sinh(x))
end
@testset "Edge cases" begin
@test isinf(log(zero(T)))
@test isnan_type(T, log(convert(T,NaN)))
@test_throws DomainError log(-one(T))
@test isinf(log1p(-one(T)))
@test isnan_type(T, log1p(convert(T,NaN)))
@test_throws DomainError log1p(convert(T,-2.0))
@test hypot(T(0), T(0)) === T(0)
@test hypot(T(Inf), T(Inf)) === T(Inf)
@test hypot(T(Inf), T(x)) === T(Inf)
@test hypot(T(Inf), T(NaN)) === T(Inf)
@test isnan_type(T, hypot(T(x), T(NaN)))
@test tanh(T(Inf)) === T(1)
end
end
@testset "Float16 expm1" begin
T=Float16
@test isequal(expm1(T(0)), T(0))
@test isequal(expm1(-floatmax(T)), -one(T))
@test isequal(expm1(floatmax(T)), T(Inf))
@test expm1(T(1)) ≈ T(ℯ)-1 atol=2*eps(T)
end
end
@testset "exponential functions" for T in (Float64, Float32, Float16)
for (func, invfunc) in ((exp2, log2), (exp, log), (exp10, log10))
@testset "$T $func accuracy" begin
minval, maxval = invfunc(floatmin(T)),prevfloat(invfunc(floatmax(T)))
# Test range and extensively test numbers near 0.
X = Iterators.flatten((minval:T(.1):maxval,
minval/100:T(.0021):maxval/100,
minval/10000:T(.000021):maxval/10000,
nextfloat(zero(T)),
T(-100):T(1):T(100) ))
for x in X
y, yb = func(x), func(widen(x))
if isfinite(eps(T(yb)))
@test abs(y-yb) <= 1.2*eps(T(yb))
end
end
end
@testset "$T $func edge cases" begin
@test func(T(-Inf)) === T(0.0)
@test func(T(Inf)) === T(Inf)
@test func(T(NaN)) === T(NaN)
@test func(T(0.0)) === T(1.0) # exact
@test func(T(5000.0)) === T(Inf)
@test func(T(-5000.0)) === T(0.0)
end
end
end
@testset "test abstractarray trig functions" begin
TAA = rand(2,2)
TAA = (TAA + TAA')/2.
STAA = Symmetric(TAA)
@test Array(atanh.(STAA)) == atanh.(TAA)
@test Array(asinh.(STAA)) == asinh.(TAA)
TAA .+= 1
@test Array(acosh.(STAA)) == acosh.(TAA)
@test Array(acsch.(STAA)) == acsch.(TAA)
@test Array(acoth.(STAA)) == acoth.(TAA)
@test sind(TAA) == sin(deg2rad.(TAA))
@test cosd(TAA) == cos(deg2rad.(TAA))
@test tand(TAA) == tan(deg2rad.(TAA))
@test asind(TAA) == rad2deg.(asin(TAA))
@test acosd(TAA) == rad2deg.(acos(TAA))
@test atand(TAA) == rad2deg.(atan(TAA))
@test asecd(TAA) == rad2deg.(asec(TAA))
@test acscd(TAA) == rad2deg.(acsc(TAA))
@test acotd(TAA) == rad2deg.(acot(TAA))
m = rand(3,2) # not square matrix
ex = @test_throws DimensionMismatch sind(m)
@test startswith(ex.value.msg, "matrix is not square")
ex = @test_throws DimensionMismatch cosd(m)
@test startswith(ex.value.msg, "matrix is not square")
ex = @test_throws DimensionMismatch tand(m)
@test startswith(ex.value.msg, "matrix is not square")
ex = @test_throws DimensionMismatch asind(m)
@test startswith(ex.value.msg, "matrix is not square")
ex = @test_throws DimensionMismatch acosd(m)
@test startswith(ex.value.msg, "matrix is not square")
ex = @test_throws DimensionMismatch atand(m)
@test startswith(ex.value.msg, "matrix is not square")
ex = @test_throws DimensionMismatch asecd(m)
@test startswith(ex.value.msg, "matrix is not square")
ex = @test_throws DimensionMismatch acscd(m)
@test startswith(ex.value.msg, "matrix is not square")
ex = @test_throws DimensionMismatch acotd(m)
@test startswith(ex.value.msg, "matrix is not square")
end
@testset "check exp2(::Integer) matches exp2(::Float)" begin
for ii in -2048:2048
expected = exp2(float(ii))
@test exp2(Int16(ii)) == expected
@test exp2(Int32(ii)) == expected
@test exp2(Int64(ii)) == expected
@test exp2(Int128(ii)) == expected
if ii >= 0
@test exp2(UInt16(ii)) == expected
@test exp2(UInt32(ii)) == expected
@test exp2(UInt64(ii)) == expected
@test exp2(UInt128(ii)) == expected
end
end
end
@testset "deg2rad/rad2deg" begin
@testset "$T" for T in (Int, Float64, BigFloat)
@test deg2rad(T(180)) ≈ 1pi
@test deg2rad.(T[45, 60]) ≈ [pi/T(4), pi/T(3)]
@test rad2deg.([pi/T(4), pi/T(3)]) ≈ [45, 60]
@test rad2deg(T(1)*pi) ≈ 180
@test rad2deg(T(1)) ≈ rad2deg(true)
@test deg2rad(T(1)) ≈ deg2rad(true)
end
@test deg2rad(180 + 60im) ≈ pi + (pi/3)*im
@test rad2deg(pi + (pi/3)*im) ≈ 180 + 60im
end
# ensure zeros are signed the same
⩲(x,y) = typeof(x) == typeof(y) && x == y && signbit(x) == signbit(y)
⩲(x::Tuple, y::Tuple) = length(x) == length(y) && all(map(⩲,x,y))
@testset "degree-based trig functions" begin
@testset "$T" for T = (Float32,Float64,Rational{Int},BigFloat)
fT = typeof(float(one(T)))
fTsc = typeof( (float(one(T)), float(one(T))) )
for x = -400:40:400
@test sind(convert(T,x))::fT ≈ sin(pi*convert(fT,x)/180) atol=eps(deg2rad(convert(fT,x)))
@test cosd(convert(T,x))::fT ≈ cos(pi*convert(fT,x)/180) atol=eps(deg2rad(convert(fT,x)))
s,c = sincosd(convert(T,x))
@test s::fT ≈ sin(pi*convert(fT,x)/180) atol=eps(deg2rad(convert(fT,x)))
@test c::fT ≈ cos(pi*convert(fT,x)/180) atol=eps(deg2rad(convert(fT,x)))
end
@testset "sind" begin
@test sind(convert(T,0.0))::fT ⩲ zero(fT)
@test sind(convert(T,180.0))::fT ⩲ zero(fT)
@test sind(convert(T,360.0))::fT ⩲ zero(fT)
T != Rational{Int} && @test sind(convert(T,-0.0))::fT ⩲ -zero(fT)
@test sind(convert(T,-180.0))::fT ⩲ -zero(fT)
@test sind(convert(T,-360.0))::fT ⩲ -zero(fT)
if T <: AbstractFloat
@test isnan(sind(T(NaN)))
end
end
@testset "cosd" begin
@test cosd(convert(T,90))::fT ⩲ zero(fT)
@test cosd(convert(T,270))::fT ⩲ zero(fT)
@test cosd(convert(T,-90))::fT ⩲ zero(fT)
@test cosd(convert(T,-270))::fT ⩲ zero(fT)
if T <: AbstractFloat
@test isnan(cosd(T(NaN)))
end
end
@testset "sincosd" begin
@test sincosd(convert(T,-360))::fTsc ⩲ ( -zero(fT), one(fT) )
@test sincosd(convert(T,-270))::fTsc ⩲ ( one(fT), zero(fT) )
@test sincosd(convert(T,-180))::fTsc ⩲ ( -zero(fT), -one(fT) )
@test sincosd(convert(T, -90))::fTsc ⩲ ( -one(fT), zero(fT) )
@test sincosd(convert(T, 0))::fTsc ⩲ ( zero(fT), one(fT) )
@test sincosd(convert(T, 90))::fTsc ⩲ ( one(fT), zero(fT) )
@test sincosd(convert(T, 180))::fTsc ⩲ ( zero(fT), -one(fT) )
@test sincosd(convert(T, 270))::fTsc ⩲ ( -one(fT), zero(fT) )
if T <: AbstractFloat
@test_throws DomainError sincosd(T(Inf))
@test all(isnan.(sincosd(T(NaN))))
end
end
@testset "$name" for (name, (sinpi, cospi)) in (
"sinpi and cospi" => (sinpi, cospi),
"sincospi" => (x->sincospi(x)[1], x->sincospi(x)[2])
)
@testset "pi * $x" for x = -3:0.3:3
@test sinpi(convert(T,x))::fT ≈ sin(pi*convert(fT,x)) atol=eps(pi*convert(fT,x))
@test cospi(convert(T,x))::fT ≈ cos(pi*convert(fT,x)) atol=eps(pi*convert(fT,x))
end
@test sinpi(convert(T,0.0))::fT ⩲ zero(fT)
@test sinpi(convert(T,1.0))::fT ⩲ zero(fT)
@test sinpi(convert(T,2.0))::fT ⩲ zero(fT)
T != Rational{Int} && @test sinpi(convert(T,-0.0))::fT ⩲ -zero(fT)
@test sinpi(convert(T,-1.0))::fT ⩲ -zero(fT)
@test sinpi(convert(T,-2.0))::fT ⩲ -zero(fT)
@test_throws DomainError sinpi(convert(T,Inf))
@test cospi(convert(T,0.5))::fT ⩲ zero(fT)
@test cospi(convert(T,1.5))::fT ⩲ zero(fT)
@test cospi(convert(T,-0.5))::fT ⩲ zero(fT)
@test cospi(convert(T,-1.5))::fT ⩲ zero(fT)
@test_throws DomainError cospi(convert(T,Inf))
end
@testset "trig pi functions accuracy" for numerator in -20:1:20
for func in (sinpi, cospi, tanpi,
x -> sincospi(x)[1],
x -> sincospi(x)[2])
x = numerator // 20
# Check that rational function works
@test func(x) ≈ func(BigFloat(x))
# Use short value so that wider values will be exactly equal
shortx = Float16(x)
# Compare to BigFloat value
bigvalue = func(BigFloat(shortx))
for T in (Float16,Float32,Float64)
@test func(T(shortx)) ≈ T(bigvalue)
end
end
end
@testset begin
# If the machine supports fma (fused multiply add), we require exact equality.
# Otherwise, we only require approximate equality.
if has_fma[T]
my_eq = (==)
@debug "On this machine, FMA is supported for $(T), so we will test for exact equality" my_eq
else
my_eq = isapprox
@debug "On this machine, FMA is not supported for $(T), so we will test for approximate equality" my_eq
end
@testset let context=(T, has_fma[T], my_eq)
@test sind(convert(T,30)) == 0.5
@test cosd(convert(T,60)) == 0.5
@test sind(convert(T,150)) == 0.5
@test my_eq(sinpi(one(T)/convert(T,6)), 0.5)
@test my_eq(sincospi(one(T)/convert(T,6))[1], 0.5)
@test_throws DomainError sind(convert(T,Inf))
@test_throws DomainError cosd(convert(T,Inf))
fT == Float64 && @test my_eq(cospi(one(T)/convert(T,3)), 0.5)
fT == Float64 && @test my_eq(sincospi(one(T)/convert(T,3))[2], 0.5)
T == Rational{Int} && @test my_eq(sinpi(5//6), 0.5)
T == Rational{Int} && @test my_eq(sincospi(5//6)[1], 0.5)
end
end
end
scdm = sincosd(missing)
@test ismissing(scdm[1])
@test ismissing(scdm[2])
end
@testset "Integer and Inf args for sinpi/cospi/tanpi/sinc/cosc" begin
for (sinpi, cospi) in ((sinpi, cospi), (x->sincospi(x)[1], x->sincospi(x)[2]))
@test sinpi(1) === 0.0
@test sinpi(-1) === -0.0
@test cospi(1) == -1
@test cospi(2) == 1
end
@test tanpi(1) === -0.0
@test tanpi(-1) === 0.0
@test tanpi(2) === 0.0
@test tanpi(-2) === -0.0
@test sinc(1) == 0
@test sinc(complex(1,0)) == 0
@test sinc(0) == 1
@test sinc(Inf) == 0
@test cosc(1) == -1
@test cosc(0) == 0
@test cosc(complex(1,0)) == -1
@test cosc(Inf) == 0
@test sinc(Inf + 3im) == 0
@test cosc(Inf + 3im) == 0
@test isequal(sinc(Inf + Inf*im), NaN + NaN*im)
@test isequal(cosc(Inf + Inf*im), NaN + NaN*im)
end
# issue #37227
@testset "sinc/cosc accuracy" begin
setprecision(256) do
for R in (BigFloat, Float16, Float32, Float64)
for T in (R, Complex{R})
for x in (0, 1e-5, 1e-20, 1e-30, 1e-40, 1e-50, 1e-60, 1e-70, 5.07138898934e-313)
if x < eps(R)
@test sinc(T(x)) == 1
end
@test cosc(T(x)) ≈ pi*(-R(x)*pi)/3 rtol=max(eps(R)*100, (pi*R(x))^2)
end
end
end
end
@test @inferred(sinc(0//1)) ⩲ 1.0
@test @inferred(cosc(0//1)) ⩲ -0.0
# test right before/after thresholds of Taylor series
@test sinc(0.001) ≈ 0.999998355066745 rtol=1e-15
@test sinc(0.00099) ≈ 0.9999983878009009 rtol=1e-15
@test sinc(0.05f0) ≈ 0.9958927352435614 rtol=1e-7
@test sinc(0.0499f0) ≈ 0.9959091277049384 rtol=1e-7
if has_fma[Float64]
@test cosc(0.14) ≈ -0.4517331883801308 rtol=1e-15
else
@test cosc(0.14) ≈ -0.4517331883801308 rtol=1e-14
end
@test cosc(0.1399) ≈ -0.45142306168781854 rtol=1e-14
@test cosc(0.26f0) ≈ -0.7996401373462212 rtol=5e-7
@test cosc(0.2599f0) ≈ -0.7993744054401625 rtol=5e-7
setprecision(256) do
@test cosc(big"0.5") ≈ big"-1.273239544735162686151070106980114896275677165923651589981338752471174381073817" rtol=1e-76
@test cosc(big"0.499") ≈ big"-1.272045747741181369948389133250213864178198918667041860771078493955590574971317" rtol=1e-76
end
end
@testset "Irrational args to sinpi/cospi/tanpi/sinc/cosc" begin
for x in (pi, ℯ, Base.MathConstants.golden)
for (sinpi, cospi) in ((sinpi, cospi), (x->sincospi(x)[1], x->sincospi(x)[2]))
@test sinpi(x) ≈ Float64(sinpi(big(x)))
@test cospi(x) ≈ Float64(cospi(big(x)))
@test sinpi(complex(x, x)) ≈ ComplexF64(sinpi(complex(big(x), big(x))))
@test cospi(complex(x, x)) ≈ ComplexF64(cospi(complex(big(x), big(x))))
end
@test tanpi(x) ≈ Float64(tanpi(big(x)))
@test sinc(x) ≈ Float64(sinc(big(x)))
@test cosc(x) ≈ Float64(cosc(big(x)))
@test sinc(complex(x, x)) ≈ ComplexF64(sinc(complex(big(x), big(x))))
@test cosc(complex(x, x)) ≈ ComplexF64(cosc(complex(big(x), big(x))))
end
end
@testset "half-integer and nan/infs for sincospi,sinpi,cospi" begin
@testset for T in (ComplexF32, ComplexF64)
@test sincospi(T(0.5, 0.0)) == (T(1.0,0.0), T(0.0, -0.0))
@test sincospi(T(1.5, 0.0)) == (T(-1.0,0.0), T(0.0, 0.0))
@test sinpi(T(1.5, 1.5)) ≈ T(-cosh(3*π/2), 0.0)
@test cospi(T(0.5, 0.5)) ≈ T(0.0, -sinh(π/2))
s, c = sincospi(T(Inf64, 0.0))
@test isnan(real(s)) && imag(s) == zero(real(T))
@test isnan(real(c)) && imag(c) == -zero(real(T))
s, c = sincospi(T(NaN, 0.0))
@test isnan(real(s)) && imag(s) == zero(real(T))
@test isnan(real(c)) && imag(c) == zero(real(T))
s, c = sincospi(T(NaN, Inf64))
@test isnan(real(s)) && isinf(imag(s))
@test isinf(real(c)) && isnan(imag(c))
s, c = sincospi(T(NaN, 2))
@test isnan(real(s)) && isnan(imag(s))
@test isnan(real(c)) && isnan(imag(c))
end
end
@testset "trig function type stability" begin
@testset "$T $f" for T = (Float32,Float64,BigFloat,Rational{Int16},Complex{Int32},ComplexF16), f = (sind,cosd,sinpi,cospi,tanpi)
@test Base.return_types(f,Tuple{T}) == [float(T)]
end
@testset "$T sincospi" for T = (Float32,Float64,BigFloat,Rational{Int16},Complex{Int32},ComplexF16)
@test Base.return_types(sincospi,Tuple{T}) == [Tuple{float(T),float(T)}]
end
end
# useful test functions for relative error, which differ from isapprox (≈)
# in that relerrc separately looks at the real and imaginary parts
relerr(z, x) = z == x ? 0.0 : abs(z - x) / abs(x)
relerrc(z, x) = max(relerr(real(z),real(x)), relerr(imag(z),imag(x)))
≅(a,b) = relerrc(a,b) ≤ 1e-13
@testset "subnormal flags" begin
# Ensure subnormal flags functions don't segfault
@test any(set_zero_subnormals(true) .== [false,true])
@test any(get_zero_subnormals() .== [false,true])
@test set_zero_subnormals(false)
@test !get_zero_subnormals()
end
@testset "evalpoly" begin
@test @evalpoly(2,3,4,5,6) == 3+2*(4+2*(5+2*6)) == @evalpoly(2+0im,3,4,5,6)
a0 = 1
a1 = 2
c = 3
@test @evalpoly(c, a0, a1) == 7
@test @evalpoly(1, 2) == 2
end
@testset "evalpoly real" begin
for x in -1.0:2.0, p1 in -3.0:3.0, p2 in -3.0:3.0, p3 in -3.0:3.0
evpm = @evalpoly(x, p1, p2, p3)
@test evalpoly(x, (p1, p2, p3)) == evpm
@test evalpoly(x, [p1, p2, p3]) == evpm
end
end
@testset "evalpoly complex" begin
for x in -1.0:2.0, y in -1.0:2.0, p1 in -3.0:3.0, p2 in -3.0:3.0, p3 in -3.0:3.0
z = x + im * y
evpm = @evalpoly(z, p1, p2, p3)
@test evalpoly(z, (p1, p2, p3)) == evpm
@test evalpoly(z, [p1, p2, p3]) == evpm
end
@test evalpoly(1+im, (2,)) == 2
@test evalpoly(1+im, [2,]) == 2
end
@testset "cis" begin
for z in (1.234, 1.234 + 5.678im)
@test cis(z) ≈ exp(im*z)
end
let z = [1.234, 5.678]
@test cis.(z) ≈ exp.(im*z)
end
end
@testset "modf" begin
@testset "$T" for T in (Float16, Float32, Float64)
@test modf(T(1.25)) === (T(0.25), T(1.0))
@test modf(T(1.0)) === (T(0.0), T(1.0))
@test modf(T(-Inf)) === (T(-0.0), T(-Inf))
@test modf(T(Inf)) === (T(0.0), T(Inf))
@test modf(T(NaN)) === (T(NaN), T(NaN))
@test modf(T(-0.0)) === (T(-0.0), T(-0.0))
@test modf(T(-1.0)) === (T(-0.0), T(-1.0))
end
end
@testset "frexp" begin
@testset "$elty" for elty in (Float16, Float32, Float64)
@test frexp( convert(elty,0.5) ) == (0.5, 0)
@test frexp( convert(elty,4.0) ) == (0.5, 3)
@test frexp( convert(elty,10.5) ) == (0.65625, 4)
end
end
@testset "log/log1p" begin
# using Tang's algorithm, should be accurate to within 0.56 ulps
X = rand(100)
for x in X
for n = -5:5
xn = ldexp(x,n)
for T in (Float32,Float64)
xt = T(x)
y = log(xt)
yb = log(big(xt))
@test abs(y-yb) <= 0.56*eps(T(yb))
y = log1p(xt)
yb = log1p(big(xt))
@test abs(y-yb) <= 0.56*eps(T(yb))
if n <= 0
y = log1p(-xt)
yb = log1p(big(-xt))
@test abs(y-yb) <= 0.56*eps(T(yb))
end
end
end
end
for n = 0:28
@test log(2,2^n) == n
end
setprecision(10_000) do
@test log(2,big(2)^100) == 100
@test log(2,big(2)^200) == 200
@test log(2,big(2)^300) == 300
@test log(2,big(2)^400) == 400
end
for T in (Float32,Float64)
@test log(zero(T)) == -Inf
@test isnan_type(T, log(T(NaN)))
@test_throws DomainError log(-one(T))
@test log1p(-one(T)) == -Inf
@test isnan_type(T, log1p(T(NaN)))
@test_throws DomainError log1p(-2*one(T))
end
@testset "log of subnormals" begin
# checked results with WolframAlpha
for (T, lr) in ((Float32, LinRange(2.f0^(-129), 2.f0^(-128), 1000)),
(Float64, LinRange(2.0^(-1025), 2.0^(-1024), 1000)))
for x in lr
@test log(x) ≈ T(log(widen(x))) rtol=2eps(T)
@test log2(x) ≈ T(log2(widen(x))) rtol=2eps(T)
@test log10(x) ≈ T(log10(widen(x))) rtol=2eps(T)
end
end
end
end
@testset "vectorization of 2-arg functions" begin
binary_math_functions = [
copysign, flipsign, log, atan, hypot, max, min,
]
@testset "$f" for f in binary_math_functions
x = y = 2
v = [f(x,y)]
@test f.([x],y) == v
@test f.(x,[y]) == v
@test f.([x],[y]) == v
end
end
@testset "issues #3024, #12822, #24240" begin
p2 = -2
p3 = -3
@test_throws DomainError 2 ^ p2
@test 2 ^ -2 == 0.25 == (2^-1)^2
@test_throws DomainError (-2)^(2.2)
@test_throws DomainError (-2.0)^(2.2)
@test_throws DomainError false ^ p2
@test false ^ -2 == Inf
@test 1 ^ -2 === (-1) ^ -2 == 1 ^ p2 === (-1) ^ p2 === 1
@test (-1) ^ -1 === (-1) ^ -3 == (-1) ^ p3 === -1
@test true ^ -2 == true ^ p2 === true
end
@testset "issue #13748" begin
let A = [1 2; 3 4]; B = [5 6; 7 8]; C = [9 10; 11 12]
@test muladd(A,B,C) == A*B + C
end
end
@testset "issue #19872" begin
f19872a(x) = x ^ 5
f19872b(x) = x ^ (-1024)
@test 0 < f19872b(2.0) < 1e-300
@test issubnormal(2.0 ^ (-1024))
@test issubnormal(f19872b(2.0))
@test !issubnormal(f19872b(0.0))
@test f19872a(2.0) === 32.0
@test !issubnormal(f19872a(2.0))
@test !issubnormal(0.0)
end
# no domain error is thrown for negative values
@test invoke(cbrt, Tuple{AbstractFloat}, -1.0) == -1.0
@testset "promote Float16 irrational #15359" begin
@test typeof(Float16(.5) * pi) == Float16
end
@testset "sincos" begin
@test sincos(1.0) === (sin(1.0), cos(1.0))
@test sincos(1f0) === (sin(1f0), cos(1f0))
@test sincos(Float16(1)) === (sin(Float16(1)), cos(Float16(1)))
@test sincos(1) === (sin(1), cos(1))
@test sincos(big(1)) == (sin(big(1)), cos(big(1)))
@test sincos(big(1.0)) == (sin(big(1.0)), cos(big(1.0)))
@test sincos(NaN) === (NaN, NaN)
@test sincos(NaN32) === (NaN32, NaN32)
@test_throws DomainError sincos(Inf32)
@test_throws DomainError sincos(Inf64)
end
@testset "test fallback definitions" begin
@test exp10(5) ≈ exp10(5.0)
@test exp10(50//10) ≈ exp10(5.0)
@test log10(exp10(ℯ)) ≈ ℯ
@test log(ℯ) === 1
@test exp2(Float16(2.0)) ≈ exp2(2.0)
@test exp2(Float16(1.0)) === Float16(exp2(1.0))
@test exp10(Float16(1.0)) === Float16(exp10(1.0))
end
@testset "isapprox" begin
# #22742: updated isapprox semantics
@test !isapprox(1.0, 1.0+1e-12, atol=1e-14)
@test isapprox(1.0, 1.0+0.5*sqrt(eps(1.0)))
@test !isapprox(1.0, 1.0+1.5*sqrt(eps(1.0)), atol=sqrt(eps(1.0)))
# #13132: Use of `norm` kwarg for scalar arguments
@test isapprox(1, 1+1.0e-12, norm=abs)
@test !isapprox(1, 1+1.0e-12, norm=x->1)
end
# test AbstractFloat fallback pr22716
struct Float22716{T<:AbstractFloat} <: AbstractFloat
x::T
end
Base.:^(x::Number, y::Float22716) = x^(y.x)
let x = 2.0
@test exp2(Float22716(x)) === 2^x
@test exp10(Float22716(x)) === 10^x
end
@testset "asin #23088" begin
for T in (Float32, Float64)
@test asin(zero(T)) === zero(T)
@test asin(-zero(T)) === -zero(T)
@test asin(nextfloat(zero(T))) === nextfloat(zero(T))
@test asin(prevfloat(zero(T))) === prevfloat(zero(T))
@test asin(one(T)) === T(pi)/2
@test asin(-one(T)) === -T(pi)/2
for x in (0.45, 0.6, 0.98)
by = asin(big(T(x)))
@test T(abs(asin(T(x)) - by))/eps(T(abs(by))) <= 1
bym = asin(big(T(-x)))
@test T(abs(asin(T(-x)) - bym))/eps(T(abs(bym))) <= 1
end
@test_throws DomainError asin(-T(Inf))
@test_throws DomainError asin(T(Inf))
@test isnan_type(T, asin(T(NaN)))
end
end
@testset "sin, cos, sincos, tan #23088" begin
for T in (Float32, Float64)
@test sin(zero(T)) === zero(T)
@test sin(-zero(T)) === -zero(T)
@test cos(zero(T)) === T(1.0)
@test cos(-zero(T)) === T(1.0)
@test sin(nextfloat(zero(T))) === nextfloat(zero(T))
@test sin(prevfloat(zero(T))) === prevfloat(zero(T))
@test cos(nextfloat(zero(T))) === T(1.0)
@test cos(prevfloat(zero(T))) === T(1.0)
for x in (0.1, 0.45, 0.6, 0.75, 0.79, 0.98)
for op in (sin, cos, tan)
by = T(op(big(x)))
@test abs(op(T(x)) - by)/eps(by) <= one(T)
bym = T(op(big(-x)))
@test abs(op(T(-x)) - bym)/eps(bym) <= one(T)
end
end
@test_throws DomainError sin(-T(Inf))
@test_throws DomainError sin(T(Inf))
@test_throws DomainError cos(-T(Inf))
@test_throws DomainError cos(T(Inf))
@test_throws DomainError tan(-T(Inf))
@test_throws DomainError tan(T(Inf))
@test sin(T(NaN)) === T(NaN)
@test cos(T(NaN)) === T(NaN)
@test tan(T(NaN)) === T(NaN)
end
end
@testset "rem_pio2 #23088" begin
vals = (2.356194490192345f0, 3.9269908169872414f0, 7.0685834705770345f0,
5.497787143782138f0, 4.216574282663131f8, 4.216574282663131f12)
for (i, x) in enumerate(vals)
for op in (prevfloat, nextfloat)
Ty = Float32(Base.Math.rem_pio2_kernel(op(vals[i]))[2].hi)
By = Float32(rem(big(op(x)), pi/2))
@test Ty ≈ By || Ty ≈ By-Float32(pi)/2
end
end
end
@testset "atan #23383" begin
for T in (Float32, Float64)
@test atan(T(NaN)) === T(NaN)
@test atan(-T(Inf)) === -T(pi)/2
@test atan(T(Inf)) === T(pi)/2
# no reduction needed |x| < 7/16
@test atan(zero(T)) === zero(T)
@test atan(prevfloat(zero(T))) === prevfloat(zero(T))
@test atan(nextfloat(zero(T))) === nextfloat(zero(T))
for x in (T(7/16), (T(7/16)+T(11/16))/2, T(11/16),
(T(11/16)+T(19/16))/2, T(19/16),
(T(19/16)+T(39/16))/2, T(39/16),
(T(39/16)+T(2)^23)/2, T(2)^23)
x = T(7/16)
by = T(atan(big(x)))
@test abs(atan(x) - by)/eps(by) <= one(T)
x = prevfloat(T(7/16))
by = T(atan(big(x)))
@test abs(atan(x) - by)/eps(by) <= one(T)
x = nextfloat(T(7/16))
by = T(atan(big(x)))
@test abs(atan(x) - by)/eps(by) <= one(T)
end
# This case was used to find a bug, but it isn't special in itself
@test atan(1.7581305072934137) ≈ 1.053644580517088
end
end
@testset "atan" begin
for T in (Float32, Float64)
@test isnan_type(T, atan(T(NaN), T(NaN)))
@test isnan_type(T, atan(T(NaN), T(0.1)))
@test isnan_type(T, atan(T(0.1), T(NaN)))
r = T(randn())
absr = abs(r)
# y zero
@test atan(T(r), one(T)) === atan(T(r))
@test atan(zero(T), absr) === zero(T)
@test atan(-zero(T), absr) === -zero(T)
@test atan(zero(T), -absr) === T(pi)
@test atan(-zero(T), -absr) === -T(pi)
# x zero and y not zero
@test atan(one(T), zero(T)) === T(pi)/2
@test atan(-one(T), zero(T)) === -T(pi)/2
# isinf(x) == true && isinf(y) == true
@test atan(T(Inf), T(Inf)) === T(pi)/4 # m == 0 (see atan code)