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isapprox: test max(atol,rtol*...) rather than atol+rtol*... #22742

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Aug 11, 2017
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isapprox: test max(atol,rtol*...) rather than atol+rtol*... #22742

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ecebffd
isapprox now tests max(atol,rtol*...) rather than atol+rtol*...
stevengj Jul 10, 2017
ecaf6ca
move news to breaking section
stevengj Jul 10, 2017
2ac767f
in isapprox, make default rtol=0 if atol>0 is specified
stevengj Jul 29, 2017
ec4b1d8
consistent isapprox for UniformScaling
stevengj Jul 29, 2017
1a600b9
whoops, remove old code
stevengj Jul 29, 2017
4b6a826
deprecate 2-arg rtoldefault
stevengj Jul 29, 2017
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4 changes: 4 additions & 0 deletions NEWS.md
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Expand Up @@ -120,6 +120,10 @@ This section lists changes that do not have deprecation warnings.
`Bidiagonal{T,V<:AbstractVector{T}}` and `SymTridiagonal{T,V<:AbstractVector{T}}`
respectively ([#22718], [#22925], [#23035]).

* `isapprox(x,y)` now tests `norm(x-y) <= max(atol, rtol*max(norm(x), norm(y)))`
rather than `norm(x-y) <= atol + ...`, and `rtol` defaults to zero
if an `atol > 0` is specified ([#22742]).

* Spaces are no longer allowed between `@` and the name of a macro in a macro call ([#22868]).

* Juxtaposition of a non-literal with a macro call (`x@macro`) is no longer valid syntax ([#22868]).
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2 changes: 2 additions & 0 deletions base/deprecated.jl
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Expand Up @@ -1661,6 +1661,8 @@ end
@deprecate num2hex(x::Union{Float16,Float32,Float64}) hex(reintepret(Unsigned, x), sizeof(x)*2)
@deprecate num2hex(n::Integer) hex(n, sizeof(n)*2)

# PR #22742: change in isapprox semantics
@deprecate rtoldefault(x,y) rtoldefault(x,y,0) false

# END 0.7 deprecations

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20 changes: 12 additions & 8 deletions base/floatfuncs.jl
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Expand Up @@ -189,15 +189,16 @@ end

# isapprox: approximate equality of numbers
"""
isapprox(x, y; rtol::Real=sqrt(eps), atol::Real=0, nans::Bool=false, norm::Function)
isapprox(x, y; rtol::Real=atol>0 ? √eps : 0, atol::Real=0, nans::Bool=false, norm::Function)

Inexact equality comparison: `true` if `norm(x-y) <= atol + rtol*max(norm(x), norm(y))`. The
Inexact equality comparison: `true` if `norm(x-y) <= max(atol, rtol*max(norm(x), norm(y)))`. The
default `atol` is zero and the default `rtol` depends on the types of `x` and `y`. The keyword
argument `nans` determines whether or not NaN values are considered equal (defaults to false).

For real or complex floating-point values, `rtol` defaults to
`sqrt(eps(typeof(real(x-y))))`. This corresponds to requiring equality of about half of the
significand digits. For other types, `rtol` defaults to zero.
For real or complex floating-point values, if an `atol > 0` is not specified, `rtol` defaults to
the square root of [`eps`](@ref) of the type of `x` or `y`, whichever is bigger (least precise).
This corresponds to requiring equality of about half of the significand digits. Otherwise,
e.g. for integer arguments or if an `atol > 0` is supplied, `rtol` defaults to zero.

`x` and `y` may also be arrays of numbers, in which case `norm` defaults to `vecnorm` but
may be changed by passing a `norm::Function` keyword argument. (For numbers, `norm` is the
Expand All @@ -220,8 +221,8 @@ julia> isapprox([10.0^9, 1.0], [10.0^9, 2.0])
true
```
"""
function isapprox(x::Number, y::Number; rtol::Real=rtoldefault(x,y), atol::Real=0, nans::Bool=false)
x == y || (isfinite(x) && isfinite(y) && abs(x-y) <= atol + rtol*max(abs(x), abs(y))) || (nans && isnan(x) && isnan(y))
function isapprox(x::Number, y::Number; atol::Real=0, rtol::Real=rtoldefault(x,y,atol), nans::Bool=false)
x == y || (isfinite(x) && isfinite(y) && abs(x-y) <= max(atol, rtol*max(abs(x), abs(y)))) || (nans && isnan(x) && isnan(y))
end

const ≈ = isapprox
Expand All @@ -230,7 +231,10 @@ const ≈ = isapprox
# default tolerance arguments
rtoldefault(::Type{T}) where {T<:AbstractFloat} = sqrt(eps(T))
rtoldefault(::Type{<:Real}) = 0
rtoldefault(x::Union{T,Type{T}}, y::Union{S,Type{S}}) where {T<:Number,S<:Number} = max(rtoldefault(real(T)),rtoldefault(real(S)))
function rtoldefault(x::Union{T,Type{T}}, y::Union{S,Type{S}}, atol::Real) where {T<:Number,S<:Number}
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enough packages are calling the two-argument version (despite it not being exported) that it should probably be deprecated

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Okay, I just pushed a deprecation.

rtol = max(rtoldefault(real(T)),rtoldefault(real(S)))
return atol > 0 ? zero(rtol) : rtol
end

# fused multiply-add
fma_libm(x::Float32, y::Float32, z::Float32) =
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7 changes: 4 additions & 3 deletions base/linalg/generic.jl
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Expand Up @@ -1296,11 +1296,12 @@ promote_leaf_eltypes(x::Union{AbstractArray,Tuple}) = mapreduce(promote_leaf_elt
# Supports nested arrays; e.g., for `a = [[1,2, [3,4]], 5.0, [6im, [7.0, 8.0]]]`
# `a ≈ a` is `true`.
function isapprox(x::AbstractArray, y::AbstractArray;
rtol::Real=Base.rtoldefault(promote_leaf_eltypes(x),promote_leaf_eltypes(y)),
atol::Real=0, nans::Bool=false, norm::Function=vecnorm)
atol::Real=0,
rtol::Real=Base.rtoldefault(promote_leaf_eltypes(x),promote_leaf_eltypes(y),atol),
nans::Bool=false, norm::Function=vecnorm)
d = norm(x - y)
if isfinite(d)
return d <= atol + rtol*max(norm(x), norm(y))
return d <= max(atol, rtol*max(norm(x), norm(y)))
else
# Fall back to a component-wise approximate comparison
return all(ab -> isapprox(ab[1], ab[2]; rtol=rtol, atol=atol, nans=nans), zip(x, y))
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10 changes: 5 additions & 5 deletions base/linalg/uniformscaling.jl
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Expand Up @@ -197,15 +197,16 @@ broadcast(::typeof(/), J::UniformScaling,x::Number) = UniformScaling(J.λ/x)
==(J1::UniformScaling,J2::UniformScaling) = (J1.λ == J2.λ)

function isapprox(J1::UniformScaling{T}, J2::UniformScaling{S};
rtol::Real=Base.rtoldefault(T,S), atol::Real=0, nans::Bool=false) where {T<:Number,S<:Number}
atol::Real=0, rtol::Real=Base.rtoldefault(T,S,atol), nans::Bool=false) where {T<:Number,S<:Number}
isapprox(J1.λ, J2.λ, rtol=rtol, atol=atol, nans=nans)
end
function isapprox(J::UniformScaling,A::AbstractMatrix;
rtol::Real=rtoldefault(promote_leaf_eltypes(A),eltype(J)),
atol::Real=0, nans::Bool=false, norm::Function=vecnorm)
atol::Real=0,
rtol::Real=rtoldefault(promote_leaf_eltypes(A),eltype(J),atol),
nans::Bool=false, norm::Function=vecnorm)
n = checksquare(A)
Jnorm = norm === vecnorm ? abs(J.λ)*sqrt(n) : (norm === Base.norm ? abs(J.λ) : norm(diagm(fill(J.λ, n))))
return norm(A - J) <= atol + rtol*max(norm(A), Jnorm)
return norm(A - J) <= max(atol, rtol*max(norm(A), Jnorm))
end
isapprox(A::AbstractMatrix,J::UniformScaling;kwargs...) = isapprox(J,A;kwargs...)

Expand Down Expand Up @@ -335,4 +336,3 @@ UniformScaling{Float64}
```
"""
chol(J::UniformScaling, args...) = ((C, info) = _chol!(J, nothing); @assertposdef C info)

5 changes: 5 additions & 0 deletions test/math.jl
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Expand Up @@ -669,6 +669,11 @@ end
@test exp10(Float16(1.0)) === Float16(exp10(1.0))
end

# #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)))

# test AbstractFloat fallback pr22716
struct Float22716{T<:AbstractFloat} <: AbstractFloat
x::T
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