fix JuliaLang#12375 (fix scaling of isapprox via more sensible default tolerances)

Author: stevengj

NEWS.md

@@ -300,6 +300,8 @@ Library improvements
 
     * The `MathConst` type has been renamed `Irrational` ([#11922]).
 
+    * `isapprox` now has simpler and more sensible default tolerances ([#12393]).
+
   * Random numbers
 
     * Streamlined random number generation APIs [#8246].
@@ -1542,3 +1544,4 @@ Too numerous to mention.
 [#12034]: https://github.com/JuliaLang/julia/issues/12034
 [#12087]: https://github.com/JuliaLang/julia/issues/12087
 [#12137]: https://github.com/JuliaLang/julia/issues/12137
+[#12393]: https://github.com/JuliaLang/julia/issues/12393

base/floatfuncs.jl

@@ -165,33 +165,12 @@ for f in (:round, :ceil, :floor, :trunc)
     end
 end
 
-# isapprox: Tolerant comparison of floating point numbers
-function isapprox(x::FloatingPoint, y::FloatingPoint; rtol::Real=rtoldefault(x,y), atol::Real=atoldefault(x,y))
-    (isinf(x) || isinf(y)) ? x == y : abs(x-y) <= atol + rtol.*max(abs(x), abs(y))
-end
-
-# promotion of non-floats
-isapprox(x::Real, y::FloatingPoint; rtol::Real=rtoldefault(x, y), atol::Real=atoldefault(x, y)) = isapprox(promote(x, y)...; rtol=rtol, atol=atol)
-isapprox(x::FloatingPoint, y::Real; rtol::Real=rtoldefault(x, y), atol::Real=atoldefault(x, y)) = isapprox(promote(x, y)...; rtol=rtol, atol=atol)
-
-# other real numbers
-isapprox(x::Real, y::Real; rtol::Real=0, atol::Real=0) = abs(x-y) <= atol
-
-# complex numbers
-isapprox(z::Complex, w::Complex; rtol::Real=rtoldefault(abs(z), abs(w)), atol::Real=atoldefault(abs(z), abs(w))) = abs(z-w) <= atol + rtol*max(abs(z), abs(w))
-
-# real-complex combinations
-isapprox(x::Real, z::Complex; rtol::Real=rtoldefault(x, abs(z)), atol::Real=atoldefault(x, abs(z))) = isapprox(complex(x), z; rtol=rtol, atol=atol)
-isapprox(z::Complex, x::Real; rtol::Real=rtoldefault(x, abs(z)), atol::Real=atoldefault(x, abs(z))) = isapprox(complex(x), z; rtol=rtol, atol=atol)
+# isapprox: approximate equality of numbers
+isapprox(x::Number, y::Number; rtol::Real=rtoldefault(x,y), atol::Real=0) =
+    x == y || (isfinite(x) && isfinite(y) && abs(x-y) <= atol + rtol*max(abs(x), abs(y)))
 
 # default tolerance arguments
-rtoldefault(x::FloatingPoint, y::FloatingPoint) = cbrt(max(eps(x), eps(y)))
-atoldefault(x::FloatingPoint, y::FloatingPoint) = sqrt(max(eps(x), eps(y)))
+rtoldefault{T<:FloatingPoint}(::Type{T}) = sqrt(eps(T))
+rtoldefault{T<:Real}(::Type{T}) = 0
+rtoldefault{T<:Number,S<:Number}(x::T, y::S) = rtoldefault(promote_type(real(T),real(S)))
 
-# promotion of non-floats
-for fun in (:rtoldefault, :atoldefault)
-    @eval begin
-        ($fun)(x::Real, y::FloatingPoint) = ($fun)(promote(x,y)...)
-        ($fun)(x::FloatingPoint, y::Real) = ($fun)(promote(x,y)...)
-    end
-end

doc/stdlib/math.rst

@@ -417,17 +417,11 @@ Mathematical Operators
 Mathematical Functions
 ----------------------
 
-.. function:: isapprox(x::Number, y::Number; rtol::Real=cbrt(maxeps), atol::Real=sqrt(maxeps))
+.. function:: isapprox(x::Number, y::Number; rtol::Real=sqrt(eps), atol::Real=0)
 
-   Inexact equality comparison - behaves slightly different depending on types of input args:
+   Inexact equality comparison: ``true`` if ``abs(x-y) <= atol + rtol*max(abs(x), abs(y))``.  The default ``atol`` is zero and the default ``rtol`` depends on the types of ``x`` and ``y``.
 
-   * For ``FloatingPoint`` numbers, ``isapprox`` returns ``true`` if ``abs(x-y) <= atol + rtol*max(abs(x), abs(y))``.
-
-   * For ``Integer`` and ``Rational`` numbers, ``isapprox`` returns ``true`` if ``abs(x-y) <= atol``. The `rtol` argument is ignored. If one of ``x`` and ``y`` is ``FloatingPoint``, the other is promoted, and the method above is called instead.
-
-   * For ``Complex`` numbers, the distance in the complex plane is compared, using the same criterion as above.
-
-   For default tolerance arguments, ``maxeps = max(eps(abs(x)), eps(abs(y)))``.
+   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.
 
 .. function:: sin(x)
 

test/fastmath.jl

@@ -138,18 +138,22 @@ for T in (Complex64, Complex128, Complex{BigFloat})
     half = (1+1im)/T(2)
     third = (1-1im)/T(3)
 
+    # some of these functions promote their result to double
+    # precision, but we want to check equality at single precision
+    rtol = Base.rtoldefault(real(T))
+
     for f in (:+, :-, :abs, :abs2, :conj, :inv, :sign,
               :acos, :acosh, :asin, :asinh, :atan, :atanh, :cos,
               :cosh, :exp10, :exp2, :exp, :expm1, :log10, :log1p,
               :log2, :log, :sin, :sinh, :sqrt, :tan, :tanh)
-        @test isapprox((@eval @fastmath $f($half)), (@eval $f($half)))
-        @test isapprox((@eval @fastmath $f($third)), (@eval $f($third)))
+        @test isapprox((@eval @fastmath $f($half)), (@eval $f($half)), rtol=rtol)
+        @test isapprox((@eval @fastmath $f($third)), (@eval $f($third)), rtol=rtol)
     end
     for f in (:+, :-, :*, :/, :(==), :!=, :^)
         @test isapprox((@eval @fastmath $f($half, $third)),
-                       (@eval $f($half, $third)))
+                       (@eval $f($half, $third)), rtol=rtol)
         @test isapprox((@eval @fastmath $f($third, $half)),
-                       (@eval $f($third, $half)))
+                       (@eval $f($third, $half)), rtol=rtol)
     end
 end
 

test/floatapprox.jl

@@ -43,3 +43,6 @@
 # Notably missing from this test suite at the moment
 # * Tests for other magnitudes of numbers - very small, very big, and combinations of small and big
 # * Tests for various odd combinations of types, e.g. isapprox(x::Integer, y::Rational)
+
+# issue #12375:
+@test !isapprox(1e17, 1)