fix accuracy of logcosh(::Union{Float16, Float32, Float64})

src/basicfuns.jl

@@ -131,9 +131,63 @@ The implementation ensures `logcosh(-x) = logcosh(x)`.
 """
 function logcosh(x::Real)
     abs_x = abs(x)
+    if (x isa Union{Float32, Float64}) && (abs_x < oftype(x, 0.7373046875))
+        return logcosh_ker(x)
+    end
     return abs_x + log1pexp(- 2 * abs_x) - IrrationalConstants.logtwo
 end
 
+"""
+    logcosh_ker(x::Union{Float32, Float64})
+
+The kernel of `logcosh`.
+
+The polynomial coefficients were found using Sollya:
+
+```sollya
+prec = 500!;
+points = 50001!;
+accurate = log(cosh(x));
+domain = [-0.125, 0.7373046875];
+constrained_part = (x^2) / 2;
+free_monomials_32 = [|4, 6, 8, 10, 12|];
+free_monomials_64 = [|4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24|];
+polynomial_32 = fpminimax(accurate, free_monomials_32, [|single...|], domain, constrained_part);
+polynomial_64 = fpminimax(accurate, free_monomials_64, [|double...|], domain, constrained_part);
+polynomial_32;
+polynomial_64;
+```
+"""
+function logcosh_ker(x::Union{Float32, Float64})
+    x² = x * x
+    if x isa Float32
+        p = (
+            5f-1,
+            -0.083333164f0,
+            0.022217678f0,
+            -0.0067060017f0,
+            0.0020296266f0,
+            -0.00044135848f0,
+        )
+    elseif x isa Float64
+        p = (
+            5e-1,
+            -0.08333333333332801,
+            0.02222222222164912,
+            -0.0067460317245250445,
+            0.0021869484500251714,
+            -0.0007385985435311435,
+            0.0002565500026777061,
+            -9.084985367586575e-5,
+            3.2348259905568986e-5,
+            -1.1058814347469105e-5,
+            3.16293199955507e-6,
+            -5.312230207322749e-7,
+        )
+    end
+    evalpoly(x², p) * x²
+end
+
 """
 $(SIGNATURES)
 

test/basicfuns.jl

@@ -107,6 +107,14 @@ end
         @test @inferred(logcosh(x)) === x
         @test @inferred(logabssinh(x)) === x
     end
+
+    @testset "accuracy of `logcosh`" begin
+        for t in (Float32, Float64)
+            for x in range(start = t(-3), stop = t(3), length = 1000)
+                @test ulp_error(logcosh, x) < 3
+            end
+        end
+    end
 end
 
 @testset "log1psq" begin

test/common/ULPError.jl

@@ -0,0 +1,57 @@
+module ULPError
+    export ulp_error, ulp_error_maximum
+    @noinline function throw_invalid()
+        throw(ArgumentError("invalid"))
+    end
+    function ulp_error(accurate::AbstractFloat, approximate::AbstractFloat)
+        # the ULP error is usually not required to great accuracy, so `Float32` should be precise enough
+        zero_return = Float32(0)
+        inf_return = Float32(Inf)
+        let accur_is_nan = isnan(accurate), approx_is_nan = isnan(approximate)
+            if accur_is_nan || approx_is_nan
+                if accur_is_nan === approx_is_nan
+                    return zero_return
+                end
+                return inf_return
+            end
+        end
+        if isinf(accurate) || iszero(accurate)  # handle floating-point edge cases
+            if isinf(accurate)
+                if isinf(approximate) && (signbit(accurate) == signbit(approximate))
+                    return zero_return
+                end
+                return inf_return
+            end
+            # `iszero(accurate)`
+            if iszero(approximate)
+                return zero_return
+            end
+            return inf_return
+        end
+        # assuming `precision(BigFloat)` is great enough
+        acc = if accurate isa BigFloat
+            accurate
+        else
+            BigFloat(accurate)::BigFloat
+        end
+        err = abs(Float32((approximate - acc) / eps(approximate))::Float32)
+        if isnan(err)
+            @noinline throw_invalid()  # unexpected
+        end
+        err
+    end
+    function ulp_error(accurate::Acc, approximate::App, x::AbstractFloat) where {Acc, App}
+        acc = accurate(x)
+        app = approximate(x)
+        ulp_error(acc, app)
+    end
+    function ulp_error(func::Func, x::AbstractFloat) where {Func}
+        ulp_error(func ∘ BigFloat, func, x)
+    end
+    function ulp_error_maximum(func::Func, iterator) where {Func}
+        function f(x::AbstractFloat)
+            ulp_error(func, x)
+        end
+        maximum(f, iterator)
+    end
+end

test/runtests.jl

@@ -10,6 +10,9 @@ using OffsetArrays
 using Random
 using Test
 
+include("common/ULPError.jl")
+using .ULPError
+
 Random.seed!(1234)
 
 include("basicfuns.jl")