Improve accuracy of logistic (#94)

Co-authored-by: John Myles White <johnmyleswhite@fb.com>

src/basicfuns.jl

@@ -21,6 +21,20 @@ Return `x * log(y)` for `y > 0` with correct limit at `x = 0`.
 xlogy(x::T, y::T) where {T<:Real} = x > zero(T) ? x * log(y) : zero(log(x))
 xlogy(x::Real, y::Real) = xlogy(promote(x, y)...)
 
+# The following bounds are precomputed versions of the following abstract
+# function, but the implicit interface for AbstractFloat doesn't uniformly
+# enforce that all floating point types implement nextfloat and prevfloat.
+# @inline function _logistic_bounds(x::AbstractFloat)
+#     (
+#         logit(nextfloat(zero(float(x)))),
+#         logit(prevfloat(one(float(x)))),
+#     )
+# end
+
+@inline _logistic_bounds(x::Float16) = (Float16(-16.64), Float16(7.625))
+@inline _logistic_bounds(x::Float32) = (-103.27893f0, 16.635532f0)
+@inline _logistic_bounds(x::Float64) = (-744.4400719213812, 36.7368005696771)
+
 """
     logistic(x::Real)
 
@@ -33,6 +47,20 @@ Its inverse is the [`logit`](@ref) function.
 """
 logistic(x::Real) = inv(exp(-x) + one(x))
 
+function logistic(x::Union{Float16, Float32, Float64})
+    e = exp(x)
+    lower, upper = _logistic_bounds(x)
+    ifelse(
+        x < lower,
+        zero(x),
+        ifelse(
+            x > upper,
+            one(x),
+            e / (one(x) + e)
+        )
+    )
+end
+
 """
     logit(p::Real)
 

test/basicfuns.jl

@@ -10,6 +10,10 @@ end
 
 @testset "logistic & logit" begin
     @test logistic(2)        ≈ 1.0 / (1.0 + exp(-2.0))
+    @test logistic(-750.0) === 0.0
+    @test logistic(-740.0) > 0.0
+    @test logistic(+36.0) < 1.0
+    @test logistic(+750.0) === 1.0
     @test iszero(logit(0.5))
     @test logit(logistic(2)) ≈ 2.0
 end