Add "a" parameter to softplus()

src/basicfuns.jl

@@ -165,9 +165,14 @@ Return `log(1+exp(x))` evaluated carefully for largish `x`.
 This is also called the ["softplus"](https://en.wikipedia.org/wiki/Rectifier_(neural_networks))
 transformation, being a smooth approximation to `max(0,x)`. Its inverse is [`logexpm1`](@ref).
 
+The generalized `softplus` function (Wiemann et al., 2024) takes an additional optional parameter `a` that control 
+the approximation error with respect to the linear spline. It defaults to `a=1.0`, in which case the softplus is 
+equivalent to `log1pexp`.
+
 See:
  * Martin Maechler (2012) [“Accurately Computing log(1 − exp(− |a|))”](http://cran.r-project.org/web/packages/Rmpfr/vignettes/log1mexp-note.pdf)
-"""
+ * Wiemann, P. F., Kneib, T., & Hambuckers, J. (2024). Using the softplus function to construct alternative link functions in generalized linear models and beyond. Statistical Papers, 65(5), 3155-3180.
+ """
 log1pexp(x::Real) = _log1pexp(float(x)) # ensures that BigInt/BigFloat, Int/Float64 etc. dispatch to the same algorithm
 
 # Approximations based on Maechler (2012)
@@ -255,10 +260,22 @@ Return `log(exp(x) - 1)` or the “invsoftplus” function.  It is the inverse o
 [`log1pexp`](@ref) (aka “softplus”).
 """
 logexpm1(x::Real) = x <= 18.0 ? log(_expm1(x)) : x <= 33.3 ? x - exp(-x) : oftype(exp(-x), x)
-logexpm1(x::Float32) = x <= 9f0 ? log(expm1(x)) : x <= 16f0 ? x - exp(-x) : oftype(exp(-x), x)
+logexpm1(x::Float32) = x <= 9.0f0 ? log(expm1(x)) : x <= 16.0f0 ? x - exp(-x) : oftype(exp(-x), x)
+
+function softplus(x; a::Real=1.0)
+    if a == 1.0
+        return log1pexp(x)
+    end
+    return log1pexp(a * x) / a
+end
+
+function invsoftplus(y; a::Real=1.0)
+    if a == 1.0
+        return logexpm1(y)
+    end
+    return logexpm1(a * y) / a
+end
 
-const softplus = log1pexp
-const invsoftplus = logexpm1
 
 """
 $(SIGNATURES)