Numerically stable log(1 + exp(x))

Computes the "softplus" function $$\mathrm{log1pexp}(x) = \log\!\bigl(1 + e^{x}\bigr)$$ without overflow. A naive log(1 + exp(x)) overflows to Inf for \(x \gtrsim 710\) even though the true value is \(\approx x\). The identity \(\log(1 + e^{x}) = x + \log(1 + e^{-x})\) for \(x > 0\) keeps every intermediate finite.

Usage

hzr_log1pexp(x)

Arguments

x

Numeric vector.

Value

Numeric vector with element-wise \(\log(1 + e^{x})\).

References

Mächler M (2012). Accurately Computing \(\log(1 - e^{-|a|})\). R package Rmpfr vignette. https://CRAN.R-project.org/package=Rmpfr/vignettes/log1mexp-note.pdf

See also

hzr_log1mexp() for the complementary \(\log(1 - e^{-x})\), hzr_clamp_prob() for boundary-safe probabilities.

Examples

hzr_log1pexp(c(-50, 0, 50))
#> [1] 1.928750e-22 6.931472e-01 5.000000e+01