positivePart

Smooth max(x, 0): 0.5(x + sqrt(x^2 + delta^2))*

FastSmoothBuildings.Utilities.positivePart, function (source)

Purpose

Smooth approximation of max(x, 0), strictly positive: a flow or a cosine that can divide or weight without a singularity at 0.

Equations

y = (x + √(x² + δ²))/2.

Infinitely differentiable (C-infinity) for δ > 0. Always above max(x, 0): y = δ/2 at x = 0, and the difference (√(x² + δ²) − |x|)/2 tends to δ²/(4|x|) for |x| >> δ: y tends to 0 for x → −∞ and to x for x → ∞.

Use

delta is the width of the rounding, in the unit of x; unlike smoothMax, the result is nowhere exact.

Revisions

• October 2026: moved from trano's FastModel.Utilities.