Function inv_norm_cdf 

pub fn inv_norm_cdf<R: Scalar>(p: R) -> R

The inverse standard-normal CDF: Acklam’s approximation at the passive value, then the inverse’s Taylor series about it written in the mode, to third order in u = (p − Φ(x₀))/φ(x₀):

Φ⁻¹(p) = x₀ + u + (x₀/2)·u² + ((2x₀² + 1)/6)·u³ + O(u⁴)

The first term is the Newton polish, which makes dx/dp = 1/φ(x) exact and the value the reference’s; the higher terms add nothing to the value (u is at rounding level) but carry the curvature, so d²x/dp² = x/φ² and d³x/dp³ = (2x² + 1)/φ³ are exact too. A Newton step alone is linear in p: its second derivative was zero, and every second-order sensitivity through a default threshold was missing the threshold’s curvature.