Function alpha_from_atm 

pub fn alpha_from_atm<R: Scalar>(
    formula: SabrFormula,
    forward: R,
    atm_vol: R,
    expiry_years: f64,
    beta: f64,
    rho: R,
    nu: R,
    shift: f64,
) -> Result<R, SabrError>

The α that reproduces atm_vol at forward: the smallest positive real root of the ATM cubic, polished by Newton on passive values, then carried into the mode by two Newton steps on the cubic with its coefficients in the mode — exact derivatives to second order (the implicit function theorem, without differentiating a root search).

Errors

SabrError::NoAlpha where no positive root exists; SabrError::Domain for a shifted forward ≤ 0 under a shifted formula.