Function temme_ik 

pub fn temme_ik<P, E, R, C, const NE: usize, const NO: usize>(
    nu: R,
    z: C,
    needed: C::Mask,
    t: &LogGamma1p<E, NE, NO>,
) -> (C, C)
where
    E: FloatElement,
    R: FloatVector<Element = E> + TranscendentalMathWithPolicy,
    C: FloatVector<Mask = R::Mask> + TranscendentalMathWithPolicy + PrimalProjection<Primal = R> + Mul<R, Output = C> + Div<R, Output = C> + Add<R, Output = C> + Sub<R, Output = C>,
    P: Policy,

$(K_\nu(x), K_{\nu+1}(x))$ unscaled, by Temme’s series, for $|x| \le 2$ and $\lvert\nu\rvert \le 1/2$.

Temme, Journal of Computational Physics vol 19, 324 (1975). Boost’s temme_ik. The structural twin of temme_y_nu: the same gamma1/gamma2 limits, the same coef chain, the same paired-Additive shape through sum_pair, combined differently and with the coef multiplier positive rather than negative, since $I$/$K$ do not oscillate.

Two of the four limits are shipped functions, not guards

$c = \sin(\pi\nu)/(\pi\nu)$ is exactly sinc_pi and $d = \sinh\sigma/\sigma$ is exactly sinhc, so neither needs the $0/0$ select Boost writes for it. Only gamma1 keeps one, and its limit is $-\gamma$.

Precondition

$\lvert\nu\rvert \le 1/2$ is not a suggestion. The series is built around $\Gamma(1\pm\nu)$ near one. The caller reduces the order and walks $K$ up, which is stable because $K$ is the dominant solution.