Function steed_jy_nu 

pub fn steed_jy_nu<P, E, V>(
    nu: V,
    n: V,
    u: V,
    x: V,
    needed: V::Mask,
) -> (V, V, V)
where
    E: FloatElement,
    V: FloatVector<Element = E>,
    P: Policy,

$(J_\nu, Y_u, Y_{u+1})$ by Steed’s method at the reduced order $u = \nu - n$, $\lvert u\rvert \le 1/2$, for the band between the two other arms.

The order reduction is not optional

Boost runs CF1 at $\nu$, recurs the $J$ ratio down to $u$, runs CF2 and the Wronskian at $u$, and walks $Y$ back up (its x > 2 branch). Running everything at $\nu$ directly is fine for $\nu \in [-1/3, 2]$ and catastrophic above: at $\nu = 7.4$, $x = 2.07$, $Y$ measured 4.3 million ULP and $J$ 678 against mpmath.

The reason is CF2. The Thompson-Barnett fraction for $H'/H$ converges for every $x > 0$, but its accuracy collapses once $\nu$ is well above $x$, the same fact that makes the modified twin reduce its order for both $K$ arms. CF1 has no such problem, so the ratio it delivers at $\nu$ is walked down instead, and the walk is the stable direction for $J$. Below $\lvert\nu\rvert \le 1/2$ the reduction is the identity and this is exactly the six-line arm.

The pieces

CF1 at $\nu$ gives $f_\nu = J_{\nu+1}/J_\nu$ and the sign of $J_\nu$. The three-term recurrence walked down from $\nu$ with a tiny seed (prev, cur proportional to $J_{k+1}$, $J_k$) reaches $u$ with two things in hand: $f_u = J_{u+1}/J_u$ and the scaling $J_\nu/J_u$. CF2 at $u$ and the Wronskian $J_u Y'_u - J'_u Y_u = 2/\pi x$ then give $\lvert J_u\rvert$, $Y_u$ and $Y_{u+1}$. The sign of $J_u$ is the sign of $J_\nu$ times the sign the walk ended on. The caller walks $Y$ up, which it does for the Temme arm anyway.

Measured

At small order the arithmetic is unchanged: notes/special/tools/model_steed.py, envelope-relative against mpmath at 60 digits, over x in [4, 30] and nu in [-1/3, 2], worst 7.62 eps for J, 5.74 for Y. The iteration counts move in opposite directions: CF1 grows with x (20 to 35 across the gap), CF2 shrinks (16 to 8), so their sum is nearly flat over the region this covers.