Function ln_near_one

fn ln_near_one<T>(y: T) -> T
where
    T: FixedPointNumber + Copy + PartialOrd + FixedPointInfo,
    T::Inner: From<u8> + Shr<u32, Output = T::Inner> + TryInto<i128> + Copy,

Computes ln(y) for a fixed-point value y near 1 using the arctanh series identity:

ln(y) = 2 * sum_{k=0}^{inf} t^(2k+1) / (2k+1)

where t = (y - 1) / (y + 1)

Converges for all y > 0, with convergence rate determined by |t|. The closer y is to 1, the smaller |t| and the faster convergence. range_reduce_sqrt ensures y is in [0.5, 1.5] before calling this function, keeping |t| <= 1/3 for fast, reliable convergence.

Arguments

• y - A fixed-point value near 1. Caller must ensure y > 0. Results are inaccurate for y far from 1.

Returns

An approximation of ln(y), accurate to within the type’s ULP for inputs in [0.5, 1.5].

Note

On unsigned types, y < 1 produces t = 0 (since y - 1 saturates to zero), returning ln(y) = 0. This is incorrect for y < 1, but the unsigned type guard in fixed_ln ensures this branch is never reached for unsigned types with y < 1.