I have implemented Carlson's algorithm for $E(k)$ from Numerical computation of real or complex elliptic integrals (available from ArXiv eprint, see also DLMF). It is essentially his formula (46) which simultaneously computes $K(k), E(k)$ for complex $k$ using the arithmetic-geometric sequence $a_m, b_m$ to get
$$K(k) = \frac{\pi}{a_n+b_n}, \quad E(k) = \left( \left(\frac{a_0+b_0}{2} \right)^2 -\sum\limits_{m=1}^{n} 2^{m-2}(a_m - b_m)^{2}\right) K(k)$$
Unfortunately with IEEE 64 or 80 bit floating point arithmetic the expression for $E(k)$ is accurate only up to $|k| \approx 1000,\,$ for larger arguments severe rounding/cancellation problems occur (e.g. for $k=10^3-10i$ the relative error in the real part is $17\epsilon,\;$for $k=10^4-10i$ it is $150\epsilon,\;$ and for $k=10^6-10i$ it is $24000\epsilon$). For very large $|k|$ I could use the logarithmic form, but is seems relative useless if there is no reliable algorithm for medium $|k|$.
Is there a known stable algorithm for this range, ie. which can compute $E(k)$ for $|k| > 100$ without using multi-precision?