\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/2
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 30.1 s
Input Error: 23.7
Output Error: 12.6
Log: ⚲
Profile: 🕒
\(\frac{(\left(\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) * \left(\log_* (1 + (e^{\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}} - 1)^*)\right) + 1.0)_*}{2.0}\)
  1. Started with
    \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    23.7
  2. Applied simplify to get
    \[\color{red}{\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}} \leadsto \color{blue}{\frac{(\left(\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}{2.0}}\]
    12.6
  3. Using strategy rm
    12.6
  4. Applied log1p-expm1-u to get
    \[\frac{(\left(\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) * \color{red}{\left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right)} + 1.0)_*}{2.0} \leadsto \frac{(\left(\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) * \color{blue}{\left(\log_* (1 + (e^{\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}} - 1)^*)\right)} + 1.0)_*}{2.0}\]
    12.6

Original test:


(lambda ((alpha default) (beta default) (i default))
  #:name "Octave 3.8, jcobi/2"
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))