\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/1
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Time: 9.2 s
Input Error: 16.4
Output Error: 16.6
Log: ⚲
Profile: 🕒
\(\frac{(\left(\beta - \alpha\right) * \left(\frac{1}{\left(\alpha + \beta\right) + 2.0}\right) + 1.0)_*}{2.0}\)
  1. Started with
    \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    16.4
  2. Using strategy rm
    16.4
  3. Applied div-inv to get
    \[\frac{\color{red}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0} \leadsto \frac{\color{blue}{\left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0}\]
    16.4
  4. Applied fma-def to get
    \[\frac{\color{red}{\left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0} + 1.0}}{2.0} \leadsto \frac{\color{blue}{(\left(\beta - \alpha\right) * \left(\frac{1}{\left(\alpha + \beta\right) + 2.0}\right) + 1.0)_*}}{2.0}\]
    16.6

Original test:


(lambda ((alpha default) (beta default))
  #:name "Octave 3.8, jcobi/1"
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))