\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/1
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Time: 21.0 s
Input Error: 6.7
Output Error: 6.5
Log: ⚲
Profile: 🕒
\(\frac{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right)}^3 - \frac{{\left(\frac{\alpha}{\left(\beta + 2.0\right) + \alpha}\right)}^3 - {1.0}^3}{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^2 + \left({1.0}^2 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}{2.0}\)
  1. Started with
    \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    6.7
  2. Using strategy rm
    6.7
  3. Applied div-sub to get
    \[\frac{\color{red}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0} \leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]
    6.7
  4. Applied associate-+l- to get
    \[\frac{\color{red}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) + 1.0}}{2.0} \leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
    6.4
  5. Using strategy rm
    6.4
  6. Applied add-cube-cbrt to get
    \[\frac{\color{red}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right)}^3} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]
    6.5
  7. Using strategy rm
    6.5
  8. Applied flip3-- to get
    \[\frac{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right)}^3 - \color{red}{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0} \leadsto \frac{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right)}^3 - \color{blue}{\frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}}{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^2 + \left({1.0}^2 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}}{2.0}\]
    6.5
  9. Applied simplify to get
    \[\frac{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right)}^3 - \frac{\color{red}{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}}}{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^2 + \left({1.0}^2 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}{2.0} \leadsto \frac{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right)}^3 - \frac{\color{blue}{{\left(\frac{\alpha}{\left(\beta + 2.0\right) + \alpha}\right)}^3 - {1.0}^3}}{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^2 + \left({1.0}^2 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}{2.0}\]
    6.5

Original test:


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