\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/1
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Time: 13.3 s
Input Error: 6.9
Output Error: 6.4
Log: ⚲
Profile: 🕒
\({\left(\frac{\sqrt[3]{\frac{\beta}{2.0}}}{\sqrt[3]{\left(\alpha + 2.0\right) + \beta}}\right)}^3 - \frac{\frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - 1.0}{2.0}\)
  1. Started with
    \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    6.9
  2. Using strategy rm
    6.9
  3. Applied pow1 to get
    \[\frac{\frac{\beta - \alpha}{\color{red}{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0} \leadsto \frac{\frac{\beta - \alpha}{\color{blue}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}}} + 1.0}{2.0}\]
    6.5
  4. Using strategy rm
    6.5
  5. Applied div-sub to get
    \[\frac{\color{red}{\frac{\beta - \alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}}} + 1.0}{2.0} \leadsto \frac{\color{blue}{\left(\frac{\beta}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - \frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}}\right)} + 1.0}{2.0}\]
    6.4
  6. Applied associate-+l- to get
    \[\frac{\color{red}{\left(\frac{\beta}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - \frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}}\right) + 1.0}}{2.0} \leadsto \frac{\color{blue}{\frac{\beta}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - \left(\frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - 1.0\right)}}{2.0}\]
    6.2
  7. Applied div-sub to get
    \[\color{red}{\frac{\frac{\beta}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - \left(\frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - 1.0\right)}{2.0}} \leadsto \color{blue}{\frac{\frac{\beta}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}}}{2.0} - \frac{\frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - 1.0}{2.0}}\]
    6.2
  8. Applied simplify to get
    \[\color{red}{\frac{\frac{\beta}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}}}{2.0}} - \frac{\frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - 1.0}{2.0} \leadsto \color{blue}{\frac{\frac{\beta}{2.0}}{\left(\alpha + 2.0\right) + \beta}} - \frac{\frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - 1.0}{2.0}\]
    6.3
  9. Using strategy rm
    6.3
  10. Applied add-cube-cbrt to get
    \[\frac{\frac{\beta}{2.0}}{\color{red}{\left(\alpha + 2.0\right) + \beta}} - \frac{\frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - 1.0}{2.0} \leadsto \frac{\frac{\beta}{2.0}}{\color{blue}{{\left(\sqrt[3]{\left(\alpha + 2.0\right) + \beta}\right)}^3}} - \frac{\frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - 1.0}{2.0}\]
    6.4
  11. Applied add-cube-cbrt to get
    \[\frac{\color{red}{\frac{\beta}{2.0}}}{{\left(\sqrt[3]{\left(\alpha + 2.0\right) + \beta}\right)}^3} - \frac{\frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - 1.0}{2.0} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{\beta}{2.0}}\right)}^3}}{{\left(\sqrt[3]{\left(\alpha + 2.0\right) + \beta}\right)}^3} - \frac{\frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - 1.0}{2.0}\]
    6.4
  12. Applied cube-undiv to get
    \[\color{red}{\frac{{\left(\sqrt[3]{\frac{\beta}{2.0}}\right)}^3}{{\left(\sqrt[3]{\left(\alpha + 2.0\right) + \beta}\right)}^3}} - \frac{\frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - 1.0}{2.0} \leadsto \color{blue}{{\left(\frac{\sqrt[3]{\frac{\beta}{2.0}}}{\sqrt[3]{\left(\alpha + 2.0\right) + \beta}}\right)}^3} - \frac{\frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - 1.0}{2.0}\]
    6.4

Original test:


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