\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/1
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Time: 10.9 s
Input Error: 6.9
Output Error: 6.3
Log: ⚲
Profile: 🕒
\(\frac{\frac{\beta}{2.0}}{\left(\alpha + 2.0\right) + \beta} - \frac{\frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - 1.0}{2.0}\)
  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. Removed slow pow expressions

Original test:


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