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Derivation

1. Initial program 16.6

   \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]

2. Initial simplification16.6

   \[\leadsto \frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0}\]
3. Using strategy rm

4. Applied div-inv16.6

   \[\leadsto \frac{1.0 + \color{blue}{\left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}}}{2.0}\]
5. Using strategy rm

6. Applied add-cbrt-cube16.6

   \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(1.0 + \left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}\right) \cdot \left(1.0 + \left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}\right)\right) \cdot \left(1.0 + \left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}\right)}}}{2.0}\]
7. Using strategy rm

8. Applied add-log-exp16.6

   \[\leadsto \frac{\sqrt[3]{\left(\color{blue}{\log \left(e^{1.0 + \left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}}\right)} \cdot \left(1.0 + \left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}\right)\right) \cdot \left(1.0 + \left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}\right)}}{2.0}\]
9. Using strategy rm

10. Applied add-exp-log16.6

    \[\leadsto \frac{\sqrt[3]{\left(\color{blue}{e^{\log \left(\log \left(e^{1.0 + \left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}}\right)\right)}} \cdot \left(1.0 + \left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}\right)\right) \cdot \left(1.0 + \left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}\right)}}{2.0}\]

11. Final simplification16.6

    \[\leadsto \frac{\sqrt[3]{\left(1.0 + \frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \left(\beta - \alpha\right)\right) \cdot \left(e^{\log \left(\log \left(e^{1.0 + \frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \left(\beta - \alpha\right)}\right)\right)} \cdot \left(1.0 + \frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \left(\beta - \alpha\right)\right)\right)}}{2.0}\]

Reproduce

herbie shell --seed 2018365 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

Details

Time bar (total: 19.2s)Debug log