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Derivation

1. Initial program 16.2

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

2. Initial simplification16.2

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

4. Applied div-inv16.2

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

6. Applied add-cbrt-cube16.2

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

8. Applied *-commutative16.2

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

10. Applied sub-neg16.2

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

11. Applied distribute-lft-in16.2

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

12. Applied associate-+r+16.2

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

13. Simplified16.2

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

14. Final simplification16.2

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

Runtime

Time bar (total: 33.3s)Debug logProfile

herbie shell --seed 2018274 
(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))