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Derivation

1. Initial program 16.4

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

2. Initial simplification16.4

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

4. Applied div-inv16.4

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

6. Applied add-cbrt-cube16.5

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

8. Applied flip3-+16.5

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

9. Applied flip3-+16.5

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

10. Applied flip3-+16.5

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

11. Applied frac-times16.5

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

12. Applied frac-times16.5

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

13. Applied cbrt-div16.5

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

14. Simplified16.5

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

15. Final simplification16.5

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

Runtime

Time bar (total: 35.7s)Debug logProfile

herbie shell --seed 2018248 
(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))