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Derivation

1. Initial program 16.3

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

2. Initial simplification16.3

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

4. Applied div-inv16.3

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

6. Applied add-sqr-sqrt16.5

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

7. Applied associate-*r*16.4

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

9. Applied sqrt-div16.4

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

10. Applied sqrt-div16.5

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

11. Applied associate-*r/16.4

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

12. Applied frac-times16.5

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

13. Simplified16.5

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

14. Simplified16.3

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

15. Final simplification16.3

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

Runtime

Time bar (total: 31.2s)Debug logProfile

herbie shell --seed 2018230 
(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))