Error

Derivation

1. Initial program 24.1

   \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

2. Initial simplification12.3

   \[\leadsto \frac{(\left(\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]
3. Using strategy rm

4. Applied add-sqr-sqrt12.4

   \[\leadsto \frac{(\left(\frac{\beta - \alpha}{\color{blue}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]

5. Applied associate-/r*12.4

   \[\leadsto \frac{(\color{blue}{\left(\frac{\frac{\beta - \alpha}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right)} \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]
6. Using strategy rm

7. Applied *-un-lft-identity12.4

   \[\leadsto \frac{(\left(\frac{\frac{\beta - \alpha}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}{\sqrt{\left(\beta + 2.0\right) + \color{blue}{1 \cdot (2 \cdot i + \alpha)_*}}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]

8. Applied *-un-lft-identity12.4

   \[\leadsto \frac{(\left(\frac{\frac{\beta - \alpha}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}{\sqrt{\color{blue}{1 \cdot \left(\beta + 2.0\right)} + 1 \cdot (2 \cdot i + \alpha)_*}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]

9. Applied distribute-lft-out12.4

   \[\leadsto \frac{(\left(\frac{\frac{\beta - \alpha}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}{\sqrt{\color{blue}{1 \cdot \left(\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*\right)}}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]

10. Applied sqrt-prod12.4

    \[\leadsto \frac{(\left(\frac{\frac{\beta - \alpha}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}{\color{blue}{\sqrt{1} \cdot \sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]

11. Applied add-sqr-sqrt12.4

    \[\leadsto \frac{(\left(\frac{\frac{\beta - \alpha}{\sqrt{\color{blue}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}}{\sqrt{1} \cdot \sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]

12. Applied sqrt-prod12.5

    \[\leadsto \frac{(\left(\frac{\frac{\beta - \alpha}{\color{blue}{\sqrt{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \sqrt{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}}{\sqrt{1} \cdot \sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]

13. Applied *-un-lft-identity12.5

    \[\leadsto \frac{(\left(\frac{\frac{\color{blue}{1 \cdot \left(\beta - \alpha\right)}}{\sqrt{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \sqrt{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}{\sqrt{1} \cdot \sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]

14. Applied times-frac12.5

    \[\leadsto \frac{(\left(\frac{\color{blue}{\frac{1}{\sqrt{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}} \cdot \frac{\beta - \alpha}{\sqrt{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}}{\sqrt{1} \cdot \sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]

15. Applied times-frac12.4

    \[\leadsto \frac{(\color{blue}{\left(\frac{\frac{1}{\sqrt{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}{\sqrt{1}} \cdot \frac{\frac{\beta - \alpha}{\sqrt{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right)} \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]
16. Using strategy rm

17. Applied add-cube-cbrt12.4

    \[\leadsto \frac{(\left(\frac{\frac{1}{\sqrt{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}{\sqrt{1}} \cdot \frac{\frac{\beta - \alpha}{\sqrt{\sqrt{\color{blue}{\left(\sqrt[3]{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \sqrt[3]{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}\right) \cdot \sqrt[3]{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}}}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]

18. Applied sqrt-prod12.4

    \[\leadsto \frac{(\left(\frac{\frac{1}{\sqrt{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}{\sqrt{1}} \cdot \frac{\frac{\beta - \alpha}{\sqrt{\color{blue}{\sqrt{\sqrt[3]{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \sqrt[3]{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \sqrt{\sqrt[3]{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}}}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]

19. Applied sqrt-prod12.4

    \[\leadsto \frac{(\left(\frac{\frac{1}{\sqrt{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}{\sqrt{1}} \cdot \frac{\frac{\beta - \alpha}{\color{blue}{\sqrt{\sqrt{\sqrt[3]{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \sqrt[3]{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}} \cdot \sqrt{\sqrt{\sqrt[3]{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}}}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]

20. Applied associate-/r*12.4

    \[\leadsto \frac{(\left(\frac{\frac{1}{\sqrt{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}{\sqrt{1}} \cdot \frac{\color{blue}{\frac{\frac{\beta - \alpha}{\sqrt{\sqrt{\sqrt[3]{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \sqrt[3]{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}}{\sqrt{\sqrt{\sqrt[3]{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}}}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]

21. Final simplification12.4

    \[\leadsto \frac{(\left(\frac{\frac{\frac{\beta - \alpha}{\sqrt{\sqrt{\sqrt[3]{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \sqrt[3]{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}}{\sqrt{\sqrt{\sqrt[3]{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} \cdot \frac{\frac{1}{\sqrt{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}}{\sqrt{1}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))