Error

Try it out

Derivation

1. Initial program 52.8

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

2. Initial simplification52.8

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

4. Applied associate-/l*39.0

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

5. Simplified39.0

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

7. Applied *-un-lft-identity39.0

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

8. Applied times-frac38.7

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

9. Applied times-frac38.7

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

10. Simplified38.7

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

11. Simplified36.5

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

13. Applied add-cube-cbrt36.8

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

14. Applied add-cube-cbrt36.6

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

15. Applied times-frac36.6

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

16. Final simplification36.6

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

Runtime

Time bar (total: 5.8m)Debug logProfile

herbie shell --seed 2018295 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))