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Derivation

1. Initial program 52.6

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

2. Initial simplification52.6

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

4. Applied add-sqr-sqrt52.6

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

5. Applied times-frac38.8

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

6. Applied times-frac38.8

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

7. Simplified38.8

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

9. Applied associate-*l/38.8

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

11. Applied *-un-lft-identity38.8

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

12. Applied times-frac38.8

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

13. Simplified38.8

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

15. Applied distribute-rgt-out38.8

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

16. Applied associate-/l*36.4

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

17. Final simplification36.4

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

Runtime

Time bar (total: 5.6m)Debug logProfile

herbie shell --seed 2018354 
(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)))