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Derivation

1. Initial program 52.2

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

2. Initial simplification46.7

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

4. Applied *-un-lft-identity46.7

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

5. Applied times-frac46.7

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

6. Simplified37.3

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

8. Applied *-un-lft-identity37.3

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

9. Applied times-frac36.4

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

10. Applied *-un-lft-identity36.4

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

11. Applied times-frac36.4

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

12. Applied associate-*r*36.3

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

14. Applied add-sqr-sqrt36.4

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

15. Applied associate-/l*36.3

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

16. Final simplification36.3

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

Reproduce

herbie shell --seed 2018360 
(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)))

Details

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