Error

Derivation

1. Initial program 51.9

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

2. Simplified51.9

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

4. Applied add-sqr-sqrt51.9

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

5. Applied difference-of-squares51.9

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

6. Applied times-frac38.0

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

7. Applied times-frac36.0

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

9. Applied add-sqr-sqrt36.1

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

10. Applied add-sqr-sqrt36.0

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

11. Applied times-frac36.0

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

13. Applied add-sqr-sqrt36.0

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

14. Applied associate-/l*36.0

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

15. Final simplification36.0

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

Reproduce

herbie shell --seed 2019091 +o rules:numerics
(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)))