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Derivation

1. Initial program 52.3

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

3. Applied times-frac39.1

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

4. Simplified39.1

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

6. Applied *-un-lft-identity39.1

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

7. Applied times-frac39.2

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

8. Simplified39.1

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

10. Applied associate-/l*36.4

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

11. Final simplification36.4

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018297 
(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)))