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Derivation

1. Initial program 52.7

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

2. Initial simplification47.4

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

4. Applied times-frac38.9

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

5. Applied associate-*l*38.9

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

6. Simplified38.9

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

8. Applied add-cube-cbrt39.3

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

9. Applied times-frac36.7

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

10. Final simplification36.7

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

Runtime

Time bar (total: 4.3m)Debug logProfile

herbie shell --seed 2018216 
(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)))