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Derivation

1. Split input into 2 regimes

2. if beta < 8.168150807681616e+148

   1. Initial program 50.4

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

   3. Applied associate-/l*35.0

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

   5. Applied add-sqr-sqrt35.0

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

   if 8.168150807681616e+148 < beta

   1. Initial program 62.5

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

   3. Applied associate-/l*56.2

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

   4. Taylor expanded around -inf 48.6

      \[\leadsto \color{blue}{0}\]
3. Recombined 2 regimes into one program.

4. Final simplification37.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 8.168150807681616 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\frac{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}}} \cdot \sqrt{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\frac{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}}}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 4.0m)Debug logProfile

herbie shell --seed 2018353 
(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)))