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Derivation

1. Split input into 2 regimes

2. if alpha < 863261300362360.1

   1. Initial program 10.9

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   2. Using strategy rm

   3. Applied associate-/l*0.2

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity0.2

      \[\leadsto \frac{\frac{\frac{\alpha + \beta}{\frac{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

   6. Applied associate-/l*0.2

      \[\leadsto \frac{\frac{\frac{\alpha + \beta}{\color{blue}{\frac{1}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity0.2

      \[\leadsto \frac{\frac{\frac{\alpha + \beta}{\frac{1}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

   9. Applied associate-/r/0.2

      \[\leadsto \frac{\frac{\frac{\alpha + \beta}{\color{blue}{\frac{1}{\beta - \alpha} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   10. Applied *-un-lft-identity0.2

       \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \left(\alpha + \beta\right)}}{\frac{1}{\beta - \alpha} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   11. Applied times-frac0.2

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\frac{1}{\beta - \alpha}} \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   12. Applied times-frac0.2

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\frac{1}{\beta - \alpha}}}{1} \cdot \frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

   13. Simplified0.2

       \[\leadsto \frac{\color{blue}{\left(\beta - \alpha\right)} \cdot \frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   14. Using strategy rm

   15. Applied add-cbrt-cube0.2

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(\beta - \alpha\right) \cdot \frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\left(\beta - \alpha\right) \cdot \frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)\right) \cdot \left(\left(\beta - \alpha\right) \cdot \frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)}}}{2.0}\]

   if 863261300362360.1 < alpha

   1. Initial program 52.1

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

   2. Taylor expanded around inf 40.8

      \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

   3. Simplified40.8

      \[\leadsto \frac{\color{blue}{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}}{2.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 863261300362360.1:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\beta - \alpha\right) \cdot \frac{\frac{\beta + \alpha}{\left(\beta + \alpha\right) + i \cdot 2}}{2.0 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} + 1.0\right) \cdot \left(\left(\left(\beta - \alpha\right) \cdot \frac{\frac{\beta + \alpha}{\left(\beta + \alpha\right) + i \cdot 2}}{2.0 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} + 1.0\right) \cdot \left(\left(\beta - \alpha\right) \cdot \frac{\frac{\beta + \alpha}{\left(\beta + \alpha\right) + i \cdot 2}}{2.0 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} + 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))

Details

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