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Derivation

1. Split input into 2 regimes

2. if alpha < 8.158565138653585e+102

   1. Initial program 14.2

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

   3. Applied *-un-lft-identity14.2

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

   4. Applied times-frac3.5

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

   5. Applied associate-/l*3.5

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

   6. Simplified3.5

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

   8. Applied flip3-+3.5

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

   if 8.158565138653585e+102 < alpha

   1. Initial program 58.5

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

   2. Taylor expanded around inf 41.6

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

   3. Simplified41.6

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

4. Final simplification12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 8.158565138653585 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\beta + \alpha}{\frac{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}}\right)}^{3} + {1.0}^{3}}{\frac{\beta + \alpha}{\frac{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}} \cdot \frac{\beta + \alpha}{\frac{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}} + \left(1.0 \cdot 1.0 - 1.0 \cdot \frac{\beta + \alpha}{\frac{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}}\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 
(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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