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Derivation

1. Split input into 2 regimes

2. if alpha < 8.467650448969669e+126

   1. Initial program 14.4

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

   3. Applied *-un-lft-identity14.4

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

   4. Applied times-frac4.2

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

   5. Simplified4.2

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

   7. Applied div-inv4.2

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

   9. Applied add-sqr-sqrt4.2

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

   10. Applied associate-/r*4.2

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

   if 8.467650448969669e+126 < alpha

   1. Initial program 60.1

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

   2. Taylor expanded around -inf 41.7

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

   3. Simplified41.7

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

4. Final simplification11.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 8.467650448969669 \cdot 10^{+126}:\\ \;\;\;\;\frac{1.0 + \frac{\frac{1}{\sqrt{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) + 2.0}}}{\sqrt{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) + 2.0}} \cdot \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018296 
(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))