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Derivation

1. Split input into 2 regimes

2. if alpha < 2.3447701514638163e+160

   1. Initial program 1.1

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
   2. Using strategy rm

   3. Applied div-inv1.1

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

   if 2.3447701514638163e+160 < alpha

   1. Initial program 17.2

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
   2. Using strategy rm

   3. Applied div-inv17.2

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

   5. Applied flip-+18.4

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

   6. Applied associate-/r/18.4

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

   7. Applied associate-/l*18.4

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

   8. Simplified18.4

      \[\leadsto \frac{\color{blue}{\frac{\frac{1.0}{2 + \left(\beta + \alpha\right)} + \frac{\alpha \cdot \beta + \left(\beta + \alpha\right)}{2 + \left(\beta + \alpha\right)}}{-4 + \left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right)}}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}{\left(\alpha + \beta\right) - 2 \cdot 1}}\]

   9. Taylor expanded around 0 7.7

      \[\leadsto \frac{\frac{\color{blue}{0.5 + \left(0.25 \cdot \beta + 0.25 \cdot \alpha\right)}}{-4 + \left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right)}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}{\left(\alpha + \beta\right) - 2 \cdot 1}}\]

   10. Simplified7.7

       \[\leadsto \frac{\frac{\color{blue}{0.5 + \left(\beta + \alpha\right) \cdot 0.25}}{-4 + \left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right)}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}{\left(\alpha + \beta\right) - 2 \cdot 1}}\]
3. Recombined 2 regimes into one program.

4. Final simplification2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.3447701514638163 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\beta + \alpha\right) + 2} \cdot \left(1.0 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)\right)}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + \alpha\right) \cdot 0.25 + 0.5}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right) + -4}}{\frac{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}{\left(\beta + \alpha\right) - 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))

Details

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