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Derivation

1. Split input into 2 regimes

2. if beta < 5.97924189304454e+188

   1. Initial program 1.7

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

   3. Applied *-un-lft-identity1.7

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

   4. Applied associate-/l*1.7

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

   if 5.97924189304454e+188 < beta

   1. Initial program 15.4

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

   3. Applied *-un-lft-identity15.4

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

   4. Applied associate-/l*15.4

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

   6. Applied div-inv15.4

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

   7. Applied add-sqr-sqrt15.4

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

   8. Applied times-frac15.4

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

   9. Applied associate-/r*15.4

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

   10. Simplified15.4

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

   11. Taylor expanded around 0 5.8

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

   12. Simplified5.8

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

4. Final simplification2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 5.97924189304454 \cdot 10^{+188}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\frac{1.0 + \left(\left(\beta + \alpha\right) + \alpha \cdot \beta\right)}{\left(\beta + \alpha\right) + 2}}}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\sqrt{2} - \left(\beta + \alpha\right) \cdot \sqrt{2}\right) \cdot 1.0 + \left(\frac{\beta \cdot 0.5}{\sqrt{2}} + \left(\alpha \cdot 0.5\right) \cdot \sqrt{\frac{1}{2}}\right)}}{\sqrt{\left(\beta + \alpha\right) + 2} \cdot \left(\left(\beta + \alpha\right) + 2\right)}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 
(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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