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Derivation

1. Split input into 2 regimes

2. if beta < 1.6041693629760017e+194

   1. Initial program 1.9

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

   3. Applied *-un-lft-identity1.9

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

   4. Applied add-sqr-sqrt2.4

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

   5. Applied add-sqr-sqrt2.0

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

   6. Applied times-frac2.0

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

   7. Applied times-frac2.0

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

   8. Simplified2.0

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

   9. Simplified2.0

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

   11. Applied div-inv2.0

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

   12. Applied sqrt-prod2.0

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

   13. Applied times-frac1.9

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

   if 1.6041693629760017e+194 < beta

   1. Initial program 17.4

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

   3. Applied *-un-lft-identity17.4

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

   4. Applied add-sqr-sqrt17.4

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

   5. Applied times-frac17.4

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

   6. Simplified17.4

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

   7. Simplified17.4

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

   8. Taylor expanded around inf 6.8

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

   9. Simplified6.8

      \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{2.0}{\beta}}{\beta} + \left(1 - \frac{1.0}{\beta}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.6041693629760017 \cdot 10^{+194}:\\ \;\;\;\;\frac{\sqrt{\frac{\alpha + \left(\alpha \cdot \beta + \left(1.0 + \beta\right)\right)}{\left(\beta + 2\right) + \alpha}}}{\sqrt{\left(\beta + 2\right) + \alpha}} \cdot \left(\frac{\sqrt{\frac{1}{\left(\alpha + \beta\right) + 2}}}{\left(\alpha + \beta\right) + \left(1.0 + 2\right)} \cdot \frac{\sqrt{\alpha + \left(\alpha \cdot \beta + \left(1.0 + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2.0}{\beta}}{\beta} + \left(1 - \frac{1.0}{\beta}\right)}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1.0}\\ \end{array}\]

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed 2018296 
(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)))