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Derivation

1. Split input into 2 regimes

2. if (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)) < 0.08340852993133803

   1. Initial program 0.1

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

   3. Applied *-un-lft-identity0.1

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

   4. Applied add-sqr-sqrt0.2

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

   5. Applied times-frac0.2

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

   6. Applied associate-/l*0.2

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

   7. Simplified0.2

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

   if 0.08340852993133803 < (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0))

   1. Initial program 60.1

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

   2. Taylor expanded around -inf 23.1

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

   3. Simplified23.1

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

4. Final simplification1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \le 0.08340852993133803:\\ \;\;\;\;\frac{\frac{\sqrt{\alpha \cdot \beta + \left(\left(\beta + 1.0\right) + \alpha\right)}}{\frac{2 + \left(\beta + \alpha\right)}{\frac{\sqrt{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}}{2 + \left(\beta + \alpha\right)}}}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1.0}{\alpha}\right) + \frac{\frac{2.0}{\alpha}}{\alpha}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\\ \end{array}\]

Runtime

Time bar (total: 8.1m)Debug logProfile

herbie shell --seed 2018351 
(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)))