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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)) < 0.08551473591693905

   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 associate-+l+0.1

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

   if 0.08551473591693905 < (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0))

   1. Initial program 61.2

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

   3. Applied associate-+l+61.2

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

   5. Applied associate-/l/61.2

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

   6. Simplified61.2

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

   7. Taylor expanded around 0 21.8

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

   8. Simplified21.8

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

4. Final simplification1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \le 0.08551473591693905:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + \left(2 + 1.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \left(\beta + \alpha\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + \left(2 + 1.0\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 7.3m)Debug logProfile

herbie shell --seed 2018252 
(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)))