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Derivation

1. Split input into 2 regimes

2. if beta < 7.218546705668313e+163

   1. Initial program 1.3

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

   2. Taylor expanded around -inf 1.3

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\beta + \left(3.0 + \alpha\right)}}\]

   if 7.218546705668313e+163 < beta

   1. Initial program 16.7

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

   2. Taylor expanded around -inf 16.7

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\beta + \left(3.0 + \alpha\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity16.7

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 \cdot \left(\beta + \left(3.0 + \alpha\right)\right)}}\]

   5. Applied div-inv16.7

      \[\leadsto \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{1 \cdot \left(\beta + \left(3.0 + \alpha\right)\right)}\]

   6. Applied times-frac17.9

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1} \cdot \frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\beta + \left(3.0 + \alpha\right)}}\]

   7. Simplified17.9

      \[\leadsto \color{blue}{\frac{\alpha \cdot \beta + \left(\alpha + \left(\beta + 1.0\right)\right)}{2 + \left(\beta + \alpha\right)}} \cdot \frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\beta + \left(3.0 + \alpha\right)}\]

   8. Simplified17.9

      \[\leadsto \frac{\alpha \cdot \beta + \left(\alpha + \left(\beta + 1.0\right)\right)}{2 + \left(\beta + \alpha\right)} \cdot \color{blue}{\frac{1}{\left(\alpha + \left(3.0 + \beta\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}}\]

   9. Taylor expanded around 0 7.6

      \[\leadsto \color{blue}{\left(0.25 \cdot \alpha + \left(0.25 \cdot \beta + 0.5\right)\right)} \cdot \frac{1}{\left(\alpha + \left(3.0 + \beta\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}\]

   10. Simplified7.6

       \[\leadsto \color{blue}{\left(0.5 + 0.25 \cdot \left(\alpha + \beta\right)\right)} \cdot \frac{1}{\left(\alpha + \left(3.0 + \beta\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 7.218546705668313 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(3.0 + \alpha\right) + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(3.0 + \beta\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)} \cdot \left(\left(\alpha + \beta\right) \cdot 0.25 + 0.5\right)\\ \end{array}\]

Runtime

Time bar (total: 6.6m)Debug logProfile

herbie shell --seed 2018273 
(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)))