Error

Derivation

1. Split input into 2 regimes

2. if alpha < 4.652297026472302e+165

   1. Initial program 1.4

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

   2. Taylor expanded around 0 1.4

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

   3. Simplified1.4

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

   if 4.652297026472302e+165 < alpha

   1. Initial program 16.4

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

   2. Taylor expanded around 0 16.4

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

   3. Simplified16.4

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

   4. Taylor expanded around -inf 7.3

      \[\leadsto \color{blue}{0}\]
3. Recombined 2 regimes into one program.

4. Final simplification2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.652297026472302 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{\frac{(\alpha \cdot \beta + \left(\beta + \alpha\right))_* + 1.0}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 10.0m)Debug logProfile

herbie shell --seed 2018353 +o rules:numerics
(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)))