Error

Derivation

1. Split input into 2 regimes

2. if alpha < 3.6280596072694005e+185

   1. Initial program 1.5

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

   3. Applied associate-+l+1.5

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

   if 3.6280596072694005e+185 < alpha

   1. Initial program 18.8

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

   3. Applied associate-+l+18.8

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

   5. Applied *-un-lft-identity18.8

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

   6. Applied div-inv18.8

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

   7. Applied times-frac19.3

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

   8. Simplified19.3

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

   9. Simplified19.3

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

   10. Taylor expanded around 0 7.2

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

   11. Simplified7.2

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

4. Final simplification2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.6280596072694005 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + \left(2 + 1.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\beta + \alpha\right) + \left(2 + 1.0\right)}}{\beta + \left(\alpha + 2\right)} \cdot (\left(\beta + \alpha\right) \cdot 0.25 + 0.5)_*\\ \end{array}\]

Runtime

Time bar (total: 6.2m)Debug logProfile

herbie shell --seed 2018274 +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)))