Error

Derivation

1. Split input into 2 regimes

2. if alpha < 2.84856999209069e+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. Using strategy rm

   3. Applied associate-+l+1.3

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

   5. Applied add-sqr-sqrt1.9

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

   6. Applied add-sqr-sqrt2.3

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

   7. Applied *-un-lft-identity2.3

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

   8. Applied times-frac2.3

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

   9. Applied times-frac2.0

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

   10. Simplified1.4

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

   11. Simplified1.3

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

   13. Applied div-inv1.4

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

   14. Applied associate-*r*1.4

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

   if 2.84856999209069e+163 < alpha

   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. Using strategy rm

   3. Applied associate-+l+16.7

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

   5. Applied add-sqr-sqrt16.7

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

   6. Applied add-sqr-sqrt16.7

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

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

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

   8. Applied times-frac16.7

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

   9. Applied times-frac16.7

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

   10. Simplified16.7

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

   11. Simplified16.7

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

   13. Applied div-inv16.7

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

   14. Applied associate-*r*16.7

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

   15. Taylor expanded around -inf 7.1

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

4. Final simplification2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.84856999209069 \cdot 10^{+163}:\\ \;\;\;\;\frac{\left(\left((\beta \cdot \alpha + \alpha)_* + \left(\beta + 1.0\right)\right) \cdot \frac{1}{\alpha + \left(2 + \beta\right)}\right) \cdot \frac{1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + \left(2 + 1.0\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 5.6m)Debug logProfile

herbie shell --seed 2018290 +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)))