Error

Derivation

1. Split input into 2 regimes

2. if i < 1.9121510033596495e+124

   1. Initial program 37.8

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

   3. Applied times-frac14.6

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

   4. Applied associate-/l*14.6

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

   6. Applied *-un-lft-identity14.6

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

   7. Applied times-frac14.5

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

   8. Applied associate-/l*14.5

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

   9. Simplified14.5

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

   11. Applied flip-+14.6

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

   12. Applied associate-/r/14.7

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

   13. Applied *-un-lft-identity14.7

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

   14. Applied add-sqr-sqrt14.7

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

   15. Applied times-frac14.7

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

   16. Applied times-frac14.7

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

   17. Applied associate-/r*14.7

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

   18. Simplified14.5

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

   if 1.9121510033596495e+124 < i

   1. Initial program 62.1

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

   3. Applied times-frac55.5

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

   4. Applied associate-/l*55.5

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

   6. Applied *-un-lft-identity55.5

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

   7. Applied times-frac55.5

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

   8. Applied associate-/l*55.5

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

   9. Simplified55.5

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

   10. Taylor expanded around 0 12.1

       \[\leadsto \frac{i}{\frac{\color{blue}{8 \cdot i + \left(4 \cdot \beta + 4 \cdot \alpha\right)}}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}}\]

   11. Simplified12.1

       \[\leadsto \frac{i}{\frac{\color{blue}{(\left(\alpha + \beta\right) \cdot 4 + \left(8 \cdot i\right))_*}}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}}\]
   12. Using strategy rm

   13. Applied add-exp-log12.1

       \[\leadsto \color{blue}{e^{\log \left(\frac{i}{\frac{(\left(\alpha + \beta\right) \cdot 4 + \left(8 \cdot i\right))_*}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}}\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1.9121510033596495 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{i}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1.0}}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(i \cdot 2\right) \cdot \left(i \cdot 2\right)}}}}{\frac{\sqrt{(\left(\beta + (2 \cdot i + \alpha)_*\right) \cdot \left(\beta + (2 \cdot i + \alpha)_*\right) + \left(-1.0\right))_*}}{(\left(\left(i + \beta\right) + \alpha\right) \cdot i + \left(\alpha \cdot \beta\right))_* \cdot \frac{(-2 \cdot i + \left(\alpha + \beta\right))_*}{\beta + (2 \cdot i + \alpha)_*}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{i}{\frac{(\left(\alpha + \beta\right) \cdot 4 + \left(i \cdot 8\right))_*}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))

Details

Time bar (total: 7.7m)Debug log