Error

Derivation

1. Split input into 2 regimes

2. if i < 8.925236338593809e+74

   1. Initial program 23.6

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

   3. Applied associate-/l*11.5

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

   5. Applied associate-/r/11.5

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

   6. Applied associate-/l*11.5

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

   7. Simplified11.5

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

   if 8.925236338593809e+74 < i

   1. Initial program 61.8

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

   3. Applied associate-/l*46.9

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

   5. Applied associate-/r/46.9

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

   6. Applied associate-/l*46.9

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

   7. Simplified46.9

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

   8. Taylor expanded around 0 14.2

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

   10. Applied add-log-exp13.0

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

4. Final simplification12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 8.925236338593809 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{\beta + \left(i + \alpha\right)}{(2 \cdot i + \alpha)_* + \beta} \cdot \frac{i}{(2 \cdot i + \alpha)_* + \beta}}{\frac{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right) - 1.0}{i \cdot \left(\left(\beta + \alpha\right) + i\right) + \beta \cdot \alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{i}{(2 \cdot i + \alpha)_* + \beta}}\right) \cdot \frac{\beta + \left(i + \alpha\right)}{(2 \cdot i + \alpha)_* + \beta}}{4}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 +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: 2.4m)Debug log