Error

Derivation

1. Split input into 2 regimes

2. if alpha < 4.275961827665933e+153

   1. Initial program 50.4

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

   2. Simplified50.4

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

   4. Applied times-frac34.6

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

   5. Applied associate-/l*34.6

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

   7. Applied div-inv34.7

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

   8. Applied div-inv34.6

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

   9. Applied times-frac34.6

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

   10. Simplified34.6

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

   if 4.275961827665933e+153 < alpha

   1. Initial program 62.5

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

   2. Simplified62.5

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

   3. Taylor expanded around -inf 48.9

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

4. Final simplification37.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.275961827665933 \cdot 10^{+153}:\\ \;\;\;\;\frac{(\left(i + \left(\beta + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\beta + \alpha\right))_* \cdot (2 \cdot i + \left(\beta + \alpha\right))_* - 1.0} \cdot \frac{\left(i + \left(\beta + \alpha\right)\right) \cdot \frac{i}{(i \cdot 2 + \left(\beta + \alpha\right))_*}}{(i \cdot 2 + \left(\beta + \alpha\right))_*}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 +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)))