Error

Derivation

1. Split input into 2 regimes

2. if beta < 7.106991561021006e+134

   1. Initial program 50.7

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

   3. Applied associate-/l*35.3

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

   5. Applied *-un-lft-identity35.3

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

   6. Applied times-frac35.3

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

   7. Simplified35.3

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

   8. Simplified35.3

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

   if 7.106991561021006e+134 < beta

   1. Initial program 62.4

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

   3. Applied associate-/l*54.8

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

   5. Applied *-un-lft-identity54.8

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

   6. Applied *-un-lft-identity54.8

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

   7. Applied times-frac54.8

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

   8. Applied times-frac54.8

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

   9. Applied times-frac54.8

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

   10. Simplified54.8

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

   11. Simplified54.8

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

   12. Taylor expanded around inf 50.0

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

4. Final simplification38.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 7.106991561021006 \cdot 10^{+134}:\\ \;\;\;\;\frac{i \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{(2 \cdot i + \left(\beta + \alpha\right))_*} \cdot (i \cdot \left(i + \left(\beta + \alpha\right)\right) + \left(\alpha \cdot \beta\right))_*}{(2 \cdot i + \left(\beta + \alpha\right))_*}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{(2 \cdot i + \left(\beta + \alpha\right))_*} \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{(\left((2 \cdot i + \left(\beta + \alpha\right))_*\right) \cdot \left((2 \cdot i + \left(\beta + \alpha\right))_*\right) + \left(-1.0\right))_*}}{\frac{2}{i}}\\ \end{array}\]

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed 2018295 +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)))