Error

Derivation

1. Split input into 2 regimes

2. if i < 4.089496632900118e+151

   1. Initial program 43.3

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

   3. Applied associate-/l*15.8

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

   5. Applied *-un-lft-identity15.8

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

   6. Applied times-frac15.9

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

   7. Applied times-frac15.8

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

   8. Simplified15.8

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

   9. Simplified15.8

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

   11. Applied *-un-lft-identity15.8

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

   12. Applied times-frac15.8

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

   13. Simplified15.8

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

   14. Simplified10.9

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

   16. Applied *-un-lft-identity10.9

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

   17. Applied times-frac11.0

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

   18. Applied associate-*r*11.0

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

   if 4.089496632900118e+151 < 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 associate-/l*61.5

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

   5. Applied *-un-lft-identity61.5

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

   6. Applied times-frac61.5

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

   7. Applied times-frac61.5

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

   8. Simplified61.5

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

   9. Simplified61.5

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

   11. Applied *-un-lft-identity61.5

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

   12. Applied times-frac61.5

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

   13. Simplified61.5

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

   14. Simplified61.3

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

   15. Taylor expanded around 0 47.1

       \[\leadsto \frac{i}{(2 \cdot i + \beta)_* + \alpha} \cdot \frac{\left(\beta + i\right) \cdot \frac{\left(\alpha + i\right) + \beta}{(2 \cdot i + \beta)_* + \alpha}}{\color{blue}{3 \cdot i + \left(2 \cdot \beta + \alpha\right)}}\]

   16. Simplified47.1

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

4. Final simplification29.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 4.089496632900118 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{\beta + \left(i + \alpha\right)}{(2 \cdot i + \beta)_* + \alpha}}{\frac{(\left((2 \cdot i + \beta)_* + \alpha\right) \cdot \left((2 \cdot i + \beta)_* + \alpha\right) + \left(-1.0\right))_*}{i + \alpha}} \cdot \left(\left(i + \beta\right) \cdot \frac{i}{(2 \cdot i + \beta)_* + \alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \beta\right) \cdot \frac{\beta + \left(i + \alpha\right)}{(2 \cdot i + \beta)_* + \alpha}}{(\beta \cdot 2 + \left((i \cdot 3 + \alpha)_*\right))_*} \cdot \frac{i}{(2 \cdot i + \beta)_* + \alpha}\\ \end{array}\]

Runtime

Time bar (total: 4.9m)Debug logProfile

herbie shell --seed 2018230 +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)))