Error

Derivation

1. Split input into 2 regimes

2. if alpha < 1.3763467476335912e+154

   1. Initial program 50.5

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

   3. Applied times-frac35.5

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

   4. Simplified35.5

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

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

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

   7. Applied times-frac35.6

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

   8. Simplified35.6

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

   9. Simplified35.6

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

   11. Applied add-sqr-sqrt35.6

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

   12. Applied *-un-lft-identity35.6

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

   13. Applied times-frac35.4

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

   14. Applied times-frac35.4

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

   15. Applied associate-*r*35.4

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

   if 1.3763467476335912e+154 < 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. Using strategy rm

   3. Applied times-frac56.7

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

   4. Simplified56.7

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

   5. Taylor expanded around inf 48.1

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

4. Final simplification37.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.3763467476335912 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{i + \alpha}{(2 \cdot i + \left(\beta + \alpha\right))_*}}{\sqrt{(\left((2 \cdot i + \left(\beta + \alpha\right))_*\right) \cdot \left((2 \cdot i + \left(\beta + \alpha\right))_*\right) + \left(-1.0\right))_*}} \cdot \left(\frac{i + \beta}{\sqrt{(\left((2 \cdot i + \left(\beta + \alpha\right))_*\right) \cdot \left((2 \cdot i + \left(\beta + \alpha\right))_*\right) + \left(-1.0\right))_*}} \cdot \frac{(i \cdot \left(i + \alpha\right) + \left(i \cdot \beta\right))_*}{\alpha + (i \cdot 2 + \beta)_*}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 4.0m)Debug logProfile

herbie shell --seed 2018227 +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)))