Error

Try it out

Derivation

1. Split input into 3 regimes

2. if alpha < 1.0531705228716414e+73

   1. Initial program 12.2

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   2. Using strategy rm

   3. Applied associate-/l*2.3

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   4. Using strategy rm

   5. Applied add-log-exp2.3

      \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}\right)}}{2.0}\]

   if 1.0531705228716414e+73 < alpha < 2.133812814160932e+123 or 7.380623390398522e+180 < alpha

   1. Initial program 57.2

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   2. Using strategy rm

   3. Applied associate-/l*44.0

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   4. Using strategy rm

   5. Applied div-inv44.3

      \[\leadsto \frac{\frac{\frac{\alpha + \beta}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \frac{1}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

   6. Applied associate-/r*44.1

      \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\frac{1}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

   7. Taylor expanded around -inf 41.1

      \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

   8. Simplified41.1

      \[\leadsto \frac{\color{blue}{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}}{2.0}\]

   if 2.133812814160932e+123 < alpha < 7.380623390398522e+180

   1. Initial program 53.3

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity53.3

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

   4. Applied add-sqr-sqrt53.3

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   5. Applied times-frac38.9

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   6. Applied times-frac38.8

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}{1} \cdot \frac{\frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

   7. Simplified38.8

      \[\leadsto \frac{\color{blue}{\frac{\beta + \alpha}{\sqrt{2 \cdot i + \left(\beta + \alpha\right)}}} \cdot \frac{\frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
3. Recombined 3 regimes into one program.

4. Final simplification12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.0531705228716414 \cdot 10^{+73}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\frac{\beta + \alpha}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\beta - \alpha}}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} + 1.0}\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 2.133812814160932 \cdot 10^{+123} \lor \neg \left(\alpha \le 7.380623390398522 \cdot 10^{+180}\right):\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0 + \frac{\beta + \alpha}{\sqrt{\left(\beta + \alpha\right) + 2 \cdot i}} \cdot \frac{\frac{\beta - \alpha}{\sqrt{\left(\beta + \alpha\right) + 2 \cdot i}}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))

Details

Time bar (total: 4.8m)Debug log