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

⬇

\[\begin{array}{l} \mathbf{if}\;\beta \le 1.9676421630074337 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\beta \cdot \alpha + i \cdot \left(\beta + \left(\alpha + i\right)\right)}{{\left(\left(i + i\right) + \left(\alpha + \beta\right)\right)}^2} \cdot \left(i \cdot \left(\beta + \left(\alpha + i\right)\right)\right)}{{\left(\left(i + i\right) + \left(\alpha + \beta\right)\right)}^2 - 1.0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Derivation

Split input into 2 regimes.
if beta < 1.9676421630074337e+154
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. Applied taylor 50.7
  \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + \left(\beta \cdot i + \left(\alpha \cdot i + {i}^2\right)\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
3. Taylor expanded around 0 50.7
  
  \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + \color{blue}{\left(\beta \cdot i + \left(\alpha \cdot i + {i}^2\right)\right)}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
4. Applied simplify 35.6
  \[\leadsto \color{blue}{\frac{\beta \cdot \alpha + i \cdot \left(\beta + \left(\alpha + i\right)\right)}{{\left(\left(i + i\right) + \left(\alpha + \beta\right)\right)}^2} \cdot \frac{i \cdot \left(\beta + \left(\alpha + i\right)\right)}{{\left(\left(i + i\right) + \left(\alpha + \beta\right)\right)}^2 - 1.0}}\]
5. Using strategy rm
6. Applied associate-*r/ 35.6
  \[\leadsto \color{blue}{\frac{\frac{\beta \cdot \alpha + i \cdot \left(\beta + \left(\alpha + i\right)\right)}{{\left(\left(i + i\right) + \left(\alpha + \beta\right)\right)}^2} \cdot \left(i \cdot \left(\beta + \left(\alpha + i\right)\right)\right)}{{\left(\left(i + i\right) + \left(\alpha + \beta\right)\right)}^2 - 1.0}}\]
if 1.9676421630074337e+154 < beta
1. Initial program 63.0
  \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
2. Applied taylor 0
  \[\leadsto 0\]
3. Taylor expanded around inf 0
  
  \[\leadsto \color{blue}{0}\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie shell --seed '#(1064524629 4159152179 2999149171 575749698 4006532819 692958815)'
(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)))

Error

Derivation

`if beta < 1.9676421630074337e+154`

`if 1.9676421630074337e+154 < beta`

Runtime