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

↓

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

Derivation

Initial program 23.9
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
Initial simplification12.4
\[\leadsto \frac{(\left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \left(\beta + 2.0\right)}\right) \cdot \left(\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]
Using strategy rm
Applied add-log-exp12.4
\[\leadsto \frac{(\left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \left(\beta + 2.0\right)}\right) \cdot \color{blue}{\left(\log \left(e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \beta}}\right)\right)} + 1.0)_*}{2.0}\]
Using strategy rm
Applied add-exp-log12.4
\[\leadsto \frac{\color{blue}{e^{\log \left((\left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \left(\beta + 2.0\right)}\right) \cdot \left(\log \left(e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \beta}}\right)\right) + 1.0)_*\right)}}}{2.0}\]
Final simplification12.4
\[\leadsto \frac{e^{\log \left((\left(\frac{\alpha + \beta}{(2 \cdot i + \alpha)_* + \left(\beta + 2.0\right)}\right) \cdot \left(\log \left(e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \beta}}\right)\right) + 1.0)_*\right)}}{2.0}\]

Runtime

herbie shell --seed 2018219 +o rules:numerics
(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))

Error

Derivation

Runtime