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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `alpha`

Derivation

Initial program 23.8
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
Applied simplify12.4
\[\leadsto \color{blue}{\frac{(\left(\frac{\beta + \alpha}{\left(2.0 + \alpha\right) + (i \cdot 2 + \beta)_*}\right) \cdot \left(\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}\right) + 1.0)_*}{2.0}}\]
Using strategy rm
Applied add-log-exp12.4
\[\leadsto \frac{\color{blue}{\log \left(e^{(\left(\frac{\beta + \alpha}{\left(2.0 + \alpha\right) + (i \cdot 2 + \beta)_*}\right) \cdot \left(\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}\right) + 1.0)_*}\right)}}{2.0}\]

Runtime

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +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))