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

↓

\[\frac{(\left((e^{\log_* (1 + \frac{\beta + \alpha}{\left(\beta + \alpha\right) + (2 \cdot i + 2.0)_*})} - 1)^*\right) \cdot \left(\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}\right) + 1.0)_*}{2.0}\]

Error

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

Derivation

Initial program 23.6
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
Applied simplify12.2
\[\leadsto \color{blue}{\frac{(\left(\frac{\beta + \alpha}{\left(\beta + \alpha\right) + (2 \cdot i + 2.0)_*}\right) \cdot \left(\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}\right) + 1.0)_*}{2.0}}\]
Using strategy rm
Applied expm1-log1p-u12.2
\[\leadsto \frac{(\color{blue}{\left((e^{\log_* (1 + \frac{\beta + \alpha}{\left(\beta + \alpha\right) + (2 \cdot i + 2.0)_*})} - 1)^*\right)} \cdot \left(\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}\right) + 1.0)_*}{2.0}\]

Runtime

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' +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))