\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/2
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 39.7 s
Input Error: 23.3
Output Error: 12.3
Log: ⚲
Profile: 🕒
\(\frac{e^{\log \left(1.0 + \frac{\alpha + \beta}{\left(\alpha + 2.0\right) + \left(\beta + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\alpha + \beta\right)}\right)}}{2.0}\)
  1. Started with
    \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    23.3
  2. Using strategy rm
    23.3
  3. Applied *-un-lft-identity to get
    \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{red}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    23.3
  4. Applied times-frac to get
    \[\frac{\frac{\color{red}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    12.3
  5. Using strategy rm
    12.3
  6. Applied add-cube-cbrt to get
    \[\frac{\frac{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\color{red}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\color{blue}{{\left(\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^3}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    12.5
  7. Applied add-cube-cbrt to get
    \[\frac{\frac{\frac{\alpha + \beta}{1} \cdot \frac{\color{red}{\beta - \alpha}}{{\left(\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^3}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\frac{\alpha + \beta}{1} \cdot \frac{\color{blue}{{\left(\sqrt[3]{\beta - \alpha}\right)}^3}}{{\left(\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^3}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    12.3
  8. Applied cube-undiv to get
    \[\frac{\frac{\frac{\alpha + \beta}{1} \cdot \color{red}{\frac{{\left(\sqrt[3]{\beta - \alpha}\right)}^3}{{\left(\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^3}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\frac{\alpha + \beta}{1} \cdot \color{blue}{{\left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}\right)}^3}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    12.3
  9. Using strategy rm
    12.3
  10. Applied add-exp-log to get
    \[\frac{\color{red}{\frac{\frac{\alpha + \beta}{1} \cdot {\left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}\right)}^3}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}}{2.0} \leadsto \frac{\color{blue}{e^{\log \left(\frac{\frac{\alpha + \beta}{1} \cdot {\left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}\right)}^3}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)}}}{2.0}\]
    12.3
  11. Applied simplify to get
    \[\frac{e^{\color{red}{\log \left(\frac{\frac{\alpha + \beta}{1} \cdot {\left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}\right)}^3}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)}}}{2.0} \leadsto \frac{e^{\color{blue}{\log \left(1.0 + \frac{\alpha + \beta}{\left(\alpha + 2.0\right) + \left(\beta + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\alpha + \beta\right)}\right)}}}{2.0}\]
    12.3

Original test:


(lambda ((alpha default) (beta default) (i default))
  #:name "Octave 3.8, jcobi/2"
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))