\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/2
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 35.0 s
Input Error: 22.6
Output Error: 12.1
Log: ⚲
Profile: 🕒
\(\frac{\frac{\frac{\alpha + \beta}{1} \cdot \sqrt[3]{{\left(\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}^3}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\)
  1. Started with
    \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    22.6
  2. Using strategy rm
    22.6
  3. Applied *-un-lft-identity to get
    \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{red}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    22.6
  4. Applied times-frac to get
    \[\frac{\frac{\color{red}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    12.1
  5. Using strategy rm
    12.1
  6. Applied add-cbrt-cube to get
    \[\frac{\frac{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\color{red}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\color{blue}{\sqrt[3]{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^3}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    24.7
  7. Applied add-cbrt-cube to get
    \[\frac{\frac{\frac{\alpha + \beta}{1} \cdot \frac{\color{red}{\beta - \alpha}}{\sqrt[3]{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^3}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\frac{\alpha + \beta}{1} \cdot \frac{\color{blue}{\sqrt[3]{{\left(\beta - \alpha\right)}^3}}}{\sqrt[3]{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^3}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    30.5
  8. Applied cbrt-undiv to get
    \[\frac{\frac{\frac{\alpha + \beta}{1} \cdot \color{red}{\frac{\sqrt[3]{{\left(\beta - \alpha\right)}^3}}{\sqrt[3]{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^3}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\frac{\alpha + \beta}{1} \cdot \color{blue}{\sqrt[3]{\frac{{\left(\beta - \alpha\right)}^3}{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^3}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    30.5
  9. Applied simplify to get
    \[\frac{\frac{\frac{\alpha + \beta}{1} \cdot \sqrt[3]{\color{red}{\frac{{\left(\beta - \alpha\right)}^3}{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^3}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\frac{\alpha + \beta}{1} \cdot \sqrt[3]{\color{blue}{{\left(\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}^3}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    12.1

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))