\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/2
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 34.5 s
Input Error: 22.6
Output Error: 12.2
Log: ⚲
Profile: 🕒
\(\frac{(\left(\sqrt[3]{{\left(\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}\right)}^3}\right) * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}{2.0}\)
  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. Applied simplify to get
    \[\color{red}{\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}} \leadsto \color{blue}{\frac{(\left(\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}{2.0}}\]
    12.1
  3. Using strategy rm
    12.1
  4. Applied add-cbrt-cube to get
    \[\frac{(\left(\frac{\beta - \alpha}{\color{red}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}}\right) * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}{2.0} \leadsto \frac{(\left(\frac{\beta - \alpha}{\color{blue}{\sqrt[3]{{\left((i * 2 + \beta)_* + \left(2.0 + \alpha\right)\right)}^3}}}\right) * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}{2.0}\]
    20.6
  5. Applied add-cbrt-cube to get
    \[\frac{(\left(\frac{\color{red}{\beta - \alpha}}{\sqrt[3]{{\left((i * 2 + \beta)_* + \left(2.0 + \alpha\right)\right)}^3}}\right) * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}{2.0} \leadsto \frac{(\left(\frac{\color{blue}{\sqrt[3]{{\left(\beta - \alpha\right)}^3}}}{\sqrt[3]{{\left((i * 2 + \beta)_* + \left(2.0 + \alpha\right)\right)}^3}}\right) * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}{2.0}\]
    26.4
  6. Applied cbrt-undiv to get
    \[\frac{(\color{red}{\left(\frac{\sqrt[3]{{\left(\beta - \alpha\right)}^3}}{\sqrt[3]{{\left((i * 2 + \beta)_* + \left(2.0 + \alpha\right)\right)}^3}}\right)} * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}{2.0} \leadsto \frac{(\color{blue}{\left(\sqrt[3]{\frac{{\left(\beta - \alpha\right)}^3}{{\left((i * 2 + \beta)_* + \left(2.0 + \alpha\right)\right)}^3}}\right)} * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}{2.0}\]
    26.4
  7. Applied simplify to get
    \[\frac{(\left(\sqrt[3]{\color{red}{\frac{{\left(\beta - \alpha\right)}^3}{{\left((i * 2 + \beta)_* + \left(2.0 + \alpha\right)\right)}^3}}}\right) * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}{2.0} \leadsto \frac{(\left(\sqrt[3]{\color{blue}{{\left(\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}\right)}^3}}\right) * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}{2.0}\]
    12.2
  8. Removed slow pow expressions

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))