\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/2
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 38.6 s
Input Error: 11.1
Output Error: 4.6
Log: ⚲
Profile: 🕒
\(\frac{\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)} \cdot {\left(\frac{\sqrt[3]{\alpha + \beta}}{\sqrt[3]{(2 * i + \left(\alpha + \beta\right))_*}}\right)}^3 + 1.0}{2.0}\)
  1. Started with
    \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    11.1
  2. Applied simplify to get
    \[\color{red}{\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}} \leadsto \color{blue}{\frac{(\left(\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}{2.0}}\]
    4.9
  3. Using strategy rm
    4.9
  4. Applied fma-udef to get
    \[\frac{\color{red}{(\left(\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}}{2.0} \leadsto \frac{\color{blue}{\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)} \cdot \frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*} + 1.0}}{2.0}\]
    4.5
  5. Using strategy rm
    4.5
  6. Applied add-cube-cbrt to get
    \[\frac{\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)} \cdot \frac{\alpha + \beta}{\color{red}{(2 * i + \left(\alpha + \beta\right))_*}} + 1.0}{2.0} \leadsto \frac{\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)} \cdot \frac{\alpha + \beta}{\color{blue}{{\left(\sqrt[3]{(2 * i + \left(\alpha + \beta\right))_*}\right)}^3}} + 1.0}{2.0}\]
    5.2
  7. Applied add-cube-cbrt to get
    \[\frac{\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)} \cdot \frac{\color{red}{\alpha + \beta}}{{\left(\sqrt[3]{(2 * i + \left(\alpha + \beta\right))_*}\right)}^3} + 1.0}{2.0} \leadsto \frac{\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)} \cdot \frac{\color{blue}{{\left(\sqrt[3]{\alpha + \beta}\right)}^3}}{{\left(\sqrt[3]{(2 * i + \left(\alpha + \beta\right))_*}\right)}^3} + 1.0}{2.0}\]
    4.6
  8. Applied cube-undiv to get
    \[\frac{\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)} \cdot \color{red}{\frac{{\left(\sqrt[3]{\alpha + \beta}\right)}^3}{{\left(\sqrt[3]{(2 * i + \left(\alpha + \beta\right))_*}\right)}^3}} + 1.0}{2.0} \leadsto \frac{\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)} \cdot \color{blue}{{\left(\frac{\sqrt[3]{\alpha + \beta}}{\sqrt[3]{(2 * i + \left(\alpha + \beta\right))_*}}\right)}^3} + 1.0}{2.0}\]
    4.6
  9. 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))