\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/2
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 41.9 s
Input Error: 23.4
Output Error: 11.9
Log: ⚲
Profile: 🕒
\(\frac{\frac{\frac{\beta}{\left(\alpha + 2.0\right) + (i * 2 + \beta)_*}}{\frac{(2 * i + \left(\alpha + \beta\right))_*}{\alpha + \beta}} - \left(\frac{\frac{\alpha}{\left(\alpha + 2.0\right) + (i * 2 + \beta)_*}}{\frac{(2 * i + \left(\alpha + \beta\right))_*}{\alpha + \beta}} - 1.0\right)}{2.0}\)
  1. Started with
    \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    23.4
  2. Applied simplify to get
    \[\color{red}{\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}} \leadsto \color{blue}{\frac{(\left(\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}{2.0}}\]
    12.2
  3. Using strategy rm
    12.2
  4. Applied clear-num to get
    \[\frac{(\color{red}{\left(\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}\right)} * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}{2.0} \leadsto \frac{(\color{blue}{\left(\frac{1}{\frac{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}{\beta - \alpha}}\right)} * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}{2.0}\]
    12.2
  5. Using strategy rm
    12.2
  6. Applied fma-udef to get
    \[\frac{\color{red}{(\left(\frac{1}{\frac{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}{\beta - \alpha}}\right) * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}}{2.0} \leadsto \frac{\color{blue}{\frac{1}{\frac{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}{\beta - \alpha}} \cdot \frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*} + 1.0}}{2.0}\]
    12.2
  7. Applied simplify to get
    \[\frac{\color{red}{\frac{1}{\frac{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}{\beta - \alpha}} \cdot \frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}} + 1.0}{2.0} \leadsto \frac{\color{blue}{\frac{\frac{\beta - \alpha}{\left(\alpha + 2.0\right) + (i * 2 + \beta)_*}}{\frac{(2 * i + \left(\alpha + \beta\right))_*}{\alpha + \beta}}} + 1.0}{2.0}\]
    12.2
  8. Using strategy rm
    12.2
  9. Applied div-sub to get
    \[\frac{\frac{\color{red}{\frac{\beta - \alpha}{\left(\alpha + 2.0\right) + (i * 2 + \beta)_*}}}{\frac{(2 * i + \left(\alpha + \beta\right))_*}{\alpha + \beta}} + 1.0}{2.0} \leadsto \frac{\frac{\color{blue}{\frac{\beta}{\left(\alpha + 2.0\right) + (i * 2 + \beta)_*} - \frac{\alpha}{\left(\alpha + 2.0\right) + (i * 2 + \beta)_*}}}{\frac{(2 * i + \left(\alpha + \beta\right))_*}{\alpha + \beta}} + 1.0}{2.0}\]
    12.2
  10. Applied div-sub to get
    \[\frac{\color{red}{\frac{\frac{\beta}{\left(\alpha + 2.0\right) + (i * 2 + \beta)_*} - \frac{\alpha}{\left(\alpha + 2.0\right) + (i * 2 + \beta)_*}}{\frac{(2 * i + \left(\alpha + \beta\right))_*}{\alpha + \beta}}} + 1.0}{2.0} \leadsto \frac{\color{blue}{\left(\frac{\frac{\beta}{\left(\alpha + 2.0\right) + (i * 2 + \beta)_*}}{\frac{(2 * i + \left(\alpha + \beta\right))_*}{\alpha + \beta}} - \frac{\frac{\alpha}{\left(\alpha + 2.0\right) + (i * 2 + \beta)_*}}{\frac{(2 * i + \left(\alpha + \beta\right))_*}{\alpha + \beta}}\right)} + 1.0}{2.0}\]
    12.2
  11. Applied associate-+l- to get
    \[\frac{\color{red}{\left(\frac{\frac{\beta}{\left(\alpha + 2.0\right) + (i * 2 + \beta)_*}}{\frac{(2 * i + \left(\alpha + \beta\right))_*}{\alpha + \beta}} - \frac{\frac{\alpha}{\left(\alpha + 2.0\right) + (i * 2 + \beta)_*}}{\frac{(2 * i + \left(\alpha + \beta\right))_*}{\alpha + \beta}}\right) + 1.0}}{2.0} \leadsto \frac{\color{blue}{\frac{\frac{\beta}{\left(\alpha + 2.0\right) + (i * 2 + \beta)_*}}{\frac{(2 * i + \left(\alpha + \beta\right))_*}{\alpha + \beta}} - \left(\frac{\frac{\alpha}{\left(\alpha + 2.0\right) + (i * 2 + \beta)_*}}{\frac{(2 * i + \left(\alpha + \beta\right))_*}{\alpha + \beta}} - 1.0\right)}}{2.0}\]
    11.9
  12. 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))