\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/2
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 42.5 s
Input Error: 22.7
Output Error: 12.2
Log: ⚲
Profile: 🕒
\(\frac{(\left(\frac{\frac{\beta - \alpha}{(2 * i + \left(\beta + \alpha\right))_*}}{\left(2.0 + \alpha\right) + (i * 2 + \beta)_*}\right) * \left(\beta + \alpha\right) + 1.0)_*}{2.0}\)
  1. Started with
    \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    22.7
  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 fma-udef to get
    \[\frac{\color{red}{(\left(\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) * \left(\frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*}\right) + 1.0)_*}}{2.0} \leadsto \frac{\color{blue}{\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)} \cdot \frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*} + 1.0}}{2.0}\]
    12.1
  5. Applied taylor to get
    \[\frac{\frac{\beta - \alpha}{(i * 2 + \beta)_* + \left(2.0 + \alpha\right)} \cdot \frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*} + 1.0}{2.0} \leadsto \frac{\frac{\beta - \alpha}{(\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\beta}\right))_* + \left(2.0 + \alpha\right)} \cdot \frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*} + 1.0}{2.0}\]
    26.9
  6. Taylor expanded around inf to get
    \[\frac{\frac{\beta - \alpha}{\color{red}{(\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\beta}\right))_* + \left(2.0 + \alpha\right)}} \cdot \frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*} + 1.0}{2.0} \leadsto \frac{\frac{\beta - \alpha}{\color{blue}{(\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\beta}\right))_* + \left(2.0 + \alpha\right)}} \cdot \frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*} + 1.0}{2.0}\]
    26.9
  7. Applied simplify to get
    \[\color{red}{\frac{\frac{\beta - \alpha}{(\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\beta}\right))_* + \left(2.0 + \alpha\right)} \cdot \frac{\alpha + \beta}{(2 * i + \left(\alpha + \beta\right))_*} + 1.0}{2.0}} \leadsto \color{blue}{\frac{(\left(\frac{\frac{\beta - \alpha}{(2 * i + \left(\beta + \alpha\right))_*}}{\left(\alpha + 2.0\right) + (\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\beta}\right))_*}\right) * \left(\beta + \alpha\right) + 1.0)_*}{2.0}}\]
    27.0
  8. Applied taylor to get
    \[\frac{(\left(\frac{\frac{\beta - \alpha}{(2 * i + \left(\beta + \alpha\right))_*}}{\left(\alpha + 2.0\right) + (\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\beta}\right))_*}\right) * \left(\beta + \alpha\right) + 1.0)_*}{2.0} \leadsto \frac{(\left(\frac{\frac{\beta - \alpha}{(2 * i + \left(\beta + \alpha\right))_*}}{\left(\alpha + 2.0\right) + (i * 2 + \beta)_*}\right) * \left(\beta + \alpha\right) + 1.0)_*}{2.0}\]
    12.2
  9. Taylor expanded around inf to get
    \[\frac{(\left(\frac{\frac{\beta - \alpha}{(2 * i + \left(\beta + \alpha\right))_*}}{\left(\alpha + 2.0\right) + \color{red}{(i * 2 + \beta)_*}}\right) * \left(\beta + \alpha\right) + 1.0)_*}{2.0} \leadsto \frac{(\left(\frac{\frac{\beta - \alpha}{(2 * i + \left(\beta + \alpha\right))_*}}{\left(\alpha + 2.0\right) + \color{blue}{(i * 2 + \beta)_*}}\right) * \left(\beta + \alpha\right) + 1.0)_*}{2.0}\]
    12.2
  10. Applied simplify to get
    \[\frac{(\left(\frac{\frac{\beta - \alpha}{(2 * i + \left(\beta + \alpha\right))_*}}{\left(\alpha + 2.0\right) + (i * 2 + \beta)_*}\right) * \left(\beta + \alpha\right) + 1.0)_*}{2.0} \leadsto \frac{(\left(\frac{\frac{\beta - \alpha}{(2 * i + \left(\beta + \alpha\right))_*}}{\left(2.0 + \alpha\right) + (i * 2 + \beta)_*}\right) * \left(\beta + \alpha\right) + 1.0)_*}{2.0}\]
    12.2
  11. Applied final simplification
  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))