\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 50.2 s
Input Error: 52.7
Output Error: 38.6
Log: ⚲
Profile: 🕒
\(\frac{\frac{i}{(i * 2 + \alpha)_* + \beta} \cdot \frac{\left(\alpha + i\right) + \beta}{(i * 2 + \alpha)_* + \beta}}{\frac{(\left((i * 2 + \alpha)_*\right) * \left((\beta * 2 + \left((i * 2 + \alpha)_*\right))_*\right) + \left({\beta}^2 - 1.0\right))_*}{(i * \left(\left(\alpha + i\right) + \beta\right) + \left(\alpha \cdot \beta\right))_*}}\)
  1. Started with
    \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
    52.7
  2. Applied simplify to get
    \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
    38.6
  3. Applied taylor to get
    \[\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0} \leadsto \frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\left(2 \cdot \left(\beta \cdot (i * 2 + \alpha)_*\right) + \left({\left((i * 2 + \alpha)_*\right)}^2 + {\beta}^2\right)\right) - 1.0}\]
    38.6
  4. Taylor expanded around 0 to get
    \[\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\color{red}{\left(2 \cdot \left(\beta \cdot (i * 2 + \alpha)_*\right) + \left({\left((i * 2 + \alpha)_*\right)}^2 + {\beta}^2\right)\right)} - 1.0} \leadsto \frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\color{blue}{\left(2 \cdot \left(\beta \cdot (i * 2 + \alpha)_*\right) + \left({\left((i * 2 + \alpha)_*\right)}^2 + {\beta}^2\right)\right)} - 1.0}\]
    38.6
  5. Applied simplify to get
    \[\color{red}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\left(2 \cdot \left(\beta \cdot (i * 2 + \alpha)_*\right) + \left({\left((i * 2 + \alpha)_*\right)}^2 + {\beta}^2\right)\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i}{(i * 2 + \alpha)_* + \beta} \cdot \frac{\left(\alpha + i\right) + \beta}{(i * 2 + \alpha)_* + \beta}}{\frac{(\left((i * 2 + \alpha)_*\right) * \left((\beta * 2 + \left((i * 2 + \alpha)_*\right))_*\right) + \left({\beta}^2 - 1.0\right))_*}{(i * \left(\left(\alpha + i\right) + \beta\right) + \left(\alpha \cdot \beta\right))_*}}}\]
    38.6

Original test:


(lambda ((alpha default) (beta default) (i default))
  #:name "Octave 3.8, jcobi/4"
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))