\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 50.9 s
Input Error: 52.7
Output Error: 39.0
Log: ⚲
Profile: 🕒
\({\left(\frac{\sqrt{\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*} \cdot \sqrt{\frac{i}{\beta + (i * 2 + \alpha)_*}}}{\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\right)}^2\)
  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
    \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
    38.8
  3. Using strategy rm
    38.8
  4. Applied add-sqr-sqrt to get
    \[\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\color{red}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}} \leadsto \frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\color{blue}{{\left(\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^2}}\]
    38.8
  5. Applied add-sqr-sqrt to get
    \[\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \color{red}{\frac{i}{\beta + (i * 2 + \alpha)_*}}}{{\left(\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^2} \leadsto \frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \color{blue}{{\left(\sqrt{\frac{i}{\beta + (i * 2 + \alpha)_*}}\right)}^2}}{{\left(\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^2}\]
    39.1
  6. Applied add-sqr-sqrt to get
    \[\frac{\color{red}{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right)} \cdot {\left(\sqrt{\frac{i}{\beta + (i * 2 + \alpha)_*}}\right)}^2}{{\left(\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^2} \leadsto \frac{\color{blue}{{\left(\sqrt{\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*}\right)}^2} \cdot {\left(\sqrt{\frac{i}{\beta + (i * 2 + \alpha)_*}}\right)}^2}{{\left(\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^2}\]
    39.1
  7. Applied square-unprod to get
    \[\frac{\color{red}{{\left(\sqrt{\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*}\right)}^2 \cdot {\left(\sqrt{\frac{i}{\beta + (i * 2 + \alpha)_*}}\right)}^2}}{{\left(\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^2} \leadsto \frac{\color{blue}{{\left(\sqrt{\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*} \cdot \sqrt{\frac{i}{\beta + (i * 2 + \alpha)_*}}\right)}^2}}{{\left(\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^2}\]
    39.0
  8. Applied square-undiv to get
    \[\color{red}{\frac{{\left(\sqrt{\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*} \cdot \sqrt{\frac{i}{\beta + (i * 2 + \alpha)_*}}\right)}^2}{{\left(\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^2}} \leadsto \color{blue}{{\left(\frac{\sqrt{\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*} \cdot \sqrt{\frac{i}{\beta + (i * 2 + \alpha)_*}}}{\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\right)}^2}\]
    39.0

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)))