\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 42.5 s
Input Error: 26.5
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(\frac{e^{\frac{0.25}{i \cdot i}}}{{4}^2}\)
  1. Started with
    \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
    26.5
  2. Applied simplify to get
    \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\]
    22.0
  3. Using strategy rm
    22.0
  4. Applied add-exp-log to get
    \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{\color{red}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{\color{blue}{e^{\log \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}}}\]
    22.8
  5. Applied add-exp-log to get
    \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{red}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{e^{\log \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{blue}{e^{\log \left(\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}\right)}}}}{e^{\log \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}}\]
    22.8
  6. Applied add-exp-log to get
    \[\frac{\frac{\color{red}{i \cdot \left(\beta + \left(i + \alpha\right)\right)}}{e^{\log \left(\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}\right)}}}{e^{\log \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}} \leadsto \frac{\frac{\color{blue}{e^{\log \left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}{e^{\log \left(\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}\right)}}}{e^{\log \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}}\]
    23.0
  7. Applied div-exp to get
    \[\frac{\color{red}{\frac{e^{\log \left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}{e^{\log \left(\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}\right)}}}}{e^{\log \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}} \leadsto \frac{\color{blue}{e^{\log \left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) - \log \left(\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}\right)}}}{e^{\log \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}}\]
    23.0
  8. Applied div-exp to get
    \[\color{red}{\frac{e^{\log \left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) - \log \left(\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}\right)}}{e^{\log \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}}} \leadsto \color{blue}{e^{\left(\log \left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) - \log \left(\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}\right)\right) - \log \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}}\]
    23.0
  9. Applied taylor to get
    \[e^{\left(\log \left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) - \log \left(\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}\right)\right) - \log \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)} \leadsto e^{0.25 \cdot \frac{1}{{i}^2} - 2 \cdot \log 4}\]
    0.0
  10. Taylor expanded around inf to get
    \[e^{\color{red}{0.25 \cdot \frac{1}{{i}^2} - 2 \cdot \log 4}} \leadsto e^{\color{blue}{0.25 \cdot \frac{1}{{i}^2} - 2 \cdot \log 4}}\]
    0.0
  11. Applied simplify to get
    \[e^{0.25 \cdot \frac{1}{{i}^2} - 2 \cdot \log 4} \leadsto \frac{e^{\frac{0.25}{i \cdot i}}}{{4}^2}\]
    0.0
  12. Applied final simplification

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)))