\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 1.0 m
Input Error: 54.9
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\frac{\frac{i}{i \cdot 2 + \left(\alpha + \beta\right)}}{8}\)
  1. Started with
    \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
    54.9
  2. Applied simplify to get
    \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\]
    44.9
  3. Using strategy rm
    44.9
  4. Applied *-un-lft-identity to get
    \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\color{red}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\color{blue}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
    44.9
  5. Applied square-mult to get
    \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{\color{red}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{\color{blue}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
    44.9
  6. Applied times-frac to get
    \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{red}{\frac{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{blue}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1} \cdot \frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
    45.0
  7. Applied times-frac to get
    \[\frac{\color{red}{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1} \cdot \frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\color{blue}{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}} \cdot \frac{\beta + \left(i + \alpha\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
    45.0
  8. Applied associate-/l* to get
    \[\color{red}{\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}} \cdot \frac{\beta + \left(i + \alpha\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}} \leadsto \color{blue}{\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}}}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}{\frac{\beta + \left(i + \alpha\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}}\]
    45.0
  9. Applied taylor to get
    \[\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}}}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}{\frac{\beta + \left(i + \alpha\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}} \leadsto \frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}}}{8}\]
    0.1
  10. Taylor expanded around 0 to get
    \[\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}}}{\color{red}{8}} \leadsto \frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}}}{\color{blue}{8}}\]
    0.1
  11. Applied simplify to get
    \[\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}}}{8} \leadsto \frac{\frac{i}{i \cdot 2 + \left(\alpha + \beta\right)}}{8}\]
    0.1
  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)))