\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4, as called
Bits:
128 bits
Bits error versus i
Time: 14.0 s
Input Error: 21.0
Output Error: 7.8
Log: ⚲
Profile: 🕒
\(\frac{i}{2} \cdot \frac{\frac{i}{2}}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}\)
  1. Started with
    \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
    21.0
  2. Applied simplify to get
    \[\color{red}{\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{{\left(\frac{i}{2}\right)}^2}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}}\]
    7.7
  3. Using strategy rm
    7.7
  4. Applied *-un-lft-identity to get
    \[\frac{{\left(\frac{i}{2}\right)}^2}{\color{red}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}} \leadsto \frac{{\left(\frac{i}{2}\right)}^2}{\color{blue}{1 \cdot \left(\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0\right)}}\]
    7.7
  5. Applied square-mult to get
    \[\frac{\color{red}{{\left(\frac{i}{2}\right)}^2}}{1 \cdot \left(\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0\right)} \leadsto \frac{\color{blue}{\frac{i}{2} \cdot \frac{i}{2}}}{1 \cdot \left(\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0\right)}\]
    7.7
  6. Applied times-frac to get
    \[\color{red}{\frac{\frac{i}{2} \cdot \frac{i}{2}}{1 \cdot \left(\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0\right)}} \leadsto \color{blue}{\frac{\frac{i}{2}}{1} \cdot \frac{\frac{i}{2}}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}}\]
    7.8
  7. Applied simplify to get
    \[\color{red}{\frac{\frac{i}{2}}{1}} \cdot \frac{\frac{i}{2}}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0} \leadsto \color{blue}{\frac{i}{2}} \cdot \frac{\frac{i}{2}}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}\]
    7.8

Original test:


(lambda ((i default))
  #:name "Octave 3.8, jcobi/4, as called"
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0)))