\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4, as called
Bits:
128 bits
Bits error versus i
Time: 10.1 s
Input Error: 20.8
Output Error: 0.5
Log: ⚲
Profile: 🕒
\(\frac{\frac{i}{2}}{\frac{{2}^3}{\frac{1}{i}} - \frac{1.0}{i} \cdot 2}\)
  1. Started with
    \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
    20.8
  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 square-mult to get
    \[\frac{\color{red}{{\left(\frac{i}{2}\right)}^2}}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0} \leadsto \frac{\color{blue}{\frac{i}{2} \cdot \frac{i}{2}}}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}\]
    7.7
  5. Applied associate-/l* to get
    \[\color{red}{\frac{\frac{i}{2} \cdot \frac{i}{2}}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i}{2}}{\frac{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}{\frac{i}{2}}}}\]
    7.8
  6. Applied simplify to get
    \[\frac{\frac{i}{2}}{\color{red}{\frac{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}{\frac{i}{2}}}} \leadsto \frac{\frac{i}{2}}{\color{blue}{\frac{{2}^3}{\frac{1}{i}} - \frac{1.0}{i} \cdot 2}}\]
    0.5
  7. Removed slow pow expressions

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)))