\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4, as called
Bits:
128 bits
Bits error versus i
Time: 7.7 s
Input Error: 21.2
Output Error: 7.8
Log: ⚲
Profile: 🕒
\(\log_* (1 + (e^{\frac{{\left(\frac{i}{2}\right)}^2}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}} - 1)^*)\)
  1. Started with
    \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
    21.2
  2. Applied simplify to get
    \[\color{red}{\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{{\left(\frac{i}{2}\right)}^2}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}}\]
    7.8
  3. Using strategy rm
    7.8
  4. Applied log1p-expm1-u to get
    \[\color{red}{\frac{{\left(\frac{i}{2}\right)}^2}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}} \leadsto \color{blue}{\log_* (1 + (e^{\frac{{\left(\frac{i}{2}\right)}^2}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}} - 1)^*)}\]
    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)))