\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4, as called
Bits:
128 bits
Bits error versus i
Time: 10.6 s
Input Error: 20.8
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{\left(\frac{i}{2}\right)}^2}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0} & \text{when } i \le 14.010054f0 \\ \left(\frac{1}{16} + \frac{0.00390625}{{i}^{4}}\right) + \frac{0.015625}{i \cdot i} & \text{otherwise} \end{cases}\)

    if i < 14.010054f0

    1. Started with
      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
      18.7
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{{\left(\frac{i}{2}\right)}^2}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}}\]
      0.0

    if 14.010054f0 < i

    1. Started with
      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
      23.0
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{{\left(\frac{i}{2}\right)}^2}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}}\]
      15.7
    3. Applied taylor to get
      \[\frac{{\left(\frac{i}{2}\right)}^2}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0} \leadsto 0.015625 \cdot \frac{1}{{i}^2} + \left(\frac{1}{16} + 0.00390625 \cdot \frac{1}{{i}^{4}}\right)\]
      0
    4. Taylor expanded around inf to get
      \[\color{red}{0.015625 \cdot \frac{1}{{i}^2} + \left(\frac{1}{16} + 0.00390625 \cdot \frac{1}{{i}^{4}}\right)} \leadsto \color{blue}{0.015625 \cdot \frac{1}{{i}^2} + \left(\frac{1}{16} + 0.00390625 \cdot \frac{1}{{i}^{4}}\right)}\]
      0
    5. Applied simplify to get
      \[\color{red}{0.015625 \cdot \frac{1}{{i}^2} + \left(\frac{1}{16} + 0.00390625 \cdot \frac{1}{{i}^{4}}\right)} \leadsto \color{blue}{\left(\frac{1}{16} + \frac{0.00390625}{{i}^{4}}\right) + \frac{0.015625}{i \cdot i}}\]
      0
  1. 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)))