Error

Derivation

1. Split input into 2 regimes

2. if i < 197.47558772186778

   1. Initial program 44.5

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

   2. Simplified0.0

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(4 \cdot \left(i \cdot i\right) - 1.0\right) \cdot 4}}\]
   3. Using strategy rm

   4. Applied times-frac0.0

      \[\leadsto \color{blue}{\frac{i}{4 \cdot \left(i \cdot i\right) - 1.0} \cdot \frac{i}{4}}\]

   if 197.47558772186778 < i

   1. Initial program 46.6

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

   2. Simplified31.4

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(4 \cdot \left(i \cdot i\right) - 1.0\right) \cdot 4}}\]
   3. Using strategy rm

   4. Applied times-frac31.5

      \[\leadsto \color{blue}{\frac{i}{4 \cdot \left(i \cdot i\right) - 1.0} \cdot \frac{i}{4}}\]

   5. Taylor expanded around -inf 0.0

      \[\leadsto \color{blue}{0.015625 \cdot \frac{1}{{i}^{2}} + \left(\frac{1}{16} + 0.00390625 \cdot \frac{1}{{i}^{4}}\right)}\]

   6. Simplified0

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{\frac{0.00390625}{i \cdot i} + 0.015625}{i \cdot i}}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 197.47558772186778:\\ \;\;\;\;\frac{i}{4} \cdot \frac{i}{4 \cdot \left(i \cdot i\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{16} + \frac{\frac{0.00390625}{i \cdot i} + 0.015625}{i \cdot i}\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0)))