Error

Derivation

1. Split input into 2 regimes

2. if i < 218.10975117139895

   1. Initial program 43.9

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

   2. Simplified0.0

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

   4. Applied *-un-lft-identity0.0

      \[\leadsto \frac{i \cdot \frac{i}{4}}{\color{blue}{1 \cdot (\left(i \cdot 4\right) \cdot i + \left(-1.0\right))_*}}\]

   5. Applied times-frac0.0

      \[\leadsto \color{blue}{\frac{i}{1} \cdot \frac{\frac{i}{4}}{(\left(i \cdot 4\right) \cdot i + \left(-1.0\right))_*}}\]

   6. Simplified0.0

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

   7. Simplified0.0

      \[\leadsto i \cdot \color{blue}{\frac{i}{(16 \cdot \left(i \cdot i\right) + \left(-4 \cdot 1.0\right))_*}}\]

   if 218.10975117139895 < i

   1. Initial program 47.4

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

   2. Simplified31.5

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

   3. Taylor expanded around inf 0.0

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

   4. Simplified0.0

      \[\leadsto \color{blue}{(\left(\frac{0.015625}{i}\right) \cdot \left(\frac{1}{i}\right) + \frac{1}{16})_* + \frac{0.00390625}{{i}^{4}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 218.10975117139895:\\ \;\;\;\;\frac{i}{(16 \cdot \left(i \cdot i\right) + \left(-4 \cdot 1.0\right))_*} \cdot i\\ \mathbf{else}:\\ \;\;\;\;\frac{0.00390625}{{i}^{4}} + (\left(\frac{0.015625}{i}\right) \cdot \left(\frac{1}{i}\right) + \frac{1}{16})_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(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)))

Details

Time bar (total: 5.0s)Debug log