Error

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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

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

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

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

   5. Applied times-frac0.0

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

   6. Simplified0.0

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

   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

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

   4. Applied *-un-lft-identity31.5

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

   5. Applied times-frac31.6

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

   6. Simplified31.6

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

   7. Taylor expanded around -inf 0.0

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

   8. Simplified0.0

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

4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019004 
(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: 6.4s)Debug log