Average Error: 46.9 → 0.0
Time: 5.7s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 218.0369401643284845704329200088977813721:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\ \end{array}\]
\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}
↓
\begin{array}{l}
\mathbf{if}\;i \le 218.0369401643284845704329200088977813721:\\
\;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\

\mathbf{else}:\\
\;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\

\end{array}
double f(double i) {
        double r47817 = i;
        double r47818 = r47817 * r47817;
        double r47819 = r47818 * r47818;
        double r47820 = 2.0;
        double r47821 = r47820 * r47817;
        double r47822 = r47821 * r47821;
        double r47823 = r47819 / r47822;
        double r47824 = 1.0;
        double r47825 = r47822 - r47824;
        double r47826 = r47823 / r47825;
        return r47826;
}
↓
double f(double i) {
        double r47827 = i;
        double r47828 = 218.03694016432848;
        bool r47829 = r47827 <= r47828;
        double r47830 = r47827 * r47827;
        double r47831 = 2.0;
        double r47832 = r47831 * r47827;
        double r47833 = r47832 * r47832;
        double r47834 = 1.0;
        double r47835 = r47833 - r47834;
        double r47836 = r47831 * r47831;
        double r47837 = r47835 * r47836;
        double r47838 = r47830 / r47837;
        double r47839 = 0.00390625;
        double r47840 = 1.0;
        double r47841 = 4.0;
        double r47842 = pow(r47827, r47841);
        double r47843 = r47840 / r47842;
        double r47844 = r47839 * r47843;
        double r47845 = 0.015625;
        double r47846 = 2.0;
        double r47847 = pow(r47827, r47846);
        double r47848 = r47840 / r47847;
        double r47849 = r47845 * r47848;
        double r47850 = 0.0625;
        double r47851 = r47849 + r47850;
        double r47852 = r47844 + r47851;
        double r47853 = r47829 ? r47838 : r47852;
        return r47853;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if i < 218.03694016432848

    1. Initial program 44.7

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\]

    if 218.03694016432848 < i

    1. Initial program 49.1

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]

    2. Simplified33.0

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\]

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 218.0369401643284845704329200088977813721:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :precision binary64
  :pre (and (> i 0.0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1)))