Average Error: 46.5 → 0.0
Time: 1.2s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 217.9947108587691388947860104963183403015:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\ \end{array}\]
\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 217.9947108587691388947860104963183403015:\\
\;\;\;\;\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 r59889 = i;
        double r59890 = r59889 * r59889;
        double r59891 = r59890 * r59890;
        double r59892 = 2.0;
        double r59893 = r59892 * r59889;
        double r59894 = r59893 * r59893;
        double r59895 = r59891 / r59894;
        double r59896 = 1.0;
        double r59897 = r59894 - r59896;
        double r59898 = r59895 / r59897;
        return r59898;
}
↓
double f(double i) {
        double r59899 = i;
        double r59900 = 217.99471085876914;
        bool r59901 = r59899 <= r59900;
        double r59902 = r59899 * r59899;
        double r59903 = 2.0;
        double r59904 = r59903 * r59899;
        double r59905 = r59904 * r59904;
        double r59906 = 1.0;
        double r59907 = r59905 - r59906;
        double r59908 = r59903 * r59903;
        double r59909 = r59907 * r59908;
        double r59910 = r59902 / r59909;
        double r59911 = 0.00390625;
        double r59912 = 1.0;
        double r59913 = 4.0;
        double r59914 = pow(r59899, r59913);
        double r59915 = r59912 / r59914;
        double r59916 = r59911 * r59915;
        double r59917 = 0.015625;
        double r59918 = 2.0;
        double r59919 = pow(r59899, r59918);
        double r59920 = r59912 / r59919;
        double r59921 = r59917 * r59920;
        double r59922 = 0.0625;
        double r59923 = r59921 + r59922;
        double r59924 = r59916 + r59923;
        double r59925 = r59901 ? r59910 : r59924;
        return r59925;
}

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 < 217.99471085876914

    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 217.99471085876914 < i

    1. Initial program 48.4

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

    2. Simplified32.6

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 217.9947108587691388947860104963183403015:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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)))