Average Error: 46.9 → 0.0
Time: 1.8s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 383.2464823959830368949042167514562606812:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\ \end{array}\]
\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 383.2464823959830368949042167514562606812:\\
\;\;\;\;\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 r120009 = i;
        double r120010 = r120009 * r120009;
        double r120011 = r120010 * r120010;
        double r120012 = 2.0;
        double r120013 = r120012 * r120009;
        double r120014 = r120013 * r120013;
        double r120015 = r120011 / r120014;
        double r120016 = 1.0;
        double r120017 = r120014 - r120016;
        double r120018 = r120015 / r120017;
        return r120018;
}
↓
double f(double i) {
        double r120019 = i;
        double r120020 = 383.24648239598304;
        bool r120021 = r120019 <= r120020;
        double r120022 = r120019 * r120019;
        double r120023 = 2.0;
        double r120024 = r120023 * r120019;
        double r120025 = r120024 * r120024;
        double r120026 = 1.0;
        double r120027 = r120025 - r120026;
        double r120028 = r120023 * r120023;
        double r120029 = r120027 * r120028;
        double r120030 = r120022 / r120029;
        double r120031 = 0.00390625;
        double r120032 = 1.0;
        double r120033 = 4.0;
        double r120034 = pow(r120019, r120033);
        double r120035 = r120032 / r120034;
        double r120036 = r120031 * r120035;
        double r120037 = 0.015625;
        double r120038 = 2.0;
        double r120039 = pow(r120019, r120038);
        double r120040 = r120032 / r120039;
        double r120041 = r120037 * r120040;
        double r120042 = 0.0625;
        double r120043 = r120041 + r120042;
        double r120044 = r120036 + r120043;
        double r120045 = r120021 ? r120030 : r120044;
        return r120045;
}

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 < 383.24648239598304

    1. Initial program 45.6

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

    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 383.24648239598304 < i

    1. Initial program 48.2

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

    2. Simplified31.5

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 383.2464823959830368949042167514562606812:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(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)))