Average Error: 46.7 → 0.0
Time: 3.3s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 248.47326712178204:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i + \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + \frac{0.015625 \cdot 1}{{i}^{2}}\right) + 0.00390625 \cdot \frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}}\\ \end{array}\]
\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 248.47326712178204:\\
\;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i + \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\

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

\end{array}
double f(double i) {
        double r108836 = i;
        double r108837 = r108836 * r108836;
        double r108838 = r108837 * r108837;
        double r108839 = 2.0;
        double r108840 = r108839 * r108836;
        double r108841 = r108840 * r108840;
        double r108842 = r108838 / r108841;
        double r108843 = 1.0;
        double r108844 = r108841 - r108843;
        double r108845 = r108842 / r108844;
        return r108845;
}
↓
double f(double i) {
        double r108846 = i;
        double r108847 = 248.47326712178204;
        bool r108848 = r108846 <= r108847;
        double r108849 = r108846 * r108846;
        double r108850 = 2.0;
        double r108851 = r108850 * r108846;
        double r108852 = 1.0;
        double r108853 = sqrt(r108852);
        double r108854 = r108851 + r108853;
        double r108855 = r108851 - r108853;
        double r108856 = r108854 * r108855;
        double r108857 = r108850 * r108850;
        double r108858 = r108856 * r108857;
        double r108859 = r108849 / r108858;
        double r108860 = 0.0625;
        double r108861 = 0.015625;
        double r108862 = r108861 * r108852;
        double r108863 = 2.0;
        double r108864 = pow(r108846, r108863);
        double r108865 = r108862 / r108864;
        double r108866 = r108860 + r108865;
        double r108867 = 0.00390625;
        double r108868 = 4.0;
        double r108869 = pow(r108853, r108868);
        double r108870 = pow(r108846, r108868);
        double r108871 = r108869 / r108870;
        double r108872 = r108867 * r108871;
        double r108873 = r108866 + r108872;
        double r108874 = r108848 ? r108859 : r108873;
        return r108874;
}

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 < 248.47326712178204

    1. Initial program 45.0

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

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.0

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

    5. Applied difference-of-squares0.0

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

    if 248.47326712178204 < i

    1. Initial program 48.5

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

    2. Simplified32.8

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt32.8

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

    5. Applied difference-of-squares32.8

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

    6. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.00390625 \cdot \frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}} + \left(0.015625 \cdot \frac{{\left(\sqrt{1}\right)}^{2}}{{i}^{2}} + 0.0625\right)}\]

    7. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 248.47326712178204:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i + \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + \frac{0.015625 \cdot 1}{{i}^{2}}\right) + 0.00390625 \cdot \frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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)))