Average Error: 46.4 → 0.0
Time: 2.4s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 219.764984409217874:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i + \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + \frac{0.015625 \cdot 1}{{i}^{2}}\right) + 0.00390625 \cdot \frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}}\\ \end{array}\]
\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 219.764984409217874:\\
\;\;\;\;\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 r63895 = i;
        double r63896 = r63895 * r63895;
        double r63897 = r63896 * r63896;
        double r63898 = 2.0;
        double r63899 = r63898 * r63895;
        double r63900 = r63899 * r63899;
        double r63901 = r63897 / r63900;
        double r63902 = 1.0;
        double r63903 = r63900 - r63902;
        double r63904 = r63901 / r63903;
        return r63904;
}
↓
double f(double i) {
        double r63905 = i;
        double r63906 = 219.76498440921787;
        bool r63907 = r63905 <= r63906;
        double r63908 = r63905 * r63905;
        double r63909 = 2.0;
        double r63910 = r63909 * r63905;
        double r63911 = 1.0;
        double r63912 = sqrt(r63911);
        double r63913 = r63910 + r63912;
        double r63914 = r63910 - r63912;
        double r63915 = r63913 * r63914;
        double r63916 = r63909 * r63909;
        double r63917 = r63915 * r63916;
        double r63918 = r63908 / r63917;
        double r63919 = 0.0625;
        double r63920 = 0.015625;
        double r63921 = r63920 * r63911;
        double r63922 = 2.0;
        double r63923 = pow(r63905, r63922);
        double r63924 = r63921 / r63923;
        double r63925 = r63919 + r63924;
        double r63926 = 0.00390625;
        double r63927 = 4.0;
        double r63928 = pow(r63912, r63927);
        double r63929 = pow(r63905, r63927);
        double r63930 = r63928 / r63929;
        double r63931 = r63926 * r63930;
        double r63932 = r63925 + r63931;
        double r63933 = r63907 ? r63918 : r63932;
        return r63933;
}

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 < 219.76498440921787

    1. Initial program 44.4

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

    2. 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 219.76498440921787 < 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. Simplified32.4

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

    4. Applied add-sqr-sqrt32.4

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

    5. Applied difference-of-squares32.4

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 219.764984409217874:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i + \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + \frac{0.015625 \cdot 1}{{i}^{2}}\right) + 0.00390625 \cdot \frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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)))