Average Error: 46.0 → 0.0
Time: 3.8s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 245.497035223853601:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \left(\frac{0.015625}{{i}^{2}} + \frac{0.00390625}{{i}^{4}}\right)\\ \end{array}\]
\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 245.497035223853601:\\
\;\;\;\;\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}:\\
\;\;\;\;0.0625 + \left(\frac{0.015625}{{i}^{2}} + \frac{0.00390625}{{i}^{4}}\right)\\

\end{array}
double f(double i) {
        double r105919 = i;
        double r105920 = r105919 * r105919;
        double r105921 = r105920 * r105920;
        double r105922 = 2.0;
        double r105923 = r105922 * r105919;
        double r105924 = r105923 * r105923;
        double r105925 = r105921 / r105924;
        double r105926 = 1.0;
        double r105927 = r105924 - r105926;
        double r105928 = r105925 / r105927;
        return r105928;
}
↓
double f(double i) {
        double r105929 = i;
        double r105930 = 245.4970352238536;
        bool r105931 = r105929 <= r105930;
        double r105932 = r105929 * r105929;
        double r105933 = 2.0;
        double r105934 = r105933 * r105929;
        double r105935 = r105934 * r105934;
        double r105936 = 1.0;
        double r105937 = r105935 - r105936;
        double r105938 = r105933 * r105933;
        double r105939 = r105937 * r105938;
        double r105940 = r105932 / r105939;
        double r105941 = 0.0625;
        double r105942 = 0.015625;
        double r105943 = 2.0;
        double r105944 = pow(r105929, r105943);
        double r105945 = r105942 / r105944;
        double r105946 = 0.00390625;
        double r105947 = 4.0;
        double r105948 = pow(r105929, r105947);
        double r105949 = r105946 / r105948;
        double r105950 = r105945 + r105949;
        double r105951 = r105941 + r105950;
        double r105952 = r105931 ? r105940 : r105951;
        return r105952;
}

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 < 245.4970352238536

    1. Initial program 43.9

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

    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 245.4970352238536 < i

    1. Initial program 48.0

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

    2. 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. Using strategy rm

    4. Applied times-frac30.6

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

    5. Taylor expanded around inf 0.0

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

    6. Simplified0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 245.497035223853601:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \left(\frac{0.015625}{{i}^{2}} + \frac{0.00390625}{{i}^{4}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(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)))