Average Error: 45.9 → 0.0
Time: 4.3s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 244.97799576821288:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\left(i \cdot i\right) \cdot 4 - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + \left(\frac{1}{16} + \frac{0.00390625}{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}\right)\\ \end{array}\]
\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) - 1.0}
↓
\begin{array}{l}
\mathbf{if}\;i \le 244.97799576821288:\\
\;\;\;\;\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\left(i \cdot i\right) \cdot 4 - 1.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{0.015625}{i \cdot i} + \left(\frac{1}{16} + \frac{0.00390625}{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}\right)\\

\end{array}
double f(double i) {
        double r946940 = i;
        double r946941 = r946940 * r946940;
        double r946942 = r946941 * r946941;
        double r946943 = 2.0;
        double r946944 = r946943 * r946940;
        double r946945 = r946944 * r946944;
        double r946946 = r946942 / r946945;
        double r946947 = 1.0;
        double r946948 = r946945 - r946947;
        double r946949 = r946946 / r946948;
        return r946949;
}
↓
double f(double i) {
        double r946950 = i;
        double r946951 = 244.97799576821288;
        bool r946952 = r946950 <= r946951;
        double r946953 = r946950 * r946950;
        double r946954 = 0.25;
        double r946955 = r946953 * r946954;
        double r946956 = 4.0;
        double r946957 = r946953 * r946956;
        double r946958 = 1.0;
        double r946959 = r946957 - r946958;
        double r946960 = r946955 / r946959;
        double r946961 = 0.015625;
        double r946962 = r946961 / r946953;
        double r946963 = 0.0625;
        double r946964 = 0.00390625;
        double r946965 = r946953 * r946953;
        double r946966 = r946964 / r946965;
        double r946967 = r946963 + r946966;
        double r946968 = r946962 + r946967;
        double r946969 = r946952 ? r946960 : r946968;
        return r946969;
}

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 < 244.97799576821288

    1. Initial program 45.0

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\left(i \cdot i\right) \cdot 4 - 1.0}}\]

    if 244.97799576821288 < i

    1. Initial program 46.8

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

    2. Simplified31.6

      \[\leadsto \color{blue}{\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\left(i \cdot i\right) \cdot 4 - 1.0}}\]

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.015625 \cdot \frac{1}{{i}^{2}} + \left(\frac{1}{16} + 0.00390625 \cdot \frac{1}{{i}^{4}}\right)}\]

    4. Simplified0.0

      \[\leadsto \color{blue}{\frac{0.015625}{i \cdot i} + \left(\frac{1}{16} + \frac{0.00390625}{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 244.97799576821288:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\left(i \cdot i\right) \cdot 4 - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + \left(\frac{1}{16} + \frac{0.00390625}{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0)))