Average Error: 45.7 → 0.0
Time: 8.8s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 213.0387272011366:\\ \;\;\;\;\left(1.0 + \left(i \cdot i\right) \cdot 4\right) \cdot \frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\left(\left(i \cdot i\right) \cdot 4\right) \cdot \left(\left(i \cdot i\right) \cdot 4\right) - 1.0 \cdot 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.00390625}{\left(i \cdot i\right) \cdot \left(i \cdot i\right)} + \left(\frac{1}{16} + \frac{0.015625}{i \cdot i}\right)\\ \end{array}\]
\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 213.0387272011366:\\
\;\;\;\;\left(1.0 + \left(i \cdot i\right) \cdot 4\right) \cdot \frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\left(\left(i \cdot i\right) \cdot 4\right) \cdot \left(\left(i \cdot i\right) \cdot 4\right) - 1.0 \cdot 1.0}\\

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

\end{array}
double f(double i) {
        double r1221926 = i;
        double r1221927 = r1221926 * r1221926;
        double r1221928 = r1221927 * r1221927;
        double r1221929 = 2.0;
        double r1221930 = r1221929 * r1221926;
        double r1221931 = r1221930 * r1221930;
        double r1221932 = r1221928 / r1221931;
        double r1221933 = 1.0;
        double r1221934 = r1221931 - r1221933;
        double r1221935 = r1221932 / r1221934;
        return r1221935;
}
↓
double f(double i) {
        double r1221936 = i;
        double r1221937 = 213.0387272011366;
        bool r1221938 = r1221936 <= r1221937;
        double r1221939 = 1.0;
        double r1221940 = r1221936 * r1221936;
        double r1221941 = 4.0;
        double r1221942 = r1221940 * r1221941;
        double r1221943 = r1221939 + r1221942;
        double r1221944 = 0.25;
        double r1221945 = r1221940 * r1221944;
        double r1221946 = r1221942 * r1221942;
        double r1221947 = r1221939 * r1221939;
        double r1221948 = r1221946 - r1221947;
        double r1221949 = r1221945 / r1221948;
        double r1221950 = r1221943 * r1221949;
        double r1221951 = 0.00390625;
        double r1221952 = r1221940 * r1221940;
        double r1221953 = r1221951 / r1221952;
        double r1221954 = 0.0625;
        double r1221955 = 0.015625;
        double r1221956 = r1221955 / r1221940;
        double r1221957 = r1221954 + r1221956;
        double r1221958 = r1221953 + r1221957;
        double r1221959 = r1221938 ? r1221950 : r1221958;
        return r1221959;
}

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 < 213.0387272011366

    1. Initial program 44.7

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

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\left(i \cdot i\right) \cdot 4 - 1.0}}\]
    3. Using strategy rm

    4. Applied flip--0.0

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

    5. Applied associate-/r/0.0

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

    if 213.0387272011366 < i

    1. Initial program 46.7

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

    2. Simplified30.9

      \[\leadsto \color{blue}{\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\left(i \cdot i\right) \cdot 4 - 1.0}}\]
    3. Using strategy rm

    4. Applied flip--46.8

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

    5. Applied associate-/r/46.8

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

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 213.0387272011366:\\ \;\;\;\;\left(1.0 + \left(i \cdot i\right) \cdot 4\right) \cdot \frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\left(\left(i \cdot i\right) \cdot 4\right) \cdot \left(\left(i \cdot i\right) \cdot 4\right) - 1.0 \cdot 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.00390625}{\left(i \cdot i\right) \cdot \left(i \cdot i\right)} + \left(\frac{1}{16} + \frac{0.015625}{i \cdot i}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +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)))