Average Error: 46.2 → 0.0
Time: 34.3s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 221.40568541809594:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(i \cdot i\right) \cdot 4 - 1.0\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.00390625}{i}}{i} + 0.015625}{i \cdot i} + \frac{1}{16}\\ \end{array}\]
\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 221.40568541809594:\\
\;\;\;\;\frac{i \cdot i}{\left(\left(i \cdot i\right) \cdot 4 - 1.0\right) \cdot 4}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{0.00390625}{i}}{i} + 0.015625}{i \cdot i} + \frac{1}{16}\\

\end{array}
double f(double i) {
        double r3462150 = i;
        double r3462151 = r3462150 * r3462150;
        double r3462152 = r3462151 * r3462151;
        double r3462153 = 2.0;
        double r3462154 = r3462153 * r3462150;
        double r3462155 = r3462154 * r3462154;
        double r3462156 = r3462152 / r3462155;
        double r3462157 = 1.0;
        double r3462158 = r3462155 - r3462157;
        double r3462159 = r3462156 / r3462158;
        return r3462159;
}
↓
double f(double i) {
        double r3462160 = i;
        double r3462161 = 221.40568541809594;
        bool r3462162 = r3462160 <= r3462161;
        double r3462163 = r3462160 * r3462160;
        double r3462164 = 4.0;
        double r3462165 = r3462163 * r3462164;
        double r3462166 = 1.0;
        double r3462167 = r3462165 - r3462166;
        double r3462168 = r3462167 * r3462164;
        double r3462169 = r3462163 / r3462168;
        double r3462170 = 0.00390625;
        double r3462171 = r3462170 / r3462160;
        double r3462172 = r3462171 / r3462160;
        double r3462173 = 0.015625;
        double r3462174 = r3462172 + r3462173;
        double r3462175 = r3462174 / r3462163;
        double r3462176 = 0.0625;
        double r3462177 = r3462175 + r3462176;
        double r3462178 = r3462162 ? r3462169 : r3462177;
        return r3462178;
}

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 < 221.40568541809594

    1. Initial program 45.4

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

    2. Simplified0.0

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

    3. Taylor expanded around 0 0.0

      \[\leadsto \frac{i \cdot i}{\left(\color{blue}{4 \cdot {i}^{2}} - 1.0\right) \cdot 4}\]

    4. Simplified0.0

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

    if 221.40568541809594 < i

    1. Initial program 46.9

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

    2. Simplified31.0

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

    3. Taylor expanded around 0 31.0

      \[\leadsto \frac{i \cdot i}{\left(\color{blue}{4 \cdot {i}^{2}} - 1.0\right) \cdot 4}\]

    4. Simplified31.0

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

    6. Applied div-inv31.1

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

    7. Taylor expanded around inf 0.0

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

    8. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 221.40568541809594:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(i \cdot i\right) \cdot 4 - 1.0\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.00390625}{i}}{i} + 0.015625}{i \cdot i} + \frac{1}{16}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 +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)))