Average Error: 46.2 → 0.0
Time: 31.8s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 221.40568541809594:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(i \cdot i\right) \cdot 4 - 1.0\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.00390625}{i \cdot i} + 0.015625}{i \cdot i} + \frac{1}{16}\\ \end{array}\]
\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{0.00390625}{i \cdot i} + 0.015625}{i \cdot i} + \frac{1}{16}\\

\end{array}
double f(double i) {
        double r4221432 = i;
        double r4221433 = r4221432 * r4221432;
        double r4221434 = r4221433 * r4221433;
        double r4221435 = 2.0;
        double r4221436 = r4221435 * r4221432;
        double r4221437 = r4221436 * r4221436;
        double r4221438 = r4221434 / r4221437;
        double r4221439 = 1.0;
        double r4221440 = r4221437 - r4221439;
        double r4221441 = r4221438 / r4221440;
        return r4221441;
}
↓
double f(double i) {
        double r4221442 = i;
        double r4221443 = 221.40568541809594;
        bool r4221444 = r4221442 <= r4221443;
        double r4221445 = r4221442 * r4221442;
        double r4221446 = 4.0;
        double r4221447 = r4221445 * r4221446;
        double r4221448 = 1.0;
        double r4221449 = r4221447 - r4221448;
        double r4221450 = r4221449 * r4221446;
        double r4221451 = r4221445 / r4221450;
        double r4221452 = 0.00390625;
        double r4221453 = r4221452 / r4221445;
        double r4221454 = 0.015625;
        double r4221455 = r4221453 + r4221454;
        double r4221456 = r4221455 / r4221445;
        double r4221457 = 0.0625;
        double r4221458 = r4221456 + r4221457;
        double r4221459 = r4221444 ? r4221451 : r4221458;
        return r4221459;
}

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 -inf 0.0

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

    4. Simplified0.0

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

    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 -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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019120 
(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)))