Average Error: 45.8 → 0.0
Time: 23.0s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 224.39059497463407:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(i \cdot i\right) \cdot 4 - 1.0\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.00390625}{i \cdot i} + 0.015625}{i \cdot i} + \frac{1}{16}\\ \end{array}\]
\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 224.39059497463407:\\
\;\;\;\;\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 r3873454 = i;
        double r3873455 = r3873454 * r3873454;
        double r3873456 = r3873455 * r3873455;
        double r3873457 = 2.0;
        double r3873458 = r3873457 * r3873454;
        double r3873459 = r3873458 * r3873458;
        double r3873460 = r3873456 / r3873459;
        double r3873461 = 1.0;
        double r3873462 = r3873459 - r3873461;
        double r3873463 = r3873460 / r3873462;
        return r3873463;
}
↓
double f(double i) {
        double r3873464 = i;
        double r3873465 = 224.39059497463407;
        bool r3873466 = r3873464 <= r3873465;
        double r3873467 = r3873464 * r3873464;
        double r3873468 = 4.0;
        double r3873469 = r3873467 * r3873468;
        double r3873470 = 1.0;
        double r3873471 = r3873469 - r3873470;
        double r3873472 = r3873471 * r3873468;
        double r3873473 = r3873467 / r3873472;
        double r3873474 = 0.00390625;
        double r3873475 = r3873474 / r3873467;
        double r3873476 = 0.015625;
        double r3873477 = r3873475 + r3873476;
        double r3873478 = r3873477 / r3873467;
        double r3873479 = 0.0625;
        double r3873480 = r3873478 + r3873479;
        double r3873481 = r3873466 ? r3873473 : r3873480;
        return r3873481;
}

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 < 224.39059497463407

    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

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

    if 224.39059497463407 < 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. Simplified31.9

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 224.39059497463407:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(i \cdot i\right) \cdot 4 - 1.0\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.00390625}{i \cdot i} + 0.015625}{i \cdot i} + \frac{1}{16}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(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)))