Average Error: 46.1 → 0.0
Time: 26.9s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 219.36859705155612:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(i \cdot i\right) \cdot 4 - 1.0\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.00390625}{i}}{i} + 0.015625}{i \cdot i} + \frac{1}{16}\\ \end{array}\]
\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 219.36859705155612:\\
\;\;\;\;\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 r6783508 = i;
        double r6783509 = r6783508 * r6783508;
        double r6783510 = r6783509 * r6783509;
        double r6783511 = 2.0;
        double r6783512 = r6783511 * r6783508;
        double r6783513 = r6783512 * r6783512;
        double r6783514 = r6783510 / r6783513;
        double r6783515 = 1.0;
        double r6783516 = r6783513 - r6783515;
        double r6783517 = r6783514 / r6783516;
        return r6783517;
}
↓
double f(double i) {
        double r6783518 = i;
        double r6783519 = 219.36859705155612;
        bool r6783520 = r6783518 <= r6783519;
        double r6783521 = r6783518 * r6783518;
        double r6783522 = 4.0;
        double r6783523 = r6783521 * r6783522;
        double r6783524 = 1.0;
        double r6783525 = r6783523 - r6783524;
        double r6783526 = r6783525 * r6783522;
        double r6783527 = r6783521 / r6783526;
        double r6783528 = 0.00390625;
        double r6783529 = r6783528 / r6783518;
        double r6783530 = r6783529 / r6783518;
        double r6783531 = 0.015625;
        double r6783532 = r6783530 + r6783531;
        double r6783533 = r6783532 / r6783521;
        double r6783534 = 0.0625;
        double r6783535 = r6783533 + r6783534;
        double r6783536 = r6783520 ? r6783527 : r6783535;
        return r6783536;
}

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 < 219.36859705155612

    1. Initial program 45.1

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

    2. Simplified0.0

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

    if 219.36859705155612 < i

    1. Initial program 47.0

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

    2. Simplified30.7

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

    4. Applied associate-/l*30.7

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

    5. Taylor expanded around -inf 0.0

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

    6. Simplified0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 219.36859705155612:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(i \cdot i\right) \cdot 4 - 1.0\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.00390625}{i}}{i} + 0.015625}{i \cdot i} + \frac{1}{16}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 +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)))