Average Error: 45.8 → 0.0
Time: 26.6s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 224.39059497463407:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(i \cdot i\right) \cdot 4 - 1.0\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.00390625}{i}}{i} + 0.015625}{i \cdot i} + \frac{1}{16}\\ \end{array}\]
\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 224.39059497463407:\\
\;\;\;\;\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 r3448360 = i;
        double r3448361 = r3448360 * r3448360;
        double r3448362 = r3448361 * r3448361;
        double r3448363 = 2.0;
        double r3448364 = r3448363 * r3448360;
        double r3448365 = r3448364 * r3448364;
        double r3448366 = r3448362 / r3448365;
        double r3448367 = 1.0;
        double r3448368 = r3448365 - r3448367;
        double r3448369 = r3448366 / r3448368;
        return r3448369;
}
↓
double f(double i) {
        double r3448370 = i;
        double r3448371 = 224.39059497463407;
        bool r3448372 = r3448370 <= r3448371;
        double r3448373 = r3448370 * r3448370;
        double r3448374 = 4.0;
        double r3448375 = r3448373 * r3448374;
        double r3448376 = 1.0;
        double r3448377 = r3448375 - r3448376;
        double r3448378 = r3448377 * r3448374;
        double r3448379 = r3448373 / r3448378;
        double r3448380 = 0.00390625;
        double r3448381 = r3448380 / r3448370;
        double r3448382 = r3448381 / r3448370;
        double r3448383 = 0.015625;
        double r3448384 = r3448382 + r3448383;
        double r3448385 = r3448384 / r3448373;
        double r3448386 = 0.0625;
        double r3448387 = r3448385 + r3448386;
        double r3448388 = r3448372 ? r3448379 : r3448387;
        return r3448388;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019119 +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)))