Average Error: 46.7 → 0.0
Time: 1.1s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 217.903437947567255:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\ \end{array}\]
\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}
↓
\begin{array}{l}
\mathbf{if}\;i \le 217.903437947567255:\\
\;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\

\mathbf{else}:\\
\;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\

\end{array}
double f(double i) {
        double r59492 = i;
        double r59493 = r59492 * r59492;
        double r59494 = r59493 * r59493;
        double r59495 = 2.0;
        double r59496 = r59495 * r59492;
        double r59497 = r59496 * r59496;
        double r59498 = r59494 / r59497;
        double r59499 = 1.0;
        double r59500 = r59497 - r59499;
        double r59501 = r59498 / r59500;
        return r59501;
}
↓
double f(double i) {
        double r59502 = i;
        double r59503 = 217.90343794756726;
        bool r59504 = r59502 <= r59503;
        double r59505 = r59502 * r59502;
        double r59506 = 2.0;
        double r59507 = r59506 * r59502;
        double r59508 = r59507 * r59507;
        double r59509 = 1.0;
        double r59510 = r59508 - r59509;
        double r59511 = r59506 * r59506;
        double r59512 = r59510 * r59511;
        double r59513 = r59505 / r59512;
        double r59514 = 0.00390625;
        double r59515 = 1.0;
        double r59516 = 4.0;
        double r59517 = pow(r59502, r59516);
        double r59518 = r59515 / r59517;
        double r59519 = r59514 * r59518;
        double r59520 = 0.015625;
        double r59521 = 2.0;
        double r59522 = pow(r59502, r59521);
        double r59523 = r59515 / r59522;
        double r59524 = r59520 * r59523;
        double r59525 = 0.0625;
        double r59526 = r59524 + r59525;
        double r59527 = r59519 + r59526;
        double r59528 = r59504 ? r59513 : r59527;
        return r59528;
}

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 < 217.90343794756726

    1. Initial program 45.3

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\]

    if 217.90343794756726 < i

    1. Initial program 48.3

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]

    2. Simplified32.9

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\]

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 217.903437947567255:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :precision binary64
  :pre (and (> i 0.0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1)))