Average Error: 47.1 → 0.0
Time: 6.2s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le \frac{1520701503786563}{274877906944}:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{256} \cdot \frac{1}{{i}^{4}} + \left(\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}\right)\\ \end{array}\]
\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 \frac{1520701503786563}{274877906944}:\\
\;\;\;\;\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}:\\
\;\;\;\;\frac{1}{256} \cdot \frac{1}{{i}^{4}} + \left(\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}\right)\\

\end{array}
double f(double i) {
        double r47581 = i;
        double r47582 = r47581 * r47581;
        double r47583 = r47582 * r47582;
        double r47584 = 2.0;
        double r47585 = r47584 * r47581;
        double r47586 = r47585 * r47585;
        double r47587 = r47583 / r47586;
        double r47588 = 1.0;
        double r47589 = r47586 - r47588;
        double r47590 = r47587 / r47589;
        return r47590;
}
↓
double f(double i) {
        double r47591 = i;
        double r47592 = 1520701503786563.0;
        double r47593 = 274877906944.0;
        double r47594 = r47592 / r47593;
        bool r47595 = r47591 <= r47594;
        double r47596 = r47591 * r47591;
        double r47597 = 2.0;
        double r47598 = r47597 * r47591;
        double r47599 = r47598 * r47598;
        double r47600 = 1.0;
        double r47601 = r47599 - r47600;
        double r47602 = r47597 * r47597;
        double r47603 = r47601 * r47602;
        double r47604 = r47596 / r47603;
        double r47605 = 256.0;
        double r47606 = r47600 / r47605;
        double r47607 = 1.0;
        double r47608 = 4.0;
        double r47609 = pow(r47591, r47608);
        double r47610 = r47607 / r47609;
        double r47611 = r47606 * r47610;
        double r47612 = 64.0;
        double r47613 = r47600 / r47612;
        double r47614 = 2.0;
        double r47615 = pow(r47591, r47614);
        double r47616 = r47607 / r47615;
        double r47617 = r47613 * r47616;
        double r47618 = 16.0;
        double r47619 = r47600 / r47618;
        double r47620 = r47617 + r47619;
        double r47621 = r47611 + r47620;
        double r47622 = r47595 ? r47604 : r47621;
        return r47622;
}

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 < 5532.2798426880145

    1. Initial program 44.4

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

    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 5532.2798426880145 < i

    1. Initial program 49.9

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

    2. Simplified33.2

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

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)}\]

    4. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{256} \cdot \frac{1}{{i}^{4}} + \left(\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le \frac{1520701503786563}{274877906944}:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{256} \cdot \frac{1}{{i}^{4}} + \left(\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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)))