Average Error: 46.4 → 0.0
Time: 2.8s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le \frac{7367253589975485}{4398046511104}:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{256} \cdot \frac{1}{{i}^{4}} + \left(\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}\right)\\ \end{array}\]
\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{7367253589975485}{4398046511104}:\\
\;\;\;\;\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 r57640 = i;
        double r57641 = r57640 * r57640;
        double r57642 = r57641 * r57641;
        double r57643 = 2.0;
        double r57644 = r57643 * r57640;
        double r57645 = r57644 * r57644;
        double r57646 = r57642 / r57645;
        double r57647 = 1.0;
        double r57648 = r57645 - r57647;
        double r57649 = r57646 / r57648;
        return r57649;
}
↓
double f(double i) {
        double r57650 = i;
        double r57651 = 7367253589975485.0;
        double r57652 = 4398046511104.0;
        double r57653 = r57651 / r57652;
        bool r57654 = r57650 <= r57653;
        double r57655 = r57650 * r57650;
        double r57656 = 2.0;
        double r57657 = r57656 * r57650;
        double r57658 = r57657 * r57657;
        double r57659 = 1.0;
        double r57660 = r57658 - r57659;
        double r57661 = r57656 * r57656;
        double r57662 = r57660 * r57661;
        double r57663 = r57655 / r57662;
        double r57664 = 256.0;
        double r57665 = r57659 / r57664;
        double r57666 = 1.0;
        double r57667 = 4.0;
        double r57668 = pow(r57650, r57667);
        double r57669 = r57666 / r57668;
        double r57670 = r57665 * r57669;
        double r57671 = 64.0;
        double r57672 = r57659 / r57671;
        double r57673 = 2.0;
        double r57674 = pow(r57650, r57673);
        double r57675 = r57666 / r57674;
        double r57676 = r57672 * r57675;
        double r57677 = 16.0;
        double r57678 = r57659 / r57677;
        double r57679 = r57676 + r57678;
        double r57680 = r57670 + r57679;
        double r57681 = r57654 ? r57663 : r57680;
        return r57681;
}

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 < 1675.1195266750722

    1. Initial program 44.1

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

    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 1675.1195266750722 < i

    1. Initial program 48.8

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

    2. Simplified32.0

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le \frac{7367253589975485}{4398046511104}:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{256} \cdot \frac{1}{{i}^{4}} + \left(\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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)))