Average Error: 45.7 → 0.0
Time: 9.6s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 8.32761208931738 \cdot 10^{-24}:\\ \;\;\;\;-\left(\left(\left(\left(i \cdot i\right) \cdot i\right) \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) \cdot 4.0 + \left(\left(i \cdot i\right) \cdot 0.25 + 1.0 \cdot \left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i} + \frac{1.0}{i \cdot i} \cdot 4\right) + 16\right) \cdot \frac{1}{2}\right) \cdot \frac{\frac{1}{2}}{64 - {\left(\frac{1.0}{i \cdot i}\right)}^{3}}\\ \end{array}\]
\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 8.32761208931738 \cdot 10^{-24}:\\
\;\;\;\;-\left(\left(\left(\left(i \cdot i\right) \cdot i\right) \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) \cdot 4.0 + \left(\left(i \cdot i\right) \cdot 0.25 + 1.0 \cdot \left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\left(\frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i} + \frac{1.0}{i \cdot i} \cdot 4\right) + 16\right) \cdot \frac{1}{2}\right) \cdot \frac{\frac{1}{2}}{64 - {\left(\frac{1.0}{i \cdot i}\right)}^{3}}\\

\end{array}
double f(double i) {
        double r1367712 = i;
        double r1367713 = r1367712 * r1367712;
        double r1367714 = r1367713 * r1367713;
        double r1367715 = 2.0;
        double r1367716 = r1367715 * r1367712;
        double r1367717 = r1367716 * r1367716;
        double r1367718 = r1367714 / r1367717;
        double r1367719 = 1.0;
        double r1367720 = r1367717 - r1367719;
        double r1367721 = r1367718 / r1367720;
        return r1367721;
}
↓
double f(double i) {
        double r1367722 = i;
        double r1367723 = 8.32761208931738e-24;
        bool r1367724 = r1367722 <= r1367723;
        double r1367725 = r1367722 * r1367722;
        double r1367726 = r1367725 * r1367722;
        double r1367727 = r1367726 * r1367726;
        double r1367728 = 4.0;
        double r1367729 = r1367727 * r1367728;
        double r1367730 = 0.25;
        double r1367731 = r1367725 * r1367730;
        double r1367732 = 1.0;
        double r1367733 = r1367725 * r1367725;
        double r1367734 = r1367732 * r1367733;
        double r1367735 = r1367731 + r1367734;
        double r1367736 = r1367729 + r1367735;
        double r1367737 = -r1367736;
        double r1367738 = r1367732 / r1367725;
        double r1367739 = r1367738 * r1367738;
        double r1367740 = 4.0;
        double r1367741 = r1367738 * r1367740;
        double r1367742 = r1367739 + r1367741;
        double r1367743 = 16.0;
        double r1367744 = r1367742 + r1367743;
        double r1367745 = 0.5;
        double r1367746 = r1367744 * r1367745;
        double r1367747 = 64.0;
        double r1367748 = 3.0;
        double r1367749 = pow(r1367738, r1367748);
        double r1367750 = r1367747 - r1367749;
        double r1367751 = r1367745 / r1367750;
        double r1367752 = r1367746 * r1367751;
        double r1367753 = r1367724 ? r1367737 : r1367752;
        return r1367753;
}

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 < 8.32761208931738e-24

    1. Initial program 48.6

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

    2. Simplified0.8

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{4 - \frac{1.0}{i \cdot i}} \cdot \frac{1}{2}}\]
    3. Using strategy rm

    4. Applied flip3--56.4

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{{4}^{3} - {\left(\frac{1.0}{i \cdot i}\right)}^{3}}{4 \cdot 4 + \left(\frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i} + 4 \cdot \frac{1.0}{i \cdot i}\right)}}} \cdot \frac{1}{2}\]

    5. Applied associate-/r/56.4

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{2}}{{4}^{3} - {\left(\frac{1.0}{i \cdot i}\right)}^{3}} \cdot \left(4 \cdot 4 + \left(\frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i} + 4 \cdot \frac{1.0}{i \cdot i}\right)\right)\right)} \cdot \frac{1}{2}\]

    6. Applied associate-*l*56.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{{4}^{3} - {\left(\frac{1.0}{i \cdot i}\right)}^{3}} \cdot \left(\left(4 \cdot 4 + \left(\frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i} + 4 \cdot \frac{1.0}{i \cdot i}\right)\right) \cdot \frac{1}{2}\right)}\]

    7. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{-\left(0.25 \cdot {i}^{2} + \left(1.0 \cdot {i}^{4} + 4.0 \cdot {i}^{6}\right)\right)}\]

    8. Simplified0.0

      \[\leadsto \color{blue}{-\left(\left(\left(i \cdot i\right) \cdot 0.25 + \left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right) \cdot 1.0\right) + \left(\left(\left(i \cdot i\right) \cdot i\right) \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) \cdot 4.0\right)}\]

    if 8.32761208931738e-24 < i

    1. Initial program 43.1

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

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{4 - \frac{1.0}{i \cdot i}} \cdot \frac{1}{2}}\]
    3. Using strategy rm

    4. Applied flip3--0.0

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{{4}^{3} - {\left(\frac{1.0}{i \cdot i}\right)}^{3}}{4 \cdot 4 + \left(\frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i} + 4 \cdot \frac{1.0}{i \cdot i}\right)}}} \cdot \frac{1}{2}\]

    5. Applied associate-/r/0.0

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{2}}{{4}^{3} - {\left(\frac{1.0}{i \cdot i}\right)}^{3}} \cdot \left(4 \cdot 4 + \left(\frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i} + 4 \cdot \frac{1.0}{i \cdot i}\right)\right)\right)} \cdot \frac{1}{2}\]

    6. Applied associate-*l*0.0

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{{4}^{3} - {\left(\frac{1.0}{i \cdot i}\right)}^{3}} \cdot \left(\left(4 \cdot 4 + \left(\frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i} + 4 \cdot \frac{1.0}{i \cdot i}\right)\right) \cdot \frac{1}{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 8.32761208931738 \cdot 10^{-24}:\\ \;\;\;\;-\left(\left(\left(\left(i \cdot i\right) \cdot i\right) \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) \cdot 4.0 + \left(\left(i \cdot i\right) \cdot 0.25 + 1.0 \cdot \left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i} + \frac{1.0}{i \cdot i} \cdot 4\right) + 16\right) \cdot \frac{1}{2}\right) \cdot \frac{\frac{1}{2}}{64 - {\left(\frac{1.0}{i \cdot i}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(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)))