Average Error: 45.8 → 0.0
Time: 13.1s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 3.714521008652235 \cdot 10^{-06}:\\ \;\;\;\;-\left(0.25 \cdot \left(i \cdot i\right) + \left(\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right) \cdot 1.0 + 4.0 \cdot \left(\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right) \cdot \left(i \cdot i\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4}}{16 - \frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i}} \cdot \left(\frac{1.0}{i \cdot i} + 4\right)\\ \end{array}\]
\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) - 1.0}
↓
\begin{array}{l}
\mathbf{if}\;i \le 3.714521008652235 \cdot 10^{-06}:\\
\;\;\;\;-\left(0.25 \cdot \left(i \cdot i\right) + \left(\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right) \cdot 1.0 + 4.0 \cdot \left(\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right) \cdot \left(i \cdot i\right)\right)\right)\right)\\

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

\end{array}
double f(double i) {
        double r1850757 = i;
        double r1850758 = r1850757 * r1850757;
        double r1850759 = r1850758 * r1850758;
        double r1850760 = 2.0;
        double r1850761 = r1850760 * r1850757;
        double r1850762 = r1850761 * r1850761;
        double r1850763 = r1850759 / r1850762;
        double r1850764 = 1.0;
        double r1850765 = r1850762 - r1850764;
        double r1850766 = r1850763 / r1850765;
        return r1850766;
}
↓
double f(double i) {
        double r1850767 = i;
        double r1850768 = 3.714521008652235e-06;
        bool r1850769 = r1850767 <= r1850768;
        double r1850770 = 0.25;
        double r1850771 = r1850767 * r1850767;
        double r1850772 = r1850770 * r1850771;
        double r1850773 = r1850771 * r1850771;
        double r1850774 = 1.0;
        double r1850775 = r1850773 * r1850774;
        double r1850776 = 4.0;
        double r1850777 = r1850773 * r1850771;
        double r1850778 = r1850776 * r1850777;
        double r1850779 = r1850775 + r1850778;
        double r1850780 = r1850772 + r1850779;
        double r1850781 = -r1850780;
        double r1850782 = 0.25;
        double r1850783 = 16.0;
        double r1850784 = r1850774 / r1850771;
        double r1850785 = r1850784 * r1850784;
        double r1850786 = r1850783 - r1850785;
        double r1850787 = r1850782 / r1850786;
        double r1850788 = 4.0;
        double r1850789 = r1850784 + r1850788;
        double r1850790 = r1850787 * r1850789;
        double r1850791 = r1850769 ? r1850781 : r1850790;
        return r1850791;
}

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 < 3.714521008652235e-06

    1. Initial program 45.5

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

    2. Simplified0.7

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

    4. Applied flip--47.2

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\frac{4 \cdot 4 - \frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i}}{4 + \frac{1.0}{i \cdot i}}}}\]

    5. Applied associate-/r/47.2

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

    6. Taylor expanded around 0 0.0

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

    7. Simplified0.0

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

    if 3.714521008652235e-06 < i

    1. Initial program 46.1

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

    2. Simplified0.0

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

    4. Applied flip--0.0

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\frac{4 \cdot 4 - \frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i}}{4 + \frac{1.0}{i \cdot i}}}}\]

    5. Applied associate-/r/0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 3.714521008652235 \cdot 10^{-06}:\\ \;\;\;\;-\left(0.25 \cdot \left(i \cdot i\right) + \left(\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right) \cdot 1.0 + 4.0 \cdot \left(\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right) \cdot \left(i \cdot i\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4}}{16 - \frac{1.0}{i \cdot i} \cdot \frac{1.0}{i \cdot i}} \cdot \left(\frac{1.0}{i \cdot i} + 4\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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)))