Average Error: 32.3 → 0.3
Time: 13.7s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03162480067046474080383333671306900214404:\\ \;\;\;\;\frac{\frac{e^{\log \left(1 - \cos x\right)}}{x}}{x}\\ \mathbf{elif}\;x \le 0.03078339684086396285667142080910707591102:\\ \;\;\;\;\left(\frac{1}{720} \cdot \left(x \cdot x\right) + \frac{-1}{24}\right) \cdot \left(x \cdot x\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\log \left(1 - \cos x\right)}}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03162480067046474080383333671306900214404:\\
\;\;\;\;\frac{\frac{e^{\log \left(1 - \cos x\right)}}{x}}{x}\\

\mathbf{elif}\;x \le 0.03078339684086396285667142080910707591102:\\
\;\;\;\;\left(\frac{1}{720} \cdot \left(x \cdot x\right) + \frac{-1}{24}\right) \cdot \left(x \cdot x\right) + \frac{1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{e^{\log \left(1 - \cos x\right)}}{x}}{x}\\

\end{array}
double f(double x) {
        double r878901 = 1.0;
        double r878902 = x;
        double r878903 = cos(r878902);
        double r878904 = r878901 - r878903;
        double r878905 = r878902 * r878902;
        double r878906 = r878904 / r878905;
        return r878906;
}
↓
double f(double x) {
        double r878907 = x;
        double r878908 = -0.03162480067046474;
        bool r878909 = r878907 <= r878908;
        double r878910 = 1.0;
        double r878911 = cos(r878907);
        double r878912 = r878910 - r878911;
        double r878913 = log(r878912);
        double r878914 = exp(r878913);
        double r878915 = r878914 / r878907;
        double r878916 = r878915 / r878907;
        double r878917 = 0.030783396840863963;
        bool r878918 = r878907 <= r878917;
        double r878919 = 0.001388888888888889;
        double r878920 = r878907 * r878907;
        double r878921 = r878919 * r878920;
        double r878922 = -0.041666666666666664;
        double r878923 = r878921 + r878922;
        double r878924 = r878923 * r878920;
        double r878925 = 0.5;
        double r878926 = r878924 + r878925;
        double r878927 = r878918 ? r878926 : r878916;
        double r878928 = r878909 ? r878916 : r878927;
        return r878928;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.03162480067046474 or 0.030783396840863963 < x

    1. Initial program 1.0

      \[\frac{1 - \cos x}{x \cdot x}\]
    2. Using strategy rm

    3. Applied associate-/r*0.5

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}}\]
    4. Using strategy rm

    5. Applied add-exp-log0.5

      \[\leadsto \frac{\frac{\color{blue}{e^{\log \left(1 - \cos x\right)}}}{x}}{x}\]

    if -0.03162480067046474 < x < 0.030783396840863963

    1. Initial program 62.3

      \[\frac{1 - \cos x}{x \cdot x}\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{1}{720} \cdot \left(x \cdot x\right) + \frac{-1}{24}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03162480067046474080383333671306900214404:\\ \;\;\;\;\frac{\frac{e^{\log \left(1 - \cos x\right)}}{x}}{x}\\ \mathbf{elif}\;x \le 0.03078339684086396285667142080910707591102:\\ \;\;\;\;\left(\frac{1}{720} \cdot \left(x \cdot x\right) + \frac{-1}{24}\right) \cdot \left(x \cdot x\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\log \left(1 - \cos x\right)}}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))