Average Error: 31.9 → 0.2
Time: 15.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03115379972147733905751820771001803223044:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \le 0.02739005695178562543867784029316680971533:\\ \;\;\;\;\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} - \frac{1}{24}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03115379972147733905751820771001803223044:\\
\;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\

\mathbf{elif}\;x \le 0.02739005695178562543867784029316680971533:\\
\;\;\;\;\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} - \frac{1}{24}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\

\end{array}
double f(double x) {
        double r715908 = 1.0;
        double r715909 = x;
        double r715910 = cos(r715909);
        double r715911 = r715908 - r715910;
        double r715912 = r715909 * r715909;
        double r715913 = r715911 / r715912;
        return r715913;
}
↓
double f(double x) {
        double r715914 = x;
        double r715915 = -0.03115379972147734;
        bool r715916 = r715914 <= r715915;
        double r715917 = 1.0;
        double r715918 = cos(r715914);
        double r715919 = r715917 - r715918;
        double r715920 = r715919 / r715914;
        double r715921 = r715920 / r715914;
        double r715922 = 0.027390056951785625;
        bool r715923 = r715914 <= r715922;
        double r715924 = 0.5;
        double r715925 = r715914 * r715914;
        double r715926 = 0.001388888888888889;
        double r715927 = r715925 * r715926;
        double r715928 = 0.041666666666666664;
        double r715929 = r715927 - r715928;
        double r715930 = r715925 * r715929;
        double r715931 = r715924 + r715930;
        double r715932 = r715923 ? r715931 : r715921;
        double r715933 = r715916 ? r715921 : r715932;
        return r715933;
}

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 < -0.03115379972147734 or 0.027390056951785625 < x

    1. Initial program 1.0

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

    3. Applied *-un-lft-identity1.0

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

    4. Applied times-frac0.5

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

    6. Applied associate-*r/0.5

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

    7. Simplified0.5

      \[\leadsto \frac{\color{blue}{\frac{1 - \cos x}{x}}}{x}\]

    if -0.03115379972147734 < x < 0.027390056951785625

    1. Initial program 62.4

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

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03115379972147733905751820771001803223044:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \le 0.02739005695178562543867784029316680971533:\\ \;\;\;\;\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} - \frac{1}{24}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))