Average Error: 30.9 → 0.3
Time: 4.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03812843131329910623117385171099158469588 \lor \neg \left(x \le 0.03662167515047411170403535152217955328524\right):\\ \;\;\;\;\frac{\frac{\log \left(e^{1 - \cos x}\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03812843131329910623117385171099158469588 \lor \neg \left(x \le 0.03662167515047411170403535152217955328524\right):\\
\;\;\;\;\frac{\frac{\log \left(e^{1 - \cos x}\right)}{x}}{x}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\

\end{array}
double f(double x) {
        double r27850 = 1.0;
        double r27851 = x;
        double r27852 = cos(r27851);
        double r27853 = r27850 - r27852;
        double r27854 = r27851 * r27851;
        double r27855 = r27853 / r27854;
        return r27855;
}
↓
double f(double x) {
        double r27856 = x;
        double r27857 = -0.038128431313299106;
        bool r27858 = r27856 <= r27857;
        double r27859 = 0.03662167515047411;
        bool r27860 = r27856 <= r27859;
        double r27861 = !r27860;
        bool r27862 = r27858 || r27861;
        double r27863 = 1.0;
        double r27864 = cos(r27856);
        double r27865 = r27863 - r27864;
        double r27866 = exp(r27865);
        double r27867 = log(r27866);
        double r27868 = r27867 / r27856;
        double r27869 = r27868 / r27856;
        double r27870 = 4.0;
        double r27871 = pow(r27856, r27870);
        double r27872 = 0.001388888888888889;
        double r27873 = 0.5;
        double r27874 = 0.041666666666666664;
        double r27875 = 2.0;
        double r27876 = pow(r27856, r27875);
        double r27877 = r27874 * r27876;
        double r27878 = r27873 - r27877;
        double r27879 = fma(r27871, r27872, r27878);
        double r27880 = r27862 ? r27869 : r27879;
        return r27880;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.038128431313299106 or 0.03662167515047411 < x

    1. Initial program 0.9

      \[\frac{1 - \cos x}{x \cdot x}\]
    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-log-exp0.6

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

    6. Applied add-log-exp0.6

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

    7. Applied diff-log0.6

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

    8. Simplified0.6

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

    if -0.038128431313299106 < x < 0.03662167515047411

    1. Initial program 62.0

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

    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

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03812843131329910623117385171099158469588 \lor \neg \left(x \le 0.03662167515047411170403535152217955328524\right):\\ \;\;\;\;\frac{\frac{\log \left(e^{1 - \cos x}\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))