Average Error: 31.2 → 0.3
Time: 11.7s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03100648570835606854245725116925314068794 \lor \neg \left(x \le 0.03254730486326439659050535624373878818005\right):\\ \;\;\;\;\frac{\left(1 - \cos x\right) \cdot \frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{2}, x, \mathsf{fma}\left(\frac{1}{720}, {x}^{5}, {x}^{3} \cdot \frac{-1}{24}\right)\right)}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03100648570835606854245725116925314068794 \lor \neg \left(x \le 0.03254730486326439659050535624373878818005\right):\\
\;\;\;\;\frac{\left(1 - \cos x\right) \cdot \frac{1}{x}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{2}, x, \mathsf{fma}\left(\frac{1}{720}, {x}^{5}, {x}^{3} \cdot \frac{-1}{24}\right)\right)}{x}\\

\end{array}
double f(double x) {
        double r20837 = 1.0;
        double r20838 = x;
        double r20839 = cos(r20838);
        double r20840 = r20837 - r20839;
        double r20841 = r20838 * r20838;
        double r20842 = r20840 / r20841;
        return r20842;
}
↓
double f(double x) {
        double r20843 = x;
        double r20844 = -0.03100648570835607;
        bool r20845 = r20843 <= r20844;
        double r20846 = 0.0325473048632644;
        bool r20847 = r20843 <= r20846;
        double r20848 = !r20847;
        bool r20849 = r20845 || r20848;
        double r20850 = 1.0;
        double r20851 = cos(r20843);
        double r20852 = r20850 - r20851;
        double r20853 = 1.0;
        double r20854 = r20853 / r20843;
        double r20855 = r20852 * r20854;
        double r20856 = r20855 / r20843;
        double r20857 = 0.5;
        double r20858 = 0.001388888888888889;
        double r20859 = 5.0;
        double r20860 = pow(r20843, r20859);
        double r20861 = 3.0;
        double r20862 = pow(r20843, r20861);
        double r20863 = -0.041666666666666664;
        double r20864 = r20862 * r20863;
        double r20865 = fma(r20858, r20860, r20864);
        double r20866 = fma(r20857, r20843, r20865);
        double r20867 = r20866 / r20843;
        double r20868 = r20849 ? r20856 : r20867;
        return r20868;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.03100648570835607 or 0.0325473048632644 < x

    1. Initial program 1.1

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

    3. Applied associate-/r*0.4

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

    5. Applied div-inv0.5

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

    if -0.03100648570835607 < x < 0.0325473048632644

    1. Initial program 62.2

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

    3. Applied associate-/r*61.3

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

    4. Taylor expanded around 0 0.0

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{720} \cdot {x}^{5}\right) - \frac{1}{24} \cdot {x}^{3}}}{x}\]

    5. Simplified0.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, \mathsf{fma}\left(\frac{1}{720}, {x}^{5}, {x}^{3} \cdot \frac{-1}{24}\right)\right)}}{x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03100648570835606854245725116925314068794 \lor \neg \left(x \le 0.03254730486326439659050535624373878818005\right):\\ \;\;\;\;\frac{\left(1 - \cos x\right) \cdot \frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{2}, x, \mathsf{fma}\left(\frac{1}{720}, {x}^{5}, {x}^{3} \cdot \frac{-1}{24}\right)\right)}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))