Average Error: 31.2 → 0.3
Time: 4.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0344878390097593454:\\ \;\;\;\;\frac{\frac{1}{x}}{x} - \frac{\frac{\cos x}{x}}{x}\\ \mathbf{elif}\;x \le 0.03176890082881835:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \cos x\right)\right)}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0344878390097593454:\\
\;\;\;\;\frac{\frac{1}{x}}{x} - \frac{\frac{\cos x}{x}}{x}\\

\mathbf{elif}\;x \le 0.03176890082881835:\\
\;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \cos x\right)\right)}{x}}{x}\\

\end{array}
double f(double x) {
        double r29862 = 1.0;
        double r29863 = x;
        double r29864 = cos(r29863);
        double r29865 = r29862 - r29864;
        double r29866 = r29863 * r29863;
        double r29867 = r29865 / r29866;
        return r29867;
}
↓
double f(double x) {
        double r29868 = x;
        double r29869 = -0.034487839009759345;
        bool r29870 = r29868 <= r29869;
        double r29871 = 1.0;
        double r29872 = r29871 / r29868;
        double r29873 = r29872 / r29868;
        double r29874 = cos(r29868);
        double r29875 = r29874 / r29868;
        double r29876 = r29875 / r29868;
        double r29877 = r29873 - r29876;
        double r29878 = 0.03176890082881835;
        bool r29879 = r29868 <= r29878;
        double r29880 = 4.0;
        double r29881 = pow(r29868, r29880);
        double r29882 = 0.001388888888888889;
        double r29883 = 0.5;
        double r29884 = 0.041666666666666664;
        double r29885 = 2.0;
        double r29886 = pow(r29868, r29885);
        double r29887 = r29884 * r29886;
        double r29888 = r29883 - r29887;
        double r29889 = fma(r29881, r29882, r29888);
        double r29890 = r29871 - r29874;
        double r29891 = log1p(r29890);
        double r29892 = expm1(r29891);
        double r29893 = r29892 / r29868;
        double r29894 = r29893 / r29868;
        double r29895 = r29879 ? r29889 : r29894;
        double r29896 = r29870 ? r29877 : r29895;
        return r29896;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.034487839009759345

    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

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

    9. Applied div-sub0.5

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

    10. Applied div-sub0.6

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

    if -0.034487839009759345 < x < 0.03176890082881835

    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

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

    if 0.03176890082881835 < x

    1. Initial program 1.1

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

    3. Applied *-un-lft-identity1.1

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

    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. Simplified0.4

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

    9. Applied expm1-log1p-u0.5

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \cos x\right)\right)}}{x}}{x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0344878390097593454:\\ \;\;\;\;\frac{\frac{1}{x}}{x} - \frac{\frac{\cos x}{x}}{x}\\ \mathbf{elif}\;x \le 0.03176890082881835:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \cos x\right)\right)}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))