Average Error: 31.0 → 0.3
Time: 7.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02927801672355929279989439351084001827985:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\\ \mathbf{elif}\;x \le 0.02874724167449887318737111741029366385192:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02927801672355929279989439351084001827985:\\
\;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\\

\mathbf{elif}\;x \le 0.02874724167449887318737111741029366385192:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r18842 = 1.0;
        double r18843 = x;
        double r18844 = cos(r18843);
        double r18845 = r18842 - r18844;
        double r18846 = r18843 * r18843;
        double r18847 = r18845 / r18846;
        return r18847;
}
↓
double f(double x) {
        double r18848 = x;
        double r18849 = -0.029278016723559293;
        bool r18850 = r18848 <= r18849;
        double r18851 = 1.0;
        double r18852 = cos(r18848);
        double r18853 = r18851 - r18852;
        double r18854 = 1.0;
        double r18855 = r18854 / r18848;
        double r18856 = r18855 / r18848;
        double r18857 = r18853 * r18856;
        double r18858 = 0.028747241674498873;
        bool r18859 = r18848 <= r18858;
        double r18860 = 0.001388888888888889;
        double r18861 = 4.0;
        double r18862 = pow(r18848, r18861);
        double r18863 = r18860 * r18862;
        double r18864 = 0.5;
        double r18865 = r18863 + r18864;
        double r18866 = 0.041666666666666664;
        double r18867 = 2.0;
        double r18868 = pow(r18848, r18867);
        double r18869 = r18866 * r18868;
        double r18870 = r18865 - r18869;
        double r18871 = exp(1.0);
        double r18872 = log(r18853);
        double r18873 = pow(r18871, r18872);
        double r18874 = r18873 / r18848;
        double r18875 = r18874 / r18848;
        double r18876 = r18859 ? r18870 : r18875;
        double r18877 = r18850 ? r18857 : r18876;
        return r18877;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -0.029278016723559293

    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 *-un-lft-identity0.4

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

    6. Applied div-inv0.5

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

    7. Applied times-frac0.5

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

    8. Simplified0.5

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

    if -0.029278016723559293 < x < 0.028747241674498873

    1. Initial program 62.2

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

    2. Taylor expanded around 0 0.0

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

    if 0.028747241674498873 < 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

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

    7. Applied pow10.5

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

    8. Applied log-pow0.5

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

    9. Applied exp-prod0.5

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

    10. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02927801672355929279989439351084001827985:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\\ \mathbf{elif}\;x \le 0.02874724167449887318737111741029366385192:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))