Average Error: 30.9 → 0.5
Time: 3.7s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0289755674093477682:\\ \;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{x \cdot x}\\ \mathbf{elif}\;x \le 0.0323653638594900775:\\ \;\;\;\;\frac{1}{720} \cdot {x}^{4} + \left(\frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{x} \cdot \frac{\sqrt{1 - \cos x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0289755674093477682:\\
\;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{x \cdot x}\\

\mathbf{elif}\;x \le 0.0323653638594900775:\\
\;\;\;\;\frac{1}{720} \cdot {x}^{4} + \left(\frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\

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

\end{array}
double f(double x) {
        double r22858 = 1.0;
        double r22859 = x;
        double r22860 = cos(r22859);
        double r22861 = r22858 - r22860;
        double r22862 = r22859 * r22859;
        double r22863 = r22861 / r22862;
        return r22863;
}
↓
double f(double x) {
        double r22864 = x;
        double r22865 = -0.028975567409347768;
        bool r22866 = r22864 <= r22865;
        double r22867 = 1.0;
        double r22868 = cos(r22864);
        double r22869 = r22867 - r22868;
        double r22870 = exp(r22869);
        double r22871 = log(r22870);
        double r22872 = r22864 * r22864;
        double r22873 = r22871 / r22872;
        double r22874 = 0.03236536385949008;
        bool r22875 = r22864 <= r22874;
        double r22876 = 0.001388888888888889;
        double r22877 = 4.0;
        double r22878 = pow(r22864, r22877);
        double r22879 = r22876 * r22878;
        double r22880 = 0.5;
        double r22881 = 0.041666666666666664;
        double r22882 = 2.0;
        double r22883 = pow(r22864, r22882);
        double r22884 = r22881 * r22883;
        double r22885 = r22880 - r22884;
        double r22886 = r22879 + r22885;
        double r22887 = sqrt(r22871);
        double r22888 = r22887 / r22864;
        double r22889 = sqrt(r22869);
        double r22890 = r22889 / r22864;
        double r22891 = r22888 * r22890;
        double r22892 = r22875 ? r22886 : r22891;
        double r22893 = r22866 ? r22873 : r22892;
        return r22893;
}

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 < -0.028975567409347768

    1. Initial program 1.0

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

    3. Applied add-log-exp1.1

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

    4. Applied add-log-exp1.1

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

    5. Applied diff-log1.3

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

    6. Simplified1.2

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

    if -0.028975567409347768 < x < 0.03236536385949008

    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. Using strategy rm

    4. Applied associate--l+0.0

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

    if 0.03236536385949008 < x

    1. Initial program 1.1

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

    3. Applied add-sqr-sqrt1.2

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

    4. Applied times-frac0.6

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

    6. Applied add-log-exp0.6

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

    7. Applied add-log-exp0.6

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

    8. Applied diff-log0.6

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

    9. Simplified0.6

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0289755674093477682:\\ \;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{x \cdot x}\\ \mathbf{elif}\;x \le 0.0323653638594900775:\\ \;\;\;\;\frac{1}{720} \cdot {x}^{4} + \left(\frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{x} \cdot \frac{\sqrt{1 - \cos x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))