Average Error: 31.4 → 0.3
Time: 8.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.023582590462374556:\\ \;\;\;\;\frac{\frac{\left(1 \cdot 1 - \frac{1}{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{x \cdot \left(1 + \cos x\right)}}{x}\\ \mathbf{elif}\;x \le 0.0302568236777068268:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.023582590462374556:\\
\;\;\;\;\frac{\frac{\left(1 \cdot 1 - \frac{1}{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{x \cdot \left(1 + \cos x\right)}}{x}\\

\mathbf{elif}\;x \le 0.0302568236777068268:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\

\end{array}
double f(double x) {
        double r21829 = 1.0;
        double r21830 = x;
        double r21831 = cos(r21830);
        double r21832 = r21829 - r21831;
        double r21833 = r21830 * r21830;
        double r21834 = r21832 / r21833;
        return r21834;
}
↓
double f(double x) {
        double r21835 = x;
        double r21836 = -0.023582590462374556;
        bool r21837 = r21835 <= r21836;
        double r21838 = 1.0;
        double r21839 = r21838 * r21838;
        double r21840 = 0.5;
        double r21841 = r21839 - r21840;
        double r21842 = 2.0;
        double r21843 = r21842 * r21835;
        double r21844 = cos(r21843);
        double r21845 = r21840 * r21844;
        double r21846 = r21841 - r21845;
        double r21847 = cos(r21835);
        double r21848 = r21838 + r21847;
        double r21849 = r21835 * r21848;
        double r21850 = r21846 / r21849;
        double r21851 = r21850 / r21835;
        double r21852 = 0.030256823677706827;
        bool r21853 = r21835 <= r21852;
        double r21854 = 0.001388888888888889;
        double r21855 = 4.0;
        double r21856 = pow(r21835, r21855);
        double r21857 = r21854 * r21856;
        double r21858 = r21857 + r21840;
        double r21859 = 0.041666666666666664;
        double r21860 = pow(r21835, r21842);
        double r21861 = r21859 * r21860;
        double r21862 = r21858 - r21861;
        double r21863 = r21838 - r21847;
        double r21864 = r21863 / r21835;
        double r21865 = r21864 / r21835;
        double r21866 = r21853 ? r21862 : r21865;
        double r21867 = r21837 ? r21851 : r21866;
        return r21867;
}

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 < -0.023582590462374556

    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 associate-*r/0.6

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

    7. Simplified0.5

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

    9. Applied flip--0.7

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

    10. Applied associate-/l/0.7

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{x \cdot \left(1 + \cos x\right)}}}{x}\]
    11. Using strategy rm

    12. Applied sqr-cos0.8

      \[\leadsto \frac{\frac{1 \cdot 1 - \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)}}{x \cdot \left(1 + \cos x\right)}}{x}\]

    13. Applied associate--r+0.7

      \[\leadsto \frac{\frac{\color{blue}{\left(1 \cdot 1 - \frac{1}{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{x \cdot \left(1 + \cos x\right)}}{x}\]

    if -0.023582590462374556 < x < 0.030256823677706827

    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 0.030256823677706827 < x

    1. Initial program 0.9

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

    3. Applied add-sqr-sqrt1.0

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

    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 associate-*r/0.6

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

    7. Simplified0.5

      \[\leadsto \frac{\color{blue}{\frac{1 - \cos x}{x}}}{x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.023582590462374556:\\ \;\;\;\;\frac{\frac{\left(1 \cdot 1 - \frac{1}{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{x \cdot \left(1 + \cos x\right)}}{x}\\ \mathbf{elif}\;x \le 0.0302568236777068268:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))