Average Error: 31.1 → 0.3
Time: 14.7s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02817374883090965551057927029887650860474 \lor \neg \left(x \le 0.03027878116855447360178388294116302859038\right):\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{\left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right) \cdot x}{{1}^{3} - {\left(\cos x\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02817374883090965551057927029887650860474 \lor \neg \left(x \le 0.03027878116855447360178388294116302859038\right):\\
\;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{\left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right) \cdot x}{{1}^{3} - {\left(\cos x\right)}^{3}}}\\

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

\end{array}
double f(double x) {
        double r21807 = 1.0;
        double r21808 = x;
        double r21809 = cos(r21808);
        double r21810 = r21807 - r21809;
        double r21811 = r21808 * r21808;
        double r21812 = r21810 / r21811;
        return r21812;
}
↓
double f(double x) {
        double r21813 = x;
        double r21814 = -0.028173748830909656;
        bool r21815 = r21813 <= r21814;
        double r21816 = 0.030278781168554474;
        bool r21817 = r21813 <= r21816;
        double r21818 = !r21817;
        bool r21819 = r21815 || r21818;
        double r21820 = 1.0;
        double r21821 = r21820 / r21813;
        double r21822 = cos(r21813);
        double r21823 = 1.0;
        double r21824 = r21823 + r21822;
        double r21825 = r21822 * r21824;
        double r21826 = r21823 * r21823;
        double r21827 = r21825 + r21826;
        double r21828 = r21827 * r21813;
        double r21829 = 3.0;
        double r21830 = pow(r21823, r21829);
        double r21831 = pow(r21822, r21829);
        double r21832 = r21830 - r21831;
        double r21833 = r21828 / r21832;
        double r21834 = r21820 / r21833;
        double r21835 = r21821 * r21834;
        double r21836 = 0.001388888888888889;
        double r21837 = 4.0;
        double r21838 = pow(r21813, r21837);
        double r21839 = r21836 * r21838;
        double r21840 = 0.5;
        double r21841 = r21839 + r21840;
        double r21842 = 0.041666666666666664;
        double r21843 = 2.0;
        double r21844 = pow(r21813, r21843);
        double r21845 = r21842 * r21844;
        double r21846 = r21841 - r21845;
        double r21847 = r21819 ? r21835 : r21846;
        return r21847;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.028173748830909656 or 0.030278781168554474 < x

    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 flip3--0.5

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

    7. Applied associate-/l/0.5

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

    8. Simplified0.5

      \[\leadsto \frac{1}{x} \cdot \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right) \cdot x}}\]
    9. Using strategy rm

    10. Applied clear-num0.6

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

    if -0.028173748830909656 < x < 0.030278781168554474

    1. Initial program 62.3

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

    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. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02817374883090965551057927029887650860474 \lor \neg \left(x \le 0.03027878116855447360178388294116302859038\right):\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{\left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right) \cdot x}{{1}^{3} - {\left(\cos x\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))