Average Error: 31.1 → 0.3
Time: 14.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02817374883090965551057927029887650860474 \lor \neg \left(x \le 0.03027878116855447360178388294116302859038\right):\\ \;\;\;\;\frac{1}{\frac{\left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right) \cdot x}{{1}^{3} - {\left(\cos x\right)}^{3}}} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]
\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}{\frac{\left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right) \cdot x}{{1}^{3} - {\left(\cos x\right)}^{3}}} \cdot \frac{1}{x}\\

\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 r21787 = 1.0;
        double r21788 = x;
        double r21789 = cos(r21788);
        double r21790 = r21787 - r21789;
        double r21791 = r21788 * r21788;
        double r21792 = r21790 / r21791;
        return r21792;
}
↓
double f(double x) {
        double r21793 = x;
        double r21794 = -0.028173748830909656;
        bool r21795 = r21793 <= r21794;
        double r21796 = 0.030278781168554474;
        bool r21797 = r21793 <= r21796;
        double r21798 = !r21797;
        bool r21799 = r21795 || r21798;
        double r21800 = 1.0;
        double r21801 = cos(r21793);
        double r21802 = 1.0;
        double r21803 = r21802 + r21801;
        double r21804 = r21801 * r21803;
        double r21805 = r21802 * r21802;
        double r21806 = r21804 + r21805;
        double r21807 = r21806 * r21793;
        double r21808 = 3.0;
        double r21809 = pow(r21802, r21808);
        double r21810 = pow(r21801, r21808);
        double r21811 = r21809 - r21810;
        double r21812 = r21807 / r21811;
        double r21813 = r21800 / r21812;
        double r21814 = r21800 / r21793;
        double r21815 = r21813 * r21814;
        double r21816 = 0.001388888888888889;
        double r21817 = 4.0;
        double r21818 = pow(r21793, r21817);
        double r21819 = r21816 * r21818;
        double r21820 = 0.5;
        double r21821 = r21819 + r21820;
        double r21822 = 0.041666666666666664;
        double r21823 = 2.0;
        double r21824 = pow(r21793, r21823);
        double r21825 = r21822 * r21824;
        double r21826 = r21821 - r21825;
        double r21827 = r21799 ? r21815 : r21826;
        return r21827;
}

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 *-un-lft-identity0.5

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

    11. Applied unpow-prod-down0.5

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

    12. Applied *-un-lft-identity0.5

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

    13. Applied unpow-prod-down0.5

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

    14. Applied distribute-lft-out--0.5

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

    15. Applied associate-/l*0.6

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

    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

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

Reproduce

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