Average Error: 31.8 → 0.3
Time: 6.2s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le \frac{-2382688094325091}{72057594037927936}:\\ \;\;\;\;\frac{\left(1 - \cos x\right) \cdot \frac{1}{x}}{x}\\ \mathbf{elif}\;x \le \frac{8486473025367923}{288230376151711744}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le \frac{-2382688094325091}{72057594037927936}:\\
\;\;\;\;\frac{\left(1 - \cos x\right) \cdot \frac{1}{x}}{x}\\

\mathbf{elif}\;x \le \frac{8486473025367923}{288230376151711744}:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot x}}{x}\\

\end{array}
double f(double x) {
        double r28854 = 1.0;
        double r28855 = x;
        double r28856 = cos(r28855);
        double r28857 = r28854 - r28856;
        double r28858 = r28855 * r28855;
        double r28859 = r28857 / r28858;
        return r28859;
}
↓
double f(double x) {
        double r28860 = x;
        double r28861 = -2382688094325091.0;
        double r28862 = 7.205759403792794e+16;
        double r28863 = r28861 / r28862;
        bool r28864 = r28860 <= r28863;
        double r28865 = 1.0;
        double r28866 = cos(r28860);
        double r28867 = r28865 - r28866;
        double r28868 = 1.0;
        double r28869 = r28868 / r28860;
        double r28870 = r28867 * r28869;
        double r28871 = r28870 / r28860;
        double r28872 = 8486473025367923.0;
        double r28873 = 2.8823037615171174e+17;
        double r28874 = r28872 / r28873;
        bool r28875 = r28860 <= r28874;
        double r28876 = 0.001388888888888889;
        double r28877 = 4.0;
        double r28878 = pow(r28860, r28877);
        double r28879 = r28876 * r28878;
        double r28880 = 0.5;
        double r28881 = r28879 + r28880;
        double r28882 = 0.041666666666666664;
        double r28883 = 2.0;
        double r28884 = pow(r28860, r28883);
        double r28885 = r28882 * r28884;
        double r28886 = r28881 - r28885;
        double r28887 = 3.0;
        double r28888 = pow(r28865, r28887);
        double r28889 = pow(r28866, r28887);
        double r28890 = r28888 - r28889;
        double r28891 = r28866 + r28865;
        double r28892 = r28866 * r28891;
        double r28893 = r28865 * r28865;
        double r28894 = r28892 + r28893;
        double r28895 = r28894 * r28860;
        double r28896 = r28890 / r28895;
        double r28897 = r28896 / r28860;
        double r28898 = r28875 ? r28886 : r28897;
        double r28899 = r28864 ? r28871 : r28898;
        return r28899;
}

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 < -0.03306643978524941

    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 div-inv0.5

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

    if -0.03306643978524941 < x < 0.029443367970699307

    1. Initial program 62.3

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

    2. Taylor expanded around 0 0.0

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

    if 0.029443367970699307 < x

    1. Initial program 0.9

      \[\frac{1 - \cos x}{x \cdot x}\]
    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 flip3--0.5

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

    6. Applied associate-/l/0.5

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

    7. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le \frac{-2382688094325091}{72057594037927936}:\\ \;\;\;\;\frac{\left(1 - \cos x\right) \cdot \frac{1}{x}}{x}\\ \mathbf{elif}\;x \le \frac{8486473025367923}{288230376151711744}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))