Average Error: 31.7 → 0.1
Time: 21.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\sin x}{x} \cdot \frac{\sin \left(\frac{x}{2}\right)}{x \cdot \cos \left(\frac{x}{2}\right)}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\sin x}{x} \cdot \frac{\sin \left(\frac{x}{2}\right)}{x \cdot \cos \left(\frac{x}{2}\right)}
double f(double x) {
        double r1132867 = 1.0;
        double r1132868 = x;
        double r1132869 = cos(r1132868);
        double r1132870 = r1132867 - r1132869;
        double r1132871 = r1132868 * r1132868;
        double r1132872 = r1132870 / r1132871;
        return r1132872;
}
↓
double f(double x) {
        double r1132873 = x;
        double r1132874 = sin(r1132873);
        double r1132875 = r1132874 / r1132873;
        double r1132876 = 2.0;
        double r1132877 = r1132873 / r1132876;
        double r1132878 = sin(r1132877);
        double r1132879 = cos(r1132877);
        double r1132880 = r1132873 * r1132879;
        double r1132881 = r1132878 / r1132880;
        double r1132882 = r1132875 * r1132881;
        return r1132882;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.7

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

  3. Applied flip--31.8

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

  4. Simplified15.5

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

  6. Applied *-un-lft-identity15.5

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

  7. Applied times-frac15.5

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

  8. Applied times-frac0.3

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

  9. Simplified0.3

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

  10. Simplified0.1

    \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{x}}\]
  11. Using strategy rm

  12. Applied tan-quot0.1

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

  13. Applied associate-/l/0.1

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

  14. Final simplification0.1

    \[\leadsto \frac{\sin x}{x} \cdot \frac{\sin \left(\frac{x}{2}\right)}{x \cdot \cos \left(\frac{x}{2}\right)}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))