Average Error: 31.0 → 0.1
Time: 37.4s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{x}}{\frac{x}{\tan \left(\frac{x}{2}\right)}}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin x}{x}}{\frac{x}{\tan \left(\frac{x}{2}\right)}}
double f(double x) {
        double r1062896 = 1.0;
        double r1062897 = x;
        double r1062898 = cos(r1062897);
        double r1062899 = r1062896 - r1062898;
        double r1062900 = r1062897 * r1062897;
        double r1062901 = r1062899 / r1062900;
        return r1062901;
}
↓
double f(double x) {
        double r1062902 = x;
        double r1062903 = sin(r1062902);
        double r1062904 = r1062903 / r1062902;
        double r1062905 = 2.0;
        double r1062906 = r1062902 / r1062905;
        double r1062907 = tan(r1062906);
        double r1062908 = r1062902 / r1062907;
        double r1062909 = r1062904 / r1062908;
        return r1062909;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.0

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

  3. Applied flip--31.1

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

  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 associate-/r*15.4

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

  7. Simplified0.1

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

  9. Applied associate-/l*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))