Average Error: 30.6 → 0.1
Time: 16.2s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{x} \cdot \tan \left(\frac{x}{2}\right)}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin x}{x} \cdot \tan \left(\frac{x}{2}\right)}{x}
double f(double x) {
        double r612088 = 1.0;
        double r612089 = x;
        double r612090 = cos(r612089);
        double r612091 = r612088 - r612090;
        double r612092 = r612089 * r612089;
        double r612093 = r612091 / r612092;
        return r612093;
}
↓
double f(double x) {
        double r612094 = x;
        double r612095 = sin(r612094);
        double r612096 = r612095 / r612094;
        double r612097 = 2.0;
        double r612098 = r612094 / r612097;
        double r612099 = tan(r612098);
        double r612100 = r612096 * r612099;
        double r612101 = r612100 / r612094;
        return r612101;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.6

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

  3. Applied flip--30.7

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

  4. Simplified15.4

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

  6. Applied associate-/r*15.3

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))