Average Error: 30.9 → 0.1
Time: 1.3m
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 r3486986 = 1.0;
        double r3486987 = x;
        double r3486988 = cos(r3486987);
        double r3486989 = r3486986 - r3486988;
        double r3486990 = r3486987 * r3486987;
        double r3486991 = r3486989 / r3486990;
        return r3486991;
}
↓
double f(double x) {
        double r3486992 = x;
        double r3486993 = sin(r3486992);
        double r3486994 = r3486993 / r3486992;
        double r3486995 = 2.0;
        double r3486996 = r3486992 / r3486995;
        double r3486997 = tan(r3486996);
        double r3486998 = r3486992 / r3486997;
        double r3486999 = r3486994 / r3486998;
        return r3486999;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.9

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

  3. Applied flip--31.0

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

  4. Applied associate-/l/31.0

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

  5. Simplified15.3

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

  6. Taylor expanded around inf 15.3

    \[\leadsto \color{blue}{\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \left(\cos x + 1\right)}}\]

  7. Simplified0.3

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

  8. Taylor expanded around -inf 0.3

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019124 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))