Average Error: 30.7 → 0.1
Time: 43.0s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\sin x}{x} \cdot \frac{1}{\frac{x}{\tan \left(\frac{x}{2}\right)}}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\sin x}{x} \cdot \frac{1}{\frac{x}{\tan \left(\frac{x}{2}\right)}}
double f(double x) {
        double r829295 = 1.0;
        double r829296 = x;
        double r829297 = cos(r829296);
        double r829298 = r829295 - r829297;
        double r829299 = r829296 * r829296;
        double r829300 = r829298 / r829299;
        return r829300;
}
↓
double f(double x) {
        double r829301 = x;
        double r829302 = sin(r829301);
        double r829303 = r829302 / r829301;
        double r829304 = 1.0;
        double r829305 = 2.0;
        double r829306 = r829301 / r829305;
        double r829307 = tan(r829306);
        double r829308 = r829301 / r829307;
        double r829309 = r829304 / r829308;
        double r829310 = r829303 * r829309;
        return r829310;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.7

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

  3. Applied flip--30.8

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

  4. Simplified14.9

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

  6. Applied *-un-lft-identity14.9

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

  7. Applied times-frac14.9

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

  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 clear-num0.1

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

  13. Final simplification0.1

    \[\leadsto \frac{\sin x}{x} \cdot \frac{1}{\frac{x}{\tan \left(\frac{x}{2}\right)}}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))