Average Error: 30.9 → 0.1
Time: 14.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{1}{\frac{x}{\sin x}} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{1}{\frac{x}{\sin x}} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}
double f(double x) {
        double r740466 = 1.0;
        double r740467 = x;
        double r740468 = cos(r740467);
        double r740469 = r740466 - r740468;
        double r740470 = r740467 * r740467;
        double r740471 = r740469 / r740470;
        return r740471;
}
↓
double f(double x) {
        double r740472 = 1.0;
        double r740473 = x;
        double r740474 = sin(r740473);
        double r740475 = r740473 / r740474;
        double r740476 = r740472 / r740475;
        double r740477 = 2.0;
        double r740478 = r740473 / r740477;
        double r740479 = tan(r740478);
        double r740480 = r740479 / r740473;
        double r740481 = r740476 * r740480;
        return r740481;
}

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. Simplified15.6

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

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

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

  7. Applied times-frac15.6

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

  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

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))