Average Error: 30.6 → 0.1
Time: 17.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \sin x}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \sin x}{x}
double f(double x) {
        double r720413 = 1.0;
        double r720414 = x;
        double r720415 = cos(r720414);
        double r720416 = r720413 - r720415;
        double r720417 = r720414 * r720414;
        double r720418 = r720416 / r720417;
        return r720418;
}
↓
double f(double x) {
        double r720419 = x;
        double r720420 = 2.0;
        double r720421 = r720419 / r720420;
        double r720422 = tan(r720421);
        double r720423 = r720422 / r720419;
        double r720424 = sin(r720419);
        double r720425 = r720423 * r720424;
        double r720426 = r720425 / r720419;
        return r720426;
}

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 *-un-lft-identity15.4

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

  7. Applied *-un-lft-identity15.4

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

  8. Applied distribute-lft-out15.4

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

  9. Applied times-frac15.4

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

  10. Applied times-frac0.3

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

  11. Simplified0.3

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

  12. Simplified0.1

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

  14. Applied associate-*l/0.1

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

  15. Final simplification0.1

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

Reproduce

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