Average Error: 31.7 → 0.1
Time: 51.4s
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 r878564 = 1.0;
        double r878565 = x;
        double r878566 = cos(r878565);
        double r878567 = r878564 - r878566;
        double r878568 = r878565 * r878565;
        double r878569 = r878567 / r878568;
        return r878569;
}
↓
double f(double x) {
        double r878570 = x;
        double r878571 = 2.0;
        double r878572 = r878570 / r878571;
        double r878573 = tan(r878572);
        double r878574 = r878573 / r878570;
        double r878575 = sin(r878570);
        double r878576 = r878574 * r878575;
        double r878577 = r878576 / r878570;
        return r878577;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.7

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

  3. Applied flip--31.8

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

  4. Simplified15.5

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

  6. Applied *-un-lft-identity15.5

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

  7. Applied times-frac15.5

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

  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 div-inv0.2

    \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(\tan \left(\frac{x}{2}\right) \cdot \frac{1}{x}\right)}\]
  13. Using strategy rm

  14. Applied associate-*l/0.2

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

  15. Simplified0.1

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

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))