Average Error: 30.9 → 0.2
Time: 49.8s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\tan \left(\frac{x}{2}\right) \cdot \frac{\frac{\sin x}{x}}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\tan \left(\frac{x}{2}\right) \cdot \frac{\frac{\sin x}{x}}{x}
double f(double x) {
        double r3800538 = 1.0;
        double r3800539 = x;
        double r3800540 = cos(r3800539);
        double r3800541 = r3800538 - r3800540;
        double r3800542 = r3800539 * r3800539;
        double r3800543 = r3800541 / r3800542;
        return r3800543;
}
↓
double f(double x) {
        double r3800544 = x;
        double r3800545 = 2.0;
        double r3800546 = r3800544 / r3800545;
        double r3800547 = tan(r3800546);
        double r3800548 = sin(r3800544);
        double r3800549 = r3800548 / r3800544;
        double r3800550 = r3800549 / r3800544;
        double r3800551 = r3800547 * r3800550;
        return r3800551;
}

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. Simplified14.7

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

  6. Taylor expanded around -inf 14.7

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

  7. Simplified0.3

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

  9. Applied associate-*l/0.3

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

  10. Applied associate-/l/0.3

    \[\leadsto \color{blue}{\frac{\sin x \cdot \frac{\sin x}{x}}{\left(\cos x + 1\right) \cdot x}}\]
  11. Using strategy rm

  12. Applied times-frac0.4

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

  13. Simplified0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))