Average Error: 30.9 → 0.1
Time: 41.2s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x}
double f(double x) {
        double r2177612 = 1.0;
        double r2177613 = x;
        double r2177614 = cos(r2177613);
        double r2177615 = r2177612 - r2177614;
        double r2177616 = r2177613 * r2177613;
        double r2177617 = r2177615 / r2177616;
        return r2177617;
}
↓
double f(double x) {
        double r2177618 = x;
        double r2177619 = 2.0;
        double r2177620 = r2177618 / r2177619;
        double r2177621 = tan(r2177620);
        double r2177622 = r2177621 / r2177618;
        double r2177623 = sin(r2177618);
        double r2177624 = r2177623 / r2177618;
        double r2177625 = r2177622 * r2177624;
        return r2177625;
}

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. Simplified15.3

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

  6. Taylor expanded around -inf 15.3

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

  7. Simplified0.3

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

  8. Taylor expanded around -inf 0.3

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))