Average Error: 31.0 → 0.1
Time: 34.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{x}}{\frac{\cos \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{x}}}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{x}}{\frac{\cos \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{x}}}
double f(double x) {
        double r2124880 = 1.0;
        double r2124881 = x;
        double r2124882 = cos(r2124881);
        double r2124883 = r2124880 - r2124882;
        double r2124884 = r2124881 * r2124881;
        double r2124885 = r2124883 / r2124884;
        return r2124885;
}
↓
double f(double x) {
        double r2124886 = x;
        double r2124887 = 0.5;
        double r2124888 = r2124886 * r2124887;
        double r2124889 = sin(r2124888);
        double r2124890 = r2124889 / r2124886;
        double r2124891 = cos(r2124888);
        double r2124892 = sin(r2124886);
        double r2124893 = r2124892 / r2124886;
        double r2124894 = r2124891 / r2124893;
        double r2124895 = r2124890 / r2124894;
        return r2124895;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.0

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

  3. Applied flip--31.1

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

  4. Applied associate-/l/31.1

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

  5. Simplified15.6

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

  7. Applied times-frac15.9

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

  8. Simplified15.8

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

  9. Taylor expanded around inf 15.5

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))