Average Error: 30.7 → 0.1
Time: 16.5s
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 r598880 = 1.0;
        double r598881 = x;
        double r598882 = cos(r598881);
        double r598883 = r598880 - r598882;
        double r598884 = r598881 * r598881;
        double r598885 = r598883 / r598884;
        return r598885;
}
↓
double f(double x) {
        double r598886 = x;
        double r598887 = 2.0;
        double r598888 = r598886 / r598887;
        double r598889 = tan(r598888);
        double r598890 = r598889 / r598886;
        double r598891 = sin(r598886);
        double r598892 = r598891 / r598886;
        double r598893 = r598890 * r598892;
        return r598893;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.7

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

  3. Applied flip--30.8

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

  4. Simplified15.9

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

  6. Applied *-un-lft-identity15.9

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

  7. Applied times-frac15.9

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

  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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019135 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))