Average Error: 30.9 → 0.6
Time: 15.0s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{1}{\left(1 + \cos x\right) \cdot \left(\frac{x}{\sin x} \cdot \frac{x}{\sin x}\right)}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{1}{\left(1 + \cos x\right) \cdot \left(\frac{x}{\sin x} \cdot \frac{x}{\sin x}\right)}
double f(double x) {
        double r923888 = 1.0;
        double r923889 = x;
        double r923890 = cos(r923889);
        double r923891 = r923888 - r923890;
        double r923892 = r923889 * r923889;
        double r923893 = r923891 / r923892;
        return r923893;
}
↓
double f(double x) {
        double r923894 = 1.0;
        double r923895 = x;
        double r923896 = cos(r923895);
        double r923897 = r923894 + r923896;
        double r923898 = sin(r923895);
        double r923899 = r923895 / r923898;
        double r923900 = r923899 * r923899;
        double r923901 = r923897 * r923900;
        double r923902 = r923894 / r923901;
        return r923902;
}

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. Simplified15.6

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

  6. Applied clear-num15.7

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

  7. Simplified0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))