Average Error: 31.2 → 0.6
Time: 15.6s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{1}{\left(\cos x + 1\right) \cdot \left(\frac{x}{\sin x} \cdot \frac{x}{\sin x}\right)}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{1}{\left(\cos x + 1\right) \cdot \left(\frac{x}{\sin x} \cdot \frac{x}{\sin x}\right)}
double f(double x) {
        double r521314 = 1.0;
        double r521315 = x;
        double r521316 = cos(r521315);
        double r521317 = r521314 - r521316;
        double r521318 = r521315 * r521315;
        double r521319 = r521317 / r521318;
        return r521319;
}
↓
double f(double x) {
        double r521320 = 1.0;
        double r521321 = x;
        double r521322 = cos(r521321);
        double r521323 = r521322 + r521320;
        double r521324 = sin(r521321);
        double r521325 = r521321 / r521324;
        double r521326 = r521325 * r521325;
        double r521327 = r521323 * r521326;
        double r521328 = r521320 / r521327;
        return r521328;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.2

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

  3. Applied flip--31.3

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

  4. Simplified15.4

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

  6. Applied clear-num15.4

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

  7. Simplified0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2019132 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))