Average Error: 30.7 → 0.1
Time: 40.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{x} \cdot \tan \left(\frac{x}{2}\right)}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin x}{x} \cdot \tan \left(\frac{x}{2}\right)}{x}
double f(double x) {
        double r1051808 = 1.0;
        double r1051809 = x;
        double r1051810 = cos(r1051809);
        double r1051811 = r1051808 - r1051810;
        double r1051812 = r1051809 * r1051809;
        double r1051813 = r1051811 / r1051812;
        return r1051813;
}
↓
double f(double x) {
        double r1051814 = x;
        double r1051815 = sin(r1051814);
        double r1051816 = r1051815 / r1051814;
        double r1051817 = 2.0;
        double r1051818 = r1051814 / r1051817;
        double r1051819 = tan(r1051818);
        double r1051820 = r1051816 * r1051819;
        double r1051821 = r1051820 / r1051814;
        return r1051821;
}

Error

Bits error versus x

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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. Simplified14.9

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

  6. Applied associate-/r*14.8

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))