Average Error: 31.0 → 0.3
Time: 1.1m
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{1 + \cos x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{1 + \cos x}
double f(double x) {
        double r5450741 = 1.0;
        double r5450742 = x;
        double r5450743 = cos(r5450742);
        double r5450744 = r5450741 - r5450743;
        double r5450745 = r5450742 * r5450742;
        double r5450746 = r5450744 / r5450745;
        return r5450746;
}
↓
double f(double x) {
        double r5450747 = x;
        double r5450748 = sin(r5450747);
        double r5450749 = r5450748 / r5450747;
        double r5450750 = r5450749 * r5450749;
        double r5450751 = 1.0;
        double r5450752 = cos(r5450747);
        double r5450753 = r5450751 + r5450752;
        double r5450754 = r5450750 / r5450753;
        return r5450754;
}

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. Simplified14.9

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

  6. Taylor expanded around -inf 14.9

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019107 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))