Average Error: 31.2 → 0.4
Time: 37.6s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\log \left(e^{1 + \cos x}\right)}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\log \left(e^{1 + \cos x}\right)}
double f(double x) {
        double r2102125 = 1.0;
        double r2102126 = x;
        double r2102127 = cos(r2102126);
        double r2102128 = r2102125 - r2102127;
        double r2102129 = r2102126 * r2102126;
        double r2102130 = r2102128 / r2102129;
        return r2102130;
}
↓
double f(double x) {
        double r2102131 = x;
        double r2102132 = sin(r2102131);
        double r2102133 = r2102132 / r2102131;
        double r2102134 = r2102133 * r2102133;
        double r2102135 = 1.0;
        double r2102136 = cos(r2102131);
        double r2102137 = r2102135 + r2102136;
        double r2102138 = exp(r2102137);
        double r2102139 = log(r2102138);
        double r2102140 = r2102134 / r2102139;
        return r2102140;
}

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. Applied associate-/l/31.3

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

  5. Simplified15.5

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

  6. Taylor expanded around inf 15.5

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

  7. Simplified0.3

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

  9. Applied add-log-exp0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))