Average Error: 31.2 → 0.4
Time: 35.4s
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 r1991115 = 1.0;
        double r1991116 = x;
        double r1991117 = cos(r1991116);
        double r1991118 = r1991115 - r1991117;
        double r1991119 = r1991116 * r1991116;
        double r1991120 = r1991118 / r1991119;
        return r1991120;
}
↓
double f(double x) {
        double r1991121 = x;
        double r1991122 = sin(r1991121);
        double r1991123 = r1991122 / r1991121;
        double r1991124 = r1991123 * r1991123;
        double r1991125 = 1.0;
        double r1991126 = cos(r1991121);
        double r1991127 = r1991125 + r1991126;
        double r1991128 = exp(r1991127);
        double r1991129 = log(r1991128);
        double r1991130 = r1991124 / r1991129;
        return r1991130;
}

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 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))