Average Error: 30.9 → 0.4
Time: 38.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\log \left(e^{\cos x} \cdot e\right)}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\log \left(e^{\cos x} \cdot e\right)}
double f(double x) {
        double r2145857 = 1.0;
        double r2145858 = x;
        double r2145859 = cos(r2145858);
        double r2145860 = r2145857 - r2145859;
        double r2145861 = r2145858 * r2145858;
        double r2145862 = r2145860 / r2145861;
        return r2145862;
}
↓
double f(double x) {
        double r2145863 = x;
        double r2145864 = sin(r2145863);
        double r2145865 = r2145864 / r2145863;
        double r2145866 = r2145865 * r2145865;
        double r2145867 = cos(r2145863);
        double r2145868 = exp(r2145867);
        double r2145869 = exp(1.0);
        double r2145870 = r2145868 * r2145869;
        double r2145871 = log(r2145870);
        double r2145872 = r2145866 / r2145871;
        return r2145872;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.9

    \[\frac{1 - \cos x}{x \cdot x}\]
  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. Simplified15.4

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

  6. Taylor expanded around inf 15.4

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

  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-exp0.3

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

  10. Applied add-log-exp0.3

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

  11. Applied sum-log0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))