Average Error: 27.0 → 2.6
Time: 1.7m
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(x \cdot 2\right)}{\left(sin \cdot x\right) \cdot cos} \cdot \frac{1}{\left(sin \cdot x\right) \cdot cos}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\cos \left(x \cdot 2\right)}{\left(sin \cdot x\right) \cdot cos} \cdot \frac{1}{\left(sin \cdot x\right) \cdot cos}
double f(double x, double cos, double sin) {
        double r12827859 = 2.0;
        double r12827860 = x;
        double r12827861 = r12827859 * r12827860;
        double r12827862 = cos(r12827861);
        double r12827863 = cos;
        double r12827864 = pow(r12827863, r12827859);
        double r12827865 = sin;
        double r12827866 = pow(r12827865, r12827859);
        double r12827867 = r12827860 * r12827866;
        double r12827868 = r12827867 * r12827860;
        double r12827869 = r12827864 * r12827868;
        double r12827870 = r12827862 / r12827869;
        return r12827870;
}
↓
double f(double x, double cos, double sin) {
        double r12827871 = x;
        double r12827872 = 2.0;
        double r12827873 = r12827871 * r12827872;
        double r12827874 = cos(r12827873);
        double r12827875 = sin;
        double r12827876 = r12827875 * r12827871;
        double r12827877 = cos;
        double r12827878 = r12827876 * r12827877;
        double r12827879 = r12827874 / r12827878;
        double r12827880 = 1.0;
        double r12827881 = r12827880 / r12827878;
        double r12827882 = r12827879 * r12827881;
        return r12827882;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 27.0

    \[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]

  2. Simplified2.6

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

  3. Taylor expanded around -inf 30.7

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

  4. Simplified2.9

    \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot sin\right) \cdot cos\right) \cdot \left(\left(x \cdot sin\right) \cdot cos\right)}}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity2.9

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

  7. Applied times-frac2.6

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

  8. Final simplification2.6

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))