Average Error: 27.0 → 2.6
Time: 1.7m
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot sin\right) \cdot cos} \cdot \frac{1}{\left(x \cdot sin\right) \cdot cos}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot sin\right) \cdot cos} \cdot \frac{1}{\left(x \cdot sin\right) \cdot cos}
double f(double x, double cos, double sin) {
        double r12885864 = 2.0;
        double r12885865 = x;
        double r12885866 = r12885864 * r12885865;
        double r12885867 = cos(r12885866);
        double r12885868 = cos;
        double r12885869 = pow(r12885868, r12885864);
        double r12885870 = sin;
        double r12885871 = pow(r12885870, r12885864);
        double r12885872 = r12885865 * r12885871;
        double r12885873 = r12885872 * r12885865;
        double r12885874 = r12885869 * r12885873;
        double r12885875 = r12885867 / r12885874;
        return r12885875;
}
↓
double f(double x, double cos, double sin) {
        double r12885876 = 2.0;
        double r12885877 = x;
        double r12885878 = r12885876 * r12885877;
        double r12885879 = cos(r12885878);
        double r12885880 = sin;
        double r12885881 = r12885877 * r12885880;
        double r12885882 = cos;
        double r12885883 = r12885881 * r12885882;
        double r12885884 = r12885879 / r12885883;
        double r12885885 = 1.0;
        double r12885886 = r12885885 / r12885883;
        double r12885887 = r12885884 * r12885886;
        return r12885887;
}

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 associate-/r*2.6

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

  8. Applied div-inv2.6

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

  9. Final simplification2.6

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

Reproduce

herbie shell --seed 2019104 
(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))))