Average Error: 26.9 → 2.6
Time: 15.8s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\sqrt[3]{\cos \left(x \cdot 2\right) \cdot \left(\cos \left(x \cdot 2\right) \cdot \cos \left(x \cdot 2\right)\right)}}{\left(x \cdot sin\right) \cdot cos}}{\left(x \cdot sin\right) \cdot cos}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{\sqrt[3]{\cos \left(x \cdot 2\right) \cdot \left(\cos \left(x \cdot 2\right) \cdot \cos \left(x \cdot 2\right)\right)}}{\left(x \cdot sin\right) \cdot cos}}{\left(x \cdot sin\right) \cdot cos}
double f(double x, double cos, double sin) {
        double r1996680 = 2.0;
        double r1996681 = x;
        double r1996682 = r1996680 * r1996681;
        double r1996683 = cos(r1996682);
        double r1996684 = cos;
        double r1996685 = pow(r1996684, r1996680);
        double r1996686 = sin;
        double r1996687 = pow(r1996686, r1996680);
        double r1996688 = r1996681 * r1996687;
        double r1996689 = r1996688 * r1996681;
        double r1996690 = r1996685 * r1996689;
        double r1996691 = r1996683 / r1996690;
        return r1996691;
}
↓
double f(double x, double cos, double sin) {
        double r1996692 = x;
        double r1996693 = 2.0;
        double r1996694 = r1996692 * r1996693;
        double r1996695 = cos(r1996694);
        double r1996696 = r1996695 * r1996695;
        double r1996697 = r1996695 * r1996696;
        double r1996698 = cbrt(r1996697);
        double r1996699 = sin;
        double r1996700 = r1996692 * r1996699;
        double r1996701 = cos;
        double r1996702 = r1996700 * r1996701;
        double r1996703 = r1996698 / r1996702;
        double r1996704 = r1996703 / r1996702;
        return r1996704;
}

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 26.9

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

  2. Simplified12.9

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

  3. Taylor expanded around inf 30.5

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

  4. Simplified2.8

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

  6. Applied associate-/r*2.5

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

  8. Applied add-cbrt-cube2.6

    \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{\left(\cos \left(x \cdot 2\right) \cdot \cos \left(x \cdot 2\right)\right) \cdot \cos \left(x \cdot 2\right)}}}{cos \cdot \left(x \cdot sin\right)}}{cos \cdot \left(x \cdot sin\right)}\]

  9. Final simplification2.6

    \[\leadsto \frac{\frac{\sqrt[3]{\cos \left(x \cdot 2\right) \cdot \left(\cos \left(x \cdot 2\right) \cdot \cos \left(x \cdot 2\right)\right)}}{\left(x \cdot sin\right) \cdot cos}}{\left(x \cdot sin\right) \cdot cos}\]

Reproduce

herbie shell --seed 2019165 +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))))