Average Error: 27.9 → 2.6
Time: 21.0s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{sin}}{cos \cdot x}}{\left(cos \cdot x\right) \cdot sin}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{sin}}{cos \cdot x}}{\left(cos \cdot x\right) \cdot sin}
double f(double x, double cos, double sin) {
        double r919695 = 2.0;
        double r919696 = x;
        double r919697 = r919695 * r919696;
        double r919698 = cos(r919697);
        double r919699 = cos;
        double r919700 = pow(r919699, r919695);
        double r919701 = sin;
        double r919702 = pow(r919701, r919695);
        double r919703 = r919696 * r919702;
        double r919704 = r919703 * r919696;
        double r919705 = r919700 * r919704;
        double r919706 = r919698 / r919705;
        return r919706;
}
↓
double f(double x, double cos, double sin) {
        double r919707 = 2.0;
        double r919708 = x;
        double r919709 = r919707 * r919708;
        double r919710 = cos(r919709);
        double r919711 = sin;
        double r919712 = r919710 / r919711;
        double r919713 = cos;
        double r919714 = r919713 * r919708;
        double r919715 = r919712 / r919714;
        double r919716 = r919714 * r919711;
        double r919717 = r919715 / r919716;
        return r919717;
}

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 27.9

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

  2. Simplified2.9

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

  3. Taylor expanded around -inf 31.4

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

  4. Simplified2.6

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

  6. Applied associate-/r*2.6

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

  7. Taylor expanded around inf 2.6

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

  8. Final simplification2.6

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

Reproduce

herbie shell --seed 2019153 
(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))))