Average Error: 27.9 → 2.6
Time: 16.4s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{sin}}{x \cdot cos}}{\left(x \cdot cos\right) \cdot sin}\]
\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(x \cdot 2\right)}{sin}}{x \cdot cos}}{\left(x \cdot cos\right) \cdot sin}
double f(double x, double cos, double sin) {
        double r1300720 = 2.0;
        double r1300721 = x;
        double r1300722 = r1300720 * r1300721;
        double r1300723 = cos(r1300722);
        double r1300724 = cos;
        double r1300725 = pow(r1300724, r1300720);
        double r1300726 = sin;
        double r1300727 = pow(r1300726, r1300720);
        double r1300728 = r1300721 * r1300727;
        double r1300729 = r1300728 * r1300721;
        double r1300730 = r1300725 * r1300729;
        double r1300731 = r1300723 / r1300730;
        return r1300731;
}
↓
double f(double x, double cos, double sin) {
        double r1300732 = x;
        double r1300733 = 2.0;
        double r1300734 = r1300732 * r1300733;
        double r1300735 = cos(r1300734);
        double r1300736 = sin;
        double r1300737 = r1300735 / r1300736;
        double r1300738 = cos;
        double r1300739 = r1300732 * r1300738;
        double r1300740 = r1300737 / r1300739;
        double r1300741 = r1300739 * r1300736;
        double r1300742 = r1300740 / r1300741;
        return r1300742;
}

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. Using strategy rm

  8. Applied *-commutative2.6

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

  9. Final simplification2.6

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

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))))