Average Error: 27.7 → 2.7
Time: 23.6s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\cos x \cdot \cos x}{cos \cdot \left(sin \cdot x\right)} - \frac{\sin x \cdot \sin x}{cos \cdot \left(sin \cdot x\right)}}{cos \cdot \left(sin \cdot x\right)}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{\cos x \cdot \cos x}{cos \cdot \left(sin \cdot x\right)} - \frac{\sin x \cdot \sin x}{cos \cdot \left(sin \cdot x\right)}}{cos \cdot \left(sin \cdot x\right)}
double f(double x, double cos, double sin) {
        double r1797688 = 2.0;
        double r1797689 = x;
        double r1797690 = r1797688 * r1797689;
        double r1797691 = cos(r1797690);
        double r1797692 = cos;
        double r1797693 = pow(r1797692, r1797688);
        double r1797694 = sin;
        double r1797695 = pow(r1797694, r1797688);
        double r1797696 = r1797689 * r1797695;
        double r1797697 = r1797696 * r1797689;
        double r1797698 = r1797693 * r1797697;
        double r1797699 = r1797691 / r1797698;
        return r1797699;
}
↓
double f(double x, double cos, double sin) {
        double r1797700 = x;
        double r1797701 = cos(r1797700);
        double r1797702 = r1797701 * r1797701;
        double r1797703 = cos;
        double r1797704 = sin;
        double r1797705 = r1797704 * r1797700;
        double r1797706 = r1797703 * r1797705;
        double r1797707 = r1797702 / r1797706;
        double r1797708 = sin(r1797700);
        double r1797709 = r1797708 * r1797708;
        double r1797710 = r1797709 / r1797706;
        double r1797711 = r1797707 - r1797710;
        double r1797712 = r1797711 / r1797706;
        return r1797712;
}

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 27.7

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

  2. Simplified2.9

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

  3. Taylor expanded around 0 31.5

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

  4. Simplified2.9

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

  6. Applied associate-/r*2.6

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

  8. Applied cos-22.7

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

  9. Applied div-sub2.7

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

  10. Final simplification2.7

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

Reproduce

herbie shell --seed 2019134 
(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))))