Average Error: 27.8 → 2.6
Time: 13.7s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{sin}}{x}}{cos}}{\left(x \cdot sin\right) \cdot cos}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{sin}}{x}}{cos}}{\left(x \cdot sin\right) \cdot cos}
double f(double x, double cos, double sin) {
        double r875159 = 2.0;
        double r875160 = x;
        double r875161 = r875159 * r875160;
        double r875162 = cos(r875161);
        double r875163 = cos;
        double r875164 = pow(r875163, r875159);
        double r875165 = sin;
        double r875166 = pow(r875165, r875159);
        double r875167 = r875160 * r875166;
        double r875168 = r875167 * r875160;
        double r875169 = r875164 * r875168;
        double r875170 = r875162 / r875169;
        return r875170;
}
↓
double f(double x, double cos, double sin) {
        double r875171 = 2.0;
        double r875172 = x;
        double r875173 = r875171 * r875172;
        double r875174 = cos(r875173);
        double r875175 = sin;
        double r875176 = r875174 / r875175;
        double r875177 = r875176 / r875172;
        double r875178 = cos;
        double r875179 = r875177 / r875178;
        double r875180 = r875172 * r875175;
        double r875181 = r875180 * r875178;
        double r875182 = r875179 / r875181;
        return r875182;
}

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 27.8

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

  2. Simplified2.7

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

  4. Applied *-un-lft-identity2.7

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

  5. Applied times-frac2.5

    \[\leadsto \color{blue}{\frac{1}{\left(sin \cdot x\right) \cdot cos} \cdot \frac{\cos \left(2 \cdot x\right)}{\left(sin \cdot x\right) \cdot cos}}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity2.5

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

  8. Applied times-frac2.6

    \[\leadsto \frac{1}{\left(sin \cdot x\right) \cdot cos} \cdot \color{blue}{\left(\frac{1}{sin \cdot x} \cdot \frac{\cos \left(2 \cdot x\right)}{cos}\right)}\]
  9. Using strategy rm

  10. Applied associate-/r*2.6

    \[\leadsto \frac{1}{\left(sin \cdot x\right) \cdot cos} \cdot \left(\color{blue}{\frac{\frac{1}{sin}}{x}} \cdot \frac{\cos \left(2 \cdot x\right)}{cos}\right)\]
  11. Using strategy rm

  12. Applied associate-*l/2.6

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

  13. Simplified2.6

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

  14. Final simplification2.6

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

Reproduce

herbie shell --seed 2019156 
(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))))