Average Error: 27.7 → 3.0
Time: 22.7s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot cos\right) \cdot sin}}{\left(x \cdot cos\right) \cdot sin}} \cdot \left(\sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot cos\right) \cdot sin}}{\left(x \cdot cos\right) \cdot sin}} \cdot \sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot cos\right) \cdot sin}}{\left(x \cdot cos\right) \cdot sin}}\right)\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot cos\right) \cdot sin}}{\left(x \cdot cos\right) \cdot sin}} \cdot \left(\sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot cos\right) \cdot sin}}{\left(x \cdot cos\right) \cdot sin}} \cdot \sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot cos\right) \cdot sin}}{\left(x \cdot cos\right) \cdot sin}}\right)
double f(double x, double cos, double sin) {
        double r1593191 = 2.0;
        double r1593192 = x;
        double r1593193 = r1593191 * r1593192;
        double r1593194 = cos(r1593193);
        double r1593195 = cos;
        double r1593196 = pow(r1593195, r1593191);
        double r1593197 = sin;
        double r1593198 = pow(r1593197, r1593191);
        double r1593199 = r1593192 * r1593198;
        double r1593200 = r1593199 * r1593192;
        double r1593201 = r1593196 * r1593200;
        double r1593202 = r1593194 / r1593201;
        return r1593202;
}
↓
double f(double x, double cos, double sin) {
        double r1593203 = 2.0;
        double r1593204 = x;
        double r1593205 = r1593203 * r1593204;
        double r1593206 = cos(r1593205);
        double r1593207 = cos;
        double r1593208 = r1593204 * r1593207;
        double r1593209 = sin;
        double r1593210 = r1593208 * r1593209;
        double r1593211 = r1593206 / r1593210;
        double r1593212 = r1593211 / r1593210;
        double r1593213 = cbrt(r1593212);
        double r1593214 = r1593213 * r1593213;
        double r1593215 = r1593213 * r1593214;
        return r1593215;
}

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. Using strategy rm

  4. Applied associate-/r*2.6

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

  6. Applied add-cube-cbrt3.0

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

  7. Final simplification3.0

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

Reproduce

herbie shell --seed 2019134 +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))))