Average Error: 27.7 → 2.9
Time: 41.6s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{\left(x \cdot sin\right) \cdot cos} \cdot \left(\sqrt[3]{\left(x \cdot sin\right) \cdot cos} \cdot \sqrt[3]{\left(x \cdot sin\right) \cdot cos}\right)} \cdot \frac{1}{\left(x \cdot sin\right) \cdot cos}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{\left(x \cdot sin\right) \cdot cos} \cdot \left(\sqrt[3]{\left(x \cdot sin\right) \cdot cos} \cdot \sqrt[3]{\left(x \cdot sin\right) \cdot cos}\right)} \cdot \frac{1}{\left(x \cdot sin\right) \cdot cos}
double f(double x, double cos, double sin) {
        double r2837520 = 2.0;
        double r2837521 = x;
        double r2837522 = r2837520 * r2837521;
        double r2837523 = cos(r2837522);
        double r2837524 = cos;
        double r2837525 = pow(r2837524, r2837520);
        double r2837526 = sin;
        double r2837527 = pow(r2837526, r2837520);
        double r2837528 = r2837521 * r2837527;
        double r2837529 = r2837528 * r2837521;
        double r2837530 = r2837525 * r2837529;
        double r2837531 = r2837523 / r2837530;
        return r2837531;
}
↓
double f(double x, double cos, double sin) {
        double r2837532 = 2.0;
        double r2837533 = x;
        double r2837534 = r2837532 * r2837533;
        double r2837535 = cos(r2837534);
        double r2837536 = sin;
        double r2837537 = r2837533 * r2837536;
        double r2837538 = cos;
        double r2837539 = r2837537 * r2837538;
        double r2837540 = cbrt(r2837539);
        double r2837541 = r2837540 * r2837540;
        double r2837542 = r2837540 * r2837541;
        double r2837543 = r2837535 / r2837542;
        double r2837544 = 1.0;
        double r2837545 = r2837544 / r2837539;
        double r2837546 = r2837543 * r2837545;
        return r2837546;
}

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. Simplified2.8

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

  4. Applied *-un-lft-identity2.8

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

  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 add-cube-cbrt2.9

    \[\leadsto \frac{1}{\left(sin \cdot x\right) \cdot cos} \cdot \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\sqrt[3]{\left(sin \cdot x\right) \cdot cos} \cdot \sqrt[3]{\left(sin \cdot x\right) \cdot cos}\right) \cdot \sqrt[3]{\left(sin \cdot x\right) \cdot cos}}}\]

  8. Final simplification2.9

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{\left(x \cdot sin\right) \cdot cos} \cdot \left(\sqrt[3]{\left(x \cdot sin\right) \cdot cos} \cdot \sqrt[3]{\left(x \cdot sin\right) \cdot cos}\right)} \cdot \frac{1}{\left(x \cdot sin\right) \cdot cos}\]

Reproduce

herbie shell --seed 2019144 +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))))