Average Error: 28.0 → 2.9
Time: 44.6s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\left(cos \cdot sin\right) \cdot x} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right)}}{x} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{cos \cdot sin}\right)\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{1}{\left(cos \cdot sin\right) \cdot x} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right)}}{x} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{cos \cdot sin}\right)
double f(double x, double cos, double sin) {
        double r1932497 = 2.0;
        double r1932498 = x;
        double r1932499 = r1932497 * r1932498;
        double r1932500 = cos(r1932499);
        double r1932501 = cos;
        double r1932502 = pow(r1932501, r1932497);
        double r1932503 = sin;
        double r1932504 = pow(r1932503, r1932497);
        double r1932505 = r1932498 * r1932504;
        double r1932506 = r1932505 * r1932498;
        double r1932507 = r1932502 * r1932506;
        double r1932508 = r1932500 / r1932507;
        return r1932508;
}
↓
double f(double x, double cos, double sin) {
        double r1932509 = 1.0;
        double r1932510 = cos;
        double r1932511 = sin;
        double r1932512 = r1932510 * r1932511;
        double r1932513 = x;
        double r1932514 = r1932512 * r1932513;
        double r1932515 = r1932509 / r1932514;
        double r1932516 = 2.0;
        double r1932517 = r1932516 * r1932513;
        double r1932518 = cos(r1932517);
        double r1932519 = cbrt(r1932518);
        double r1932520 = r1932519 * r1932519;
        double r1932521 = r1932519 * r1932520;
        double r1932522 = cbrt(r1932521);
        double r1932523 = r1932522 / r1932513;
        double r1932524 = r1932520 / r1932512;
        double r1932525 = r1932523 * r1932524;
        double r1932526 = r1932515 * r1932525;
        return r1932526;
}

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 28.0

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

  2. Simplified2.8

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

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

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

  5. Applied times-frac2.6

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

  7. Applied add-cube-cbrt2.7

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

  8. Applied times-frac2.9

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

  10. Applied add-cbrt-cube2.9

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

  11. Final simplification2.9

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

Reproduce

herbie shell --seed 2019130 +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))))