Average Error: 27.2 → 2.9
Time: 16.4s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\cos \left(2 \cdot x\right)}{\left(\left(\left(cos \cdot x\right) \cdot \sqrt[3]{sin}\right) \cdot \sqrt[3]{sin}\right) \cdot \sqrt[3]{sin}}}{\left(cos \cdot x\right) \cdot sin}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{\cos \left(2 \cdot x\right)}{\left(\left(\left(cos \cdot x\right) \cdot \sqrt[3]{sin}\right) \cdot \sqrt[3]{sin}\right) \cdot \sqrt[3]{sin}}}{\left(cos \cdot x\right) \cdot sin}
double f(double x, double cos, double sin) {
        double r970369 = 2.0;
        double r970370 = x;
        double r970371 = r970369 * r970370;
        double r970372 = cos(r970371);
        double r970373 = cos;
        double r970374 = pow(r970373, r970369);
        double r970375 = sin;
        double r970376 = pow(r970375, r970369);
        double r970377 = r970370 * r970376;
        double r970378 = r970377 * r970370;
        double r970379 = r970374 * r970378;
        double r970380 = r970372 / r970379;
        return r970380;
}
↓
double f(double x, double cos, double sin) {
        double r970381 = 2.0;
        double r970382 = x;
        double r970383 = r970381 * r970382;
        double r970384 = cos(r970383);
        double r970385 = cos;
        double r970386 = r970385 * r970382;
        double r970387 = sin;
        double r970388 = cbrt(r970387);
        double r970389 = r970386 * r970388;
        double r970390 = r970389 * r970388;
        double r970391 = r970390 * r970388;
        double r970392 = r970384 / r970391;
        double r970393 = r970386 * r970387;
        double r970394 = r970392 / r970393;
        return r970394;
}

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 27.2

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

  2. Simplified2.8

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

  4. Applied associate-/r*2.6

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

  6. Applied add-cube-cbrt2.9

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

  7. Applied associate-*l*2.9

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

  9. Applied associate-*l*2.9

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

  10. Final simplification2.9

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

Reproduce

herbie shell --seed 2019128 +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))))