Average Error: 28.0 → 3.0
Time: 33.4s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot sin\right) \cdot cos}} \cdot \sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot sin\right) \cdot cos}}}{\frac{\left(x \cdot sin\right) \cdot cos}{\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{\left(x \cdot sin\right) \cdot cos} \cdot \left(\sqrt[3]{\left(x \cdot sin\right) \cdot cos} \cdot \sqrt[3]{\left(x \cdot sin\right) \cdot cos}\right)}}}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot sin\right) \cdot cos}} \cdot \sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot sin\right) \cdot cos}}}{\frac{\left(x \cdot sin\right) \cdot cos}{\sqrt[3]{\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)}}}}
double f(double x, double cos, double sin) {
        double r2378476 = 2.0;
        double r2378477 = x;
        double r2378478 = r2378476 * r2378477;
        double r2378479 = cos(r2378478);
        double r2378480 = cos;
        double r2378481 = pow(r2378480, r2378476);
        double r2378482 = sin;
        double r2378483 = pow(r2378482, r2378476);
        double r2378484 = r2378477 * r2378483;
        double r2378485 = r2378484 * r2378477;
        double r2378486 = r2378481 * r2378485;
        double r2378487 = r2378479 / r2378486;
        return r2378487;
}
↓
double f(double x, double cos, double sin) {
        double r2378488 = 2.0;
        double r2378489 = x;
        double r2378490 = r2378488 * r2378489;
        double r2378491 = cos(r2378490);
        double r2378492 = sin;
        double r2378493 = r2378489 * r2378492;
        double r2378494 = cos;
        double r2378495 = r2378493 * r2378494;
        double r2378496 = r2378491 / r2378495;
        double r2378497 = cbrt(r2378496);
        double r2378498 = r2378497 * r2378497;
        double r2378499 = cbrt(r2378495);
        double r2378500 = r2378499 * r2378499;
        double r2378501 = r2378499 * r2378500;
        double r2378502 = r2378491 / r2378501;
        double r2378503 = cbrt(r2378502);
        double r2378504 = r2378495 / r2378503;
        double r2378505 = r2378498 / r2378504;
        return r2378505;
}

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. Simplified2.9

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

  4. Applied associate-/r*2.6

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

  6. Applied add-cube-cbrt3.0

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

  7. Applied associate-/l*3.0

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

  9. Applied add-cube-cbrt3.0

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

  10. Final simplification3.0

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

Reproduce

herbie shell --seed 2019151 +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))))