Average Error: 28.0 → 6.0
Time: 52.8s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(x \cdot 2\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right) \cdot \left(\left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right) \cdot {cos}^{\left(\frac{2}{2}\right)}\right)\right)}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\cos \left(x \cdot 2\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right) \cdot \left(\left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right) \cdot {cos}^{\left(\frac{2}{2}\right)}\right)\right)}
double f(double x, double cos, double sin) {
        double r3151442 = 2.0;
        double r3151443 = x;
        double r3151444 = r3151442 * r3151443;
        double r3151445 = cos(r3151444);
        double r3151446 = cos;
        double r3151447 = pow(r3151446, r3151442);
        double r3151448 = sin;
        double r3151449 = pow(r3151448, r3151442);
        double r3151450 = r3151443 * r3151449;
        double r3151451 = r3151450 * r3151443;
        double r3151452 = r3151447 * r3151451;
        double r3151453 = r3151445 / r3151452;
        return r3151453;
}
↓
double f(double x, double cos, double sin) {
        double r3151454 = x;
        double r3151455 = 2.0;
        double r3151456 = r3151454 * r3151455;
        double r3151457 = cos(r3151456);
        double r3151458 = cos;
        double r3151459 = 2.0;
        double r3151460 = r3151455 / r3151459;
        double r3151461 = pow(r3151458, r3151460);
        double r3151462 = sin;
        double r3151463 = pow(r3151462, r3151460);
        double r3151464 = r3151463 * r3151454;
        double r3151465 = r3151464 * r3151461;
        double r3151466 = r3151464 * r3151465;
        double r3151467 = r3151461 * r3151466;
        double r3151468 = r3151457 / r3151467;
        return r3151468;
}

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. Using strategy rm

  3. Applied sqr-pow28.0

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

  4. Applied associate-*l*23.4

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

  6. Applied sqr-pow23.4

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

  7. Applied associate-*r*16.6

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

  9. Applied associate-*l*13.6

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

  11. Applied associate-*r*6.0

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

  12. Final simplification6.0

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  (/ (cos (* 2.0 x)) (* (pow cos 2.0) (* (* x (pow sin 2.0)) x))))