Average Error: 27.8 → 2.6
Time: 41.7s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot cos\right) \cdot sin\right) \cdot \left(\left(x \cdot cos\right) \cdot sin\right)}\]
\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)}{\left(\left(x \cdot cos\right) \cdot sin\right) \cdot \left(\left(x \cdot cos\right) \cdot sin\right)}
double f(double x, double cos, double sin) {
        double r2953451 = 2.0;
        double r2953452 = x;
        double r2953453 = r2953451 * r2953452;
        double r2953454 = cos(r2953453);
        double r2953455 = cos;
        double r2953456 = pow(r2953455, r2953451);
        double r2953457 = sin;
        double r2953458 = pow(r2953457, r2953451);
        double r2953459 = r2953452 * r2953458;
        double r2953460 = r2953459 * r2953452;
        double r2953461 = r2953456 * r2953460;
        double r2953462 = r2953454 / r2953461;
        return r2953462;
}
↓
double f(double x, double cos, double sin) {
        double r2953463 = 2.0;
        double r2953464 = x;
        double r2953465 = r2953463 * r2953464;
        double r2953466 = cos(r2953465);
        double r2953467 = cos;
        double r2953468 = r2953464 * r2953467;
        double r2953469 = sin;
        double r2953470 = r2953468 * r2953469;
        double r2953471 = r2953470 * r2953470;
        double r2953472 = r2953466 / r2953471;
        return r2953472;
}

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 27.8

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

  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. Taylor expanded around 0 31.0

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

  4. Simplified2.6

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

  6. Applied *-un-lft-identity2.6

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

  7. Applied associate-/l*2.6

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

  8. Taylor expanded around -inf 31.0

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

  9. Simplified2.6

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

  10. Final simplification2.6

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

Reproduce

herbie shell --seed 2019143 +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))))