Average Error: 26.9 → 2.6
Time: 20.3s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\left(\cos x + \sin x\right) \cdot \left(\cos x - \sin x\right)}{\left(x \cdot sin\right) \cdot cos}}{\left(x \cdot sin\right) \cdot cos}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{\left(\cos x + \sin x\right) \cdot \left(\cos x - \sin x\right)}{\left(x \cdot sin\right) \cdot cos}}{\left(x \cdot sin\right) \cdot cos}
double f(double x, double cos, double sin) {
        double r2486294 = 2.0;
        double r2486295 = x;
        double r2486296 = r2486294 * r2486295;
        double r2486297 = cos(r2486296);
        double r2486298 = cos;
        double r2486299 = pow(r2486298, r2486294);
        double r2486300 = sin;
        double r2486301 = pow(r2486300, r2486294);
        double r2486302 = r2486295 * r2486301;
        double r2486303 = r2486302 * r2486295;
        double r2486304 = r2486299 * r2486303;
        double r2486305 = r2486297 / r2486304;
        return r2486305;
}
↓
double f(double x, double cos, double sin) {
        double r2486306 = x;
        double r2486307 = cos(r2486306);
        double r2486308 = sin(r2486306);
        double r2486309 = r2486307 + r2486308;
        double r2486310 = r2486307 - r2486308;
        double r2486311 = r2486309 * r2486310;
        double r2486312 = sin;
        double r2486313 = r2486306 * r2486312;
        double r2486314 = cos;
        double r2486315 = r2486313 * r2486314;
        double r2486316 = r2486311 / r2486315;
        double r2486317 = r2486316 / r2486315;
        return r2486317;
}

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 26.9

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

  2. Simplified12.9

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

  4. Applied div-inv12.9

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

  5. Applied associate-/l*13.1

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

  6. Simplified2.8

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

  8. Applied associate-/r*2.5

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

  10. Applied cos-22.6

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

  12. Applied difference-of-squares2.6

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

  13. Final simplification2.6

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

Reproduce

herbie shell --seed 2019165 
(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))))