Average Error: 27.8 → 2.7
Time: 34.7s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{1}{sin \cdot x} \cdot \frac{\cos \left(2 \cdot x\right)}{cos}}{\left(sin \cdot x\right) \cdot cos}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{1}{sin \cdot x} \cdot \frac{\cos \left(2 \cdot x\right)}{cos}}{\left(sin \cdot x\right) \cdot cos}
double f(double x, double cos, double sin) {
        double r2494980 = 2.0;
        double r2494981 = x;
        double r2494982 = r2494980 * r2494981;
        double r2494983 = cos(r2494982);
        double r2494984 = cos;
        double r2494985 = pow(r2494984, r2494980);
        double r2494986 = sin;
        double r2494987 = pow(r2494986, r2494980);
        double r2494988 = r2494981 * r2494987;
        double r2494989 = r2494988 * r2494981;
        double r2494990 = r2494985 * r2494989;
        double r2494991 = r2494983 / r2494990;
        return r2494991;
}
↓
double f(double x, double cos, double sin) {
        double r2494992 = 1.0;
        double r2494993 = sin;
        double r2494994 = x;
        double r2494995 = r2494993 * r2494994;
        double r2494996 = r2494992 / r2494995;
        double r2494997 = 2.0;
        double r2494998 = r2494997 * r2494994;
        double r2494999 = cos(r2494998);
        double r2495000 = cos;
        double r2495001 = r2494999 / r2495000;
        double r2495002 = r2494996 * r2495001;
        double r2495003 = r2494995 * r2495000;
        double r2495004 = r2495002 / r2495003;
        return r2495004;
}

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. Simplified2.8

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

  4. Applied associate-/r*2.5

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

  6. Applied *-un-lft-identity2.5

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

  7. Applied times-frac2.7

    \[\leadsto \frac{\color{blue}{\frac{1}{sin \cdot x} \cdot \frac{\cos \left(2 \cdot x\right)}{cos}}}{\left(sin \cdot x\right) \cdot cos}\]
  8. Using strategy rm

  9. Applied associate-/r*2.7

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

  10. Taylor expanded around -inf 2.7

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

  11. Final simplification2.7

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

Reproduce

herbie shell --seed 2019142 
(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))))