Average Error: 27.3 → 2.6
Time: 3.2m
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\left(cos \cdot x\right) \cdot sin} \cdot \frac{1}{\frac{cos \cdot x}{\frac{\cos \left(x \cdot 2\right)}{sin}}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{1}{\left(cos \cdot x\right) \cdot sin} \cdot \frac{1}{\frac{cos \cdot x}{\frac{\cos \left(x \cdot 2\right)}{sin}}}
double f(double x, double cos, double sin) {
        double r23094087 = 2.0;
        double r23094088 = x;
        double r23094089 = r23094087 * r23094088;
        double r23094090 = cos(r23094089);
        double r23094091 = cos;
        double r23094092 = pow(r23094091, r23094087);
        double r23094093 = sin;
        double r23094094 = pow(r23094093, r23094087);
        double r23094095 = r23094088 * r23094094;
        double r23094096 = r23094095 * r23094088;
        double r23094097 = r23094092 * r23094096;
        double r23094098 = r23094090 / r23094097;
        return r23094098;
}
↓
double f(double x, double cos, double sin) {
        double r23094099 = 1.0;
        double r23094100 = cos;
        double r23094101 = x;
        double r23094102 = r23094100 * r23094101;
        double r23094103 = sin;
        double r23094104 = r23094102 * r23094103;
        double r23094105 = r23094099 / r23094104;
        double r23094106 = 2.0;
        double r23094107 = r23094101 * r23094106;
        double r23094108 = cos(r23094107);
        double r23094109 = r23094108 / r23094103;
        double r23094110 = r23094102 / r23094109;
        double r23094111 = r23094099 / r23094110;
        double r23094112 = r23094105 * r23094111;
        return r23094112;
}

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 27.3

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

  2. Simplified2.8

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

  4. Applied associate-/r*2.6

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

  6. Applied associate-/r*2.5

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

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

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

  9. Applied associate-/l*2.6

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

  11. Applied div-inv2.6

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

  12. Final simplification2.6

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

Reproduce

herbie shell --seed 2019121 
(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))))