Average Error: 27.2 → 2.7
Time: 4.3m
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\frac{1}{cos}}{x} \cdot \frac{\cos \left(x \cdot 2\right)}{sin}}{\left(x \cdot cos\right) \cdot sin}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{\frac{1}{cos}}{x} \cdot \frac{\cos \left(x \cdot 2\right)}{sin}}{\left(x \cdot cos\right) \cdot sin}
double f(double x, double cos, double sin) {
        double r31835300 = 2.0;
        double r31835301 = x;
        double r31835302 = r31835300 * r31835301;
        double r31835303 = cos(r31835302);
        double r31835304 = cos;
        double r31835305 = pow(r31835304, r31835300);
        double r31835306 = sin;
        double r31835307 = pow(r31835306, r31835300);
        double r31835308 = r31835301 * r31835307;
        double r31835309 = r31835308 * r31835301;
        double r31835310 = r31835305 * r31835309;
        double r31835311 = r31835303 / r31835310;
        return r31835311;
}
↓
double f(double x, double cos, double sin) {
        double r31835312 = 1.0;
        double r31835313 = cos;
        double r31835314 = r31835312 / r31835313;
        double r31835315 = x;
        double r31835316 = r31835314 / r31835315;
        double r31835317 = 2.0;
        double r31835318 = r31835315 * r31835317;
        double r31835319 = cos(r31835318);
        double r31835320 = sin;
        double r31835321 = r31835319 / r31835320;
        double r31835322 = r31835316 * r31835321;
        double r31835323 = r31835315 * r31835313;
        double r31835324 = r31835323 * r31835320;
        double r31835325 = r31835322 / r31835324;
        return r31835325;
}

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 27.2

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

  2. Simplified2.8

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

  4. Applied associate-/r*2.6

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

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

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

  7. Applied times-frac2.7

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

  9. Applied *-un-lft-identity2.7

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

  10. Applied times-frac2.7

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

  12. Applied associate-*l/2.7

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

  13. Simplified2.7

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

  14. Final simplification2.7

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

Reproduce

herbie shell --seed 2019128 
(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))))