Average Error: 28.8 → 8.6
Time: 27.5s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left({cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right) \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot x\right)}\]
\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)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left({cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right) \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot x\right)}
double f(double x, double cos, double sin) {
        double r69241 = 2.0;
        double r69242 = x;
        double r69243 = r69241 * r69242;
        double r69244 = cos(r69243);
        double r69245 = cos;
        double r69246 = pow(r69245, r69241);
        double r69247 = sin;
        double r69248 = pow(r69247, r69241);
        double r69249 = r69242 * r69248;
        double r69250 = r69249 * r69242;
        double r69251 = r69246 * r69250;
        double r69252 = r69244 / r69251;
        return r69252;
}
↓
double f(double x, double cos, double sin) {
        double r69253 = 2.0;
        double r69254 = x;
        double r69255 = r69253 * r69254;
        double r69256 = cos(r69255);
        double r69257 = cos;
        double r69258 = 2.0;
        double r69259 = r69253 / r69258;
        double r69260 = pow(r69257, r69259);
        double r69261 = sin;
        double r69262 = pow(r69261, r69259);
        double r69263 = r69254 * r69262;
        double r69264 = r69260 * r69263;
        double r69265 = r69264 * r69262;
        double r69266 = r69265 * r69254;
        double r69267 = r69260 * r69266;
        double r69268 = r69256 / r69267;
        return r69268;
}

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 28.8

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

  3. Applied sqr-pow28.8

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

  4. Applied associate-*r*22.6

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

  6. Applied sqr-pow22.6

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

  7. Applied associate-*l*17.3

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

  8. Simplified20.1

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

  10. Applied sqr-pow20.1

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

  11. Applied associate-*r*14.1

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

  13. Applied associate-*r*8.6

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

  14. Final simplification8.6

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

Reproduce

herbie shell --seed 2019195 
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  (/ (cos (* 2.0 x)) (* (pow cos 2.0) (* (* x (pow sin 2.0)) x))))