Average Error: 28.2 → 2.6
Time: 27.6s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}
double f(double x, double cos, double sin) {
        double r3670050 = 2.0;
        double r3670051 = x;
        double r3670052 = r3670050 * r3670051;
        double r3670053 = cos(r3670052);
        double r3670054 = cos;
        double r3670055 = pow(r3670054, r3670050);
        double r3670056 = sin;
        double r3670057 = pow(r3670056, r3670050);
        double r3670058 = r3670051 * r3670057;
        double r3670059 = r3670058 * r3670051;
        double r3670060 = r3670055 * r3670059;
        double r3670061 = r3670053 / r3670060;
        return r3670061;
}
↓
double f(double x, double cos, double sin) {
        double r3670062 = 2.0;
        double r3670063 = x;
        double r3670064 = r3670062 * r3670063;
        double r3670065 = cos(r3670064);
        double r3670066 = cos;
        double r3670067 = 2.0;
        double r3670068 = r3670062 / r3670067;
        double r3670069 = pow(r3670066, r3670068);
        double r3670070 = sin;
        double r3670071 = pow(r3670070, r3670068);
        double r3670072 = r3670063 * r3670071;
        double r3670073 = r3670069 * r3670072;
        double r3670074 = r3670065 / r3670073;
        double r3670075 = r3670074 / r3670073;
        return r3670075;
}

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 28.2

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

  3. Applied sqr-pow28.2

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

  4. Applied associate-*r*21.7

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

  6. Applied associate-*l*19.7

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

  7. Simplified19.7

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

  9. Applied sqr-pow19.7

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

  10. Applied unswap-sqr2.8

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

  12. Applied associate-/r*2.6

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

  13. Final simplification2.6

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

Reproduce

herbie shell --seed 2019171 
(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))))