Average Error: 28.0 → 2.7
Time: 32.6s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(2 \cdot x\right)}{\left({cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)}\]
\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)}{\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)}
double f(double x, double cos, double sin) {
        double r4283067 = 2.0;
        double r4283068 = x;
        double r4283069 = r4283067 * r4283068;
        double r4283070 = cos(r4283069);
        double r4283071 = cos;
        double r4283072 = pow(r4283071, r4283067);
        double r4283073 = sin;
        double r4283074 = pow(r4283073, r4283067);
        double r4283075 = r4283068 * r4283074;
        double r4283076 = r4283075 * r4283068;
        double r4283077 = r4283072 * r4283076;
        double r4283078 = r4283070 / r4283077;
        return r4283078;
}
↓
double f(double x, double cos, double sin) {
        double r4283079 = 2.0;
        double r4283080 = x;
        double r4283081 = r4283079 * r4283080;
        double r4283082 = cos(r4283081);
        double r4283083 = cos;
        double r4283084 = 2.0;
        double r4283085 = r4283079 / r4283084;
        double r4283086 = pow(r4283083, r4283085);
        double r4283087 = sin;
        double r4283088 = pow(r4283087, r4283085);
        double r4283089 = r4283080 * r4283088;
        double r4283090 = r4283086 * r4283089;
        double r4283091 = r4283090 * r4283090;
        double r4283092 = r4283082 / r4283091;
        return r4283092;
}

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 28.0

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

  3. Applied sqr-pow28.0

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

  4. Applied associate-*r*21.8

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

  6. Applied sqr-pow21.8

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

  7. Applied associate-*l*16.6

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

  8. Simplified13.6

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

  10. Applied associate-*l*6.0

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

  12. Applied associate-*r*2.7

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

  13. Final simplification2.7

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(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))))