Average Error: 28.1 → 6.6
Time: 20.4s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(2 \cdot x\right)}{\left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)}\]
\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)}{\left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)}
double f(double x, double cos, double sin) {
        double r56798 = 2.0;
        double r56799 = x;
        double r56800 = r56798 * r56799;
        double r56801 = cos(r56800);
        double r56802 = cos;
        double r56803 = pow(r56802, r56798);
        double r56804 = sin;
        double r56805 = pow(r56804, r56798);
        double r56806 = r56799 * r56805;
        double r56807 = r56806 * r56799;
        double r56808 = r56803 * r56807;
        double r56809 = r56801 / r56808;
        return r56809;
}
↓
double f(double x, double cos, double sin) {
        double r56810 = 2.0;
        double r56811 = x;
        double r56812 = r56810 * r56811;
        double r56813 = cos(r56812);
        double r56814 = cos;
        double r56815 = 2.0;
        double r56816 = r56810 / r56815;
        double r56817 = pow(r56814, r56816);
        double r56818 = sin;
        double r56819 = pow(r56818, r56816);
        double r56820 = r56811 * r56819;
        double r56821 = r56817 * r56820;
        double r56822 = r56817 * r56821;
        double r56823 = r56819 * r56811;
        double r56824 = r56822 * r56823;
        double r56825 = r56813 / r56824;
        return r56825;
}

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 28.1

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

  3. Applied sqr-pow28.1

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

  4. Applied associate-*r*21.9

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

  6. Applied associate-*l*19.9

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

  8. Applied associate-*r*15.9

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

  10. Applied sqr-pow15.9

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

  11. Applied associate-*l*6.6

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

  12. Final simplification6.6

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  :precision binary64
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))