Average Error: 27.3 → 2.8
Time: 27.5s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\frac{\left(\left(x \cdot sin\right) \cdot cos\right) \cdot \left(\left(x \cdot sin\right) \cdot cos\right)}{\cos \left(2 \cdot x\right)}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{1}{\frac{\left(\left(x \cdot sin\right) \cdot cos\right) \cdot \left(\left(x \cdot sin\right) \cdot cos\right)}{\cos \left(2 \cdot x\right)}}
double f(double x, double cos, double sin) {
        double r1745760 = 2.0;
        double r1745761 = x;
        double r1745762 = r1745760 * r1745761;
        double r1745763 = cos(r1745762);
        double r1745764 = cos;
        double r1745765 = pow(r1745764, r1745760);
        double r1745766 = sin;
        double r1745767 = pow(r1745766, r1745760);
        double r1745768 = r1745761 * r1745767;
        double r1745769 = r1745768 * r1745761;
        double r1745770 = r1745765 * r1745769;
        double r1745771 = r1745763 / r1745770;
        return r1745771;
}
↓
double f(double x, double cos, double sin) {
        double r1745772 = 1.0;
        double r1745773 = x;
        double r1745774 = sin;
        double r1745775 = r1745773 * r1745774;
        double r1745776 = cos;
        double r1745777 = r1745775 * r1745776;
        double r1745778 = r1745777 * r1745777;
        double r1745779 = 2.0;
        double r1745780 = r1745779 * r1745773;
        double r1745781 = cos(r1745780);
        double r1745782 = r1745778 / r1745781;
        double r1745783 = r1745772 / r1745782;
        return r1745783;
}

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 27.3

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

  2. Simplified2.8

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

  4. Applied clear-num2.8

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

  5. Final simplification2.8

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(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))))