Average Error: 27.8 → 2.6
Time: 13.4s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot sin\right) \cdot cos} \cdot \left(\frac{1}{cos} \cdot \frac{1}{x \cdot sin}\right)\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot sin\right) \cdot cos} \cdot \left(\frac{1}{cos} \cdot \frac{1}{x \cdot sin}\right)
double f(double x, double cos, double sin) {
        double r730764 = 2.0;
        double r730765 = x;
        double r730766 = r730764 * r730765;
        double r730767 = cos(r730766);
        double r730768 = cos;
        double r730769 = pow(r730768, r730764);
        double r730770 = sin;
        double r730771 = pow(r730770, r730764);
        double r730772 = r730765 * r730771;
        double r730773 = r730772 * r730765;
        double r730774 = r730769 * r730773;
        double r730775 = r730767 / r730774;
        return r730775;
}
↓
double f(double x, double cos, double sin) {
        double r730776 = x;
        double r730777 = 2.0;
        double r730778 = r730776 * r730777;
        double r730779 = cos(r730778);
        double r730780 = sin;
        double r730781 = r730776 * r730780;
        double r730782 = cos;
        double r730783 = r730781 * r730782;
        double r730784 = r730779 / r730783;
        double r730785 = 1.0;
        double r730786 = r730785 / r730782;
        double r730787 = r730785 / r730781;
        double r730788 = r730786 * r730787;
        double r730789 = r730784 * r730788;
        return r730789;
}

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 27.8

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

  2. Simplified2.7

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

  4. Applied *-un-lft-identity2.7

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

  5. Applied times-frac2.5

    \[\leadsto \color{blue}{\frac{1}{\left(sin \cdot x\right) \cdot cos} \cdot \frac{\cos \left(2 \cdot x\right)}{\left(sin \cdot x\right) \cdot cos}}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity2.5

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

  8. Applied times-frac2.6

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

  9. Final simplification2.6

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

Reproduce

herbie shell --seed 2019156 +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))))