Average Error: 27.0 → 2.6
Time: 21.1s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{cos}}{x}}{sin} \cdot \frac{1}{sin \cdot \left(x \cdot cos\right)}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{cos}}{x}}{sin} \cdot \frac{1}{sin \cdot \left(x \cdot cos\right)}
double f(double x, double cos, double sin) {
        double r1510747 = 2.0;
        double r1510748 = x;
        double r1510749 = r1510747 * r1510748;
        double r1510750 = cos(r1510749);
        double r1510751 = cos;
        double r1510752 = pow(r1510751, r1510747);
        double r1510753 = sin;
        double r1510754 = pow(r1510753, r1510747);
        double r1510755 = r1510748 * r1510754;
        double r1510756 = r1510755 * r1510748;
        double r1510757 = r1510752 * r1510756;
        double r1510758 = r1510750 / r1510757;
        return r1510758;
}
↓
double f(double x, double cos, double sin) {
        double r1510759 = 2.0;
        double r1510760 = x;
        double r1510761 = r1510759 * r1510760;
        double r1510762 = cos(r1510761);
        double r1510763 = cos;
        double r1510764 = r1510762 / r1510763;
        double r1510765 = r1510764 / r1510760;
        double r1510766 = sin;
        double r1510767 = r1510765 / r1510766;
        double r1510768 = 1.0;
        double r1510769 = r1510760 * r1510763;
        double r1510770 = r1510766 * r1510769;
        double r1510771 = r1510768 / r1510770;
        double r1510772 = r1510767 * r1510771;
        return r1510772;
}

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 27.0

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

  2. Simplified2.9

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

  4. Applied *-un-lft-identity2.9

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

  5. Applied times-frac2.6

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

  7. Applied *-un-lft-identity2.6

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

  8. Applied associate-/l*2.6

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

  10. Applied *-un-lft-identity2.6

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

  11. Applied times-frac2.6

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

  12. Applied add-cube-cbrt2.6

    \[\leadsto \frac{1}{sin \cdot \left(x \cdot cos\right)} \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{sin}{1} \cdot \frac{x \cdot cos}{\cos \left(2 \cdot x\right)}}\]

  13. Applied times-frac2.7

    \[\leadsto \frac{1}{sin \cdot \left(x \cdot cos\right)} \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{sin}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{x \cdot cos}{\cos \left(2 \cdot x\right)}}\right)}\]

  14. Simplified2.7

    \[\leadsto \frac{1}{sin \cdot \left(x \cdot cos\right)} \cdot \left(\color{blue}{\frac{1}{sin}} \cdot \frac{\sqrt[3]{1}}{\frac{x \cdot cos}{\cos \left(2 \cdot x\right)}}\right)\]

  15. Simplified2.7

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

  17. Applied associate-*l/2.7

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

  18. Simplified2.6

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

  19. Final simplification2.6

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

Reproduce

herbie shell --seed 2019135 
(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))))