cos(2*x)/(cos^2(x)*sin^2(x))

Average Error: 28.2 → 2.8

Time: 28.6s

Precision: 64

Math

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

↓

\[\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)} \cdot \frac{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{x \cdot {sin}^{\left(\frac{2}{2}\right)}}}{{cos}^{\left(\frac{2}{2}\right)}}\]

C

double f(double x, double cos, double sin) {
        double r3186964 = 2.0;
        double r3186965 = x;
        double r3186966 = r3186964 * r3186965;
        double r3186967 = cos(r3186966);
        double r3186968 = cos;
        double r3186969 = pow(r3186968, r3186964);
        double r3186970 = sin;
        double r3186971 = pow(r3186970, r3186964);
        double r3186972 = r3186965 * r3186971;
        double r3186973 = r3186972 * r3186965;
        double r3186974 = r3186969 * r3186973;
        double r3186975 = r3186967 / r3186974;
        return r3186975;
}

↓

double f(double x, double cos, double sin) {
        double r3186976 = 2.0;
        double r3186977 = x;
        double r3186978 = r3186976 * r3186977;
        double r3186979 = cos(r3186978);
        double r3186980 = cbrt(r3186979);
        double r3186981 = r3186980 * r3186980;
        double r3186982 = cos;
        double r3186983 = 2.0;
        double r3186984 = r3186976 / r3186983;
        double r3186985 = pow(r3186982, r3186984);
        double r3186986 = sin;
        double r3186987 = pow(r3186986, r3186984);
        double r3186988 = r3186977 * r3186987;
        double r3186989 = r3186985 * r3186988;
        double r3186990 = r3186981 / r3186989;
        double r3186991 = r3186980 / r3186988;
        double r3186992 = r3186991 / r3186985;
        double r3186993 = r3186990 * r3186992;
        return r3186993;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

1. Initial program 28.2

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

3. Applied sqr-pow 28.2

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

4. Applied associate-*r* 21.7

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

6. Applied sqr-pow 21.7

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

7. Applied associate-*l* 16.1

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

8. Simplified 6.1

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

10. Applied associate-/r* 5.9

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

12. Applied *-un-lft-identity 5.9

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

13. Applied unpow-prod-down 5.9

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

14. Applied add-cube-cbrt 6.1

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

15. Applied times-frac 6.1

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

16. Applied times-frac 2.7

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

17. Simplified 2.7

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

18. Simplified 2.8

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

19. Final simplification 2.8

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

Reproduce

herbie shell --seed 2019171 +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))))