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

Average Error: 28.0 → 2.7

Time: 28.0s

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 r2736009 = 2.0;
        double r2736010 = x;
        double r2736011 = r2736009 * r2736010;
        double r2736012 = cos(r2736011);
        double r2736013 = cos;
        double r2736014 = pow(r2736013, r2736009);
        double r2736015 = sin;
        double r2736016 = pow(r2736015, r2736009);
        double r2736017 = r2736010 * r2736016;
        double r2736018 = r2736017 * r2736010;
        double r2736019 = r2736014 * r2736018;
        double r2736020 = r2736012 / r2736019;
        return r2736020;
}

↓

double f(double x, double cos, double sin) {
        double r2736021 = 2.0;
        double r2736022 = x;
        double r2736023 = r2736021 * r2736022;
        double r2736024 = cos(r2736023);
        double r2736025 = cbrt(r2736024);
        double r2736026 = r2736025 * r2736025;
        double r2736027 = cos;
        double r2736028 = 2.0;
        double r2736029 = r2736021 / r2736028;
        double r2736030 = pow(r2736027, r2736029);
        double r2736031 = sin;
        double r2736032 = pow(r2736031, r2736029);
        double r2736033 = r2736022 * r2736032;
        double r2736034 = r2736030 * r2736033;
        double r2736035 = r2736026 / r2736034;
        double r2736036 = r2736025 / r2736033;
        double r2736037 = r2736036 / r2736030;
        double r2736038 = r2736035 * r2736037;
        return r2736038;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

1. Initial program 28.0

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

3. Applied sqr-pow 28.0

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

4. Applied associate-*r* 21.8

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

6. Applied sqr-pow 21.8

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

7. Applied associate-*l* 16.6

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

8. Simplified 6.0

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

10. Applied associate-/r* 5.8

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

12. Applied *-un-lft-identity 5.8

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

13. Applied unpow-prod-down 5.8

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

14. Applied add-cube-cbrt 5.9

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

15. Applied times-frac 5.9

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

16. Applied times-frac 2.6

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

17. Simplified 2.6

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

18. Simplified 2.7

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

19. Final simplification 2.7

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

Reproduce

herbie shell --seed 2019174 +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))))