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

Average Error: 27.8 → 1.5

Time: 23.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot {sin}^{2}\right) \cdot x\right) \cdot {cos}^{2}} \le -0.0:\\ \;\;\;\;\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \left(\frac{\frac{1}{cos}}{sin \cdot x} \cdot \left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \frac{\frac{1}{cos}}{sin \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(sin \cdot cos\right)\right) \cdot \left(x \cdot \left(sin \cdot cos\right)\right)}\\ \end{array}\]

C

double f(double x, double cos, double sin) {
        double r1917482 = 2.0;
        double r1917483 = x;
        double r1917484 = r1917482 * r1917483;
        double r1917485 = cos(r1917484);
        double r1917486 = cos;
        double r1917487 = pow(r1917486, r1917482);
        double r1917488 = sin;
        double r1917489 = pow(r1917488, r1917482);
        double r1917490 = r1917483 * r1917489;
        double r1917491 = r1917490 * r1917483;
        double r1917492 = r1917487 * r1917491;
        double r1917493 = r1917485 / r1917492;
        return r1917493;
}

↓

double f(double x, double cos, double sin) {
        double r1917494 = 2.0;
        double r1917495 = x;
        double r1917496 = r1917494 * r1917495;
        double r1917497 = cos(r1917496);
        double r1917498 = sin;
        double r1917499 = pow(r1917498, r1917494);
        double r1917500 = r1917495 * r1917499;
        double r1917501 = r1917500 * r1917495;
        double r1917502 = cos;
        double r1917503 = pow(r1917502, r1917494);
        double r1917504 = r1917501 * r1917503;
        double r1917505 = r1917497 / r1917504;
        double r1917506 = -0.0;
        bool r1917507 = r1917505 <= r1917506;
        double r1917508 = cbrt(r1917497);
        double r1917509 = r1917508 * r1917508;
        double r1917510 = 1.0;
        double r1917511 = r1917510 / r1917502;
        double r1917512 = r1917498 * r1917495;
        double r1917513 = r1917511 / r1917512;
        double r1917514 = r1917508 * r1917513;
        double r1917515 = r1917513 * r1917514;
        double r1917516 = r1917509 * r1917515;
        double r1917517 = r1917498 * r1917502;
        double r1917518 = r1917495 * r1917517;
        double r1917519 = r1917518 * r1917518;
        double r1917520 = r1917497 / r1917519;
        double r1917521 = r1917507 ? r1917516 : r1917520;
        return r1917521;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

1. Split input into 2 regimes

2. if (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))) < -0.0

   1. Initial program 17.0

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

   2. Simplified 1.7

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

   4. Applied div-inv 1.7

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

   6. Applied pow1 1.7

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

   7. Applied pow1 1.7

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

   8. Applied pow-prod-down 1.7

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

   9. Applied pow1 1.7

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

   10. Applied pow-prod-down 1.7

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

   11. Applied pow1 1.7

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

   12. Applied pow1 1.7

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

   13. Applied pow-prod-down 1.7

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

   14. Applied pow1 1.7

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

   15. Applied pow-prod-down 1.7

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

   16. Applied pow-prod-up 1.7

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

   17. Applied pow-flip 1.4

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

   18. Simplified 1.4

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

   19. Taylor expanded around inf 2.6

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

   21. Applied add-cube-cbrt 2.7

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

   22. Applied associate-*l* 2.7

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

   23. Simplified 0.4

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

   if -0.0 < (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))

   1. Initial program 45.2

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

   2. Simplified 4.6

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

   4. Applied div-inv 4.6

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

   6. Applied pow1 4.6

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

   7. Applied pow1 4.6

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

   8. Applied pow-prod-down 4.6

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

   9. Applied pow1 4.6

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

   10. Applied pow-prod-down 4.6

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

   11. Applied pow1 4.6

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

   12. Applied pow1 4.6

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

   13. Applied pow-prod-down 4.6

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

   14. Applied pow1 4.6

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

   15. Applied pow-prod-down 4.6

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

   16. Applied pow-prod-up 4.6

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

   17. Applied pow-flip 4.5

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

   18. Simplified 4.5

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

   19. Taylor expanded around inf 3.0

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

   20. Taylor expanded around inf 48.4

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

   21. Simplified 3.2

       \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(cos \cdot sin\right)\right) \cdot \left(x \cdot \left(cos \cdot sin\right)\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot {sin}^{2}\right) \cdot x\right) \cdot {cos}^{2}} \le -0.0:\\ \;\;\;\;\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \left(\frac{\frac{1}{cos}}{sin \cdot x} \cdot \left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \frac{\frac{1}{cos}}{sin \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(sin \cdot cos\right)\right) \cdot \left(x \cdot \left(sin \cdot cos\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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))))