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

Average Error: 27.8 → 2.7

Time: 29.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2020.583662756933335913345217704772949219:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)} \cdot \frac{\frac{1}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(\frac{2}{2}\right)} \cdot x}\\ \mathbf{elif}\;x \le 7.441590536785484532779866860185941272842 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\cos \left(2 \cdot x\right)}}{{sin}^{\left(\frac{2}{2}\right)} \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot x\right)} \cdot \frac{\sqrt{\cos \left(2 \cdot x\right)}}{{sin}^{\left(\frac{2}{2}\right)} \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)} \cdot \frac{\frac{1}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(\frac{2}{2}\right)} \cdot x}\\ \end{array}\]

C

double f(double x, double cos, double sin) {
        double r2959599 = 2.0;
        double r2959600 = x;
        double r2959601 = r2959599 * r2959600;
        double r2959602 = cos(r2959601);
        double r2959603 = cos;
        double r2959604 = pow(r2959603, r2959599);
        double r2959605 = sin;
        double r2959606 = pow(r2959605, r2959599);
        double r2959607 = r2959600 * r2959606;
        double r2959608 = r2959607 * r2959600;
        double r2959609 = r2959604 * r2959608;
        double r2959610 = r2959602 / r2959609;
        return r2959610;
}

↓

double f(double x, double cos, double sin) {
        double r2959611 = x;
        double r2959612 = -2020.5836627569333;
        bool r2959613 = r2959611 <= r2959612;
        double r2959614 = 2.0;
        double r2959615 = r2959614 * r2959611;
        double r2959616 = cos(r2959615);
        double r2959617 = cos;
        double r2959618 = 2.0;
        double r2959619 = r2959614 / r2959618;
        double r2959620 = pow(r2959617, r2959619);
        double r2959621 = sin;
        double r2959622 = pow(r2959621, r2959619);
        double r2959623 = r2959622 * r2959611;
        double r2959624 = r2959620 * r2959623;
        double r2959625 = r2959616 / r2959624;
        double r2959626 = 1.0;
        double r2959627 = r2959626 / r2959620;
        double r2959628 = r2959627 / r2959623;
        double r2959629 = r2959625 * r2959628;
        double r2959630 = 7.441590536785485e-128;
        bool r2959631 = r2959611 <= r2959630;
        double r2959632 = sqrt(r2959616);
        double r2959633 = r2959620 * r2959611;
        double r2959634 = r2959622 * r2959633;
        double r2959635 = r2959632 / r2959634;
        double r2959636 = r2959635 * r2959635;
        double r2959637 = r2959631 ? r2959636 : r2959629;
        double r2959638 = r2959613 ? r2959629 : r2959637;
        return r2959638;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

1. Split input into 2 regimes

2. if x < -2020.5836627569333 or 7.441590536785485e-128 < x

   1. Initial program 24.9

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

   3. Applied sqr-pow 24.9

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

   4. Applied associate-*r* 19.7

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

   6. Applied sqr-pow 19.7

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

   7. Applied associate-*l* 14.3

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

   8. Simplified 5.6

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

   10. Applied associate-/r* 5.4

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

   12. Applied div-inv 5.4

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

   13. Applied times-frac 2.2

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

   if -2020.5836627569333 < x < 7.441590536785485e-128

   1. Initial program 35.6

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

   3. Applied sqr-pow 35.6

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

   4. Applied associate-*r* 28.2

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

   6. Applied sqr-pow 28.2

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

   7. Applied associate-*l* 23.0

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

   8. Simplified 8.9

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

   10. Applied associate-/r* 8.6

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

   12. Applied *-un-lft-identity 8.6

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

   13. Applied unpow-prod-down 8.6

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

   14. Applied add-sqr-sqrt 8.8

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

   15. Applied times-frac 8.8

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

   16. Applied times-frac 4.1

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

   17. Simplified 7.2

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

   18. Simplified 3.9

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

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2020.583662756933335913345217704772949219:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)} \cdot \frac{\frac{1}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(\frac{2}{2}\right)} \cdot x}\\ \mathbf{elif}\;x \le 7.441590536785484532779866860185941272842 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\cos \left(2 \cdot x\right)}}{{sin}^{\left(\frac{2}{2}\right)} \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot x\right)} \cdot \frac{\sqrt{\cos \left(2 \cdot x\right)}}{{sin}^{\left(\frac{2}{2}\right)} \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)} \cdot \frac{\frac{1}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(\frac{2}{2}\right)} \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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))))