Average Error: 28.1 → 2.5
Time: 7.9s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;cos \le 4.19899225278698649 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\left|\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right|}\\ \mathbf{elif}\;cos \le 7.57685492731950486 \cdot 10^{145}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left|{cos}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot \left({cos}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)\right| \cdot \left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \end{array}\]
\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}\;cos \le 4.19899225278698649 \cdot 10^{-308}:\\
\;\;\;\;\frac{1}{\left|\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right|}\\

\mathbf{elif}\;cos \le 7.57685492731950486 \cdot 10^{145}:\\
\;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left|{cos}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot \left({cos}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)\right| \cdot \left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}\\

\mathbf{else}:\\
\;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\

\end{array}
double f(double x, double cos, double sin) {
        double r52531 = 2.0;
        double r52532 = x;
        double r52533 = r52531 * r52532;
        double r52534 = cos(r52533);
        double r52535 = cos;
        double r52536 = pow(r52535, r52531);
        double r52537 = sin;
        double r52538 = pow(r52537, r52531);
        double r52539 = r52532 * r52538;
        double r52540 = r52539 * r52532;
        double r52541 = r52536 * r52540;
        double r52542 = r52534 / r52541;
        return r52542;
}
↓
double f(double x, double cos, double sin) {
        double r52543 = cos;
        double r52544 = 4.1989922527869865e-308;
        bool r52545 = r52543 <= r52544;
        double r52546 = 1.0;
        double r52547 = 1.0;
        double r52548 = pow(r52543, r52547);
        double r52549 = sin;
        double r52550 = pow(r52549, r52547);
        double r52551 = r52548 * r52550;
        double r52552 = pow(r52551, r52547);
        double r52553 = x;
        double r52554 = r52552 * r52553;
        double r52555 = fabs(r52554);
        double r52556 = fabs(r52555);
        double r52557 = r52546 / r52556;
        double r52558 = 2.0;
        double r52559 = r52558 * r52553;
        double r52560 = cos(r52559);
        double r52561 = r52560 / r52556;
        double r52562 = r52557 * r52561;
        double r52563 = 7.576854927319505e+145;
        bool r52564 = r52543 <= r52563;
        double r52565 = 2.0;
        double r52566 = r52558 / r52565;
        double r52567 = r52566 / r52565;
        double r52568 = pow(r52543, r52567);
        double r52569 = pow(r52549, r52566);
        double r52570 = r52553 * r52569;
        double r52571 = r52568 * r52570;
        double r52572 = r52568 * r52571;
        double r52573 = fabs(r52572);
        double r52574 = pow(r52543, r52566);
        double r52575 = r52574 * r52570;
        double r52576 = fabs(r52575);
        double r52577 = r52573 * r52576;
        double r52578 = r52560 / r52577;
        double r52579 = pow(r52555, r52565);
        double r52580 = r52560 / r52579;
        double r52581 = r52564 ? r52578 : r52580;
        double r52582 = r52545 ? r52562 : r52581;
        return r52582;
}

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. Split input into 3 regimes

  2. if cos < 4.1989922527869865e-308

    1. Initial program 28.3

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

    3. Applied sqr-pow28.3

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

    4. Applied associate-*r*22.6

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

    6. Applied add-sqr-sqrt22.7

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

    7. Simplified22.6

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

    8. Simplified3.0

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

    9. Taylor expanded around 0 2.8

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

    10. Simplified2.8

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

    12. Applied add-sqr-sqrt2.8

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

    13. Applied *-un-lft-identity2.8

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

    14. Applied times-frac2.8

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

    15. Simplified2.8

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

    16. Simplified2.6

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

    if 4.1989922527869865e-308 < cos < 7.576854927319505e+145

    1. Initial program 29.7

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

    3. Applied sqr-pow29.7

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

    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 add-sqr-sqrt21.8

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

    7. Simplified21.8

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

    8. Simplified2.4

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

    10. Applied sqr-pow2.5

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

    11. Applied associate-*l*2.4

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

    if 7.576854927319505e+145 < cos

    1. Initial program 24.7

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

    3. Applied sqr-pow24.7

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

    4. Applied associate-*r*21.1

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

    6. Applied add-sqr-sqrt21.1

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

    7. Simplified21.1

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

    8. Simplified3.4

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

    9. Taylor expanded around 0 2.4

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

    10. Simplified2.4

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;cos \le 4.19899225278698649 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\left|\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right|}\\ \mathbf{elif}\;cos \le 7.57685492731950486 \cdot 10^{145}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left|{cos}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot \left({cos}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)\right| \cdot \left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  :precision binary64
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))