Average Error: 28.2 → 2.4
Time: 8.9s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;cos \le 4.24708257629406067 \cdot 10^{-274}:\\ \;\;\;\;\cos \left(2 \cdot x\right) \cdot \frac{1}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \mathbf{elif}\;cos \le 6.4244675320823765 \cdot 10^{214}:\\ \;\;\;\;\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|sin \cdot \left(x \cdot cos\right)\right|\right)}^{2}}\\ \end{array}\]
\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 4.24708257629406067 \cdot 10^{-274}:\\
\;\;\;\;\cos \left(2 \cdot x\right) \cdot \frac{1}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\

\mathbf{elif}\;cos \le 6.4244675320823765 \cdot 10^{214}:\\
\;\;\;\;\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\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 \frac{\sqrt[3]{\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|}\\

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

\end{array}
double f(double x, double cos, double sin) {
        double r69545 = 2.0;
        double r69546 = x;
        double r69547 = r69545 * r69546;
        double r69548 = cos(r69547);
        double r69549 = cos;
        double r69550 = pow(r69549, r69545);
        double r69551 = sin;
        double r69552 = pow(r69551, r69545);
        double r69553 = r69546 * r69552;
        double r69554 = r69553 * r69546;
        double r69555 = r69550 * r69554;
        double r69556 = r69548 / r69555;
        return r69556;
}
↓
double f(double x, double cos, double sin) {
        double r69557 = cos;
        double r69558 = 4.2470825762940607e-274;
        bool r69559 = r69557 <= r69558;
        double r69560 = 2.0;
        double r69561 = x;
        double r69562 = r69560 * r69561;
        double r69563 = cos(r69562);
        double r69564 = 1.0;
        double r69565 = 1.0;
        double r69566 = pow(r69557, r69565);
        double r69567 = sin;
        double r69568 = pow(r69567, r69565);
        double r69569 = r69566 * r69568;
        double r69570 = pow(r69569, r69565);
        double r69571 = r69570 * r69561;
        double r69572 = fabs(r69571);
        double r69573 = 2.0;
        double r69574 = pow(r69572, r69573);
        double r69575 = r69564 / r69574;
        double r69576 = r69563 * r69575;
        double r69577 = 6.4244675320823765e+214;
        bool r69578 = r69557 <= r69577;
        double r69579 = cbrt(r69563);
        double r69580 = r69579 * r69579;
        double r69581 = r69560 / r69573;
        double r69582 = pow(r69557, r69581);
        double r69583 = pow(r69567, r69581);
        double r69584 = r69561 * r69583;
        double r69585 = r69582 * r69584;
        double r69586 = fabs(r69585);
        double r69587 = r69580 / r69586;
        double r69588 = r69579 / r69586;
        double r69589 = r69587 * r69588;
        double r69590 = r69561 * r69557;
        double r69591 = r69567 * r69590;
        double r69592 = fabs(r69591);
        double r69593 = pow(r69592, r69573);
        double r69594 = r69563 / r69593;
        double r69595 = r69578 ? r69589 : r69594;
        double r69596 = r69559 ? r69576 : r69595;
        return r69596;
}

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 < 4.2470825762940607e-274

    1. Initial program 28.9

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

    3. Applied sqr-pow28.9

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

    4. Applied associate-*r*23.2

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

    6. Applied add-sqr-sqrt23.3

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

    7. Simplified23.2

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

    8. Simplified3.2

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

    9. Taylor expanded around inf 2.8

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

    11. Applied div-inv2.8

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

    if 4.2470825762940607e-274 < cos < 6.4244675320823765e+214

    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-pow28.2

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

    4. Applied associate-*r*20.9

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

    6. Applied add-sqr-sqrt20.9

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

    7. Simplified20.9

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

    8. Simplified1.9

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

    10. Applied add-cube-cbrt2.1

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

    11. Applied times-frac1.8

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

    if 6.4244675320823765e+214 < cos

    1. Initial program 24.4

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

    3. Applied sqr-pow24.4

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

    4. Applied associate-*r*21.2

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

    6. Applied add-sqr-sqrt21.2

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

    7. Simplified21.2

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

    8. Simplified4.5

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

    9. Taylor expanded around inf 2.2

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

    10. Taylor expanded around 0 2.6

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;cos \le 4.24708257629406067 \cdot 10^{-274}:\\ \;\;\;\;\cos \left(2 \cdot x\right) \cdot \frac{1}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \mathbf{elif}\;cos \le 6.4244675320823765 \cdot 10^{214}:\\ \;\;\;\;\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|sin \cdot \left(x \cdot cos\right)\right|\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(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))))