Average Error: 28.2 → 9.1
Time: 59.7s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;{cos}^{2} \le 3.591898879565668310505104221865211871992 \cdot 10^{-101}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left({sin}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot \left(x \cdot {cos}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot {cos}^{\left(\frac{2}{2}\right)}}\\ \mathbf{elif}\;{cos}^{2} \le 5.979953044827646933715475705077781566284 \cdot 10^{146}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left({sin}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot \left(x \cdot {cos}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot {cos}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]
\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}^{2} \le 3.591898879565668310505104221865211871992 \cdot 10^{-101}:\\
\;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left({sin}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot \left(x \cdot {cos}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot {cos}^{\left(\frac{2}{2}\right)}}\\

\mathbf{elif}\;{cos}^{2} \le 5.979953044827646933715475705077781566284 \cdot 10^{146}:\\
\;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)}\\

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

\end{array}
double f(double x, double cos, double sin) {
        double r3318625 = 2.0;
        double r3318626 = x;
        double r3318627 = r3318625 * r3318626;
        double r3318628 = cos(r3318627);
        double r3318629 = cos;
        double r3318630 = pow(r3318629, r3318625);
        double r3318631 = sin;
        double r3318632 = pow(r3318631, r3318625);
        double r3318633 = r3318626 * r3318632;
        double r3318634 = r3318633 * r3318626;
        double r3318635 = r3318630 * r3318634;
        double r3318636 = r3318628 / r3318635;
        return r3318636;
}
↓
double f(double x, double cos, double sin) {
        double r3318637 = cos;
        double r3318638 = 2.0;
        double r3318639 = pow(r3318637, r3318638);
        double r3318640 = 3.591898879565668e-101;
        bool r3318641 = r3318639 <= r3318640;
        double r3318642 = x;
        double r3318643 = r3318642 * r3318638;
        double r3318644 = cos(r3318643);
        double r3318645 = sin;
        double r3318646 = 2.0;
        double r3318647 = r3318638 / r3318646;
        double r3318648 = pow(r3318645, r3318647);
        double r3318649 = r3318642 * r3318648;
        double r3318650 = pow(r3318637, r3318647);
        double r3318651 = r3318642 * r3318650;
        double r3318652 = r3318649 * r3318651;
        double r3318653 = r3318648 * r3318652;
        double r3318654 = r3318653 * r3318650;
        double r3318655 = r3318644 / r3318654;
        double r3318656 = 5.979953044827647e+146;
        bool r3318657 = r3318639 <= r3318656;
        double r3318658 = r3318649 * r3318649;
        double r3318659 = r3318639 * r3318658;
        double r3318660 = r3318644 / r3318659;
        double r3318661 = r3318657 ? r3318660 : r3318655;
        double r3318662 = r3318641 ? r3318655 : r3318661;
        return r3318662;
}

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 2 regimes

  2. if (pow cos 2.0) < 3.591898879565668e-101 or 5.979953044827647e+146 < (pow cos 2.0)

    1. Initial program 30.7

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

    3. Applied sqr-pow30.7

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

    4. Applied associate-*r*25.8

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

    6. Applied sqr-pow25.8

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

    7. Applied associate-*l*19.0

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

    8. Simplified20.4

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

    10. Applied sqr-pow20.4

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

    11. Applied associate-*r*15.3

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

    13. Applied associate-*r*10.5

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

    if 3.591898879565668e-101 < (pow cos 2.0) < 5.979953044827647e+146

    1. Initial program 19.3

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

    3. Applied sqr-pow19.3

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

    4. Applied associate-*r*8.9

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

    6. Applied associate-*l*4.1

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;{cos}^{2} \le 3.591898879565668310505104221865211871992 \cdot 10^{-101}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left({sin}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot \left(x \cdot {cos}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot {cos}^{\left(\frac{2}{2}\right)}}\\ \mathbf{elif}\;{cos}^{2} \le 5.979953044827646933715475705077781566284 \cdot 10^{146}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left({sin}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot \left(x \cdot {cos}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot {cos}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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))))