Average Error: 28.0 → 4.4
Time: 30.1s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;cos \le -5.981995449898381218390393230545955863716 \cdot 10^{208}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)\right) \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\\ \mathbf{elif}\;cos \le -2.950166798107946823792822167133127542143 \cdot 10^{-147}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)\right) \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)}\\ \mathbf{elif}\;cos \le 5.98497475444158792752950739670094938722 \cdot 10^{-216}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)\right) \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\\ \mathbf{elif}\;cos \le 2.089088039096389298494294320304309164377 \cdot 10^{199}:\\ \;\;\;\;\left(\frac{1}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)} \cdot \frac{1}{{cos}^{\left(\frac{2}{2}\right)}}\right) \cdot \frac{\cos \left(2 \cdot x\right)}{{sin}^{\left(\frac{2}{2}\right)} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)\right) \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\\ \end{array}\]
\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 \le -5.981995449898381218390393230545955863716 \cdot 10^{208}:\\
\;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)\right) \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\\

\mathbf{elif}\;cos \le -2.950166798107946823792822167133127542143 \cdot 10^{-147}:\\
\;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)\right) \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)}\\

\mathbf{elif}\;cos \le 5.98497475444158792752950739670094938722 \cdot 10^{-216}:\\
\;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)\right) \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\\

\mathbf{elif}\;cos \le 2.089088039096389298494294320304309164377 \cdot 10^{199}:\\
\;\;\;\;\left(\frac{1}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)} \cdot \frac{1}{{cos}^{\left(\frac{2}{2}\right)}}\right) \cdot \frac{\cos \left(2 \cdot x\right)}{{sin}^{\left(\frac{2}{2}\right)} \cdot x}\\

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

\end{array}
double f(double x, double cos, double sin) {
        double r3215445 = 2.0;
        double r3215446 = x;
        double r3215447 = r3215445 * r3215446;
        double r3215448 = cos(r3215447);
        double r3215449 = cos;
        double r3215450 = pow(r3215449, r3215445);
        double r3215451 = sin;
        double r3215452 = pow(r3215451, r3215445);
        double r3215453 = r3215446 * r3215452;
        double r3215454 = r3215453 * r3215446;
        double r3215455 = r3215450 * r3215454;
        double r3215456 = r3215448 / r3215455;
        return r3215456;
}
↓
double f(double x, double cos, double sin) {
        double r3215457 = cos;
        double r3215458 = -5.981995449898381e+208;
        bool r3215459 = r3215457 <= r3215458;
        double r3215460 = 2.0;
        double r3215461 = x;
        double r3215462 = r3215460 * r3215461;
        double r3215463 = cos(r3215462);
        double r3215464 = 2.0;
        double r3215465 = r3215460 / r3215464;
        double r3215466 = pow(r3215457, r3215465);
        double r3215467 = sin;
        double r3215468 = pow(r3215467, r3215465);
        double r3215469 = r3215468 * r3215461;
        double r3215470 = r3215466 * r3215469;
        double r3215471 = r3215461 * r3215470;
        double r3215472 = r3215471 * r3215468;
        double r3215473 = r3215466 * r3215472;
        double r3215474 = r3215463 / r3215473;
        double r3215475 = -2.950166798107947e-147;
        bool r3215476 = r3215457 <= r3215475;
        double r3215477 = r3215466 * r3215470;
        double r3215478 = r3215477 * r3215469;
        double r3215479 = r3215463 / r3215478;
        double r3215480 = 5.984974754441588e-216;
        bool r3215481 = r3215457 <= r3215480;
        double r3215482 = 2.0890880390963893e+199;
        bool r3215483 = r3215457 <= r3215482;
        double r3215484 = 1.0;
        double r3215485 = r3215484 / r3215470;
        double r3215486 = r3215484 / r3215466;
        double r3215487 = r3215485 * r3215486;
        double r3215488 = r3215463 / r3215469;
        double r3215489 = r3215487 * r3215488;
        double r3215490 = r3215483 ? r3215489 : r3215474;
        double r3215491 = r3215481 ? r3215474 : r3215490;
        double r3215492 = r3215476 ? r3215479 : r3215491;
        double r3215493 = r3215459 ? r3215474 : r3215492;
        return r3215493;
}

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 < -5.981995449898381e+208 or -2.950166798107947e-147 < cos < 5.984974754441588e-216 or 2.0890880390963893e+199 < cos

    1. Initial program 36.0

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

    3. Applied sqr-pow36.0

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

    4. Applied associate-*r*33.1

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

    6. Applied sqr-pow33.1

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

    7. Applied associate-*l*23.5

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

    8. Simplified9.1

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

    10. Applied associate-*r*9.5

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

    if -5.981995449898381e+208 < cos < -2.950166798107947e-147

    1. Initial program 21.7

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

    3. Applied sqr-pow21.7

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

    4. Applied associate-*r*14.0

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

    6. Applied sqr-pow14.0

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

    7. Applied associate-*l*12.0

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

    8. Simplified4.6

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

    10. Applied associate-*r*1.7

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

    if 5.984974754441588e-216 < cos < 2.0890880390963893e+199

    1. Initial program 25.2

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

    3. Applied sqr-pow25.2

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

    4. Applied associate-*r*17.7

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

    6. Applied sqr-pow17.7

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

    7. Applied associate-*l*13.6

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

    8. Simplified4.7

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

    10. Applied *-un-lft-identity4.7

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

    11. Applied times-frac4.8

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

    13. Applied *-un-lft-identity4.8

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

    14. Applied times-frac4.6

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

    15. Applied associate-*r*1.5

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

  4. Final simplification4.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;cos \le -5.981995449898381218390393230545955863716 \cdot 10^{208}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)\right) \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\\ \mathbf{elif}\;cos \le -2.950166798107946823792822167133127542143 \cdot 10^{-147}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)\right) \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)}\\ \mathbf{elif}\;cos \le 5.98497475444158792752950739670094938722 \cdot 10^{-216}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)\right) \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\\ \mathbf{elif}\;cos \le 2.089088039096389298494294320304309164377 \cdot 10^{199}:\\ \;\;\;\;\left(\frac{1}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)} \cdot \frac{1}{{cos}^{\left(\frac{2}{2}\right)}}\right) \cdot \frac{\cos \left(2 \cdot x\right)}{{sin}^{\left(\frac{2}{2}\right)} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)\right) \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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))))