Average Error: 28.7 → 5.3
Time: 41.4s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;{cos}^{2} \le 1.305111268867258864826361619975638845439 \cdot 10^{-288}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left({cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{elif}\;{cos}^{2} \le 6.070872255868253113071126070766490033787 \cdot 10^{199}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left({cos}^{2} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left({cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \end{array}\]
\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 1.305111268867258864826361619975638845439 \cdot 10^{-288}:\\
\;\;\;\;\frac{\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)}\\

\mathbf{elif}\;{cos}^{2} \le 6.070872255868253113071126070766490033787 \cdot 10^{199}:\\
\;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left({cos}^{2} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\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)}\\

\end{array}
double f(double x, double cos, double sin) {
        double r6545434 = 2.0;
        double r6545435 = x;
        double r6545436 = r6545434 * r6545435;
        double r6545437 = cos(r6545436);
        double r6545438 = cos;
        double r6545439 = pow(r6545438, r6545434);
        double r6545440 = sin;
        double r6545441 = pow(r6545440, r6545434);
        double r6545442 = r6545435 * r6545441;
        double r6545443 = r6545442 * r6545435;
        double r6545444 = r6545439 * r6545443;
        double r6545445 = r6545437 / r6545444;
        return r6545445;
}
↓
double f(double x, double cos, double sin) {
        double r6545446 = cos;
        double r6545447 = 2.0;
        double r6545448 = pow(r6545446, r6545447);
        double r6545449 = 1.3051112688672589e-288;
        bool r6545450 = r6545448 <= r6545449;
        double r6545451 = x;
        double r6545452 = r6545447 * r6545451;
        double r6545453 = cos(r6545452);
        double r6545454 = 2.0;
        double r6545455 = r6545447 / r6545454;
        double r6545456 = pow(r6545446, r6545455);
        double r6545457 = sin;
        double r6545458 = pow(r6545457, r6545455);
        double r6545459 = r6545451 * r6545458;
        double r6545460 = r6545456 * r6545459;
        double r6545461 = r6545460 * r6545459;
        double r6545462 = r6545456 * r6545461;
        double r6545463 = r6545453 / r6545462;
        double r6545464 = 6.070872255868253e+199;
        bool r6545465 = r6545448 <= r6545464;
        double r6545466 = r6545448 * r6545459;
        double r6545467 = r6545466 * r6545459;
        double r6545468 = r6545453 / r6545467;
        double r6545469 = r6545465 ? r6545468 : r6545463;
        double r6545470 = r6545450 ? r6545463 : r6545469;
        return r6545470;
}

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) < 1.3051112688672589e-288 or 6.070872255868253e+199 < (pow cos 2.0)

    1. Initial program 33.8

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

    3. Applied sqr-pow33.8

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

    4. Applied associate-*r*30.1

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

    6. Applied sqr-pow30.1

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

    7. Applied associate-*l*21.2

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

    8. Simplified18.9

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

    10. Applied associate-*r*8.3

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

    if 1.3051112688672589e-288 < (pow cos 2.0) < 6.070872255868253e+199

    1. Initial program 21.1

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

    3. Applied sqr-pow21.1

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

    4. Applied associate-*r*10.5

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

    6. Applied sqr-pow10.5

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

    7. Applied associate-*l*10.5

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

    8. Simplified7.0

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

    10. Applied associate-*r*3.6

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

    12. Applied associate-*r*0.7

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

    13. Simplified0.7

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

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;{cos}^{2} \le 1.305111268867258864826361619975638845439 \cdot 10^{-288}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left({cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{elif}\;{cos}^{2} \le 6.070872255868253113071126070766490033787 \cdot 10^{199}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left({cos}^{2} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left({cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(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))))