Average Error: 27.8 → 9.2
Time: 29.7s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;{sin}^{2} \le 1.375411136237947412717068778069207808383 \cdot 10^{-257}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot {cos}^{\left(\frac{2}{2}\right)}\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)\right)}\\ \mathbf{elif}\;{sin}^{2} \le 4.691137056220841244900674601205005981837 \cdot 10^{253}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot \left({sin}^{2} \cdot \left(x \cdot {cos}^{\left(\frac{2}{2}\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot {cos}^{\left(\frac{2}{2}\right)}\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)\right)}\\ \end{array}\]
\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}\;{sin}^{2} \le 1.375411136237947412717068778069207808383 \cdot 10^{-257}:\\
\;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot {cos}^{\left(\frac{2}{2}\right)}\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)\right)}\\

\mathbf{elif}\;{sin}^{2} \le 4.691137056220841244900674601205005981837 \cdot 10^{253}:\\
\;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot \left({sin}^{2} \cdot \left(x \cdot {cos}^{\left(\frac{2}{2}\right)}\right)\right)\right)}\\

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

\end{array}
double f(double x, double cos, double sin) {
        double r3175088 = 2.0;
        double r3175089 = x;
        double r3175090 = r3175088 * r3175089;
        double r3175091 = cos(r3175090);
        double r3175092 = cos;
        double r3175093 = pow(r3175092, r3175088);
        double r3175094 = sin;
        double r3175095 = pow(r3175094, r3175088);
        double r3175096 = r3175089 * r3175095;
        double r3175097 = r3175096 * r3175089;
        double r3175098 = r3175093 * r3175097;
        double r3175099 = r3175091 / r3175098;
        return r3175099;
}
↓
double f(double x, double cos, double sin) {
        double r3175100 = sin;
        double r3175101 = 2.0;
        double r3175102 = pow(r3175100, r3175101);
        double r3175103 = 1.3754111362379474e-257;
        bool r3175104 = r3175102 <= r3175103;
        double r3175105 = x;
        double r3175106 = r3175105 * r3175101;
        double r3175107 = cos(r3175106);
        double r3175108 = cos;
        double r3175109 = 2.0;
        double r3175110 = r3175101 / r3175109;
        double r3175111 = pow(r3175108, r3175110);
        double r3175112 = pow(r3175100, r3175110);
        double r3175113 = r3175105 * r3175111;
        double r3175114 = r3175105 * r3175112;
        double r3175115 = r3175113 * r3175114;
        double r3175116 = r3175112 * r3175115;
        double r3175117 = r3175111 * r3175116;
        double r3175118 = r3175107 / r3175117;
        double r3175119 = 4.691137056220841e+253;
        bool r3175120 = r3175102 <= r3175119;
        double r3175121 = r3175102 * r3175113;
        double r3175122 = r3175105 * r3175121;
        double r3175123 = r3175111 * r3175122;
        double r3175124 = r3175107 / r3175123;
        double r3175125 = r3175120 ? r3175124 : r3175118;
        double r3175126 = r3175104 ? r3175118 : r3175125;
        return r3175126;
}

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 sin 2.0) < 1.3754111362379474e-257 or 4.691137056220841e+253 < (pow sin 2.0)

    1. Initial program 33.0

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

    3. Applied sqr-pow33.0

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

    4. Applied associate-*r*22.7

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

    6. Applied sqr-pow22.7

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

    7. Applied associate-*l*17.9

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

    8. Simplified29.4

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

    10. Applied sqr-pow29.4

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

    11. Applied associate-*r*19.1

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

    13. Applied associate-*r*11.8

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

    if 1.3754111362379474e-257 < (pow sin 2.0) < 4.691137056220841e+253

    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*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 sqr-pow21.2

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

    7. Applied associate-*l*15.1

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

    8. Simplified7.2

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

    10. Applied sqr-pow7.2

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

    11. Applied associate-*r*7.2

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

    13. Applied pow17.2

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

    14. Applied pow17.2

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

    15. Applied pow-prod-down7.2

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

    16. Simplified6.0

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;{sin}^{2} \le 1.375411136237947412717068778069207808383 \cdot 10^{-257}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot {cos}^{\left(\frac{2}{2}\right)}\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)\right)}\\ \mathbf{elif}\;{sin}^{2} \le 4.691137056220841244900674601205005981837 \cdot 10^{253}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot \left({sin}^{2} \cdot \left(x \cdot {cos}^{\left(\frac{2}{2}\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot {cos}^{\left(\frac{2}{2}\right)}\right) \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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))))