Average Error: 27.8 → 15.0
Time: 8.6s
Precision: binary64
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;{c}^{2} \le 1.2069967661290195 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot {c}^{\left(\frac{2}{2}\right)}}}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot {\left(\sqrt[3]{s} \cdot \sqrt[3]{s}\right)}^{2}\right) \cdot {\left(\sqrt[3]{s}\right)}^{2}\right)}\\ \mathbf{elif}\;{c}^{2} \le 5.4405840560867589 \cdot 10^{247}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left({s}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot {c}^{\left(\frac{2}{2}\right)}} \cdot \frac{\cos \left(2 \cdot x\right)}{x \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{2}\right)}\\ \end{array}\]
\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}
↓
\begin{array}{l}
\mathbf{if}\;{c}^{2} \le 1.2069967661290195 \cdot 10^{-118}:\\
\;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot {c}^{\left(\frac{2}{2}\right)}}}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot {\left(\sqrt[3]{s} \cdot \sqrt[3]{s}\right)}^{2}\right) \cdot {\left(\sqrt[3]{s}\right)}^{2}\right)}\\

\mathbf{elif}\;{c}^{2} \le 5.4405840560867589 \cdot 10^{247}:\\
\;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left({s}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x \cdot {c}^{\left(\frac{2}{2}\right)}} \cdot \frac{\cos \left(2 \cdot x\right)}{x \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{2}\right)}\\

\end{array}
double code(double x, double c, double s) {
	return ((double) (((double) cos(((double) (2.0 * x)))) / ((double) (((double) pow(c, 2.0)) * ((double) (((double) (x * ((double) pow(s, 2.0)))) * x))))));
}
↓
double code(double x, double c, double s) {
	double VAR;
	if ((((double) pow(c, 2.0)) <= 1.2069967661290195e-118)) {
		VAR = ((double) (((double) (((double) cos(((double) (2.0 * x)))) / ((double) (x * ((double) pow(c, ((double) (2.0 / 2.0)))))))) / ((double) (((double) pow(c, ((double) (2.0 / 2.0)))) * ((double) (((double) (x * ((double) pow(((double) (((double) cbrt(s)) * ((double) cbrt(s)))), 2.0)))) * ((double) pow(((double) cbrt(s)), 2.0))))))));
	} else {
		double VAR_1;
		if ((((double) pow(c, 2.0)) <= 5.440584056086759e+247)) {
			VAR_1 = ((double) (((double) cos(((double) (2.0 * x)))) / ((double) (((double) pow(c, 2.0)) * ((double) (x * ((double) (((double) pow(s, ((double) (2.0 / 2.0)))) * ((double) (x * ((double) pow(s, ((double) (2.0 / 2.0))))))))))))));
		} else {
			VAR_1 = ((double) (((double) (1.0 / ((double) (x * ((double) pow(c, ((double) (2.0 / 2.0)))))))) * ((double) (((double) cos(((double) (2.0 * x)))) / ((double) (x * ((double) (((double) pow(c, ((double) (2.0 / 2.0)))) * ((double) pow(s, 2.0))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus c

Bits error versus s

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (pow c 2.0) < 1.2069967661290195e-118

    1. Initial program 44.0

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

    3. Applied sqr-pow44.0

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

    4. Applied associate-*l*33.2

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

    5. Simplified26.1

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

    7. Applied associate-*r*25.0

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

    8. Simplified25.0

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

    10. Applied associate-/r*25.0

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

    11. Simplified25.0

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

    13. Applied add-cube-cbrt25.5

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

    14. Applied unpow-prod-down25.5

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

    15. Applied associate-*r*21.4

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

    if 1.2069967661290195e-118 < (pow c 2.0) < 5.4405840560867589e247

    1. Initial program 18.4

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

    3. Applied sqr-pow18.4

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

    4. Applied associate-*r*8.3

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

    if 5.4405840560867589e247 < (pow c 2.0)

    1. Initial program 24.6

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

    3. Applied sqr-pow24.6

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

    4. Applied associate-*l*20.6

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

    5. Simplified17.2

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

    7. Applied associate-*r*16.7

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

    8. Simplified16.7

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

    10. Applied *-un-lft-identity16.7

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

    11. Applied times-frac16.6

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

    12. Simplified16.2

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

  4. Final simplification15.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;{c}^{2} \le 1.2069967661290195 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot {c}^{\left(\frac{2}{2}\right)}}}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot {\left(\sqrt[3]{s} \cdot \sqrt[3]{s}\right)}^{2}\right) \cdot {\left(\sqrt[3]{s}\right)}^{2}\right)}\\ \mathbf{elif}\;{c}^{2} \le 5.4405840560867589 \cdot 10^{247}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left({s}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot {c}^{\left(\frac{2}{2}\right)}} \cdot \frac{\cos \left(2 \cdot x\right)}{x \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))