Average Error: 28.3 → 1.9
Time: 8.0s
Precision: binary64
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;{c}^{2} \le 0.0 \lor \neg \left({c}^{2} \le 1.4929315053770008 \cdot 10^{246}\right):\\ \;\;\;\;\frac{1}{\left|{\left({s}^{1} \cdot {c}^{1}\right)}^{1} \cdot x\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|{\left({s}^{1} \cdot {c}^{1}\right)}^{1} \cdot x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|}}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|}\\ \end{array}\]
\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 0.0 \lor \neg \left({c}^{2} \le 1.4929315053770008 \cdot 10^{246}\right):\\
\;\;\;\;\frac{1}{\left|{\left({s}^{1} \cdot {c}^{1}\right)}^{1} \cdot x\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|{\left({s}^{1} \cdot {c}^{1}\right)}^{1} \cdot x\right|}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|}}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\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)) <= 0.0) || !(((double) pow(c, 2.0)) <= 1.4929315053770008e+246))) {
		VAR = ((double) (((double) (1.0 / ((double) fabs(((double) (((double) pow(((double) (((double) pow(s, 1.0)) * ((double) pow(c, 1.0)))), 1.0)) * x)))))) * ((double) (((double) cos(((double) (2.0 * x)))) / ((double) fabs(((double) (((double) pow(((double) (((double) pow(s, 1.0)) * ((double) pow(c, 1.0)))), 1.0)) * x))))))));
	} else {
		VAR = ((double) (((double) (((double) cos(((double) (2.0 * x)))) / ((double) fabs(((double) (((double) pow(c, ((double) (2.0 / 2.0)))) * ((double) (x * ((double) pow(s, ((double) (2.0 / 2.0)))))))))))) / ((double) fabs(((double) (((double) pow(c, ((double) (2.0 / 2.0)))) * ((double) (x * ((double) pow(s, ((double) (2.0 / 2.0))))))))))));
	}
	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 2 regimes

  2. if (pow c 2.0) < 0.0 or 1.4929315053770008e246 < (pow c 2.0)

    1. Initial program 34.1

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

    3. Applied sqr-pow34.1

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

    4. Applied associate-*r*30.8

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

    6. Applied add-sqr-sqrt30.8

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

    7. Simplified30.8

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

    8. Simplified4.7

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

    9. Taylor expanded around inf 3.3

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({s}^{1} \cdot {c}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}}\]
    10. Using strategy rm

    11. Applied add-sqr-sqrt3.4

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

    12. Applied unpow-prod-down3.4

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

    13. Applied *-un-lft-identity3.4

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

    14. Applied times-frac3.2

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

    15. Simplified3.1

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

    16. Simplified3.1

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

    if 0.0 < (pow c 2.0) < 1.4929315053770008e246

    1. Initial program 21.6

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

    3. Applied sqr-pow21.6

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

    4. Applied associate-*r*11.4

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

    6. Applied add-sqr-sqrt11.5

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

    7. Simplified11.5

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

    8. Simplified0.8

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

    10. Applied associate-/r*0.5

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;{c}^{2} \le 0.0 \lor \neg \left({c}^{2} \le 1.4929315053770008 \cdot 10^{246}\right):\\ \;\;\;\;\frac{1}{\left|{\left({s}^{1} \cdot {c}^{1}\right)}^{1} \cdot x\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|{\left({s}^{1} \cdot {c}^{1}\right)}^{1} \cdot x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|}}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|}\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))