Average Error: 28.2 → 2.7
Time: 7.5s
Precision: binary64
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot \frac{1}{2}\right)}} \cdot \frac{\cos \left(x \cdot 2\right)}{x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot \frac{1}{2}\right)}}\]
\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}
↓
\frac{1}{x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot \frac{1}{2}\right)}} \cdot \frac{\cos \left(x \cdot 2\right)}{x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot \frac{1}{2}\right)}}
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) {
	return ((double) (((double) (1.0 / ((double) (x * ((double) pow(((double) (c * s)), ((double) (2.0 * 0.5)))))))) * ((double) (((double) cos(((double) (x * 2.0)))) / ((double) (x * ((double) pow(((double) (c * s)), ((double) (2.0 * 0.5))))))))));
}

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. Initial program 28.2

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

  2. Simplified28.2

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

  4. Applied pow-prod-down13.3

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

  6. Applied sqr-pow13.3

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

  7. Applied associate-*r*5.9

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

  8. Simplified5.9

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

  10. Applied associate-/r*5.7

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

  11. Simplified5.7

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

  13. Applied *-un-lft-identity5.7

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

  14. Applied *-un-lft-identity5.7

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

  15. Applied times-frac5.7

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

  16. Applied times-frac2.7

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

  17. Simplified2.7

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

  18. Simplified2.7

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

  19. Final simplification2.7

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

Reproduce

herbie shell --seed 2020184 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))