Average Error: 27.7 → 2.6
Time: 7.7s
Precision: binary64
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot 0.5\right)}} \cdot \frac{\cos \left(x \cdot 2\right)}{x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot 0.5\right)}}\]
\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 0.5\right)}} \cdot \frac{\cos \left(x \cdot 2\right)}{x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot 0.5\right)}}
double code(double x, double c, double s) {
	return (((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) ((1.0 / ((double) (x * ((double) pow(((double) (c * s)), ((double) (2.0 * 0.5))))))) * (((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 Error: 27.7 bits

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

  2. SimplifiedError: 27.8 bits

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

  4. Applied pow-prod-downError: 13.2 bits

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

  6. Applied sqr-powError: 13.2 bits

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

  7. Applied associate-*r*Error: 5.9 bits

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

  8. SimplifiedError: 5.9 bits

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\color{blue}{\left(x \cdot {\left(c \cdot s\right)}^{\left(0.5 \cdot 2\right)}\right)} \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}\right)}\]
  9. Using strategy rm

  10. Applied associate-/r*Error: 5.7 bits

    \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(x \cdot {\left(c \cdot s\right)}^{\left(0.5 \cdot 2\right)}\right) \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}}\]

  11. SimplifiedError: 5.7 bits

    \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 2\right)}{x}}}{\left(x \cdot {\left(c \cdot s\right)}^{\left(0.5 \cdot 2\right)}\right) \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\]
  12. Using strategy rm

  13. Applied *-un-lft-identityError: 5.7 bits

    \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\color{blue}{1 \cdot x}}}{\left(x \cdot {\left(c \cdot s\right)}^{\left(0.5 \cdot 2\right)}\right) \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\]

  14. Applied *-un-lft-identityError: 5.7 bits

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \cos \left(x \cdot 2\right)}}{1 \cdot x}}{\left(x \cdot {\left(c \cdot s\right)}^{\left(0.5 \cdot 2\right)}\right) \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\]

  15. Applied times-fracError: 5.7 bits

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{\cos \left(x \cdot 2\right)}{x}}}{\left(x \cdot {\left(c \cdot s\right)}^{\left(0.5 \cdot 2\right)}\right) \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\]

  16. Applied times-fracError: 2.5 bits

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{x \cdot {\left(c \cdot s\right)}^{\left(0.5 \cdot 2\right)}} \cdot \frac{\frac{\cos \left(x \cdot 2\right)}{x}}{{\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}}\]

  17. SimplifiedError: 2.5 bits

    \[\leadsto \color{blue}{\frac{1}{x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot 0.5\right)}}} \cdot \frac{\frac{\cos \left(x \cdot 2\right)}{x}}{{\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\]

  18. SimplifiedError: 2.6 bits

    \[\leadsto \frac{1}{x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot 0.5\right)}} \cdot \color{blue}{\frac{\cos \left(x \cdot 2\right)}{x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot 0.5\right)}}}\]

  19. Final simplificationError: 2.6 bits

    \[\leadsto \frac{1}{x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot 0.5\right)}} \cdot \frac{\cos \left(x \cdot 2\right)}{x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot 0.5\right)}}\]

Reproduce

herbie shell --seed 2020200 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))