Average Error: 28.0 → 2.7
Time: 7.9s
Precision: binary64
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)} \cdot \frac{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)}\]
\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}
↓
\frac{1}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)} \cdot \frac{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)}
(FPCore (x c s)
 :precision binary64
 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))
↓
(FPCore (x c s)
 :precision binary64
 (*
  (/ 1.0 (* (pow c (/ 2.0 2.0)) (* x (pow s (/ 2.0 2.0)))))
  (/ (cos (* 2.0 x)) (* (pow c (/ 2.0 2.0)) (* x (pow s (/ 2.0 2.0)))))))
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) (((double) pow(c, (2.0 / 2.0))) * ((double) (x * ((double) pow(s, (2.0 / 2.0)))))))) * (((double) cos(((double) (2.0 * x)))) / ((double) (((double) pow(c, (2.0 / 2.0))) * ((double) (x * ((double) pow(s, (2.0 / 2.0))))))))));
}

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 28.0

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

  3. Applied sqr-pow_binary6428.0

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

  4. Applied associate-*r*_binary6421.8

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

  6. Applied associate-*l*_binary6419.7

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

  7. Simplified19.7

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

  9. Applied sqr-pow_binary6419.7

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

  10. Applied unswap-sqr_binary643.1

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

  11. Simplified3.1

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

  12. Simplified3.1

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

  14. Applied *-un-lft-identity_binary643.1

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

  15. Applied times-frac_binary642.7

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

  16. Simplified2.7

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

  17. Simplified2.7

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

  18. Final simplification2.7

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

Reproduce

herbie shell --seed 2020204 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))