Average Error: 28.8 → 2.8
Time: 10.1s
Precision: binary64
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\left|\left(x \cdot s\right) \cdot c\right|} \cdot \left(\frac{1}{\sqrt{\left|\left(x \cdot s\right) \cdot c\right|}} \cdot \frac{\cos \left(x \cdot 2\right)}{\sqrt{\sqrt{\left|\left(x \cdot s\right) \cdot c\right|}} \cdot \sqrt{\sqrt{\left|\left(x \cdot s\right) \cdot c\right|}}}\right)\]
\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}
↓
\frac{1}{\left|\left(x \cdot s\right) \cdot c\right|} \cdot \left(\frac{1}{\sqrt{\left|\left(x \cdot s\right) \cdot c\right|}} \cdot \frac{\cos \left(x \cdot 2\right)}{\sqrt{\sqrt{\left|\left(x \cdot s\right) \cdot c\right|}} \cdot \sqrt{\sqrt{\left|\left(x \cdot s\right) \cdot c\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 (fabs (* (* x s) c)))
  (*
   (/ 1.0 (sqrt (fabs (* (* x s) c))))
   (/
    (cos (* x 2.0))
    (*
     (sqrt (sqrt (fabs (* (* x s) c))))
     (sqrt (sqrt (fabs (* (* x s) c)))))))))
double code(double x, double c, double s) {
	return cos(2.0 * x) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x));
}
↓
double code(double x, double c, double s) {
	return (1.0 / fabs((x * s) * c)) * ((1.0 / sqrt(fabs((x * s) * c))) * (cos(x * 2.0) / (sqrt(sqrt(fabs((x * s) * c))) * sqrt(sqrt(fabs((x * s) * c))))));
}

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 28.8

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

  3. Applied add-sqr-sqrt_binary6428.8

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

  4. Simplified28.8

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

  5. Simplified3.0

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

  7. Applied *-un-lft-identity_binary643.0

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

  8. Applied times-frac_binary642.7

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

  9. Simplified2.7

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

  11. Applied add-sqr-sqrt_binary642.8

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

  12. Applied *-un-lft-identity_binary642.8

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

  13. Applied times-frac_binary642.8

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

  15. Applied add-sqr-sqrt_binary642.8

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

  16. Final simplification2.8

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

Reproduce

herbie shell --seed 2020268 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))