Average Error: 27.9 → 2.7
Time: 8.4s
Precision: binary64
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|}\]
\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}
↓
\frac{1}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|} \cdot \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|}
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) fabs(((double) (((double) pow(c, ((double) (2.0 / 2.0)))) * ((double) (x * ((double) pow(s, ((double) (2.0 / 2.0)))))))))))) * ((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))))))))))))));
}

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 27.9

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

  3. Applied sqr-pow27.9

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

  4. Applied associate-*r*21.9

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

  6. Applied add-sqr-sqrt22.0

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

  7. Simplified21.9

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

  8. Simplified3.0

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

  10. Applied *-un-lft-identity3.0

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

  11. Applied times-frac2.7

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

  12. Final simplification2.7

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

Reproduce

herbie shell --seed 2020157 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))