Average Error: 28.0 → 2.9
Time: 13.0s
Precision: binary64
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}\]
↓
\[\cos \left(2 \cdot x\right) \cdot \frac{1}{{\left(\left|{\left({s}^{1} \cdot {c}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\]
\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}
↓
\cos \left(2 \cdot x\right) \cdot \frac{1}{{\left(\left|{\left({s}^{1} \cdot {c}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}
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) cos(((double) (2.0 * x)))) * ((double) (1.0 / ((double) pow(((double) fabs(((double) (((double) pow(((double) (((double) pow(s, 1.0)) * ((double) pow(c, 1.0)))), 1.0)) * x)))), 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-pow28.0

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

  4. Applied associate-*r*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-sqrt21.9

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

  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. Simplified2.8

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

  9. Taylor expanded around 0 2.9

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

  11. Applied div-inv2.9

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

  12. Final simplification2.9

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

Reproduce

herbie shell --seed 2020163 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))