Average Error: 27.8 → 2.9
Time: 7.7s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\frac{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}{\cos \left(2 \cdot x\right)}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{1}{\frac{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}{\cos \left(2 \cdot x\right)}}
double code(double x, double cos, double sin) {
	return (cos((2.0 * x)) / (pow(cos, 2.0) * ((x * pow(sin, 2.0)) * x)));
}
↓
double code(double x, double cos, double sin) {
	return (1.0 / (pow(fabs((pow((pow(cos, 1.0) * pow(sin, 1.0)), 1.0) * x)), 2.0) / cos((2.0 * x))));
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 27.8

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

  3. Applied sqr-pow27.8

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

  4. Applied associate-*r*21.8

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

  6. Applied add-sqr-sqrt21.8

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

  7. Simplified21.8

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

  8. Simplified3.0

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

  9. Taylor expanded around 0 2.9

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

  10. Simplified2.9

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

  12. Applied clear-num2.9

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

  13. Final simplification2.9

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

Reproduce

herbie shell --seed 2020075 
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  :precision binary64
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))