Average Error: 28.1 → 2.7
Time: 7.4s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(\frac{2}{2}\right)} \cdot x} \cdot \frac{1}{\left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right) \cdot {cos}^{\left(\frac{2}{2}\right)}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(\frac{2}{2}\right)} \cdot x} \cdot \frac{1}{\left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right) \cdot {cos}^{\left(\frac{2}{2}\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 (((cos((2.0 * x)) / pow(cos, (2.0 / 2.0))) / (pow(sin, (2.0 / 2.0)) * x)) * (1.0 / ((pow(sin, (2.0 / 2.0)) * x) * pow(cos, (2.0 / 2.0)))));
}

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 28.1

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

  3. Applied sqr-pow28.1

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

  4. Applied associate-*r*22.0

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

  5. Applied associate-*l*19.9

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

  6. Applied associate-*r*16.0

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

  7. Applied associate-/r*15.8

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

  9. Applied sqr-pow15.8

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

  10. Applied associate-*l*6.9

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

  11. Applied associate-/r*6.7

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

  13. Applied *-commutative6.7

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

  14. Applied associate-/r*6.7

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

  15. Applied associate-/l/2.6

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

  17. Applied div-inv2.8

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

  18. Applied times-frac2.8

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

  19. Simplified2.7

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

  20. Final simplification2.7

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

Reproduce

herbie shell --seed 2020071 
(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))))