Average Error: 28.5 → 6.8
Time: 18.6s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(2 \cdot x\right)}{\left({cos}^{\left(\frac{2}{2}\right)} \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\cos \left(2 \cdot x\right)}{\left({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)\right) \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)}
double f(double x, double cos, double sin) {
        double r63403 = 2.0;
        double r63404 = x;
        double r63405 = r63403 * r63404;
        double r63406 = cos(r63405);
        double r63407 = cos;
        double r63408 = pow(r63407, r63403);
        double r63409 = sin;
        double r63410 = pow(r63409, r63403);
        double r63411 = r63404 * r63410;
        double r63412 = r63411 * r63404;
        double r63413 = r63408 * r63412;
        double r63414 = r63406 / r63413;
        return r63414;
}
↓
double f(double x, double cos, double sin) {
        double r63415 = 2.0;
        double r63416 = x;
        double r63417 = r63415 * r63416;
        double r63418 = cos(r63417);
        double r63419 = cos;
        double r63420 = 2.0;
        double r63421 = r63415 / r63420;
        double r63422 = pow(r63419, r63421);
        double r63423 = sin;
        double r63424 = pow(r63423, r63421);
        double r63425 = r63416 * r63424;
        double r63426 = r63422 * r63425;
        double r63427 = r63422 * r63426;
        double r63428 = r63424 * r63416;
        double r63429 = r63427 * r63428;
        double r63430 = r63418 / r63429;
        return r63430;
}

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 28.5

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

  3. Applied sqr-pow28.5

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

  4. Applied associate-*r*22.6

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

  6. Applied associate-*l*20.5

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

  8. Applied associate-*r*16.4

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

  10. Applied sqr-pow16.4

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

  11. Applied associate-*l*6.8

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

  12. Final simplification6.8

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

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(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))))