Average Error: 28.3 → 7.0
Time: 13.5s
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 r71326 = 2.0;
        double r71327 = x;
        double r71328 = r71326 * r71327;
        double r71329 = cos(r71328);
        double r71330 = cos;
        double r71331 = pow(r71330, r71326);
        double r71332 = sin;
        double r71333 = pow(r71332, r71326);
        double r71334 = r71327 * r71333;
        double r71335 = r71334 * r71327;
        double r71336 = r71331 * r71335;
        double r71337 = r71329 / r71336;
        return r71337;
}
↓
double f(double x, double cos, double sin) {
        double r71338 = 2.0;
        double r71339 = x;
        double r71340 = r71338 * r71339;
        double r71341 = cos(r71340);
        double r71342 = cos;
        double r71343 = 2.0;
        double r71344 = r71338 / r71343;
        double r71345 = pow(r71342, r71344);
        double r71346 = sin;
        double r71347 = pow(r71346, r71344);
        double r71348 = r71339 * r71347;
        double r71349 = r71345 * r71348;
        double r71350 = r71345 * r71349;
        double r71351 = r71347 * r71339;
        double r71352 = r71350 * r71351;
        double r71353 = r71341 / r71352;
        return r71353;
}

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 28.3

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

  3. Applied sqr-pow28.3

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

  4. Applied associate-*r*22.2

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

  6. Applied associate-*l*20.0

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

  8. Applied associate-*r*16.1

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

  10. Applied sqr-pow16.1

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

  11. Applied associate-*l*7.0

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

  12. Final simplification7.0

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

Reproduce

herbie shell --seed 2019350 +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))))