Average Error: 27.8 → 2.6
Time: 42.0s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\frac{\left(sin \cdot \left(cos \cdot x\right)\right) \cdot \left(sin \cdot \left(cos \cdot x\right)\right)}{\cos \left(x \cdot 2\right)}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{1}{\frac{\left(sin \cdot \left(cos \cdot x\right)\right) \cdot \left(sin \cdot \left(cos \cdot x\right)\right)}{\cos \left(x \cdot 2\right)}}
double f(double x, double cos, double sin) {
        double r3212923 = 2.0;
        double r3212924 = x;
        double r3212925 = r3212923 * r3212924;
        double r3212926 = cos(r3212925);
        double r3212927 = cos;
        double r3212928 = pow(r3212927, r3212923);
        double r3212929 = sin;
        double r3212930 = pow(r3212929, r3212923);
        double r3212931 = r3212924 * r3212930;
        double r3212932 = r3212931 * r3212924;
        double r3212933 = r3212928 * r3212932;
        double r3212934 = r3212926 / r3212933;
        return r3212934;
}
↓
double f(double x, double cos, double sin) {
        double r3212935 = 1.0;
        double r3212936 = sin;
        double r3212937 = cos;
        double r3212938 = x;
        double r3212939 = r3212937 * r3212938;
        double r3212940 = r3212936 * r3212939;
        double r3212941 = r3212940 * r3212940;
        double r3212942 = 2.0;
        double r3212943 = r3212938 * r3212942;
        double r3212944 = cos(r3212943);
        double r3212945 = r3212941 / r3212944;
        double r3212946 = r3212935 / r3212945;
        return r3212946;
}

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. Simplified2.9

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

  4. Applied associate-/r*2.6

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

  6. Applied div-inv2.6

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

  8. Applied associate-/l*2.9

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

  9. Simplified2.6

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

  11. Applied clear-num2.6

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

  12. Final simplification2.6

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

Reproduce

herbie shell --seed 2019143 
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))