Average Error: 27.3 → 3.0
Time: 23.2s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(2 \cdot x\right)}{\left(\left(cos \cdot x\right) \cdot sin\right) \cdot \left(\left(cos \cdot x\right) \cdot sin\right)}\]
\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(\left(cos \cdot x\right) \cdot sin\right) \cdot \left(\left(cos \cdot x\right) \cdot sin\right)}
double f(double x, double cos, double sin) {
        double r951976 = 2.0;
        double r951977 = x;
        double r951978 = r951976 * r951977;
        double r951979 = cos(r951978);
        double r951980 = cos;
        double r951981 = pow(r951980, r951976);
        double r951982 = sin;
        double r951983 = pow(r951982, r951976);
        double r951984 = r951977 * r951983;
        double r951985 = r951984 * r951977;
        double r951986 = r951981 * r951985;
        double r951987 = r951979 / r951986;
        return r951987;
}
↓
double f(double x, double cos, double sin) {
        double r951988 = 2.0;
        double r951989 = x;
        double r951990 = r951988 * r951989;
        double r951991 = cos(r951990);
        double r951992 = cos;
        double r951993 = r951992 * r951989;
        double r951994 = sin;
        double r951995 = r951993 * r951994;
        double r951996 = r951995 * r951995;
        double r951997 = r951991 / r951996;
        return r951997;
}

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 27.3

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

  2. Simplified3.0

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

  4. Applied associate-/r*2.8

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

  6. Applied associate-/r*2.8

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

  7. Taylor expanded around -inf 31.0

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

  8. Simplified3.0

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

  9. Final simplification3.0

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(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))))