Average Error: 27.5 → 2.9
Time: 36.7s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\frac{sin \cdot \left(x \cdot cos\right)}{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{1}{\frac{sin \cdot \left(x \cdot cos\right)}{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}}
double f(double x, double cos, double sin) {
        double r1399142 = 2.0;
        double r1399143 = x;
        double r1399144 = r1399142 * r1399143;
        double r1399145 = cos(r1399144);
        double r1399146 = cos;
        double r1399147 = pow(r1399146, r1399142);
        double r1399148 = sin;
        double r1399149 = pow(r1399148, r1399142);
        double r1399150 = r1399143 * r1399149;
        double r1399151 = r1399150 * r1399143;
        double r1399152 = r1399147 * r1399151;
        double r1399153 = r1399145 / r1399152;
        return r1399153;
}
↓
double f(double x, double cos, double sin) {
        double r1399154 = 1.0;
        double r1399155 = sin;
        double r1399156 = x;
        double r1399157 = cos;
        double r1399158 = r1399156 * r1399157;
        double r1399159 = r1399155 * r1399158;
        double r1399160 = 2.0;
        double r1399161 = r1399160 * r1399156;
        double r1399162 = cos(r1399161);
        double r1399163 = r1399162 / r1399159;
        double r1399164 = r1399159 / r1399163;
        double r1399165 = r1399154 / r1399164;
        return r1399165;
}

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 27.5

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

  2. Simplified2.8

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

  4. Applied *-un-lft-identity2.8

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

  5. Applied associate-/l*2.9

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

  7. Applied *-un-lft-identity2.9

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

  8. Applied associate-/l*2.9

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

  9. Simplified2.9

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

  10. Final simplification2.9

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

Reproduce

herbie shell --seed 2019133 +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))))