Average Error: 28.0 → 2.8
Time: 41.2s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(x \cdot 2\right)}{\left(cos \cdot sin\right) \cdot x} \cdot \frac{1}{\left(\left(cos \cdot sin\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\cos \left(x \cdot 2\right)}{\left(cos \cdot sin\right) \cdot x} \cdot \frac{1}{\left(\left(cos \cdot sin\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x}}
double f(double x, double cos, double sin) {
        double r1620035 = 2.0;
        double r1620036 = x;
        double r1620037 = r1620035 * r1620036;
        double r1620038 = cos(r1620037);
        double r1620039 = cos;
        double r1620040 = pow(r1620039, r1620035);
        double r1620041 = sin;
        double r1620042 = pow(r1620041, r1620035);
        double r1620043 = r1620036 * r1620042;
        double r1620044 = r1620043 * r1620036;
        double r1620045 = r1620040 * r1620044;
        double r1620046 = r1620038 / r1620045;
        return r1620046;
}
↓
double f(double x, double cos, double sin) {
        double r1620047 = x;
        double r1620048 = 2.0;
        double r1620049 = r1620047 * r1620048;
        double r1620050 = cos(r1620049);
        double r1620051 = cos;
        double r1620052 = sin;
        double r1620053 = r1620051 * r1620052;
        double r1620054 = r1620053 * r1620047;
        double r1620055 = r1620050 / r1620054;
        double r1620056 = 1.0;
        double r1620057 = cbrt(r1620047);
        double r1620058 = r1620057 * r1620057;
        double r1620059 = r1620053 * r1620058;
        double r1620060 = r1620059 * r1620057;
        double r1620061 = r1620056 / r1620060;
        double r1620062 = r1620055 * r1620061;
        return r1620062;
}

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 28.0

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

  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 times-frac2.5

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

  7. Applied add-cube-cbrt2.8

    \[\leadsto \frac{1}{\left(sin \cdot cos\right) \cdot \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}} \cdot \frac{\cos \left(2 \cdot x\right)}{\left(sin \cdot cos\right) \cdot x}\]

  8. Applied associate-*r*2.8

    \[\leadsto \frac{1}{\color{blue}{\left(\left(sin \cdot cos\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x}}} \cdot \frac{\cos \left(2 \cdot x\right)}{\left(sin \cdot cos\right) \cdot x}\]

  9. Final simplification2.8

    \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(cos \cdot sin\right) \cdot x} \cdot \frac{1}{\left(\left(cos \cdot sin\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x}}\]

Reproduce

herbie shell --seed 2019132 +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))))