Average Error: 27.8 → 2.9
Time: 1.7m
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}}{\left(\sqrt[3]{sin} \cdot \sqrt[3]{sin}\right) \cdot \left(\sqrt[3]{sin} \cdot \left(cos \cdot x\right)\right)}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}}{\left(\sqrt[3]{sin} \cdot \sqrt[3]{sin}\right) \cdot \left(\sqrt[3]{sin} \cdot \left(cos \cdot x\right)\right)}
double f(double x, double cos, double sin) {
        double r12385284 = 2.0;
        double r12385285 = x;
        double r12385286 = r12385284 * r12385285;
        double r12385287 = cos(r12385286);
        double r12385288 = cos;
        double r12385289 = pow(r12385288, r12385284);
        double r12385290 = sin;
        double r12385291 = pow(r12385290, r12385284);
        double r12385292 = r12385285 * r12385291;
        double r12385293 = r12385292 * r12385285;
        double r12385294 = r12385289 * r12385293;
        double r12385295 = r12385287 / r12385294;
        return r12385295;
}
↓
double f(double x, double cos, double sin) {
        double r12385296 = 2.0;
        double r12385297 = x;
        double r12385298 = r12385296 * r12385297;
        double r12385299 = cos(r12385298);
        double r12385300 = cos;
        double r12385301 = r12385300 * r12385297;
        double r12385302 = sin;
        double r12385303 = r12385301 * r12385302;
        double r12385304 = r12385299 / r12385303;
        double r12385305 = cbrt(r12385302);
        double r12385306 = r12385305 * r12385305;
        double r12385307 = r12385305 * r12385301;
        double r12385308 = r12385306 * r12385307;
        double r12385309 = r12385304 / r12385308;
        return r12385309;
}

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. Simplified2.8

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

  4. Applied associate-/r*2.5

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

  6. Applied add-cube-cbrt2.9

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

  7. Applied associate-*l*2.9

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

  8. Final simplification2.9

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

Reproduce

herbie shell --seed 2019119 +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))))