Average Error: 27.0 → 2.7
Time: 32.3s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\cos x \cdot \cos x}{\left(sin \cdot x\right) \cdot cos} - \frac{\sin x \cdot \sin x}{\left(sin \cdot x\right) \cdot cos}}{\left(sin \cdot x\right) \cdot cos}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{\cos x \cdot \cos x}{\left(sin \cdot x\right) \cdot cos} - \frac{\sin x \cdot \sin x}{\left(sin \cdot x\right) \cdot cos}}{\left(sin \cdot x\right) \cdot cos}
double f(double x, double cos, double sin) {
        double r2688345 = 2.0;
        double r2688346 = x;
        double r2688347 = r2688345 * r2688346;
        double r2688348 = cos(r2688347);
        double r2688349 = cos;
        double r2688350 = pow(r2688349, r2688345);
        double r2688351 = sin;
        double r2688352 = pow(r2688351, r2688345);
        double r2688353 = r2688346 * r2688352;
        double r2688354 = r2688353 * r2688346;
        double r2688355 = r2688350 * r2688354;
        double r2688356 = r2688348 / r2688355;
        return r2688356;
}
↓
double f(double x, double cos, double sin) {
        double r2688357 = x;
        double r2688358 = cos(r2688357);
        double r2688359 = r2688358 * r2688358;
        double r2688360 = sin;
        double r2688361 = r2688360 * r2688357;
        double r2688362 = cos;
        double r2688363 = r2688361 * r2688362;
        double r2688364 = r2688359 / r2688363;
        double r2688365 = sin(r2688357);
        double r2688366 = r2688365 * r2688365;
        double r2688367 = r2688366 / r2688363;
        double r2688368 = r2688364 - r2688367;
        double r2688369 = r2688368 / r2688363;
        return r2688369;
}

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 27.0

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

  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*2.7

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

  6. Applied cos-22.7

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

  7. Applied div-sub2.7

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

  8. Final simplification2.7

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

Reproduce

herbie shell --seed 2019146 +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))))