Average Error: 27.3 → 2.9
Time: 32.7s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\frac{\left(cos \cdot x\right) \cdot sin}{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{1}{\frac{\left(cos \cdot x\right) \cdot sin}{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}}}
double f(double x, double cos, double sin) {
        double r2283495 = 2.0;
        double r2283496 = x;
        double r2283497 = r2283495 * r2283496;
        double r2283498 = cos(r2283497);
        double r2283499 = cos;
        double r2283500 = pow(r2283499, r2283495);
        double r2283501 = sin;
        double r2283502 = pow(r2283501, r2283495);
        double r2283503 = r2283496 * r2283502;
        double r2283504 = r2283503 * r2283496;
        double r2283505 = r2283500 * r2283504;
        double r2283506 = r2283498 / r2283505;
        return r2283506;
}
↓
double f(double x, double cos, double sin) {
        double r2283507 = 1.0;
        double r2283508 = cos;
        double r2283509 = x;
        double r2283510 = r2283508 * r2283509;
        double r2283511 = sin;
        double r2283512 = r2283510 * r2283511;
        double r2283513 = 2.0;
        double r2283514 = r2283513 * r2283509;
        double r2283515 = cos(r2283514);
        double r2283516 = r2283515 / r2283512;
        double r2283517 = r2283512 / r2283516;
        double r2283518 = r2283507 / r2283517;
        return r2283518;
}

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. 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*2.6

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

  6. Applied associate-/r*2.8

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

  8. Applied clear-num3.0

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

  9. Simplified2.9

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

  10. Final simplification2.9

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

Reproduce

herbie shell --seed 2019149 +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))))