Average Error: 28.3 → 2.7
Time: 12.3s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\]
\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)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}
double f(double x, double cos, double sin) {
        double r63632 = 2.0;
        double r63633 = x;
        double r63634 = r63632 * r63633;
        double r63635 = cos(r63634);
        double r63636 = cos;
        double r63637 = pow(r63636, r63632);
        double r63638 = sin;
        double r63639 = pow(r63638, r63632);
        double r63640 = r63633 * r63639;
        double r63641 = r63640 * r63633;
        double r63642 = r63637 * r63641;
        double r63643 = r63635 / r63642;
        return r63643;
}
↓
double f(double x, double cos, double sin) {
        double r63644 = 2.0;
        double r63645 = x;
        double r63646 = r63644 * r63645;
        double r63647 = cos(r63646);
        double r63648 = cos;
        double r63649 = 2.0;
        double r63650 = r63644 / r63649;
        double r63651 = pow(r63648, r63650);
        double r63652 = sin;
        double r63653 = pow(r63652, r63650);
        double r63654 = r63645 * r63653;
        double r63655 = r63651 * r63654;
        double r63656 = r63647 / r63655;
        double r63657 = r63656 / r63655;
        return r63657;
}

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 28.3

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

  3. Applied sqr-pow28.3

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

  4. Applied associate-*r*22.3

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

  6. Applied associate-*l*20.2

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

  7. Simplified20.2

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

  9. Applied sqr-pow20.2

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

  10. Applied unswap-sqr2.9

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

  12. Applied associate-/r*2.7

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

  13. Final simplification2.7

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  :precision binary64
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))