Average Error: 28.2 → 2.7
Time: 18.6s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\cos \left(2 \cdot x\right)}{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|}}{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|}\]
\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|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|}}{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|}
double f(double x, double cos, double sin) {
        double r53601 = 2.0;
        double r53602 = x;
        double r53603 = r53601 * r53602;
        double r53604 = cos(r53603);
        double r53605 = cos;
        double r53606 = pow(r53605, r53601);
        double r53607 = sin;
        double r53608 = pow(r53607, r53601);
        double r53609 = r53602 * r53608;
        double r53610 = r53609 * r53602;
        double r53611 = r53606 * r53610;
        double r53612 = r53604 / r53611;
        return r53612;
}
↓
double f(double x, double cos, double sin) {
        double r53613 = 2.0;
        double r53614 = x;
        double r53615 = r53613 * r53614;
        double r53616 = cos(r53615);
        double r53617 = cos;
        double r53618 = 1.0;
        double r53619 = pow(r53617, r53618);
        double r53620 = sin;
        double r53621 = pow(r53620, r53618);
        double r53622 = r53619 * r53621;
        double r53623 = pow(r53622, r53618);
        double r53624 = r53623 * r53614;
        double r53625 = fabs(r53624);
        double r53626 = r53616 / r53625;
        double r53627 = r53626 / r53625;
        return r53627;
}

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 28.2

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

  3. Applied sqr-pow28.2

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

  4. Applied associate-*r*22.2

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

  6. Applied add-sqr-sqrt22.2

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

  7. Simplified22.2

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

  8. Simplified2.8

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

  9. Taylor expanded around inf 2.9

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

  11. Applied sqr-pow2.9

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

  12. Applied associate-/r*2.7

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

  13. Simplified2.7

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

  14. Final simplification2.7

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

Reproduce

herbie shell --seed 2019291 
(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))))