Average Error: 28.1 → 2.4
Time: 8.0s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{1}{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|}
double f(double x, double cos, double sin) {
        double r71351 = 2.0;
        double r71352 = x;
        double r71353 = r71351 * r71352;
        double r71354 = cos(r71353);
        double r71355 = cos;
        double r71356 = pow(r71355, r71351);
        double r71357 = sin;
        double r71358 = pow(r71357, r71351);
        double r71359 = r71352 * r71358;
        double r71360 = r71359 * r71352;
        double r71361 = r71356 * r71360;
        double r71362 = r71354 / r71361;
        return r71362;
}
↓
double f(double x, double cos, double sin) {
        double r71363 = 1.0;
        double r71364 = cos;
        double r71365 = 1.0;
        double r71366 = pow(r71364, r71365);
        double r71367 = sin;
        double r71368 = pow(r71367, r71365);
        double r71369 = r71366 * r71368;
        double r71370 = pow(r71369, r71365);
        double r71371 = x;
        double r71372 = r71370 * r71371;
        double r71373 = fabs(r71372);
        double r71374 = r71363 / r71373;
        double r71375 = 2.0;
        double r71376 = r71375 * r71371;
        double r71377 = cos(r71376);
        double r71378 = r71377 / r71373;
        double r71379 = r71374 * r71378;
        return r71379;
}

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 28.1

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

  3. Applied sqr-pow28.1

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

  4. Applied associate-*r*21.9

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

  6. Applied add-sqr-sqrt21.9

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

  7. Simplified21.9

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

  8. Simplified3.0

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

  9. Taylor expanded around inf 2.6

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

  11. Applied add-sqr-sqrt2.8

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

  12. Applied unpow-prod-down2.8

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

  13. Applied *-un-lft-identity2.8

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

  14. Applied times-frac2.6

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

  15. Simplified2.5

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

  16. Simplified2.4

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

  17. Final simplification2.4

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

Reproduce

herbie shell --seed 2020003 +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))))