Average Error: 28.7 → 3.2
Time: 17.4s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(2 \cdot x\right)}{{\left(\left|sin \cdot \left(x \cdot cos\right)\right|\right)}^{2}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\cos \left(2 \cdot x\right)}{{\left(\left|sin \cdot \left(x \cdot cos\right)\right|\right)}^{2}}
double f(double x, double cos, double sin) {
        double r48213 = 2.0;
        double r48214 = x;
        double r48215 = r48213 * r48214;
        double r48216 = cos(r48215);
        double r48217 = cos;
        double r48218 = pow(r48217, r48213);
        double r48219 = sin;
        double r48220 = pow(r48219, r48213);
        double r48221 = r48214 * r48220;
        double r48222 = r48221 * r48214;
        double r48223 = r48218 * r48222;
        double r48224 = r48216 / r48223;
        return r48224;
}
↓
double f(double x, double cos, double sin) {
        double r48225 = 2.0;
        double r48226 = x;
        double r48227 = r48225 * r48226;
        double r48228 = cos(r48227);
        double r48229 = sin;
        double r48230 = cos;
        double r48231 = r48226 * r48230;
        double r48232 = r48229 * r48231;
        double r48233 = fabs(r48232);
        double r48234 = 2.0;
        double r48235 = pow(r48233, r48234);
        double r48236 = r48228 / r48235;
        return r48236;
}

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 28.7

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

  3. Applied sqr-pow28.7

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

  4. Applied associate-*r*22.7

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

  6. Applied add-sqr-sqrt22.7

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

  7. Simplified22.7

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

  8. Simplified2.9

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

  9. Taylor expanded around 0 2.7

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

  10. Simplified2.7

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

  11. Taylor expanded around 0 3.2

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

  12. Final simplification3.2

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

Reproduce

herbie shell --seed 2019298 
(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))))