Average Error: 27.6 → 2.8
Time: 22.4s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\left(\left(cos \cdot sin\right) \cdot x\right) \cdot \left(\left(cos \cdot sin\right) \cdot x\right)} \cdot \cos \left(x \cdot 2\right)\]
\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 \cdot sin\right) \cdot x\right) \cdot \left(\left(cos \cdot sin\right) \cdot x\right)} \cdot \cos \left(x \cdot 2\right)
double f(double x, double cos, double sin) {
        double r2184196 = 2.0;
        double r2184197 = x;
        double r2184198 = r2184196 * r2184197;
        double r2184199 = cos(r2184198);
        double r2184200 = cos;
        double r2184201 = pow(r2184200, r2184196);
        double r2184202 = sin;
        double r2184203 = pow(r2184202, r2184196);
        double r2184204 = r2184197 * r2184203;
        double r2184205 = r2184204 * r2184197;
        double r2184206 = r2184201 * r2184205;
        double r2184207 = r2184199 / r2184206;
        return r2184207;
}
↓
double f(double x, double cos, double sin) {
        double r2184208 = 1.0;
        double r2184209 = cos;
        double r2184210 = sin;
        double r2184211 = r2184209 * r2184210;
        double r2184212 = x;
        double r2184213 = r2184211 * r2184212;
        double r2184214 = r2184213 * r2184213;
        double r2184215 = r2184208 / r2184214;
        double r2184216 = 2.0;
        double r2184217 = r2184212 * r2184216;
        double r2184218 = cos(r2184217);
        double r2184219 = r2184215 * r2184218;
        return r2184219;
}

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 27.6

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

  2. Simplified2.7

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

  4. Applied *-un-lft-identity2.7

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

  5. Applied times-frac2.5

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

  7. Applied associate-/r*2.5

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

  8. Taylor expanded around inf 30.9

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

  9. Simplified2.8

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

  11. Applied div-inv2.8

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

  12. Final simplification2.8

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

Reproduce

herbie shell --seed 2019164 
(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))))