Average Error: 27.3 → 2.8
Time: 2.0m
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(sin \cdot cos\right) \cdot x} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(sin \cdot cos\right) \cdot x}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(sin \cdot cos\right) \cdot x} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(sin \cdot cos\right) \cdot x}
double f(double x, double cos, double sin) {
        double r15483594 = 2.0;
        double r15483595 = x;
        double r15483596 = r15483594 * r15483595;
        double r15483597 = cos(r15483596);
        double r15483598 = cos;
        double r15483599 = pow(r15483598, r15483594);
        double r15483600 = sin;
        double r15483601 = pow(r15483600, r15483594);
        double r15483602 = r15483595 * r15483601;
        double r15483603 = r15483602 * r15483595;
        double r15483604 = r15483599 * r15483603;
        double r15483605 = r15483597 / r15483604;
        return r15483605;
}
↓
double f(double x, double cos, double sin) {
        double r15483606 = 2.0;
        double r15483607 = x;
        double r15483608 = r15483606 * r15483607;
        double r15483609 = cos(r15483608);
        double r15483610 = cbrt(r15483609);
        double r15483611 = r15483610 * r15483610;
        double r15483612 = sin;
        double r15483613 = cos;
        double r15483614 = r15483612 * r15483613;
        double r15483615 = r15483614 * r15483607;
        double r15483616 = r15483611 / r15483615;
        double r15483617 = r15483610 / r15483615;
        double r15483618 = r15483616 * r15483617;
        return r15483618;
}

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 27.3

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

  2. Simplified2.8

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

  3. Taylor expanded around -inf 31.0

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

  4. Simplified3.1

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

  5. Taylor expanded around inf 30.9

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

  6. Simplified3.0

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

  8. Applied add-cube-cbrt3.2

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}{\left(\left(cos \cdot sin\right) \cdot x\right) \cdot \left(\left(cos \cdot sin\right) \cdot x\right)}\]

  9. Applied times-frac2.8

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(cos \cdot sin\right) \cdot x} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(cos \cdot sin\right) \cdot x}}\]

  10. Final simplification2.8

    \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(sin \cdot cos\right) \cdot x} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(sin \cdot cos\right) \cdot x}\]

Reproduce

herbie shell --seed 2019112 
(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))))