Average Error: 27.4 → 2.7
Time: 21.0s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin} \cdot \frac{1}{\left(cos \cdot x\right) \cdot sin}\]
\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(cos \cdot x\right) \cdot sin} \cdot \frac{1}{\left(cos \cdot x\right) \cdot sin}
double f(double x, double cos, double sin) {
        double r3236727 = 2.0;
        double r3236728 = x;
        double r3236729 = r3236727 * r3236728;
        double r3236730 = cos(r3236729);
        double r3236731 = cos;
        double r3236732 = pow(r3236731, r3236727);
        double r3236733 = sin;
        double r3236734 = pow(r3236733, r3236727);
        double r3236735 = r3236728 * r3236734;
        double r3236736 = r3236735 * r3236728;
        double r3236737 = r3236732 * r3236736;
        double r3236738 = r3236730 / r3236737;
        return r3236738;
}
↓
double f(double x, double cos, double sin) {
        double r3236739 = 2.0;
        double r3236740 = x;
        double r3236741 = r3236739 * r3236740;
        double r3236742 = cos(r3236741);
        double r3236743 = cos;
        double r3236744 = r3236743 * r3236740;
        double r3236745 = sin;
        double r3236746 = r3236744 * r3236745;
        double r3236747 = r3236742 / r3236746;
        double r3236748 = 1.0;
        double r3236749 = r3236748 / r3236746;
        double r3236750 = r3236747 * r3236749;
        return r3236750;
}

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 27.4

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

  2. Simplified3.0

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

  4. Applied associate-/r*2.7

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

  6. Applied div-inv2.7

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

  7. Final simplification2.7

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

Reproduce

herbie shell --seed 2019168 
(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))))