Average Error: 27.8 → 2.8
Time: 3.5m
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\frac{sin \cdot \left(x \cdot cos\right)}{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{cos} \cdot \frac{1}{sin}}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{1}{\frac{sin \cdot \left(x \cdot cos\right)}{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{cos} \cdot \frac{1}{sin}}}
double f(double x, double cos, double sin) {
        double r30198104 = 2.0;
        double r30198105 = x;
        double r30198106 = r30198104 * r30198105;
        double r30198107 = cos(r30198106);
        double r30198108 = cos;
        double r30198109 = pow(r30198108, r30198104);
        double r30198110 = sin;
        double r30198111 = pow(r30198110, r30198104);
        double r30198112 = r30198105 * r30198111;
        double r30198113 = r30198112 * r30198105;
        double r30198114 = r30198109 * r30198113;
        double r30198115 = r30198107 / r30198114;
        return r30198115;
}
↓
double f(double x, double cos, double sin) {
        double r30198116 = 1.0;
        double r30198117 = sin;
        double r30198118 = x;
        double r30198119 = cos;
        double r30198120 = r30198118 * r30198119;
        double r30198121 = r30198117 * r30198120;
        double r30198122 = 2.0;
        double r30198123 = r30198122 * r30198118;
        double r30198124 = cos(r30198123);
        double r30198125 = r30198124 / r30198118;
        double r30198126 = r30198125 / r30198119;
        double r30198127 = r30198116 / r30198117;
        double r30198128 = r30198126 * r30198127;
        double r30198129 = r30198121 / r30198128;
        double r30198130 = r30198116 / r30198129;
        return r30198130;
}

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 27.8

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

  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 associate-/r*2.5

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

  6. Applied *-un-lft-identity2.5

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

  7. Applied times-frac2.6

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

  9. Applied associate-/r*2.6

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

  11. Applied clear-num2.8

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

  12. Final simplification2.8

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

Reproduce

herbie shell --seed 2019120 
(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))))