Average Error: 27.2 → 2.6
Time: 1.9m
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\cos x \cdot \cos x}{\left(cos \cdot x\right) \cdot sin} - \left(\frac{\frac{1}{2}}{\left(cos \cdot x\right) \cdot sin} - \frac{\cos \left(x \cdot 2\right) \cdot \frac{1}{2}}{\left(cos \cdot x\right) \cdot sin}\right)}{\left(cos \cdot x\right) \cdot sin}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{\cos x \cdot \cos x}{\left(cos \cdot x\right) \cdot sin} - \left(\frac{\frac{1}{2}}{\left(cos \cdot x\right) \cdot sin} - \frac{\cos \left(x \cdot 2\right) \cdot \frac{1}{2}}{\left(cos \cdot x\right) \cdot sin}\right)}{\left(cos \cdot x\right) \cdot sin}
double f(double x, double cos, double sin) {
        double r13514211 = 2.0;
        double r13514212 = x;
        double r13514213 = r13514211 * r13514212;
        double r13514214 = cos(r13514213);
        double r13514215 = cos;
        double r13514216 = pow(r13514215, r13514211);
        double r13514217 = sin;
        double r13514218 = pow(r13514217, r13514211);
        double r13514219 = r13514212 * r13514218;
        double r13514220 = r13514219 * r13514212;
        double r13514221 = r13514216 * r13514220;
        double r13514222 = r13514214 / r13514221;
        return r13514222;
}
↓
double f(double x, double cos, double sin) {
        double r13514223 = x;
        double r13514224 = cos(r13514223);
        double r13514225 = r13514224 * r13514224;
        double r13514226 = cos;
        double r13514227 = r13514226 * r13514223;
        double r13514228 = sin;
        double r13514229 = r13514227 * r13514228;
        double r13514230 = r13514225 / r13514229;
        double r13514231 = 0.5;
        double r13514232 = r13514231 / r13514229;
        double r13514233 = 2.0;
        double r13514234 = r13514223 * r13514233;
        double r13514235 = cos(r13514234);
        double r13514236 = r13514235 * r13514231;
        double r13514237 = r13514236 / r13514229;
        double r13514238 = r13514232 - r13514237;
        double r13514239 = r13514230 - r13514238;
        double r13514240 = r13514239 / r13514229;
        return r13514240;
}

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 27.2

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

  2. Simplified2.8

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

  4. Applied associate-/r*2.6

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

  6. Applied cos-22.6

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

  7. Applied div-sub2.6

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

  9. Applied sqr-sin2.6

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

  10. Applied div-sub2.6

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

  11. Final simplification2.6

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(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))))