Average Error: 28.7 → 3.0
Time: 41.2s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot {cos}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot {cos}^{\left(\frac{2}{2}\right)}\right)}\]
\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(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot {cos}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot {cos}^{\left(\frac{2}{2}\right)}\right)}
double f(double x, double cos, double sin) {
        double r6406147 = 2.0;
        double r6406148 = x;
        double r6406149 = r6406147 * r6406148;
        double r6406150 = cos(r6406149);
        double r6406151 = cos;
        double r6406152 = pow(r6406151, r6406147);
        double r6406153 = sin;
        double r6406154 = pow(r6406153, r6406147);
        double r6406155 = r6406148 * r6406154;
        double r6406156 = r6406155 * r6406148;
        double r6406157 = r6406152 * r6406156;
        double r6406158 = r6406150 / r6406157;
        return r6406158;
}
↓
double f(double x, double cos, double sin) {
        double r6406159 = 2.0;
        double r6406160 = x;
        double r6406161 = r6406159 * r6406160;
        double r6406162 = cos(r6406161);
        double r6406163 = sin;
        double r6406164 = 2.0;
        double r6406165 = r6406159 / r6406164;
        double r6406166 = pow(r6406163, r6406165);
        double r6406167 = r6406160 * r6406166;
        double r6406168 = cos;
        double r6406169 = pow(r6406168, r6406165);
        double r6406170 = r6406167 * r6406169;
        double r6406171 = r6406170 * r6406170;
        double r6406172 = r6406162 / r6406171;
        return r6406172;
}

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 28.7

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

  3. Applied sqr-pow28.7

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

  4. Applied associate-*r*22.3

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

  6. Applied sqr-pow22.3

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

  7. Applied associate-*l*16.9

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

  8. Simplified14.1

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

  10. Applied associate-*l*6.5

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

  12. Applied associate-*r*3.0

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

  13. Final simplification3.0

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

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  (/ (cos (* 2.0 x)) (* (pow cos 2.0) (* (* x (pow sin 2.0)) x))))