Average Error: 28.4 → 19.1
Time: 14.5s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot x\right)}}{x}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({cos}^{\left(\frac{2}{2}\right)} \cdot x\right)}}{x}
double f(double x, double cos, double sin) {
        double r62189 = 2.0;
        double r62190 = x;
        double r62191 = r62189 * r62190;
        double r62192 = cos(r62191);
        double r62193 = cos;
        double r62194 = pow(r62193, r62189);
        double r62195 = sin;
        double r62196 = pow(r62195, r62189);
        double r62197 = r62190 * r62196;
        double r62198 = r62197 * r62190;
        double r62199 = r62194 * r62198;
        double r62200 = r62192 / r62199;
        return r62200;
}
↓
double f(double x, double cos, double sin) {
        double r62201 = 2.0;
        double r62202 = x;
        double r62203 = r62201 * r62202;
        double r62204 = cos(r62203);
        double r62205 = cos;
        double r62206 = 2.0;
        double r62207 = r62201 / r62206;
        double r62208 = pow(r62205, r62207);
        double r62209 = r62204 / r62208;
        double r62210 = sin;
        double r62211 = r62206 * r62207;
        double r62212 = pow(r62210, r62211);
        double r62213 = r62208 * r62202;
        double r62214 = r62212 * r62213;
        double r62215 = r62209 / r62214;
        double r62216 = r62215 / r62202;
        return r62216;
}

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 28.4

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

  3. Applied sqr-pow28.4

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

  4. Applied associate-*l*24.1

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

  6. Applied associate-*r*20.4

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

  8. Applied associate-*r*19.5

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

  10. Applied sqr-pow19.5

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

  11. Applied associate-*r*10.7

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

  12. Final simplification19.1

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

Reproduce

herbie shell --seed 2019304 
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  :precision binary64
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))