cos(2*x)/(cos^2(x)*sin^2(x))

Average Error: 27.8 → 2.9

Time: 23.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;cos \le -9.175790255859891 \cdot 10^{-110}:\\ \;\;\;\;\left({\left(x \cdot \left(sin \cdot cos\right)\right)}^{-1} \cdot \cos \left(2 \cdot x\right)\right) \cdot {\left(x \cdot \left(sin \cdot cos\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(sin \cdot \left(x \cdot cos\right)\right) \cdot \left(sin \cdot \left(x \cdot cos\right)\right)}{\cos \left(2 \cdot x\right)}}\\ \end{array}\]

C

double f(double x, double cos, double sin) {
        double r2187128 = 2.0;
        double r2187129 = x;
        double r2187130 = r2187128 * r2187129;
        double r2187131 = cos(r2187130);
        double r2187132 = cos;
        double r2187133 = pow(r2187132, r2187128);
        double r2187134 = sin;
        double r2187135 = pow(r2187134, r2187128);
        double r2187136 = r2187129 * r2187135;
        double r2187137 = r2187136 * r2187129;
        double r2187138 = r2187133 * r2187137;
        double r2187139 = r2187131 / r2187138;
        return r2187139;
}

↓

double f(double x, double cos, double sin) {
        double r2187140 = cos;
        double r2187141 = -9.175790255859891e-110;
        bool r2187142 = r2187140 <= r2187141;
        double r2187143 = x;
        double r2187144 = sin;
        double r2187145 = r2187144 * r2187140;
        double r2187146 = r2187143 * r2187145;
        double r2187147 = -1.0;
        double r2187148 = pow(r2187146, r2187147);
        double r2187149 = 2.0;
        double r2187150 = r2187149 * r2187143;
        double r2187151 = cos(r2187150);
        double r2187152 = r2187148 * r2187151;
        double r2187153 = r2187152 * r2187148;
        double r2187154 = 1.0;
        double r2187155 = r2187143 * r2187140;
        double r2187156 = r2187144 * r2187155;
        double r2187157 = r2187156 * r2187156;
        double r2187158 = r2187157 / r2187151;
        double r2187159 = r2187154 / r2187158;
        double r2187160 = r2187142 ? r2187153 : r2187159;
        return r2187160;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

1. Split input into 2 regimes

2. if cos < -9.175790255859891e-110

   1. Initial program 21.9

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

   2. Simplified 2.2

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

   4. Applied div-inv 2.2

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

   6. Applied pow1 2.2

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

   7. Applied pow1 2.2

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

   8. Applied pow-prod-down 2.2

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

   9. Applied pow1 2.2

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

   10. Applied pow-prod-down 2.2

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

   11. Applied pow1 2.2

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

   12. Applied pow1 2.2

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

   13. Applied pow-prod-down 2.2

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

   14. Applied pow1 2.2

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

   15. Applied pow-prod-down 2.2

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

   16. Applied pow-prod-up 2.2

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

   17. Applied pow-flip 2.0

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

   18. Simplified 2.0

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

   19. Taylor expanded around inf 2.3

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

   21. Applied sqr-pow 2.3

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

   22. Applied associate-*r* 2.3

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

   if -9.175790255859891e-110 < cos

   1. Initial program 31.8

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

   2. Simplified 3.2

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

   4. Applied clear-num 3.3

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(sin \cdot \left(x \cdot cos\right)\right) \cdot \left(sin \cdot \left(x \cdot cos\right)\right)}{\cos \left(2 \cdot x\right)}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;cos \le -9.175790255859891 \cdot 10^{-110}:\\ \;\;\;\;\left({\left(x \cdot \left(sin \cdot cos\right)\right)}^{-1} \cdot \cos \left(2 \cdot x\right)\right) \cdot {\left(x \cdot \left(sin \cdot cos\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(sin \cdot \left(x \cdot cos\right)\right) \cdot \left(sin \cdot \left(x \cdot cos\right)\right)}{\cos \left(2 \cdot x\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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))))