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

Average Error: 28.6 → 2.2

Time: 18.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;cos \le -5.658624829795034525283840639142347785327 \cdot 10^{-292} \lor \neg \left(cos \le 5.254045133843250338248612619311844129892 \cdot 10^{-192}\right):\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \end{array}\]

C

double f(double x, double cos, double sin) {
        double r53701 = 2.0;
        double r53702 = x;
        double r53703 = r53701 * r53702;
        double r53704 = cos(r53703);
        double r53705 = cos;
        double r53706 = pow(r53705, r53701);
        double r53707 = sin;
        double r53708 = pow(r53707, r53701);
        double r53709 = r53702 * r53708;
        double r53710 = r53709 * r53702;
        double r53711 = r53706 * r53710;
        double r53712 = r53704 / r53711;
        return r53712;
}

↓

double f(double x, double cos, double sin) {
        double r53713 = cos;
        double r53714 = -5.6586248297950345e-292;
        bool r53715 = r53713 <= r53714;
        double r53716 = 5.25404513384325e-192;
        bool r53717 = r53713 <= r53716;
        double r53718 = !r53717;
        bool r53719 = r53715 || r53718;
        double r53720 = 2.0;
        double r53721 = x;
        double r53722 = r53720 * r53721;
        double r53723 = cos(r53722);
        double r53724 = 2.0;
        double r53725 = r53720 / r53724;
        double r53726 = pow(r53713, r53725);
        double r53727 = sin;
        double r53728 = pow(r53727, r53725);
        double r53729 = r53721 * r53728;
        double r53730 = r53726 * r53729;
        double r53731 = fabs(r53730);
        double r53732 = r53723 / r53731;
        double r53733 = r53732 / r53731;
        double r53734 = 1.0;
        double r53735 = pow(r53713, r53734);
        double r53736 = pow(r53727, r53734);
        double r53737 = r53735 * r53736;
        double r53738 = pow(r53737, r53734);
        double r53739 = r53738 * r53721;
        double r53740 = fabs(r53739);
        double r53741 = pow(r53740, r53724);
        double r53742 = r53723 / r53741;
        double r53743 = r53719 ? r53733 : r53742;
        return r53743;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

1. Split input into 2 regimes

2. if cos < -5.6586248297950345e-292 or 5.25404513384325e-192 < cos

   1. Initial program 26.4

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

   3. Applied sqr-pow 26.4

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

   4. Applied associate-*r* 19.9

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

   6. Applied add-sqr-sqrt 20.0

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

   7. Simplified 19.9

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

   8. Simplified 2.3

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

   10. Applied associate-/r* 2.0

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

   if -5.6586248297950345e-292 < cos < 5.25404513384325e-192

   1. Initial program 64.0

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

   3. Applied sqr-pow 64.0

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

   4. Applied associate-*r* 64.0

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

   6. Applied add-sqr-sqrt 64.0

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

   7. Simplified 64.0

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

   8. Simplified 10.8

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

   9. Taylor expanded around inf 6.2

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

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;cos \le -5.658624829795034525283840639142347785327 \cdot 10^{-292} \lor \neg \left(cos \le 5.254045133843250338248612619311844129892 \cdot 10^{-192}\right):\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(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))))