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

Average Error: 28.3 → 2.5

Time: 7.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;cos \le 2.2959203971383585 \cdot 10^{-210}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}}}{\frac{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}{\frac{\cos \left(2 \cdot x\right)}{\sqrt{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}}}}\\ \end{array}\]

C

double f(double x, double cos, double sin) {
        double r57103 = 2.0;
        double r57104 = x;
        double r57105 = r57103 * r57104;
        double r57106 = cos(r57105);
        double r57107 = cos;
        double r57108 = pow(r57107, r57103);
        double r57109 = sin;
        double r57110 = pow(r57109, r57103);
        double r57111 = r57104 * r57110;
        double r57112 = r57111 * r57104;
        double r57113 = r57108 * r57112;
        double r57114 = r57106 / r57113;
        return r57114;
}

↓

double f(double x, double cos, double sin) {
        double r57115 = cos;
        double r57116 = 2.2959203971383585e-210;
        bool r57117 = r57115 <= r57116;
        double r57118 = 2.0;
        double r57119 = x;
        double r57120 = r57118 * r57119;
        double r57121 = cos(r57120);
        double r57122 = 1.0;
        double r57123 = pow(r57115, r57122);
        double r57124 = sin;
        double r57125 = pow(r57124, r57122);
        double r57126 = r57123 * r57125;
        double r57127 = pow(r57126, r57122);
        double r57128 = r57127 * r57119;
        double r57129 = fabs(r57128);
        double r57130 = 2.0;
        double r57131 = pow(r57129, r57130);
        double r57132 = r57121 / r57131;
        double r57133 = 1.0;
        double r57134 = r57118 / r57130;
        double r57135 = pow(r57115, r57134);
        double r57136 = pow(r57124, r57134);
        double r57137 = r57119 * r57136;
        double r57138 = r57135 * r57137;
        double r57139 = fabs(r57138);
        double r57140 = sqrt(r57139);
        double r57141 = r57133 / r57140;
        double r57142 = r57121 / r57140;
        double r57143 = r57139 / r57142;
        double r57144 = r57141 / r57143;
        double r57145 = r57117 ? r57132 : r57144;
        return r57145;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

1. Split input into 2 regimes

2. if cos < 2.2959203971383585e-210

   1. Initial program 31.1

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

   3. Applied sqr-pow 31.1

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

   4. Applied associate-*r* 25.3

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

   6. Applied add-sqr-sqrt 25.3

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

   7. Simplified 25.3

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

   8. Simplified 3.7

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

   9. Taylor expanded around 0 3.1

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

   10. Simplified 3.1

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

   if 2.2959203971383585e-210 < cos

   1. Initial program 24.9

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

   3. Applied sqr-pow 24.9

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

   4. Applied associate-*r* 18.1

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

   6. Applied add-sqr-sqrt 18.1

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

   7. Simplified 18.1

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

   8. Simplified 2.0

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

   10. Applied associate-/r* 1.7

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

   12. Applied add-sqr-sqrt 1.8

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

   13. Applied *-un-lft-identity 1.8

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

   14. Applied times-frac 1.8

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

   15. Applied associate-/l* 1.8

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;cos \le 2.2959203971383585 \cdot 10^{-210}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}}}{\frac{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}{\frac{\cos \left(2 \cdot x\right)}{\sqrt{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))