mixedcos

Average Error: 28.1 → 2.4

Time: 23.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.694771603218296 \cdot 10^{247}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right| \cdot \left|{\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sqrt[3]{c}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)\right|}\\ \mathbf{elif}\;x \le -3.9412058536221336 \cdot 10^{-237}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({s}^{1} \cdot {c}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|}\\ \end{array}\]

C

double code(double x, double c, double s) {
	return ((double) (((double) cos(((double) (2.0 * x)))) / ((double) (((double) pow(c, 2.0)) * ((double) (((double) (x * ((double) pow(s, 2.0)))) * x))))));
}

↓

double code(double x, double c, double s) {
	double VAR;
	if ((x <= -4.694771603218296e+247)) {
		VAR = ((double) (((double) cos(((double) (2.0 * x)))) / ((double) (((double) fabs(((double) (((double) pow(c, ((double) (2.0 / 2.0)))) * ((double) (x * ((double) pow(s, ((double) (2.0 / 2.0)))))))))) * ((double) fabs(((double) (((double) pow(((double) (((double) cbrt(c)) * ((double) cbrt(c)))), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(((double) cbrt(c)), ((double) (2.0 / 2.0)))) * ((double) (x * ((double) pow(s, ((double) (2.0 / 2.0))))))))))))))));
	} else {
		double VAR_1;
		if ((x <= -3.9412058536221336e-237)) {
			VAR_1 = ((double) (((double) cos(((double) (2.0 * x)))) / ((double) pow(((double) fabs(((double) (((double) pow(((double) (((double) pow(s, 1.0)) * ((double) pow(c, 1.0)))), 1.0)) * x)))), 2.0))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) cbrt(((double) cos(((double) (2.0 * x)))))) * ((double) cbrt(((double) cos(((double) (2.0 * x)))))))) / ((double) fabs(((double) (((double) pow(c, ((double) (2.0 / 2.0)))) * ((double) (x * ((double) pow(s, ((double) (2.0 / 2.0)))))))))))) * ((double) (((double) cbrt(((double) cos(((double) (2.0 * x)))))) / ((double) fabs(((double) (((double) pow(c, ((double) (2.0 / 2.0)))) * ((double) (x * ((double) pow(s, ((double) (2.0 / 2.0))))))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus c

Bits error versus s

Derivation

1. Split input into 3 regimes

2. if x < -4.694771603218296e247

   1. Initial program 25.9

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

   3. Applied sqr-pow 25.9

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

   4. Applied associate-*r* 16.1

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

   6. Applied add-sqr-sqrt 16.1

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

   7. Simplified 16.1

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

   8. Simplified 2.3

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

   10. Applied add-cube-cbrt 2.5

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right| \cdot \left|{\color{blue}{\left(\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \sqrt[3]{c}\right)}}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|}\]

   11. Applied unpow-prod-down 2.5

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right| \cdot \left|\color{blue}{\left({\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{c}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|}\]

   12. Applied associate-*l* 2.5

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right| \cdot \left|\color{blue}{{\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sqrt[3]{c}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)}\right|}\]

   if -4.694771603218296e247 < x < -3.9412058536221336e-237

   1. Initial program 27.3

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

   3. Applied sqr-pow 27.3

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

   4. Applied associate-*r* 22.5

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

   6. Applied add-sqr-sqrt 22.5

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

   7. Simplified 22.5

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

   8. Simplified 2.9

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

   9. Taylor expanded around inf 1.9

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

   if -3.9412058536221336e-237 < x

   1. Initial program 29.0

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

   3. Applied sqr-pow 29.0

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

   4. Applied associate-*r* 22.0

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

   6. Applied add-sqr-sqrt 22.0

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

   7. Simplified 22.0

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

   8. Simplified 3.0

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

   10. Applied add-cube-cbrt 3.1

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right| \cdot \left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|}\]

   11. Applied times-frac 2.9

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.694771603218296 \cdot 10^{247}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right| \cdot \left|{\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sqrt[3]{c}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)\right|}\\ \mathbf{elif}\;x \le -3.9412058536221336 \cdot 10^{-237}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({s}^{1} \cdot {c}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|}\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))