mixedcos

Average Error: 27.4 → 2.6

Time: 8.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.84998239742904699 \cdot 10^{218}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|}}{\left|\left({c}^{\left(\frac{2}{2}\right)} \cdot x\right) \cdot {s}^{\left(\frac{2}{2}\right)}\right|}\\ \mathbf{elif}\;x \le -2.42422156316495082 \cdot 10^{-275}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({s}^{1} \cdot {c}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right| \cdot \left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|}\\ \end{array}\]

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 <= -2.849982397429047e+218)) {
		VAR = ((double) (((double) (((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) fabs(((double) (((double) (((double) pow(c, ((double) (2.0 / 2.0)))) * x)) * ((double) pow(s, ((double) (2.0 / 2.0))))))))));
	} else {
		double VAR_1;
		if ((x <= -2.4242215631649508e-275)) {
			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) 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(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 < -2.84998239742904699e218

   1. Initial program 27.1

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

   3. Applied sqr-pow 27.1

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

   4. Applied associate-*r* 17.2

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

   6. Applied add-sqr-sqrt 17.2

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

   7. Simplified 17.2

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

   8. Simplified 2.6

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

   10. Applied associate-/r* 2.5

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

   12. Applied associate-*r* 3.9

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

   if -2.84998239742904699e218 < x < -2.42422156316495082e-275

   1. Initial program 27.2

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

   3. Applied sqr-pow 27.2

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

   4. Applied associate-*r* 22.3

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

   6. Applied add-sqr-sqrt 22.3

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

   7. Simplified 22.3

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

   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 0 1.8

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

   if -2.42422156316495082e-275 < x

   1. Initial program 27.7

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

   3. Applied sqr-pow 27.7

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

   4. Applied associate-*r* 21.5

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

   6. Applied add-sqr-sqrt 21.5

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

   7. Simplified 21.5

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

   8. Simplified 2.9

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

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.84998239742904699 \cdot 10^{218}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|}}{\left|\left({c}^{\left(\frac{2}{2}\right)} \cdot x\right) \cdot {s}^{\left(\frac{2}{2}\right)}\right|}\\ \mathbf{elif}\;x \le -2.42422156316495082 \cdot 10^{-275}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({s}^{1} \cdot {c}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right| \cdot \left|{c}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right|}\\ \end{array}\]

Reproduce

herbie shell --seed 2020155 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))