mixedcos

Average Error: 28.1 → 7.6

Time: 9.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;{s}^{2} \leq 0:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{elif}\;{s}^{2} \leq 1.30542547701306 \cdot 10^{+247}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {c}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)}}}{x \cdot \left(\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \end{array}\]

C

double code(double x, double c, double s) {
	return (((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 ((((double) pow(s, 2.0)) <= 0.0)) {
		VAR = (((double) cos(((double) (2.0 * x)))) / ((double) (((double) pow(c, (2.0 / 2.0))) * ((double) (((double) (x * ((double) (x * ((double) pow(s, (2.0 / 2.0))))))) * ((double) (((double) pow(c, (2.0 / 2.0))) * ((double) pow(s, (2.0 / 2.0))))))))));
	} else {
		double VAR_1;
		if ((((double) pow(s, 2.0)) <= 1.30542547701306e+247)) {
			VAR_1 = (((double) cos(((double) (2.0 * x)))) / ((double) (((double) (x * ((double) pow(c, (2.0 / 2.0))))) * ((double) (((double) (x * ((double) pow(s, (2.0 / 2.0))))) * ((double) (((double) pow(c, (2.0 / 2.0))) * ((double) pow(s, (2.0 / 2.0))))))))));
		} else {
			VAR_1 = ((((double) cos(((double) (2.0 * x)))) / ((double) pow(c, (2.0 / 2.0)))) / ((double) (x * ((double) (((double) (x * ((double) pow(s, (2.0 / 2.0))))) * ((double) (((double) pow(c, (2.0 / 2.0))) * ((double) pow(s, (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 (pow s 2.0) < 0.0

   1. Initial program 64.0

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

   3. Applied sqr-pow 64.0

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

   4. Applied associate-*l* 64.0

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

   5. Simplified 64.0

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

   7. Applied sqr-pow 64.0

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

   8. Applied associate-*r* 36.8

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

   10. Applied associate-*l* 21.2

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

   11. Simplified 21.2

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

   13. Applied associate-*r* 18.9

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

   if 0.0 < (pow s 2.0) < 1.30542547701306e247

   1. Initial program 21.6

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

   3. Applied sqr-pow 21.6

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

   4. Applied associate-*l* 15.7

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

   5. Simplified 8.3

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

   7. Applied sqr-pow 8.3

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

   8. Applied associate-*r* 8.1

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

   10. Applied associate-*l* 7.8

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

   11. Simplified 7.8

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

   13. Applied associate-*r* 3.9

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

   14. Simplified 3.9

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

   if 1.30542547701306e247 < (pow s 2.0)

   1. Initial program 24.0

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

   3. Applied sqr-pow 24.0

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

   4. Applied associate-*l* 19.2

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

   5. Simplified 19.0

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

   7. Applied sqr-pow 19.0

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

   8. Applied associate-*r* 13.1

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

   10. Applied associate-*l* 8.5

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

   11. Simplified 8.5

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

   13. Applied associate-/r* 8.4

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

4. Final simplification 7.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;{s}^{2} \leq 0:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(\left(x \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{elif}\;{s}^{2} \leq 1.30542547701306 \cdot 10^{+247}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {c}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)}}}{x \cdot \left(\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))