mixedcos

Average Error: 28.6 → 7.6

Time: 9.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;{s}^{2} \leq 0:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)}}}{x \cdot \left(\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{elif}\;{s}^{2} \leq 8.28441953518293 \cdot 10^{+303}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(x \cdot {c}^{\left(\frac{2}{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(\left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot \left(x \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)\right)}\\ \end{array}\]

FPCore

(FPCore (x c s)
 :precision binary64
 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))

↓

(FPCore (x c s)
 :precision binary64
 (if (<= (pow s 2.0) 0.0)
   (/
    (/ (cos (* 2.0 x)) (pow c (/ 2.0 2.0)))
    (*
     x
     (*
      (* x (pow s (/ 2.0 2.0)))
      (* (pow c (/ 2.0 2.0)) (pow s (/ 2.0 2.0))))))
   (if (<= (pow s 2.0) 8.28441953518293e+303)
     (/
      (cos (* 2.0 x))
      (*
       (*
        (* x (pow s (/ 2.0 2.0)))
        (* (pow c (/ 2.0 2.0)) (pow s (/ 2.0 2.0))))
       (* x (pow c (/ 2.0 2.0)))))
     (/
      (cos (* 2.0 x))
      (*
       (pow c (/ 2.0 2.0))
       (*
        (* (pow c (/ 2.0 2.0)) (pow s (/ 2.0 2.0)))
        (* x (* x (pow s (/ 2.0 2.0))))))))))

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 tmp;
	if ((((double) pow(s, 2.0)) <= 0.0)) {
		tmp = ((((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))))))))));
	} else {
		double tmp_1;
		if ((((double) pow(s, 2.0)) <= 8.28441953518293e+303)) {
			tmp_1 = (((double) cos(((double) (2.0 * x)))) / ((double) (((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))))))) * ((double) (x * ((double) pow(c, (2.0 / 2.0))))))));
		} else {
			tmp_1 = (((double) cos(((double) (2.0 * x)))) / ((double) (((double) pow(c, (2.0 / 2.0))) * ((double) (((double) (((double) pow(c, (2.0 / 2.0))) * ((double) pow(s, (2.0 / 2.0))))) * ((double) (x * ((double) (x * ((double) pow(s, (2.0 / 2.0))))))))))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

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 Error: 64.0 bits

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

   3. Applied sqr-pow Error: 64.0 bits

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

   4. Applied associate-*l* Error: 64.0 bits

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

   5. Simplified Error: 64.0 bits

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

   7. Applied sqr-pow Error: 64.0 bits

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

   8. Applied associate-*r* Error: 36.1 bits

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

   10. Applied associate-*l* Error: 21.3 bits

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

   11. Simplified Error: 21.3 bits

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

   13. Applied associate-/r* Error: 20.9 bits

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

   if 0.0 < (pow s 2.0) < 8.28441953518293013e303

   1. Initial program Error: 21.8 bits

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

   3. Applied sqr-pow Error: 21.8 bits

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

   4. Applied associate-*l* Error: 15.8 bits

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

   5. Simplified Error: 8.1 bits

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

   7. Applied sqr-pow Error: 8.1 bits

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

   8. Applied associate-*r* Error: 7.9 bits

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

   10. Applied associate-*l* Error: 7.5 bits

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

   11. Simplified Error: 7.5 bits

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

   13. Applied associate-*r* Error: 3.7 bits

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

   14. Simplified Error: 3.7 bits

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

   if 8.28441953518293013e303 < (pow s 2.0)

   1. Initial program Error: 25.1 bits

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

   3. Applied sqr-pow Error: 25.1 bits

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

   4. Applied associate-*l* Error: 20.7 bits

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

   5. Simplified Error: 20.7 bits

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

   7. Applied sqr-pow Error: 20.7 bits

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

   8. Applied associate-*r* Error: 13.5 bits

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

   10. Applied associate-*l* Error: 8.3 bits

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

   11. Simplified Error: 8.3 bits

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

   13. Applied associate-*r* Error: 8.3 bits

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

4. Final simplification Error: 7.6 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;{s}^{2} \leq 0:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)}}}{x \cdot \left(\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{elif}\;{s}^{2} \leq 8.28441953518293 \cdot 10^{+303}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(x \cdot {c}^{\left(\frac{2}{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(\left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot \left(x \cdot \left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))