Average Error: 27.6 → 7.9
Time: 8.9s
Precision: binary64
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \leq 7.440582287819774 \cdot 10^{-222}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {c}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(\left|\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot {c}^{\left(\frac{2}{4}\right)}\right| \cdot \left|\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot {c}^{\left(\frac{2}{4}\right)}\right|\right)}\\ \end{array}\]
\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}\;c \leq 7.440582287819774 \cdot 10^{-222}:\\
\;\;\;\;\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{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(\left|\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot {c}^{\left(\frac{2}{4}\right)}\right| \cdot \left|\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot {c}^{\left(\frac{2}{4}\right)}\right|\right)}\\

\end{array}
(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 (<= c 7.440582287819774e-222)
   (/
    (cos (* 2.0 x))
    (*
     (* x (pow c (/ 2.0 2.0)))
     (*
      (* x (pow s (/ 2.0 2.0)))
      (* (pow c (/ 2.0 2.0)) (pow s (/ 2.0 2.0))))))
   (/
    (cos (* 2.0 x))
    (*
     (pow c (/ 2.0 2.0))
     (*
      (fabs (* (* x (pow s (/ 2.0 2.0))) (pow c (/ 2.0 4.0))))
      (fabs (* (* x (pow s (/ 2.0 2.0))) (pow c (/ 2.0 4.0)))))))))
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 ((c <= 7.440582287819774e-222)) {
		VAR = (((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 = (((double) cos(((double) (2.0 * x)))) / ((double) (((double) pow(c, (2.0 / 2.0))) * ((double) (((double) fabs(((double) (((double) (x * ((double) pow(s, (2.0 / 2.0))))) * ((double) pow(c, (2.0 / 4.0))))))) * ((double) fabs(((double) (((double) (x * ((double) pow(s, (2.0 / 2.0))))) * ((double) pow(c, (2.0 / 4.0))))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus c

Bits error versus s

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if c < 7.4405822878197738e-222

    1. Initial program 29.8

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

    3. Applied sqr-pow29.8

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

    4. Applied associate-*l*24.2

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

    5. Simplified19.8

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

    7. Applied sqr-pow19.8

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

    8. Applied associate-*r*14.6

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

    10. Applied associate-*l*10.3

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

    11. Simplified10.3

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

    13. Applied associate-*r*10.0

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

    14. Simplified10.0

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

    if 7.4405822878197738e-222 < c

    1. Initial program 25.1

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

    3. Applied sqr-pow25.1

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

    4. Applied associate-*l*21.6

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

    5. Simplified18.6

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

    7. Applied sqr-pow18.6

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

    8. Applied associate-*r*12.1

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

    10. Applied add-sqr-sqrt12.2

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

    11. Simplified12.2

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(\color{blue}{\left|\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot {c}^{\left(\frac{2}{4}\right)}\right|} \cdot \sqrt{x \cdot \left(\left(\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot {c}^{\left(\frac{2}{2}\right)}\right)}\right)}\]

    12. Simplified5.4

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(\left|\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot {c}^{\left(\frac{2}{4}\right)}\right| \cdot \color{blue}{\left|\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot {c}^{\left(\frac{2}{4}\right)}\right|}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 7.440582287819774 \cdot 10^{-222}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {c}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(\left|\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot {c}^{\left(\frac{2}{4}\right)}\right| \cdot \left|\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot {c}^{\left(\frac{2}{4}\right)}\right|\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))