Average Error: 27.2 → 4.5
Time: 9.7s
Precision: binary64
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.1028652727373659 \cdot 10^{-195} \lor \neg \left(x \le 2.00202347433805179 \cdot 10^{-202}\right) \land x \le 1.1274878712839377 \cdot 10^{158}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot \frac{1}{2}\right)}\right)}}{{\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x}}{\left(x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot \frac{1}{2}\right)}\right) \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]
\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}\;x \le -1.1028652727373659 \cdot 10^{-195} \lor \neg \left(x \le 2.00202347433805179 \cdot 10^{-202}\right) \land x \le 1.1274878712839377 \cdot 10^{158}:\\
\;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot \frac{1}{2}\right)}\right)}}{{\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x}}{\left(x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot \frac{1}{2}\right)}\right) \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\\

\end{array}
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 <= -1.1028652727373659e-195) || (!(x <= 2.0020234743380518e-202) && (x <= 1.1274878712839377e+158)))) {
		VAR = ((double) (((double) (((double) cos(((double) (x * 2.0)))) / ((double) (x * ((double) (x * ((double) pow(((double) (c * s)), ((double) (2.0 * 0.5)))))))))) / ((double) pow(((double) (c * s)), ((double) (2.0 / 2.0))))));
	} else {
		VAR = ((double) (((double) (((double) cos(((double) (x * 2.0)))) / x)) / ((double) (((double) (x * ((double) pow(((double) (c * s)), ((double) (2.0 * 0.5)))))) * ((double) pow(((double) (c * s)), ((double) (2.0 / 2.0))))))));
	}
	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 x < -1.1028652727373659e-195 or 2.00202347433805179e-202 < x < 1.1274878712839377e158

    1. Initial program 25.0

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

    2. Simplified25.3

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

    4. Applied pow-prod-down10.2

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

    6. Applied sqr-pow10.2

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

    7. Applied associate-*r*4.7

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

    8. Simplified4.7

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\color{blue}{\left(x \cdot {\left(c \cdot s\right)}^{\left(\frac{1}{2} \cdot 2\right)}\right)} \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}\right)}\]
    9. Using strategy rm

    10. Applied associate-*r*3.0

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot \left(x \cdot {\left(c \cdot s\right)}^{\left(\frac{1}{2} \cdot 2\right)}\right)\right) \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}}\]
    11. Using strategy rm

    12. Applied associate-/r*2.9

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

    13. Simplified2.9

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

    if -1.1028652727373659e-195 < x < 2.00202347433805179e-202 or 1.1274878712839377e158 < x

    1. Initial program 33.6

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

    2. Simplified33.4

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

    4. Applied pow-prod-down23.1

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

    6. Applied sqr-pow23.1

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

    7. Applied associate-*r*9.5

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

    8. Simplified9.5

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\color{blue}{\left(x \cdot {\left(c \cdot s\right)}^{\left(\frac{1}{2} \cdot 2\right)}\right)} \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}\right)}\]
    9. Using strategy rm

    10. Applied associate-/r*9.2

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

    11. Simplified9.2

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 2\right)}{x}}}{\left(x \cdot {\left(c \cdot s\right)}^{\left(\frac{1}{2} \cdot 2\right)}\right) \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.1028652727373659 \cdot 10^{-195} \lor \neg \left(x \le 2.00202347433805179 \cdot 10^{-202}\right) \land x \le 1.1274878712839377 \cdot 10^{158}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot \frac{1}{2}\right)}\right)}}{{\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x}}{\left(x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot \frac{1}{2}\right)}\right) \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))