Average Error: 27.7 → 7.8
Time: 8.1s
Precision: binary64
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \leq 1.0954292840385535 \cdot 10^{-275}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {c}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(\left|\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot {c}^{\left(\frac{2}{4}\right)}\right| \cdot \left|\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot {c}^{\left(\frac{2}{4}\right)}\right|\right)}\\ \end{array}\]
\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 1.0954292840385535 \cdot 10^{-275}:\\
\;\;\;\;\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}
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 <= 1.0954292840385535e-275)) {
		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 < 1.0954292840385535e-275

    1. Initial program 28.7

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

    3. Applied sqr-pow28.7

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

    4. Applied associate-*l*24.0

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

    5. Simplified19.9

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

    7. Applied sqr-pow19.9

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

    8. Applied associate-*r*13.7

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

    10. Applied associate-*l*9.9

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

    11. Simplified9.9

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

    13. Applied associate-*r*9.9

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

    14. Simplified9.9

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

    if 1.0954292840385535e-275 < c

    1. Initial program 26.7

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

    3. Applied sqr-pow26.7

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

    4. Applied associate-*l*22.4

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

    5. Simplified18.9

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

    7. Applied sqr-pow18.9

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

    8. Applied associate-*r*12.7

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

    10. Applied add-sqr-sqrt12.8

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

    11. Simplified12.8

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

    12. Simplified5.7

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

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.0954292840385535 \cdot 10^{-275}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {c}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot \left({c}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{c}^{\left(\frac{2}{2}\right)} \cdot \left(\left|\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot {c}^{\left(\frac{2}{4}\right)}\right| \cdot \left|\left(x \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot {c}^{\left(\frac{2}{4}\right)}\right|\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))