Average Error: 30.7 → 15.6
Time: 4.3s
Precision: binary64
\[\sqrt{2 \cdot {x}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;{x}^{2} \le 1.32805 \cdot 10^{-320} \lor \neg \left({x}^{2} \le 2.06962326861426283 \cdot 10^{306}\right):\\ \;\;\;\;\left({x}^{1} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \sqrt[3]{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot {x}^{2}}\\ \end{array}\]
\sqrt{2 \cdot {x}^{2}}
↓
\begin{array}{l}
\mathbf{if}\;{x}^{2} \le 1.32805 \cdot 10^{-320} \lor \neg \left({x}^{2} \le 2.06962326861426283 \cdot 10^{306}\right):\\
\;\;\;\;\left({x}^{1} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \sqrt[3]{\sqrt{2}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2 \cdot {x}^{2}}\\

\end{array}
double code(double x) {
	return ((double) sqrt(((double) (2.0 * ((double) pow(x, 2.0))))));
}
↓
double code(double x) {
	double VAR;
	if (((((double) pow(x, 2.0)) <= 1.3280484560213e-320) || !(((double) pow(x, 2.0)) <= 2.0696232686142628e+306))) {
		VAR = ((double) (((double) (((double) pow(x, 1.0)) * ((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) cbrt(((double) sqrt(2.0)))))))) * ((double) cbrt(((double) sqrt(2.0))))));
	} else {
		VAR = ((double) sqrt(((double) (2.0 * ((double) pow(x, 2.0))))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (pow x 2.0) < 1.32805e-320 or 2.06962326861426283e306 < (pow x 2.0)

    1. Initial program 61.5

      \[\sqrt{2 \cdot {x}^{2}}\]

    2. Taylor expanded around 0 35.0

      \[\leadsto \color{blue}{\sqrt{2} \cdot e^{1 \cdot \left(\log 1 + \log x\right)}}\]

    3. Simplified31.0

      \[\leadsto \color{blue}{{x}^{1} \cdot \sqrt{2}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt31.0

      \[\leadsto {x}^{1} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}\right)}\]

    6. Applied associate-*r*31.0

      \[\leadsto \color{blue}{\left({x}^{1} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \sqrt[3]{\sqrt{2}}}\]

    if 1.32805e-320 < (pow x 2.0) < 2.06962326861426283e306

    1. Initial program 0.5

      \[\sqrt{2 \cdot {x}^{2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification15.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{2} \le 1.32805 \cdot 10^{-320} \lor \neg \left({x}^{2} \le 2.06962326861426283 \cdot 10^{306}\right):\\ \;\;\;\;\left({x}^{1} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \sqrt[3]{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot {x}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020175 
(FPCore (x)
  :name "sqrt D"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))