Average Error: 30.6 → 15.7
Time: 5.9s
Precision: binary64
\[\sqrt{2 \cdot {x}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.485172571465744 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2 \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({x}^{1} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\\ \end{array}\]
\sqrt{2 \cdot {x}^{2}}
↓
\begin{array}{l}
\mathbf{if}\;x \le 2.485172571465744 \cdot 10^{-310}:\\
\;\;\;\;\sqrt{2 \cdot {x}^{2}}\\

\mathbf{else}:\\
\;\;\;\;\left(\left({x}^{1} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{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 ((x <= 2.48517257146574e-310)) {
		VAR = ((double) sqrt(((double) (2.0 * ((double) pow(x, 2.0))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) pow(x, 1.0)) * ((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) cbrt(((double) sqrt(2.0)))))))) * ((double) (((double) cbrt(((double) cbrt(((double) sqrt(2.0)))))) * ((double) cbrt(((double) cbrt(((double) sqrt(2.0)))))))))) * ((double) cbrt(((double) cbrt(((double) sqrt(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 x < 2.485172571465744e-310

    1. Initial program 31.2

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

    if 2.485172571465744e-310 < x

    1. Initial program 30.0

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

    2. Taylor expanded around 0 5.7

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

    3. Simplified0.4

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

    5. Applied add-cube-cbrt0.4

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

    6. Applied associate-*r*0.4

      \[\leadsto \color{blue}{\left({x}^{1} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \sqrt[3]{\sqrt{2}}}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt0.4

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

    9. Applied associate-*r*0.4

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

  4. Final simplification15.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.485172571465744 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2 \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({x}^{1} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(FPCore (x)
  :name "sqrt D"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))