Average Error: 30.5 → 15.6
Time: 4.5s
Precision: binary64
\[\sqrt{2 \cdot {x}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 2.5365652372159 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{2 \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left({x}^{1} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)\right)\\ \end{array}\]
\sqrt{2 \cdot {x}^{2}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 2.5365652372159 \cdot 10^{-311}:\\
\;\;\;\;\sqrt{2 \cdot {x}^{2}}\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left({x}^{1} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)\right)\\

\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 <= 2.5365652372159e-311)) {
		VAR = ((double) sqrt(((double) (2.0 * ((double) pow(x, 2.0))))));
	} else {
		VAR = ((double) (((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) cbrt(((double) sqrt(2.0)))))) * ((double) (((double) cbrt(((double) cbrt(((double) sqrt(2.0)))))) * ((double) (((double) pow(x, 1.0)) * ((double) (((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 < 2.53656523721594e-311

    1. Initial program 30.6

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

    if 2.53656523721594e-311 < x

    1. Initial program 30.3

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

    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

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

    5. Applied add-cube-cbrt0.4

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

    6. Applied associate-*l*0.4

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

    7. Simplified0.4

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

    9. Applied add-cube-cbrt0.4

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

    10. Applied associate-*r*0.4

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

  4. Final simplification15.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5365652372159 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{2 \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left({x}^{1} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x)
  :name "sqrt D"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))