Average Error: 30.5 → 0.4
Time: 2.2s
Precision: binary64
\[\sqrt{\left(2 \cdot x\right) \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 1.18320777133784 \cdot 10^{-310}:\\ \;\;\;\;\left(x \cdot \left(\sqrt{\sqrt{2}} \cdot \left(-\left|\sqrt[3]{\sqrt{2}}\right|\right)\right)\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2} \cdot \sqrt{x}\\ \end{array}\]
\sqrt{\left(2 \cdot x\right) \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 1.18320777133784 \cdot 10^{-310}:\\
\;\;\;\;\left(x \cdot \left(\sqrt{\sqrt{2}} \cdot \left(-\left|\sqrt[3]{\sqrt{2}}\right|\right)\right)\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}}}\\

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

\end{array}
double code(double x) {
	return ((double) sqrt(((double) (((double) (2.0 * x)) * x))));
}
↓
double code(double x) {
	double VAR;
	if ((x <= 1.18320777133784e-310)) {
		VAR = ((double) (((double) (x * ((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) -(((double) fabs(((double) cbrt(((double) sqrt(2.0)))))))))))) * ((double) sqrt(((double) cbrt(((double) sqrt(2.0))))))));
	} else {
		VAR = ((double) (((double) sqrt(((double) (x * 2.0)))) * ((double) sqrt(x))));
	}
	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 < 1.18320777133784e-310

    1. Initial program Error: 30.7 bits

      \[\sqrt{\left(2 \cdot x\right) \cdot x}\]

    2. Taylor expanded around -inf Error: 0.4 bits

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

    3. SimplifiedError: 0.4 bits

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

    5. Applied add-sqr-sqrtError: 0.4 bits

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

    6. Applied sqrt-prodError: 0.6 bits

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

    7. Applied distribute-lft-neg-inError: 0.6 bits

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

    8. Applied associate-*r*Error: 0.5 bits

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

    10. Applied add-cube-cbrtError: 0.5 bits

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

    11. Applied sqrt-prodError: 0.7 bits

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

    12. Applied associate-*r*Error: 0.4 bits

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

    13. SimplifiedError: 0.3 bits

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

    if 1.18320777133784e-310 < x

    1. Initial program Error: 30.3 bits

      \[\sqrt{\left(2 \cdot x\right) \cdot x}\]
    2. Using strategy rm

    3. Applied sqrt-prodError: 0.4 bits

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

  4. Final simplificationError: 0.4 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.18320777133784 \cdot 10^{-310}:\\ \;\;\;\;\left(x \cdot \left(\sqrt{\sqrt{2}} \cdot \left(-\left|\sqrt[3]{\sqrt{2}}\right|\right)\right)\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2} \cdot \sqrt{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))