Average Error: 30.5 → 15.5
Time: 4.5s
Precision: binary64
\[\sqrt{2 \cdot {x}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 1.18320777133784 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2 \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left(\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left({x}^{1} \cdot {\left(\sqrt{\sqrt{\sqrt{2}}}\right)}^{3}\right)\right)\\ \end{array}\]
\sqrt{2 \cdot {x}^{2}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 1.18320777133784 \cdot 10^{-310}:\\
\;\;\;\;\sqrt{2 \cdot {x}^{2}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left(\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left({x}^{1} \cdot {\left(\sqrt{\sqrt{\sqrt{2}}}\right)}^{3}\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 <= 1.18320777133784e-310)) {
		VAR = ((double) sqrt(((double) (2.0 * ((double) pow(x, 2.0))))));
	} else {
		VAR = ((double) (((double) sqrt(((double) sqrt(((double) sqrt(((double) sqrt(2.0)))))))) * ((double) (((double) sqrt(((double) sqrt(((double) sqrt(((double) sqrt(2.0)))))))) * ((double) (((double) pow(x, 1.0)) * ((double) pow(((double) sqrt(((double) sqrt(((double) sqrt(2.0)))))), 3.0))))))));
	}
	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 < 1.18320777133784e-310

    1. Initial program Error: 30.7 bits

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

    if 1.18320777133784e-310 < x

    1. Initial program Error: 30.3 bits

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

    2. Taylor expanded around 0 Error: 5.7 bits

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

    3. SimplifiedError: 0.4 bits

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

    5. Applied add-sqr-sqrtError: 0.4 bits

      \[\leadsto \sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot {x}^{1}\]

    6. Applied sqrt-prodError: 0.6 bits

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

    7. Applied associate-*l*Error: 0.4 bits

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

    8. SimplifiedError: 0.4 bits

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

    10. Applied add-sqr-sqrtError: 0.4 bits

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

    11. Applied sqrt-prodError: 0.4 bits

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

    12. Applied sqrt-prodError: 0.4 bits

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

    13. Applied associate-*l*Error: 0.4 bits

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

    14. SimplifiedError: 0.4 bits

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

    16. Applied add-sqr-sqrtError: 0.4 bits

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

    17. Applied sqrt-prodError: 0.4 bits

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

    18. Applied sqrt-prodError: 0.4 bits

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

    19. Applied sqrt-prodError: 1.0 bits

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

    20. Applied associate-*l*Error: 1.0 bits

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

    21. SimplifiedError: 0.3 bits

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

  4. Final simplificationError: 15.5 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.18320777133784 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2 \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left(\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left({x}^{1} \cdot {\left(\sqrt{\sqrt{\sqrt{2}}}\right)}^{3}\right)\right)\\ \end{array}\]

Reproduce

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