Average Error: 30.4 → 0.3
Time: 2.1s
Precision: binary64
\[\sqrt{\left(2 \cdot x\right) \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.74804693269183 \cdot 10^{-310}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left(\left(\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot {\left(\sqrt{\sqrt{\sqrt{2}}}\right)}^{3}\right) \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot x} \cdot \sqrt{x}\\ \end{array}\]
\sqrt{\left(2 \cdot x\right) \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le 1.74804693269183 \cdot 10^{-310}:\\
\;\;\;\;-1 \cdot \left(\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left(\left(\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot {\left(\sqrt{\sqrt{\sqrt{2}}}\right)}^{3}\right) \cdot x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2 \cdot x} \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 <= 1.74804693269183e-310)) {
		VAR = ((double) (-1.0 * ((double) (((double) sqrt(((double) sqrt(((double) sqrt(((double) sqrt(2.0)))))))) * ((double) (((double) (((double) sqrt(((double) sqrt(((double) sqrt(((double) sqrt(2.0)))))))) * ((double) pow(((double) sqrt(((double) sqrt(((double) sqrt(2.0)))))), 3.0)))) * x))))));
	} else {
		VAR = ((double) (((double) sqrt(((double) (2.0 * x)))) * ((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 < 1.74804693269183e-310

    1. Initial program 30.5

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

    2. Taylor expanded around -inf 0.4

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

    4. Applied add-sqr-sqrt0.4

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

    5. Applied sqrt-prod0.6

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

    6. Applied associate-*l*0.4

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

    8. Applied add-sqr-sqrt0.4

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

    9. Applied sqrt-prod0.4

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

    10. Applied sqrt-prod0.4

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

    11. Applied associate-*l*0.4

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

    13. Applied add-sqr-sqrt0.4

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

    14. Applied sqrt-prod0.4

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

    15. Applied sqrt-prod0.4

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

    16. Applied sqrt-prod1.0

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

    17. Applied associate-*l*1.0

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

    18. Simplified0.3

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

    if 1.74804693269183e-310 < x

    1. Initial program 30.3

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

    3. Applied sqrt-prod0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.74804693269183 \cdot 10^{-310}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left(\left(\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot {\left(\sqrt{\sqrt{\sqrt{2}}}\right)}^{3}\right) \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot x} \cdot \sqrt{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020156 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))