Average Error: 30.5 → 15.6
Time: 4.3s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 2.5365652372159 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{{x}^{2} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{2}\right)}^{0.6666666666666666} \cdot {x}^{1}\right) \cdot \sqrt[3]{\sqrt{2}}\\ \end{array}\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 2.5365652372159 \cdot 10^{-311}:\\
\;\;\;\;\sqrt{{x}^{2} \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\left({\left(\sqrt{2}\right)}^{0.6666666666666666} \cdot {x}^{1}\right) \cdot \sqrt[3]{\sqrt{2}}\\

\end{array}
double code(double x) {
	return ((double) sqrt(((double) (((double) pow(x, 2.0)) + ((double) pow(x, 2.0))))));
}
↓
double code(double x) {
	double VAR;
	if ((x <= 2.5365652372159e-311)) {
		VAR = ((double) sqrt(((double) (((double) pow(x, 2.0)) * 2.0))));
	} else {
		VAR = ((double) (((double) (((double) pow(((double) sqrt(2.0)), 0.6666666666666666)) * ((double) pow(x, 1.0)))) * ((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

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

    2. Simplified30.6

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

    if 2.53656523721594e-311 < x

    1. Initial program 30.3

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

    2. Simplified30.3

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

    3. Taylor expanded around 0 5.7

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

    4. Simplified0.4

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

    6. Applied add-cube-cbrt0.4

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

    7. Applied associate-*l*0.4

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

    8. Simplified0.4

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

    10. Applied associate-*r*0.4

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

    11. Simplified0.4

      \[\leadsto \color{blue}{\left({\left(\sqrt{2}\right)}^{0.6666666666666666} \cdot {x}^{1}\right)} \cdot \sqrt[3]{\sqrt{2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification15.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5365652372159 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{{x}^{2} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{2}\right)}^{0.6666666666666666} \cdot {x}^{1}\right) \cdot \sqrt[3]{\sqrt{2}}\\ \end{array}\]

Reproduce

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