Average Error: 30.4 → 15.7
Time: 3.9s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.02232331549204 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{{x}^{2}} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot {x}^{1}\right)\\ \end{array}\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.02232331549204 \cdot 10^{-311}:\\
\;\;\;\;\sqrt{{x}^{2}} \cdot \sqrt{2}\\

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

\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 <= -2.022323315492e-311)) {
		VAR = ((double) (((double) sqrt(((double) pow(x, 2.0)))) * ((double) sqrt(2.0))));
	} else {
		VAR = ((double) (((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) cbrt(((double) sqrt(2.0)))))) * ((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) pow(x, 1.0))))));
	}
	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 < -2.02232331549204e-311

    1. Initial program 30.8

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

    2. Simplified30.8

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

    4. Applied sqrt-prod31.0

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

    if -2.02232331549204e-311 < x

    1. Initial program 30.1

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

    2. Simplified30.1

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

    3. Taylor expanded around 0 5.6

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

    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

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

  4. Final simplification15.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.02232331549204 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{{x}^{2}} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot {x}^{1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))