Average Error: 30.2 → 16.1
Time: 4.3s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;{x}^{2} \le 0.0 \lor \neg \left({x}^{2} \le 3.2212222080674436 \cdot 10^{295}\right):\\ \;\;\;\;\sqrt{2} \cdot {x}^{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{2} \cdot 2}\\ \end{array}\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\begin{array}{l}
\mathbf{if}\;{x}^{2} \le 0.0 \lor \neg \left({x}^{2} \le 3.2212222080674436 \cdot 10^{295}\right):\\
\;\;\;\;\sqrt{2} \cdot {x}^{1}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{{x}^{2} \cdot 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 (((((double) pow(x, 2.0)) <= 0.0) || !(((double) pow(x, 2.0)) <= 3.2212222080674436e+295))) {
		VAR = ((double) (((double) sqrt(2.0)) * ((double) pow(x, 1.0))));
	} else {
		VAR = ((double) sqrt(((double) (((double) pow(x, 2.0)) * 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 (pow x 2.0) < 0.0 or 3.2212222080674436e295 < (pow x 2.0)

    1. Initial program 60.5

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

    2. Simplified60.5

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

    3. Taylor expanded around 0 34.2

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

    4. Simplified31.7

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

    if 0.0 < (pow x 2.0) < 3.2212222080674436e295

    1. Initial program 0.7

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

    2. Simplified0.7

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

  4. Final simplification16.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{2} \le 0.0 \lor \neg \left({x}^{2} \le 3.2212222080674436 \cdot 10^{295}\right):\\ \;\;\;\;\sqrt{2} \cdot {x}^{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{2} \cdot 2}\\ \end{array}\]

Reproduce

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