Average Error: 30.5 → 0.4
Time: 4.3s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.741586352025499 \cdot 10^{-310}:\\ \;\;\;\;-1 \cdot \left(x \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{{x}^{\left(\frac{2}{2}\right)} + {x}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.741586352025499 \cdot 10^{-310}:\\
\;\;\;\;-1 \cdot \left(x \cdot \sqrt{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{{x}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{{x}^{\left(\frac{2}{2}\right)} + {x}^{\left(\frac{2}{2}\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 <= -1.7415863520255e-310)) {
		VAR = ((double) (-1.0 * ((double) (x * ((double) sqrt(2.0))))));
	} else {
		VAR = ((double) (((double) sqrt(((double) pow(x, ((double) (2.0 / 2.0)))))) * ((double) sqrt(((double) (((double) pow(x, ((double) (2.0 / 2.0)))) + ((double) pow(x, ((double) (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 x < -1.741586352025499e-310

    1. Initial program 30.6

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

    2. Taylor expanded around -inf 0.4

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

    if -1.741586352025499e-310 < x

    1. Initial program 30.4

      \[\sqrt{{x}^{2} + {x}^{2}}\]
    2. Using strategy rm

    3. Applied sqr-pow30.4

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

    4. Applied sqr-pow30.4

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

    5. Applied distribute-lft-out30.4

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

    6. Applied sqrt-prod0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.741586352025499 \cdot 10^{-310}:\\ \;\;\;\;-1 \cdot \left(x \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{{x}^{\left(\frac{2}{2}\right)} + {x}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020150 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))