Average Error: 30.3 → 0.4
Time: 2.1s
Precision: binary64
\[\sqrt{2 \cdot {x}^{2}} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 6.07190567874543 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot {\left(x \cdot 2\right)}^{0.5}\\ \end{array} \]
\sqrt{2 \cdot {x}^{2}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 6.07190567874543 \cdot 10^{-310}:\\
\;\;\;\;-x \cdot \sqrt{2}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{x} \cdot {\left(x \cdot 2\right)}^{0.5}\\


\end{array}

(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x 6.07190567874543e-310)
   (- (* x (sqrt 2.0)))
   (* (sqrt x) (pow (* x 2.0) 0.5))))
double code(double x) {
	return sqrt((2.0 * pow(x, 2.0)));
}
↓
double code(double x) {
	double tmp;
	if (x <= 6.07190567874543e-310) {
		tmp = -(x * sqrt(2.0));
	} else {
		tmp = sqrt(x) * pow((x * 2.0), 0.5);
	}
	return tmp;
}

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 < 6.071905678745428e-310

    1. Initial program 30.4

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

    2. Taylor expanded in x around -inf 0.4

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

    3. Simplified0.4

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

    if 6.071905678745428e-310 < x

    1. Initial program 30.3

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

    2. Applied egg-rr0.3

      \[\leadsto \color{blue}{\sqrt{x} \cdot {\left(x \cdot 2\right)}^{0.5}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.07190567874543 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot {\left(x \cdot 2\right)}^{0.5}\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x)
  :name "sqrt D"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))