Average Error: 30.7 → 0.3
Time: 3.8s
Precision: binary64
\[\sqrt{2 \cdot {x}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -3.89140349991603 \cdot 10^{-310}:\\ \;\;\;\;-\left(x \cdot {\left(\sqrt{\sqrt{2}}\right)}^{1.5}\right) \cdot \sqrt{\sqrt{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array}\]
\sqrt{2 \cdot {x}^{2}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -3.89140349991603 \cdot 10^{-310}:\\
\;\;\;\;-\left(x \cdot {\left(\sqrt{\sqrt{2}}\right)}^{1.5}\right) \cdot \sqrt{\sqrt{\sqrt{2}}}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \sqrt{2}\\

\end{array}
(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -3.89140349991603e-310)
   (- (* (* x (pow (sqrt (sqrt 2.0)) 1.5)) (sqrt (sqrt (sqrt 2.0)))))
   (* x (sqrt 2.0))))
double code(double x) {
	return sqrt(2.0 * pow(x, 2.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -3.89140349991603e-310) {
		tmp = -((x * pow(sqrt(sqrt(2.0)), 1.5)) * sqrt(sqrt(sqrt(2.0))));
	} else {
		tmp = x * sqrt(2.0);
	}
	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 < -3.89140349991603e-310

    1. Initial program 31.2

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

    2. Taylor expanded around -inf 0.4

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

    3. Simplified0.4

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

    5. Applied add-sqr-sqrt_binary640.6

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

    6. Applied associate-*r*_binary640.4

      \[\leadsto -\color{blue}{\left(x \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt_binary640.4

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

    9. Applied associate-*r*_binary640.4

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

    10. Simplified0.3

      \[\leadsto -\color{blue}{\left(x \cdot {\left(\sqrt{\sqrt{2}}\right)}^{1.5}\right)} \cdot \sqrt{\sqrt{\sqrt{2}}}\]

    if -3.89140349991603e-310 < x

    1. Initial program 30.1

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

    2. Taylor expanded around 0 0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.89140349991603 \cdot 10^{-310}:\\ \;\;\;\;-\left(x \cdot {\left(\sqrt{\sqrt{2}}\right)}^{1.5}\right) \cdot \sqrt{\sqrt{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x)
  :name "sqrt D"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))