Average Error: 30.4 → 0.4
Time: 1.5s
Precision: binary64
\[\sqrt{\left(2 \cdot x\right) \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 3.8265285002463 \cdot 10^{-311}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2} \cdot \sqrt{x}\\ \end{array}\]
\sqrt{\left(2 \cdot x\right) \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 3.8265285002463 \cdot 10^{-311}:\\
\;\;\;\;-x \cdot \sqrt{2}\\

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

\end{array}
(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x 3.8265285002463e-311)
   (- (* x (sqrt 2.0)))
   (* (sqrt (* x 2.0)) (sqrt x))))
double code(double x) {
	return ((double) sqrt(((double) (((double) (2.0 * x)) * x))));
}
↓
double code(double x) {
	double tmp;
	if ((x <= 3.8265285002463e-311)) {
		tmp = ((double) -(((double) (x * ((double) sqrt(2.0))))));
	} else {
		tmp = ((double) (((double) sqrt(((double) (x * 2.0)))) * ((double) sqrt(x))));
	}
	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 < 3.82652850024632e-311

    1. Initial program 30.2

      \[\sqrt{\left(2 \cdot x\right) \cdot x}\]

    2. Taylor expanded around -inf 0.4

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

    3. Simplified0.4

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

    if 3.82652850024632e-311 < x

    1. Initial program 30.6

      \[\sqrt{\left(2 \cdot x\right) \cdot x}\]
    2. Using strategy rm

    3. Applied sqrt-prod_binary640.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8265285002463 \cdot 10^{-311}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2} \cdot \sqrt{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))