Average Error: 29.5 → 0.4
Time: 2.6s
Precision: binary64
\[\sqrt{\left(2 \cdot x\right) \cdot x} \]
↓
\[\begin{array}{l} t_0 := x \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -2.45058024836507 \cdot 10^{-310}:\\ \;\;\;\;-t_0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
\sqrt{\left(2 \cdot x\right) \cdot x}
↓
\begin{array}{l}
t_0 := x \cdot \sqrt{2}\\
\mathbf{if}\;x \leq -2.45058024836507 \cdot 10^{-310}:\\
\;\;\;\;-t_0\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (sqrt 2.0))))
   (if (<= x -2.45058024836507e-310) (- t_0) t_0)))
double code(double x) {
	return sqrt((2.0 * x) * x);
}
↓
double code(double x) {
	double t_0 = x * sqrt(2.0);
	double tmp;
	if (x <= -2.45058024836507e-310) {
		tmp = -t_0;
	} else {
		tmp = t_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 < -2.45058024836507e-310

    1. Initial program 29.5

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

    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 -2.45058024836507e-310 < x

    1. Initial program 29.5

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

    2. Taylor expanded in x around 0 0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45058024836507 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))