Average Error: 30.4 → 0.4
Time: 1.4s
Precision: binary64
\[\sqrt{\left(2 \cdot x\right) \cdot x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -3.2220197881532 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt[3]{\sqrt{2}}\\ \left(t_0 \cdot t_0\right) \cdot \left(x \cdot t_0\right) \end{array}\\ \end{array} \]
\sqrt{\left(2 \cdot x\right) \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -3.2220197881532 \cdot 10^{-310}:\\
\;\;\;\;-x \cdot \sqrt{2}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := \sqrt[3]{\sqrt{2}}\\
\left(t_0 \cdot t_0\right) \cdot \left(x \cdot t_0\right)
\end{array}\\


\end{array}

(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -3.2220197881532e-310)
   (- (* x (sqrt 2.0)))
   (let* ((t_0 (cbrt (sqrt 2.0)))) (* (* t_0 t_0) (* x t_0)))))
double code(double x) {
	return sqrt((2.0 * x) * x);
}
↓
double code(double x) {
	double tmp;
	if (x <= -3.2220197881532e-310) {
		tmp = -(x * sqrt(2.0));
	} else {
		double t_0 = cbrt(sqrt(2.0));
		tmp = (t_0 * t_0) * (x * 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 < -3.222019788153194e-310

    1. Initial program 30.9

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

    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 -3.222019788153194e-310 < x

    1. Initial program 29.9

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

    2. Taylor expanded in x around 0 0.4

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

    3. Applied add-cube-cbrt_binary640.4

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot x \]

    4. Applied associate-*l*_binary640.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2220197881532 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(x \cdot \sqrt[3]{\sqrt{2}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022104 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))