Average Error: 31.1 → 0.4
Time: 1.9s
Precision: binary64
\[\sqrt{\left(2 \cdot x\right) \cdot x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.7583548792704 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt{\sqrt{2}}\\ t_1 := \sqrt{t_0}\\ t_1 \cdot \left(t_1 \cdot \left(x \cdot t_0\right)\right) \end{array}\\ \end{array} \]
\sqrt{\left(2 \cdot x\right) \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.7583548792704 \cdot 10^{-310}:\\
\;\;\;\;-x \cdot \sqrt{2}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := \sqrt{\sqrt{2}}\\
t_1 := \sqrt{t_0}\\
t_1 \cdot \left(t_1 \cdot \left(x \cdot t_0\right)\right)
\end{array}\\


\end{array}

(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.7583548792704e-310)
   (- (* x (sqrt 2.0)))
   (let* ((t_0 (sqrt (sqrt 2.0))) (t_1 (sqrt t_0)))
     (* t_1 (* t_1 (* x t_0))))))
double code(double x) {
	return sqrt((2.0 * x) * x);
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.7583548792704e-310) {
		tmp = -(x * sqrt(2.0));
	} else {
		double t_0 = sqrt(sqrt(2.0));
		double t_1 = sqrt(t_0);
		tmp = t_1 * (t_1 * (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 < -1.758354879270385e-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 -1.758354879270385e-310 < x

    1. Initial program 31.3

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

    2. Taylor expanded in x around 0 0.4

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

    3. Applied add-sqr-sqrt_binary640.5

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

    4. Applied associate-*l*_binary640.4

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

    5. Applied add-sqr-sqrt_binary640.4

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

    6. Applied associate-*l*_binary640.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7583548792704 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(x \cdot \sqrt{\sqrt{2}}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))