Average Error: 29.9 → 0.4
Time: 2.0s
Precision: binary64
\[\sqrt{\left(2 \cdot x\right) \cdot x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 4.495787152736 \cdot 10^{-310}:\\ \;\;\;\;-\left(x \cdot \sqrt[3]{\sqrt{2}}\right) \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array} \]
\sqrt{\left(2 \cdot x\right) \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 4.495787152736 \cdot 10^{-310}:\\
\;\;\;\;-\left(x \cdot \sqrt[3]{\sqrt{2}}\right) \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666}\\

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


\end{array}

(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x 4.495787152736e-310)
   (- (* (* x (cbrt (sqrt 2.0))) (pow (sqrt 2.0) 0.6666666666666666)))
   (* x (sqrt 2.0))))
double code(double x) {
	return sqrt((2.0 * x) * x);
}
↓
double code(double x) {
	double tmp;
	if (x <= 4.495787152736e-310) {
		tmp = -((x * cbrt(sqrt(2.0))) * pow(sqrt(2.0), 0.6666666666666666));
	} 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 < 4.495787152735977e-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

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

    4. Applied add-cube-cbrt_binary640.4

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

    5. Applied associate-*l*_binary640.4

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

    6. Applied associate-*l*_binary640.6

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

    7. Simplified0.4

      \[\leadsto -\sqrt[3]{\sqrt{2}} \cdot \color{blue}{\left(x \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666}\right)} \]

    8. Applied associate-*r*_binary640.4

      \[\leadsto -\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot x\right) \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666}} \]

    9. Simplified0.4

      \[\leadsto -\color{blue}{\left(x \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666} \]

    if 4.495787152735977e-310 < x

    1. Initial program 30.4

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.495787152736 \cdot 10^{-310}:\\ \;\;\;\;-\left(x \cdot \sqrt[3]{\sqrt{2}}\right) \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))